{"id":455,"date":"2024-03-20T14:26:22","date_gmt":"2024-03-20T06:26:22","guid":{"rendered":"https:\/\/qwq.cafe\/?p=455"},"modified":"2024-03-21T11:02:44","modified_gmt":"2024-03-21T03:02:44","slug":"the-basic-of-category-theory","status":"publish","type":"post","link":"https:\/\/qwq.cafe\/?p=455","title":{"rendered":"The Basic of Category Theory"},"content":{"rendered":"\n<p>Here is my note of the category theory, which is the basic of modern mathematics. While it&#39;s very useful, it&#39;s too hard and abstract to learn. I want to record my understanding of category theory, and so I write this note.<\/p>\n<p>I&#39;m sorry for my poor English and math, and therefore here may be amounts of faults in my note. If you find something wrong in my note, please leave a comment to let me know. If you do so, I couldn&#39;t be more appreciated to you.<\/p>\n<p>References:<\/p>\n<ul>\n<li><em>Advanced Modern Algebra<\/em>, Joseph J. Rotman<\/li>\n<li><em>Algebra: Chapter 0<\/em>, Paolo Aluffi<\/li>\n<li><em>\u4ee3\u6570\u5b66\u65b9\u6cd5 \u5377\u4e00\uff1a\u57fa\u7840\u67b6\u6784<\/em>, \u674e\u6587\u5a01<\/li>\n<li><em>A Term of Commutative Algebra<\/em>, Allen Altman and Steven Kleiman<\/li>\n<li><em>An Introduction to Algebraic Topology<\/em>, Joseph J. Rotman<\/li>\n<li><em>An Introduction to Homological Topology<\/em>, Joseph J. Rotman<\/li>\n<li><a href='https:\/\/ncatlab.org\/nlab\/show\/HomePage'><em>nlab<\/em><\/a><\/li>\n<li><a href='https:\/\/www.bananaspace.org\/wiki\/%E9%A6%96%E9%A1%B5'><em>\u9999\u8549\u7a7a\u95f4<\/em><\/a><\/li>\n\n<\/ul>\n<h2 id='1-categories'>1. Categories<\/h2>\n<p>Before we begin, let&#39;s discuss why category theory is worth learning.<\/p>\n<p>In my view, category theory serves as a language for mathematics, offering a way to describe abstract concepts and theorems succinctly and clearly. The following text will include examples to illustrate this point.<\/p>\n<h3 id='11-definition'>1.1 Definition<\/h3>\n<p>A <strong>category<\/strong> <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-168-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-168-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr{C}<\/script> consists of:<\/p>\n<ul>\n<li><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.324ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3237 1000\" 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347L182 339V98Z\"><\/path><path id=\"MJX-176-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-176-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-176-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-176-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-176-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-176-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-176-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-176-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2029,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-176-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2848,0)\"><use data-c=\"29\" xlink:href=\"#MJX-176-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr{C})<\/script>:  the class of the objects (It may not be a set because it may be too large to be a set, which will be talked later);<\/li>\n<li><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.706ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3406 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-170-TEX-N-4D\" d=\"M132 622Q125 629 121 631T105 634T62 637H29V683H135Q221 683 232 682T249 675Q250 674 354 398L458 124L562 398Q666 674 668 675Q671 681 683 682T781 683H887V637H854Q814 636 803 634T785 622V61Q791 51 802 49T854 46H887V0H876Q855 3 736 3Q605 3 596 0H585V46H618Q660 47 669 49T688 61V347Q688 424 688 461T688 546T688 613L687 632Q454 14 450 7Q446 1 430 1T410 7Q409 9 292 316L176 624V606Q175 588 175 543T175 463T175 356L176 86Q187 50 261 46H278V0H269Q254 3 154 3Q52 3 37 0H29V46H46Q78 48 98 56T122 69T132 86V622Z\"><\/path><path id=\"MJX-170-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-170-TEX-N-72\" d=\"M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z\"><\/path><path id=\"MJX-170-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-170-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-170-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4D\" xlink:href=\"#MJX-170-TEX-N-4D\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-170-TEX-N-6F\" transform=\"translate(917,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-170-TEX-N-72\" transform=\"translate(1417,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1809,0)\"><use data-c=\"28\" xlink:href=\"#MJX-170-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2198,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-170-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3017,0)\"><use data-c=\"29\" xlink:href=\"#MJX-170-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Mor}(\\mathscr C)<\/script>: the class of morphisms. For every morphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.716ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5178.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-171-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-171-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-171-TEX-N-4D\" d=\"M132 622Q125 629 121 631T105 634T62 637H29V683H135Q221 683 232 682T249 675Q250 674 354 398L458 124L562 398Q666 674 668 675Q671 681 683 682T781 683H887V637H854Q814 636 803 634T785 622V61Q791 51 802 49T854 46H887V0H876Q855 3 736 3Q605 3 596 0H585V46H618Q660 47 669 49T688 61V347Q688 424 688 461T688 546T688 613L687 632Q454 14 450 7Q446 1 430 1T410 7Q409 9 292 316L176 624V606Q175 588 175 543T175 463T175 356L176 86Q187 50 261 46H278V0H269Q254 3 154 3Q52 3 37 0H29V46H46Q78 48 98 56T122 69T132 86V622Z\"><\/path><path 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741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-171-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-171-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-171-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-171-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1772.6,0)\"><use data-c=\"4D\" xlink:href=\"#MJX-171-TEX-N-4D\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-171-TEX-N-6F\" transform=\"translate(917,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-171-TEX-N-72\" transform=\"translate(1417,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3581.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-171-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3970.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-171-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4789.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-171-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f\\in \\operatorname{Mor}(\\mathscr C)<\/script>, it has a <strong>source<\/strong> <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.066ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1797 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-174-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-174-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-174-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-174-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D460\" xlink:href=\"#MJX-174-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(469,0)\"><use data-c=\"28\" xlink:href=\"#MJX-174-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(858,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-174-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1408,0)\"><use data-c=\"29\" xlink:href=\"#MJX-174-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">s(f)<\/script> and a <strong>target<\/strong> <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.821ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1689 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-175-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-175-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-175-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-175-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-175-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(361,0)\"><use data-c=\"28\" xlink:href=\"#MJX-175-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-175-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1300,0)\"><use data-c=\"29\" xlink:href=\"#MJX-175-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">t(f)<\/script>, where both <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.066ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1797 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-174-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-174-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-174-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-174-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D460\" xlink:href=\"#MJX-174-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(469,0)\"><use data-c=\"28\" xlink:href=\"#MJX-174-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(858,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-174-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1408,0)\"><use data-c=\"29\" xlink:href=\"#MJX-174-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">s(f)<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.821ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1689 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-175-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-175-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-175-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-175-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-175-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(361,0)\"><use data-c=\"28\" xlink:href=\"#MJX-175-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-175-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1300,0)\"><use data-c=\"29\" xlink:href=\"#MJX-175-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">t(f)<\/script> are elements of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.324ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3237 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-176-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-176-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-176-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-176-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-176-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-176-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-176-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-176-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-176-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-176-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2029,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-176-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2848,0)\"><use data-c=\"29\" xlink:href=\"#MJX-176-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr{C})<\/script>. Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-177-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-177-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">X<\/script> be the source of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.244ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 550 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-1183-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1183-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-186-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-186-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">Y<\/script> be the target of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.244ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 550 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-1183-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1183-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f<\/script> we can denote <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.244ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 550 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-1183-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1183-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f<\/script> as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.303ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4554.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-234-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-234-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-234-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-234-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-234-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-234-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-234-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1383.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-234-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2513.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-234-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3791.1,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-234-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:X\\to Y<\/script>. And we define <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.631ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5582.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-183-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-183-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-183-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-183-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-183-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-183-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-183-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-183-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-183-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-183-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-183-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-183-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-183-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2745.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-183-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3134.1,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-183-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3986.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-183-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4430.8,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-183-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5193.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-183-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}_\\mathscr{C}(X,Y)<\/script> (also <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.133ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4920.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-184-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-184-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-184-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-184-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-184-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 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data-c=\"1D44C\" xlink:href=\"#MJX-184-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4531.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-184-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}(X,Y)<\/script> for short) as the class of morphisms with source <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-177-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 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class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.726ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 763 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-186-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 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xlink:href=\"#MJX-187-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10502.2,0)\"><use data-c=\"2229\" xlink:href=\"#MJX-187-TEX-N-2229\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(11391.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-187-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(394,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-187-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-187-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12739.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-187-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13128.2,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-187-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13891.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-187-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}_\\mathscr{C}(X,Y):=s^{-1}(X)\\cap t^{-1}(Y)<\/script>.<\/li>\n\n<\/ul>\n<p>Additionally, objects and morphisms should satisfy these properties:<\/p>\n<ul>\n<li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"18.649ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8242.9 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-188-TEX-N-2200\" d=\"M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z\"><\/path><path id=\"MJX-188-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 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xlink:href=\"#MJX-188-TEX-N-2200\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(556,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-188-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1408,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-188-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1852.7,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-188-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2615.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-188-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3060.3,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-188-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4061.1,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-188-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5005.9,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-188-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-188-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-188-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6645.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-188-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(7034.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-188-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7853.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-188-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\forall X,Y,Z\\in\\operatorname{Obj}(\\mathscr C)<\/script>, there is a <strong>composition<\/strong> <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"42.318ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 18704.6 1000\" 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\\operatorname{Hom}(X,Z)<\/script>, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.069ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5776.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-190-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-190-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 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230H95V155Z\"><\/path><path id=\"MJX-190-TEX-N-2218\" d=\"M55 251Q55 328 112 386T249 444T386 388T444 249Q444 171 388 113T250 55Q170 55 113 112T55 251ZM245 403Q188 403 142 361T96 250Q96 183 141 140T250 96Q284 96 313 109T354 135T375 160Q403 197 403 250Q403 313 360 358T245 403Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-190-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-190-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(866,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-190-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1310.7,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-190-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1860.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-190-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2527.4,0)\"><use data-c=\"21A6\" xlink:href=\"#MJX-190-TEX-N-21A6\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3805.2,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-190-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4504.4,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-190-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5226.7,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-190-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(g,f)\\mapsto g\\circ f<\/script>. We also abbreviate <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.46ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1971.4 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-191-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-191-TEX-N-2218\" d=\"M55 251Q55 328 112 386T249 444T386 388T444 249Q444 171 388 113T250 55Q170 55 113 112T55 251ZM245 403Q188 403 142 361T96 250Q96 183 141 140T250 96Q284 96 313 109T354 135T375 160Q403 197 403 250Q403 313 360 358T245 403Z\"><\/path><path id=\"MJX-191-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-191-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(699.2,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-191-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1421.4,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-191-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">g\\circ f<\/script> as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.324ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1027 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-192-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-192-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-192-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(477,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-192-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">gf<\/script>\u200b\u200b.<\/p>\n<p>And we can use a commutative diagram<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n37\" cid=\"n37\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container 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y2=\"49.51\" stroke-dasharray=\"\"><\/line><g transform=\"translate(51.14,49.51) rotate(44.07) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(-11.76,12.15) \"><\/g><g transform=\"translate(62.52,0) \"><\/g><line x1=\"62.52\" y1=\"11.06\" x2=\"62.52\" y2=\"49.46\" stroke-dasharray=\"\"><\/line><g transform=\"translate(62.52,49.46) rotate(90) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(8.96,0) \"><\/g><g transform=\"translate(0,60.53) \"><\/g><g transform=\"translate(62.52,60.53) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.426\" data-y=\"0.3415\" data-xypic-id=\"0\" transform=\"translate(400,-91.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-1379-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.7592956351736995\" data-y=\"-0.5363782463805518\" data-xypic-id=\"1\" transform=\"translate(2585.3,786.4)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1379-TEX-I-1D453\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"0.6789035539297239\" data-y=\"2.8274352760034063\" data-xypic-id=\"2\" transform=\"translate(1504.9,-2577.4)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-1379-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(477,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-1379-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(977,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-1379-TEX-I-1D453\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.5260000000000007\" data-y=\"0.3415\" data-xypic-id=\"3\" transform=\"translate(4352,-91.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-1379-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.298921356237309\" data-y=\"1.9752921535706058\" data-xypic-id=\"4\" transform=\"translate(5124.9,-1725.3)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-1379-TEX-I-1D454\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.5460000000000003\" data-y=\"4.1245\" data-xypic-id=\"6\" transform=\"translate(4372,-3874.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1379-TEX-I-1D44D\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>to describe.<\/p>\n<p>You can see the definition of commutative diagram in <a href='https:\/\/en.wikipedia.org\/wiki\/Commutative_diagram'>wikipedia<\/a> or <a href='https:\/\/www.bananaspace.org\/wiki\/%E4%BA%A4%E6%8D%A2%E5%9B%BE'>\u9999\u8549\u7a7a\u95f4<\/a>.<\/p>\n<\/li>\n<li><p>The composition we defined above satisfies the <strong>association law<\/strong>; that is <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"21.372ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 9446.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-193-TEX-N-2200\" d=\"M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z\"><\/path><path id=\"MJX-193-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-193-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 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391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-193-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-193-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-193-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-193-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"2200\" xlink:href=\"#MJX-193-TEX-N-2200\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(556,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-193-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1306,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-193-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1750.7,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-193-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2509.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-193-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2954.3,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-193-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3714.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-193-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4159,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-193-TEX-I-1D437\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5264.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-193-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6209.6,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-193-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-193-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-193-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7849.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-193-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(8238.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-193-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9057.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-193-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\forall A,B,C,D\\in\\operatorname{Obj}(\\mathscr C)<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"32.298ex\" height=\"2.084ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 14275.7 921\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-194-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 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xlink:href=\"#MJX-194-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-194-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1383.6,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-194-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2411.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-194-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3689.1,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-194-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4448.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-194-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4892.8,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-194-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5647.6,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-194-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6203.3,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-194-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7240.1,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-194-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8517.9,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-194-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9277.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-194-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9722.6,0)\"><use data-c=\"210E\" xlink:href=\"#MJX-194-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10576.3,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-194-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11132.1,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-194-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12169.9,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-194-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13447.7,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-194-TEX-I-1D437\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:A\\to B,g:B\\to C,h:C\\to D<\/script>, we have<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n42\" cid=\"n42\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"22.338ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 9873.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1380-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-1380-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"210E\" xlink:href=\"#MJX-1380-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1187.2,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-1380-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1909.4,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-1380-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2386.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1380-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2997.7,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-1380-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3719.9,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-1380-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4547.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1380-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5603.4,0)\"><use data-c=\"210E\" xlink:href=\"#MJX-1380-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6401.7,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-1380-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7123.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-1380-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7512.9,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-1380-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8212.1,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-1380-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8934.3,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-1380-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9484.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1380-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>And we can use a commutative diagram<\/p>\n<div contenteditable=\"false\" 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data-c=\"2218\" xlink:href=\"#MJX-1381-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(977,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-1381-TEX-I-1D453\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.5145000000000004\" data-y=\"0.3415\" data-xypic-id=\"3\" transform=\"translate(5327.1,-91.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1381-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.285421356237309\" data-y=\"2.0836376080001107\" data-xypic-id=\"4\" transform=\"translate(6098,-1833.6)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-1381-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(576,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-1381-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1076,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-1381-TEX-I-1D454\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.38000000000000006\" data-y=\"4.162999999999999\" data-xypic-id=\"5\" transform=\"translate(1432.6,-3913)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-1381-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.7433532470182742\" data-y=\"4.703653462380773\" data-xypic-id=\"6\" transform=\"translate(3555.9,-4453.7)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-1381-TEX-I-210E\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.4800000000000004\" data-y=\"4.162999999999999\" data-xypic-id=\"7\" transform=\"translate(5292.6,-3913)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-1381-TEX-I-1D437\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>to describe.<\/p>\n<\/li>\n<li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.275ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5867.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-195-TEX-N-2200\" d=\"M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z\"><\/path><path id=\"MJX-195-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-195-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-195-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 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-79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-195-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-195-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-195-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"2200\" xlink:href=\"#MJX-195-TEX-N-2200\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(556,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-195-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1685.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-195-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2630.6,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-195-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-195-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-195-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4270.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-195-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4659.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-195-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5478.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-195-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\forall X\\in\\operatorname{Obj}(\\mathscr C)<\/script>, there is a <strong>identity<\/strong> <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.782ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7417.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-196-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-196-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-196-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-196-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-196-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-196-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-196-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-196-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-196-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-196-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-196-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1463.2,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-196-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2408,0)\"><use data-c=\"48\" xlink:href=\"#MJX-196-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-196-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-196-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4491,0)\"><use data-c=\"28\" xlink:href=\"#MJX-196-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4880,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-196-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5732,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-196-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6176.7,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-196-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7028.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-196-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">1_X\\in \\operatorname{Hom}(X,X)<\/script>. For all <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.458ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6390.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-197-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 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250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-197-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-197-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-197-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-197-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-197-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-197-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-197-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(763,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-197-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1207.7,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-197-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2208.4,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-197-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3153.2,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-197-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-197-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-197-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4793.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-197-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5182.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-197-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6001.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-197-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">Y,Z\\in\\operatorname{Obj}(\\mathscr C)<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.303ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4554.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-234-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-234-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-234-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-234-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-234-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-234-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-234-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1383.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-234-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2513.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-234-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3791.1,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-234-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:X\\to Y<\/script> ,<mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.048ex\" height=\"2.009ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 4441.1 888\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-199-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-199-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-199-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-199-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-199-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-199-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(754.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-199-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1310.6,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-199-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2311.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-199-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3589.1,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-199-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">g: Z\\to X<\/script>, we have<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n48\" cid=\"n48\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.954ex\" height=\"5.656ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1500 4841.5 2500\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.262ex;\"><defs><path id=\"MJX-1382-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-1382-TEX-N-2218\" d=\"M55 251Q55 328 112 386T249 444T386 388T444 249Q444 171 388 113T250 55Q170 55 113 112T55 251ZM245 403Q188 403 142 361T96 250Q96 183 141 140T250 96Q284 96 313 109T354 135T375 160Q403 197 403 250Q403 313 360 358T245 403Z\"><\/path><path id=\"MJX-1382-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1382-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 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153Z\"><\/path><path id=\"MJX-1382-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1382-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-1382-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\"><g data-mml-node=\"mtr\" transform=\"translate(0,750)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1382-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(772.2,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-1382-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1494.4,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-1382-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-1382-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2957.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1382-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4013.5,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-1382-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4563.5,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1382-TEX-N-2C\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-750)\"><g data-mml-node=\"mtd\" transform=\"translate(415.7,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-1382-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-1382-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1444.4,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-1382-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2199.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1382-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3255,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-1382-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3732,0)\"><use data-c=\"2E\" xlink:href=\"#MJX-1382-TEX-N-2E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<\/li>\n\n<\/ul>\n<p>The definition of category is simple but abstract. Let&#39;s understand it with some examples.<\/p>\n<h3 id='12-example'>1.2 Example<\/h3>\n<ul>\n<li>Let&#39;s begin with a simple example, the category of sets, denoted as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.649ex\" height=\"1.602ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 1613 708\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-239-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-239-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-239-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-239-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-239-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-239-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}<\/script>. <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.12ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4031 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path 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639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-201-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-201-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-201-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-201-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-201-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-201-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-201-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2029,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-201-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-201-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-201-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3642,0)\"><use data-c=\"29\" xlink:href=\"#MJX-201-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathbf{Set})<\/script> is the class of all sets (as we all know, it can&#39;t be a set because of Russell&#39;s paradox). <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.661ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" 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46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-202-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-202-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-202-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-202-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-202-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-202-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-202-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3306.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-202-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3695.6,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-202-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4445.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-202-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4890.2,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-202-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5649.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-202-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}_{\\mathbf{Set}}(A,B)<\/script> are all maps from <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1424-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1424-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A<\/script> to <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1418-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1418-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">B<\/script>. It&#39;s easy to check <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.649ex\" height=\"1.602ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 1613 708\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-239-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-239-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-239-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-239-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-239-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-239-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}<\/script> satisfies the concept of category.<\/li>\n<li>The category of topological spaces, denoted as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.557ex\" height=\"1.966ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -675 2014 869\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-245-TEX-B-1D413\" d=\"M41 425Q41 426 51 545T62 669V675H737V669Q738 665 748 546T758 425V419H696V425Q687 517 669 555T595 607Q578 612 522 613H478V62H631V0H615Q585 3 399 3Q214 3 184 0H168V62H321V613H277H263Q164 613 134 561Q113 527 103 425V419H41V425Z\"><\/path><path id=\"MJX-245-TEX-B-1D428\" d=\"M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z\"><\/path><path id=\"MJX-245-TEX-B-1D429\" d=\"M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D413\" xlink:href=\"#MJX-245-TEX-B-1D413\"><\/use><use data-c=\"1D428\" xlink:href=\"#MJX-245-TEX-B-1D428\" transform=\"translate(800,0)\"><\/use><use data-c=\"1D429\" xlink:href=\"#MJX-245-TEX-B-1D429\" transform=\"translate(1375,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf {Top}<\/script>. The objects of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.557ex\" height=\"1.966ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -675 2014 869\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-245-TEX-B-1D413\" d=\"M41 425Q41 426 51 545T62 669V675H737V669Q738 665 748 546T758 425V419H696V425Q687 517 669 555T595 607Q578 612 522 613H478V62H631V0H615Q585 3 399 3Q214 3 184 0H168V62H321V613H277H263Q164 613 134 561Q113 527 103 425V419H41V425Z\"><\/path><path id=\"MJX-245-TEX-B-1D428\" d=\"M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z\"><\/path><path id=\"MJX-245-TEX-B-1D429\" d=\"M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D413\" xlink:href=\"#MJX-245-TEX-B-1D413\"><\/use><use data-c=\"1D428\" xlink:href=\"#MJX-245-TEX-B-1D428\" transform=\"translate(800,0)\"><\/use><use data-c=\"1D429\" xlink:href=\"#MJX-245-TEX-B-1D429\" transform=\"translate(1375,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf {Top}<\/script> are all topological spaces, and the morphisms are continuous functions.<\/li>\n<li>The category of groups, denoted as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.563ex\" height=\"2.016ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 2017 891\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-241-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-241-TEX-B-1D42B\" d=\"M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z\"><\/path><path id=\"MJX-241-TEX-B-1D429\" d=\"M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-241-TEX-B-1D406\"><\/use><use data-c=\"1D42B\" xlink:href=\"#MJX-241-TEX-B-1D42B\" transform=\"translate(904,0)\"><\/use><use data-c=\"1D429\" xlink:href=\"#MJX-241-TEX-B-1D429\" transform=\"translate(1378,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Grp}<\/script>, in which objects are all groups and the morphisms are group homomorphisms; similarly, the category of abelian groups, denoted as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.412ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -698 1508 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-242-TEX-B-1D400\" d=\"M296 0Q278 3 164 3Q58 3 49 0H40V62H92Q144 62 144 64Q388 682 397 689Q403 698 434 698Q463 698 471 689Q475 686 538 530T663 218L724 64Q724 62 776 62H828V0H817Q796 3 658 3Q509 3 485 0H472V62H517Q561 62 561 63L517 175H262L240 120Q218 65 217 64Q217 62 261 62H306V0H296ZM390 237L492 238L440 365Q390 491 388 491Q287 239 287 237H390Z\"><\/path><path id=\"MJX-242-TEX-B-1D41B\" d=\"M32 686L123 690Q214 694 215 694H221V409Q289 450 378 450Q479 450 539 387T600 221Q600 122 535 58T358 -6H355Q272 -6 203 53L160 1L129 0H98V301Q98 362 98 435T99 525Q99 591 97 604T83 620Q69 624 42 624H29V686H32ZM227 105L232 99Q237 93 242 87T258 73T280 59T306 49T339 45Q380 45 411 66T451 131Q457 160 457 230Q457 264 456 284T448 329T430 367T396 389T343 398Q282 398 235 355L227 348V105Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D400\" xlink:href=\"#MJX-242-TEX-B-1D400\"><\/use><use data-c=\"1D41B\" xlink:href=\"#MJX-242-TEX-B-1D41B\" transform=\"translate(869,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Ab}<\/script>\u200b, in which objects are all  abelian groups and the morphisms are group homomorphisms.<\/li>\n<li>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-213-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-213-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">k<\/script> be a field. The category of the vector spaces on <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-213-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-213-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">k<\/script> is denoted as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.347ex\" height=\"1.909ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 2805.4 843.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-243-TEX-B-1D415\" d=\"M592 686H604Q615 685 631 685T666 684T700 684T724 683Q829 683 835 686H843V624H744L611 315Q584 254 546 165Q492 40 482 19T461 -6L460 -7H409Q398 -4 391 9Q385 20 257 315L124 624H25V686H36Q57 683 190 683Q340 683 364 686H377V624H289L384 403L480 185L492 212Q504 240 529 298T575 405L670 624H582V686H592Z\"><\/path><path id=\"MJX-243-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-243-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-243-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-243-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D415\" xlink:href=\"#MJX-243-TEX-B-1D415\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-243-TEX-B-1D41E\" transform=\"translate(869,0)\"><\/use><use data-c=\"1D41C\" xlink:href=\"#MJX-243-TEX-B-1D41C\" transform=\"translate(1396,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-243-TEX-B-1D42D\" transform=\"translate(1907,0)\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(2387,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-243-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Vect}_k<\/script>, in which objects are all vector spaces on <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-213-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-213-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">k<\/script> and morphisms are linear maps.<\/li>\n<li>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-1064-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1064-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R<\/script> be a ring with units. The category of the left modules on <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-1064-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1064-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R<\/script> is denoted as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.619ex\" height=\"1.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2925.7 858.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.373ex;\"><defs><path id=\"MJX-244-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-244-TEX-B-1D40C\" d=\"M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H305Q316 679 323 667Q330 653 434 414L546 157L658 414Q766 662 773 674Q778 681 788 686H1052V624H944V62H1052V0H1040Q1016 3 874 3T708 0H696V62H804V341L803 618L786 580Q770 543 735 462T671 315Q540 13 536 9Q528 1 507 1Q485 1 477 9Q472 14 408 162T281 457T217 603Q215 603 215 334V62H323V0H314Z\"><\/path><path id=\"MJX-244-TEX-B-1D428\" d=\"M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z\"><\/path><path id=\"MJX-244-TEX-B-1D41D\" d=\"M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><\/g><g data-mml-node=\"mi\" transform=\"translate(33,-150) scale(0.707)\"><use data-c=\"1D445\" xlink:href=\"#MJX-244-TEX-I-1D445\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(619.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D40C\" xlink:href=\"#MJX-244-TEX-B-1D40C\"><\/use><use data-c=\"1D428\" xlink:href=\"#MJX-244-TEX-B-1D428\" transform=\"translate(1092,0)\"><\/use><use data-c=\"1D41D\" xlink:href=\"#MJX-244-TEX-B-1D41D\" transform=\"translate(1667,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">{}_R\\mathbf{Mod}<\/script>, in which objects are all modules on <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-1064-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1064-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R<\/script> and morphisms are <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-1064-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1064-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R<\/script>-module homomorphisms. Similarly, we have the category of right modules <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.619ex\" height=\"1.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2925.7 858.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.373ex;\"><defs><path id=\"MJX-219-TEX-B-1D40C\" d=\"M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H305Q316 679 323 667Q330 653 434 414L546 157L658 414Q766 662 773 674Q778 681 788 686H1052V624H944V62H1052V0H1040Q1016 3 874 3T708 0H696V62H804V341L803 618L786 580Q770 543 735 462T671 315Q540 13 536 9Q528 1 507 1Q485 1 477 9Q472 14 408 162T281 457T217 603Q215 603 215 334V62H323V0H314Z\"><\/path><path id=\"MJX-219-TEX-B-1D428\" d=\"M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z\"><\/path><path id=\"MJX-219-TEX-B-1D41D\" d=\"M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z\"><\/path><path id=\"MJX-219-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D40C\" xlink:href=\"#MJX-219-TEX-B-1D40C\"><\/use><use data-c=\"1D428\" xlink:href=\"#MJX-219-TEX-B-1D428\" transform=\"translate(1092,0)\"><\/use><use data-c=\"1D41D\" xlink:href=\"#MJX-219-TEX-B-1D41D\" transform=\"translate(1667,0)\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(2339,-150) scale(0.707)\"><use data-c=\"1D445\" xlink:href=\"#MJX-219-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Mod}_R<\/script>\u200b.<\/li>\n\n<\/ul>\n<h3 id='13-example'>1.3 Example<\/h3>\n<p>As we can see, all the morphisms are maps in the examples above. However, morphisms may not be maps. Let me show you in this example.<\/p>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.454ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2852.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-220-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-220-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-220-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-220-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-220-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-220-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-220-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1241,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-220-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1685.7,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-220-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2463.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-220-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(X,\\le)<\/script> be a poset. We can code the partial order relation of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-177-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-177-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">X<\/script> into a category <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>.<\/p>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.268ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5422.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-223-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-223-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-223-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-223-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-223-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-223-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-223-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-223-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-223-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-223-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-223-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-223-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2029,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-223-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2848,0)\"><use data-c=\"29\" xlink:href=\"#MJX-223-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3514.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-223-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4570.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-223-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr{C})=X<\/script>, and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"34.396ex\" height=\"5.428ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1449.5 15202.9 2399\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.148ex;\"><defs><path id=\"MJX-224-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-224-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 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stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44E\" xlink:href=\"#MJX-225-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(806.8,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-225-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1862.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-225-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2569.3,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-225-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3625.1,0)\"><use data-c=\"1D450\" xlink:href=\"#MJX-225-TEX-I-1D450\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">a\\le b\\le c<\/script> define <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"19.747ex\" height=\"2.262ex\" role=\"img\" 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id=\"MJX-226-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-226-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-226-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-226-TEX-N-2218\" d=\"M55 251Q55 328 112 386T249 444T386 388T444 249Q444 171 388 113T250 55Q170 55 113 112T55 251ZM245 403Q188 403 142 361T96 250Q96 183 141 140T250 96Q284 96 313 109T354 135T375 160Q403 197 403 250Q403 313 360 358T245 403Z\"><\/path><path id=\"MJX-226-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-226-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-226-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-226-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(818,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-226-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1262.7,0)\"><use data-c=\"1D450\" xlink:href=\"#MJX-226-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1695.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-226-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2306.9,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-226-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3029.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-226-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3418.1,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-226-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3947.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-226-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4391.8,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-226-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4820.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-226-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5487.6,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-226-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6543.3,0)\"><use data-c=\"28\" xlink:href=\"#MJX-226-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6932.3,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-226-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7461.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-226-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7906,0)\"><use data-c=\"1D450\" xlink:href=\"#MJX-226-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8339,0)\"><use data-c=\"29\" xlink:href=\"#MJX-226-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(b,c)\\circ(a,b)=(a,c)<\/script>. Then it&#39;s easy to check <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>\u200b is a category.<\/p>\n<p>Similarly, we can code the equivalence relation into a category.<\/p>\n<h3 id='14-definition'>1.4 Definition<\/h3>\n<p>A morphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.303ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4554.1 910\" 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18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-234-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-234-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-234-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 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xlink:href=\"#MJX-234-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:X\\to Y<\/script> is called a <strong>monomorphism<\/strong> if <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"32.466ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 14350.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-229-TEX-N-2200\" d=\"M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z\"><\/path><path id=\"MJX-229-TEX-I-1D6FC\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 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-186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-229-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-229-TEX-N-21D2\" d=\"M580 514Q580 525 596 525Q601 525 604 525T609 525T613 524T615 523T617 520T619 517T622 512Q659 438 720 381T831 300T927 263Q944 258 944 250T935 239T898 228T840 204Q696 134 622 -12Q618 -21 615 -22T600 -24Q580 -24 580 -17Q580 -13 585 0Q620 69 671 123L681 133H70Q56 140 56 153Q56 168 72 173H725L735 181Q774 211 852 250Q851 251 834 259T789 283T735 319L725 327H72Q56 332 56 347Q56 360 70 367H681L671 377Q638 412 609 458T580 514Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"2200\" xlink:href=\"#MJX-229-TEX-N-2200\"><\/use><\/g><g 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data-mml-node=\"mi\" transform=\"translate(6615.4,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-229-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7165.4,0)\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-229-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8083.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-229-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9139,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-229-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9689,0)\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-229-TEX-I-1D6FD\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10532.8,0)\"><use data-c=\"21D2\" xlink:href=\"#MJX-229-TEX-N-21D2\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11810.6,0)\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-229-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12728.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-229-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13784.1,0)\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-229-TEX-I-1D6FD\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\forall \\alpha,\\beta:Z\\to X,f\\alpha=f\\beta\\Rightarrow\\alpha=\\beta<\/script>. And we can use <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.588ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4680.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-230-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-230-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-230-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-230-TEX-N-21AA\" d=\"M55 347Q55 380 72 404T113 441T159 458T197 464Q222 464 222 444Q222 429 204 426T157 417T110 387Q95 369 95 347Q95 339 98 330T111 307T142 283T196 270H961Q845 357 818 493Q818 494 818 496T817 499Q817 511 834 511H837Q846 511 849 510T855 506T858 497T861 481T869 456Q891 389 942 336T1061 261Q1070 258 1070 250Q1070 244 1065 241T1041 231T1003 212Q962 186 932 152T887 85T866 35T858 4Q856 -6 853 -8T837 -11Q817 -11 817 0Q817 7 822 25Q854 151 961 230H196Q149 230 102 260T55 347Z\"><\/path><path id=\"MJX-230-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-230-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-230-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1383.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-230-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2513.3,0)\"><use data-c=\"21AA\" xlink:href=\"#MJX-230-TEX-N-21AA\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3917.1,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-230-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:X\\hookrightarrow Y<\/script> to describe a monomorphism.<\/p>\n<p>A morphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.303ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4554.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-234-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-234-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-234-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 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244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-234-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-234-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-234-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1383.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-234-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2513.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-234-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3791.1,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-234-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:X\\to Y<\/script> is called an <strong>epimorphism<\/strong> if <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"32.265ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 14261.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-232-TEX-N-2200\" d=\"M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z\"><\/path><path id=\"MJX-232-TEX-I-1D6FC\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path id=\"MJX-232-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-232-TEX-I-1D6FD\" d=\"M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z\"><\/path><path id=\"MJX-232-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-232-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-232-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-232-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-232-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-232-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-232-TEX-N-21D2\" d=\"M580 514Q580 525 596 525Q601 525 604 525T609 525T613 524T615 523T617 520T619 517T622 512Q659 438 720 381T831 300T927 263Q944 258 944 250T935 239T898 228T840 204Q696 134 622 -12Q618 -21 615 -22T600 -24Q580 -24 580 -17Q580 -13 585 0Q620 69 671 123L681 133H70Q56 140 56 153Q56 168 72 173H725L735 181Q774 211 852 250Q851 251 834 259T789 283T735 319L725 327H72Q56 332 56 347Q56 360 70 367H681L671 377Q638 412 609 458T580 514Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"2200\" xlink:href=\"#MJX-232-TEX-N-2200\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(556,0)\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-232-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1196,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-232-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1640.7,0)\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-232-TEX-I-1D6FD\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2484.4,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-232-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3040.2,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-232-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4081,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-232-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5358.8,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-232-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6081.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-232-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6526.4,0)\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-232-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7166.4,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-232-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7994.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-232-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9050,0)\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-232-TEX-I-1D6FD\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9616,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-232-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10443.8,0)\"><use data-c=\"21D2\" xlink:href=\"#MJX-232-TEX-N-21D2\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11721.6,0)\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-232-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12639.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-232-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13695.1,0)\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-232-TEX-I-1D6FD\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\forall \\alpha,\\beta:Y\\to Z,\\alpha f=\\beta f\\Rightarrow\\alpha=\\beta<\/script>. And we can use <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.303ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4554.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-233-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-233-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-233-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-233-TEX-N-21A0\" d=\"M943 250Q895 221 856 177T801 83H778Q755 83 755 84Q755 86 762 103T791 156T841 222L848 230H723L714 223Q677 194 650 155T611 83H565Q565 87 575 108T606 162T650 221L659 230H365Q66 230 62 234L61 235Q55 241 55 250T61 265L62 266Q66 270 365 270H659L650 279Q627 304 606 337T575 392T565 417H611Q622 384 649 345T714 277L723 270H848L841 278Q813 309 792 344T763 396T755 416Q755 417 778 417H801Q817 367 856 323T943 250Z\"><\/path><path id=\"MJX-233-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-233-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-233-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1383.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-233-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2513.3,0)\"><use data-c=\"21A0\" xlink:href=\"#MJX-233-TEX-N-21A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3791.1,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-233-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:X\\twoheadrightarrow Y<\/script> to describe an epimorphism.<\/p>\n<p>A morphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.303ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4554.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-234-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-234-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-234-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-234-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-234-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-234-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-234-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1383.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-234-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2513.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-234-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3791.1,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-234-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:X\\to Y<\/script> is called an <strong>isomorphism<\/strong> if <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"29.311ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 12955.5 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-235-TEX-N-2203\" d=\"M56 661T56 674T70 694H487Q497 686 500 679V15Q497 10 487 1L279 0H70Q56 7 56 20T70 40H460V327H84Q70 334 70 347T84 367H460V654H70Q56 661 56 674Z\"><\/path><path id=\"MJX-235-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-235-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-235-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 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transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"2203\" xlink:href=\"#MJX-235-TEX-N-2203\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(556,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-235-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1310.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-235-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1866.6,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-235-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2907.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-235-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4185.1,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-235-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5037.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-235-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5481.8,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-235-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5958.8,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-235-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6786.6,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-235-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7842.3,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-235-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-235-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9027.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-235-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9472.5,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-235-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10022.5,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-235-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10777.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-235-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(11833,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-235-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-235-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\exists g:Y\\to X,gf=1_X,fg=1_Y<\/script>. We call <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.079ex\" height=\"1.464ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 477 647\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-728-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-728-TEX-I-1D454\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">g<\/script> is the inverse of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.244ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 550 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-1183-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1183-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f<\/script> and denote it as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.597ex\" height=\"2.351ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 1589.7 1038.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-238-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-238-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-238-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-238-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(636,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-238-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-238-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f^{-1}<\/script>.<\/p>\n<p>Monomorphism is a concept like an injection, and epimorphism is a concept like a surjection. Actually, in many categories such as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.649ex\" height=\"1.602ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 1613 708\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-239-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-239-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-239-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-239-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-239-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-239-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}<\/script>, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.557ex\" height=\"1.966ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -675 2014 869\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-245-TEX-B-1D413\" d=\"M41 425Q41 426 51 545T62 669V675H737V669Q738 665 748 546T758 425V419H696V425Q687 517 669 555T595 607Q578 612 522 613H478V62H631V0H615Q585 3 399 3Q214 3 184 0H168V62H321V613H277H263Q164 613 134 561Q113 527 103 425V419H41V425Z\"><\/path><path id=\"MJX-245-TEX-B-1D428\" d=\"M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z\"><\/path><path id=\"MJX-245-TEX-B-1D429\" d=\"M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D413\" xlink:href=\"#MJX-245-TEX-B-1D413\"><\/use><use data-c=\"1D428\" xlink:href=\"#MJX-245-TEX-B-1D428\" transform=\"translate(800,0)\"><\/use><use data-c=\"1D429\" xlink:href=\"#MJX-245-TEX-B-1D429\" transform=\"translate(1375,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf {Top}<\/script>, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.563ex\" height=\"2.016ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 2017 891\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-241-TEX-B-1D406\" d=\"M465 -10Q281 -10 173 88T64 343Q64 413 85 471T143 568T217 631T298 670Q371 697 449 697Q452 697 459 697T470 696Q502 696 531 690T582 675T618 658T644 641T656 632L732 695Q734 697 745 697Q758 697 761 692T765 668V627V489V449Q765 428 761 424T741 419H731H724Q705 419 702 422T695 444Q683 520 631 577T495 635Q364 635 295 563Q261 528 247 477T232 343Q232 296 236 260T256 185T296 120T366 76T472 52Q481 51 498 51Q544 51 573 67T607 108Q608 111 608 164V214H464V276H479Q506 273 680 273Q816 273 834 276H845V214H765V113V51Q765 16 763 8T750 0Q742 2 709 16T658 40L648 46Q592 -10 465 -10Z\"><\/path><path id=\"MJX-241-TEX-B-1D42B\" d=\"M405 293T374 293T324 312T305 361Q305 378 312 394Q315 397 315 399Q305 399 294 394T266 375T238 329T222 249Q221 241 221 149V62H308V0H298Q280 3 161 3Q47 3 38 0H29V62H98V210V303Q98 353 96 363T83 376Q69 380 42 380H29V442H32L118 446Q204 450 205 450H210V414L211 378Q247 449 315 449H321Q384 449 413 422T442 360Q442 332 424 313Z\"><\/path><path id=\"MJX-241-TEX-B-1D429\" d=\"M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D406\" xlink:href=\"#MJX-241-TEX-B-1D406\"><\/use><use data-c=\"1D42B\" xlink:href=\"#MJX-241-TEX-B-1D42B\" transform=\"translate(904,0)\"><\/use><use data-c=\"1D429\" xlink:href=\"#MJX-241-TEX-B-1D429\" transform=\"translate(1378,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Grp}<\/script>, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.412ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -698 1508 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.014ex;\"><defs><path id=\"MJX-242-TEX-B-1D400\" d=\"M296 0Q278 3 164 3Q58 3 49 0H40V62H92Q144 62 144 64Q388 682 397 689Q403 698 434 698Q463 698 471 689Q475 686 538 530T663 218L724 64Q724 62 776 62H828V0H817Q796 3 658 3Q509 3 485 0H472V62H517Q561 62 561 63L517 175H262L240 120Q218 65 217 64Q217 62 261 62H306V0H296ZM390 237L492 238L440 365Q390 491 388 491Q287 239 287 237H390Z\"><\/path><path id=\"MJX-242-TEX-B-1D41B\" d=\"M32 686L123 690Q214 694 215 694H221V409Q289 450 378 450Q479 450 539 387T600 221Q600 122 535 58T358 -6H355Q272 -6 203 53L160 1L129 0H98V301Q98 362 98 435T99 525Q99 591 97 604T83 620Q69 624 42 624H29V686H32ZM227 105L232 99Q237 93 242 87T258 73T280 59T306 49T339 45Q380 45 411 66T451 131Q457 160 457 230Q457 264 456 284T448 329T430 367T396 389T343 398Q282 398 235 355L227 348V105Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D400\" xlink:href=\"#MJX-242-TEX-B-1D400\"><\/use><use data-c=\"1D41B\" xlink:href=\"#MJX-242-TEX-B-1D41B\" transform=\"translate(869,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Ab}<\/script>, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.347ex\" height=\"1.909ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 2805.4 843.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-243-TEX-B-1D415\" d=\"M592 686H604Q615 685 631 685T666 684T700 684T724 683Q829 683 835 686H843V624H744L611 315Q584 254 546 165Q492 40 482 19T461 -6L460 -7H409Q398 -4 391 9Q385 20 257 315L124 624H25V686H36Q57 683 190 683Q340 683 364 686H377V624H289L384 403L480 185L492 212Q504 240 529 298T575 405L670 624H582V686H592Z\"><\/path><path id=\"MJX-243-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-243-TEX-B-1D41C\" d=\"M447 131H458Q478 131 478 117Q478 112 471 95T439 51T377 9Q330 -6 286 -6Q196 -6 135 35Q39 96 39 222Q39 324 101 384Q169 453 286 453Q359 453 411 431T464 353Q464 319 445 302T395 284Q360 284 343 305T325 353Q325 380 338 396H333Q317 398 295 398H292Q280 398 271 397T245 390T218 373T197 338T183 283Q182 275 182 231Q182 199 184 180T193 132T220 85T270 57Q289 50 317 50H326Q385 50 414 115Q419 127 423 129T447 131Z\"><\/path><path id=\"MJX-243-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-243-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D415\" xlink:href=\"#MJX-243-TEX-B-1D415\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-243-TEX-B-1D41E\" transform=\"translate(869,0)\"><\/use><use data-c=\"1D41C\" xlink:href=\"#MJX-243-TEX-B-1D41C\" transform=\"translate(1396,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-243-TEX-B-1D42D\" transform=\"translate(1907,0)\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(2387,-150) scale(0.707)\"><use data-c=\"1D458\" xlink:href=\"#MJX-243-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Vect}_k<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.619ex\" height=\"1.943ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2925.7 858.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.373ex;\"><defs><path id=\"MJX-244-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-244-TEX-B-1D40C\" d=\"M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H305Q316 679 323 667Q330 653 434 414L546 157L658 414Q766 662 773 674Q778 681 788 686H1052V624H944V62H1052V0H1040Q1016 3 874 3T708 0H696V62H804V341L803 618L786 580Q770 543 735 462T671 315Q540 13 536 9Q528 1 507 1Q485 1 477 9Q472 14 408 162T281 457T217 603Q215 603 215 334V62H323V0H314Z\"><\/path><path id=\"MJX-244-TEX-B-1D428\" d=\"M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z\"><\/path><path id=\"MJX-244-TEX-B-1D41D\" d=\"M351 686L442 690Q533 694 534 694H540V389Q540 327 540 253T539 163Q539 97 541 83T555 66Q569 62 596 62H609V31Q609 0 608 0Q588 0 510 -3T412 -6Q411 -6 411 16V38L401 31Q337 -6 265 -6Q159 -6 99 58T38 224Q38 265 51 303T92 375T165 429T272 449Q359 449 417 412V507V555Q417 597 415 607T402 620Q388 624 361 624H348V686H351ZM411 350Q362 399 291 399Q278 399 256 392T218 371Q195 351 189 320T182 238V221Q182 179 183 159T191 115T212 74Q241 46 288 46Q358 46 404 100L411 109V350Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><\/g><g data-mml-node=\"mi\" transform=\"translate(33,-150) scale(0.707)\"><use data-c=\"1D445\" xlink:href=\"#MJX-244-TEX-I-1D445\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(619.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D40C\" xlink:href=\"#MJX-244-TEX-B-1D40C\"><\/use><use data-c=\"1D428\" xlink:href=\"#MJX-244-TEX-B-1D428\" transform=\"translate(1092,0)\"><\/use><use data-c=\"1D41D\" xlink:href=\"#MJX-244-TEX-B-1D41D\" transform=\"translate(1667,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">{}_R\\mathbf{Mod}<\/script>, a monomorphism is equivalence to an injection as well as an epimorphism is equivalence to a surjection.<\/p>\n<p>And we should pay attention to that a morphism may not be an isomorphism although it is both a monomorphism and an epimorphism. For example, in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.557ex\" height=\"1.966ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -675 2014 869\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-245-TEX-B-1D413\" d=\"M41 425Q41 426 51 545T62 669V675H737V669Q738 665 748 546T758 425V419H696V425Q687 517 669 555T595 607Q578 612 522 613H478V62H631V0H615Q585 3 399 3Q214 3 184 0H168V62H321V613H277H263Q164 613 134 561Q113 527 103 425V419H41V425Z\"><\/path><path id=\"MJX-245-TEX-B-1D428\" d=\"M287 -5Q228 -5 182 10T109 48T63 102T39 161T32 219Q32 272 50 314T94 382T154 423T214 446T265 452H279Q319 452 326 451Q428 439 485 376T542 221Q542 156 514 108T442 33Q384 -5 287 -5ZM399 230V250Q399 280 398 298T391 338T372 372T338 392T282 401Q241 401 212 380Q190 363 183 334T175 230Q175 202 175 189T177 153T183 118T195 91T215 68T245 56T287 50Q348 50 374 84Q388 101 393 132T399 230Z\"><\/path><path id=\"MJX-245-TEX-B-1D429\" d=\"M32 442L123 446Q214 450 215 450H221V409Q222 409 229 413T251 423T284 436T328 446T382 450Q480 450 540 388T600 223Q600 128 539 61T361 -6H354Q292 -6 236 28L227 34V-132H296V-194H287Q269 -191 163 -191Q56 -191 38 -194H29V-132H98V113V284Q98 330 97 348T93 370T83 376Q69 380 42 380H29V442H32ZM457 224Q457 303 427 349T350 395Q282 395 235 352L227 345V104L233 97Q274 45 337 45Q383 45 420 86T457 224Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D413\" xlink:href=\"#MJX-245-TEX-B-1D413\"><\/use><use data-c=\"1D428\" xlink:href=\"#MJX-245-TEX-B-1D428\" transform=\"translate(800,0)\"><\/use><use data-c=\"1D429\" xlink:href=\"#MJX-245-TEX-B-1D429\" transform=\"translate(1375,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf {Top}<\/script>, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.244ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 550 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-1183-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1183-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f<\/script>\u200b is bijective may not have a continuous inverse.<\/p>\n<h3 id='15-example'>1.5 Example<\/h3>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category. We call <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> is a <strong>groupoid<\/strong> if all morphisms in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> are isomorphisms.<\/p>\n<p>For example, give a topological space <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-177-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-177-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">X<\/script>. Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.268ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5422.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-251-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-251-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-251-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-251-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-251-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-251-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-251-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-251-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-251-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-251-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-251-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-251-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2029,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-251-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2848,0)\"><use data-c=\"29\" xlink:href=\"#MJX-251-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3514.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-251-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4570.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-251-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr C)=X<\/script> and the morphisms from <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.282ex\" height=\"1.375ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1008.6 607.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.375ex;\"><defs><path id=\"MJX-254-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-254-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-254-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"30\" xlink:href=\"#MJX-254-TEX-N-30\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">x_0<\/script> to <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.282ex\" height=\"1.339ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1008.6 592\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-255-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-255-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-255-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-255-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">x_1<\/script> are the homotopy classes of path from <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.282ex\" height=\"1.375ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1008.6 607.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.375ex;\"><defs><path id=\"MJX-254-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-254-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-254-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"30\" xlink:href=\"#MJX-254-TEX-N-30\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">x_0<\/script> to <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.282ex\" height=\"1.339ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1008.6 592\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-255-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-255-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-255-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-255-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">x_1<\/script>. We call <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> is the fundamental groupoid of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-177-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-177-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">X<\/script>.<\/p>\n<p>And we can regard a group as a groupoid with only one object.<\/p>\n<h3 id='16-example'>1.6 Example<\/h3>\n<p>In this part, let me show you some special examples of categories.<\/p>\n<ul>\n<li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.637ex\" height=\"1.728ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -752.9 2049.6 763.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-319-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-319-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-319-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-319-TEX-N-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-319-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-319-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-319-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1646,424.1) scale(0.707)\"><use data-c=\"2217\" xlink:href=\"#MJX-319-TEX-N-2217\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}^*<\/script> is the category of sets with <strong>based point<\/strong>. <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.108ex\" height=\"2.269ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -752.9 4467.6 1002.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-259-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-259-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-259-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-259-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-259-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 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-6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-259-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-259-TEX-N-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><path id=\"MJX-259-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-259-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-259-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-259-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-259-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2029,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-259-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-259-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-259-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1646,424.1) scale(0.707)\"><use data-c=\"2217\" xlink:href=\"#MJX-259-TEX-N-2217\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4078.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-259-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathbf{Set}^*)<\/script> consists of all ordered pairs like <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.52ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2439.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-260-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-260-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-260-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-260-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 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data-c=\"29\" xlink:href=\"#MJX-260-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(S,x)<\/script>, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-261-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-261-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">S<\/script> is a set and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.519ex\" height=\"1.686ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2439.6 745\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-262-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-262-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-262-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-262-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(849.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-262-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1794.6,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-262-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">x\\in S<\/script>. The morphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"21.495ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 9500.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-263-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 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427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-263-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-263-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-263-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-263-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-263-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-263-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-263-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-263-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1383.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-263-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1772.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-263-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-263-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2822.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-263-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3266.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-263-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-263-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4275.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-263-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4942.1,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-263-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6219.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-263-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6608.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-263-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-263-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7658.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-263-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8103.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-263-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-263-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9111.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-263-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:(S_1,x_1)\\to (S_2,x_2)<\/script> is a map from <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.375ex\" height=\"1.934ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1049.6 855\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-273-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-273-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-273-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-273-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">S_1<\/script> to <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.375ex\" height=\"1.934ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1049.6 855\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-274-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-274-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-274-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-274-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">S_2<\/script> satisfied <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.585ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4678.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-266-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-266-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-266-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-266-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-266-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-266-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-266-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-266-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(550,0)\"><use data-c=\"28\" xlink:href=\"#MJX-266-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(939,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-266-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-266-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1947.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-266-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2614.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-266-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3670.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-266-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-266-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f(x_1)=x_2<\/script>.<\/p>\n<\/li>\n<li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.637ex\" height=\"2.05ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -895 2049.6 906\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-320-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-320-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-320-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-320-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-320-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-320-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-320-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(1646,424.1) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-320-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}^2<\/script>. The Objects of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.637ex\" height=\"2.05ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -895 2049.6 906\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-320-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-320-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-320-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-320-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-320-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-320-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-320-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(1646,424.1) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-320-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}^2<\/script> consists of all ordered pairs like <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.818ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2571.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-269-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-269-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-269-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-269-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-269-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-269-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-269-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1034,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-269-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1478.7,0)\"><use data-c=\"1D447\" xlink:href=\"#MJX-269-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2182.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-269-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(S,T)<\/script>, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.058ex\" height=\"2.034ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1793.7 899\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-270-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-270-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-270-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-270-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(645,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-270-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1089.7,0)\"><use data-c=\"1D447\" xlink:href=\"#MJX-270-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">S,T<\/script> are sets and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.069ex\" height=\"1.686ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2682.6 745\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-271-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-271-TEX-N-2282\" d=\"M84 250Q84 372 166 450T360 539Q361 539 370 539T395 539T430 540T475 540T524 540H679Q694 532 694 520Q694 511 681 501L522 500H470H441Q366 500 338 496T266 472Q244 461 224 446T179 404T139 337T124 250V245Q124 157 185 89Q244 25 328 7Q348 2 366 2T522 0H681Q694 -10 694 -20Q694 -32 679 -40H526Q510 -40 480 -40T434 -41Q350 -41 289 -25T172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-271-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-271-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(981.8,0)\"><use data-c=\"2282\" xlink:href=\"#MJX-271-TEX-N-2282\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2037.6,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-271-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">T\\subset S<\/script>. The morphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"21.549ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 9524.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-272-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-272-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-272-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-272-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-272-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-272-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-272-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-272-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-272-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-272-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-272-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-272-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1383.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-272-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1772.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-272-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-272-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2822.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-272-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3266.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-272-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-272-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4287.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-272-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4954.1,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-272-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6231.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-272-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6620.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-272-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-272-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7670.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-272-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8115.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-272-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-272-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9135.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-272-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:(S_1,T_1)\\to (S_2,T_2)<\/script> is a map from <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.375ex\" height=\"1.934ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1049.6 855\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-273-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-273-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-273-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-273-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">S_1<\/script> to <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.375ex\" height=\"1.934ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1049.6 855\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-274-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-274-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-274-TEX-I-1D446\"><\/use><\/g><g 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435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-275-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-275-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-275-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-275-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-275-TEX-N-2282\" d=\"M84 250Q84 372 166 450T360 539Q361 539 370 539T395 539T430 540T475 540T524 540H679Q694 532 694 520Q694 511 681 501L522 500H470H441Q366 500 338 496T266 472Q244 461 224 446T179 404T139 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xlink:href=\"#MJX-275-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(939,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-275-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-275-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1959.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-275-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2626.3,0)\"><use data-c=\"2282\" xlink:href=\"#MJX-275-TEX-N-2282\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3682.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-275-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-275-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f(T_1)\\subset T_2<\/script>. We can describe this use a commutative diagram<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n85\" cid=\"n85\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container math-jax-postprocess\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.205ex\" height=\"15.144ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1576.3 6720.5 6693.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -11.578ex;\"><defs><path id=\"MJX-1383-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-1383-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1383-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 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429Z\"><\/path><path id=\"MJX-1383-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-1383-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 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x2=\"0\" y2=\"12.44\" stroke-dasharray=\"\"><\/line><g transform=\"translate(0,50.84) rotate(-90) \"><path d=\"M0,0 A 4,4 0 0,1 0,-8\"><\/path><\/g><g transform=\"translate(0,12.44) rotate(-90) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(-10.68,0) \"><\/g><line x1=\"13.76\" y1=\"63.06\" x2=\"52.63\" y2=\"63.06\" stroke-dasharray=\"\"><\/line><g transform=\"translate(52.63,63.06) rotate(0) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(0,11.41) \"><\/g><g transform=\"translate(66.39,63.06) \"><\/g><line x1=\"66.39\" y1=\"50.84\" x2=\"66.39\" y2=\"12.44\" stroke-dasharray=\"\"><\/line><g transform=\"translate(66.39,50.84) rotate(-90) \"><path d=\"M0,0 A 4,4 0 0,1 0,-8\"><\/path><\/g><g transform=\"translate(66.39,12.44) rotate(-90) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 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xlink:href=\"#MJX-1383-TEX-I-1D453\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.624776695296637\" data-y=\"0.2775\" data-xypic-id=\"2\" transform=\"translate(4910.3,-27.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-1383-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(646,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1383-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.5102766952966369\" data-y=\"4.2044999999999995\" data-xypic-id=\"3\" transform=\"translate(775.2,-3954.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1383-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1383-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.9440630585851519\" data-y=\"2.151165782593163\" data-xypic-id=\"4\" transform=\"translate(341.4,-1901.2)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-1383-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(378,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1383-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.8803223304703358\" data-y=\"4.830931636973825\" data-xypic-id=\"5\" transform=\"translate(3165.8,-4580.9)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1383-TEX-I-1D453\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.639276695296637\" data-y=\"4.2044999999999995\" data-xypic-id=\"6\" transform=\"translate(4924.8,-3954.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1383-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1383-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.5409747468305826\" data-y=\"2.151165782593163\" data-xypic-id=\"7\" transform=\"translate(5826.5,-1901.2)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-1383-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(378,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1383-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.542ex\" height=\"1.934ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -661 2007.8 855\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-276-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-276-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-276-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-276-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-276-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(378,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-276-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(781.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-276-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1226.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-276-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(378,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-276-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">i_1,i_2<\/script>\u200b are inclusions.<\/p>\n<p>We can also consider the objects of this category are some diagrams like<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n88\" cid=\"n88\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container math-jax-postprocess\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.876ex\" height=\"3.455ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -963.5 5249 1527\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.275ex;\"><defs><path id=\"MJX-1384-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-1384-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"xypic\" data-cmd-id=\"0\" data-text-mml-ids=\"[0,1]\"><g fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-width=\"0.64\" transform=\"translate(722.5,250) scale(62.5, -62.5)\"><g transform=\"translate(0,0) \"><\/g><line x1=\"10.76\" y1=\"0\" x2=\"49.16\" y2=\"0\" stroke-dasharray=\"\"><\/line><g transform=\"translate(10.76,0) rotate(0) \"><path d=\"M0,0 A 4,4 0 0,1 0,-8\"><\/path><\/g><g transform=\"translate(49.16,0) rotate(0) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(60.39,0) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.32249999999999995\" data-y=\"0.3415\" data-xypic-id=\"0\" transform=\"translate(400,-91.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-1384-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.4225\" data-y=\"0.3385\" data-xypic-id=\"1\" transform=\"translate(4145,-88.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1384-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>and morphisms are commutative diagrams like the one above.<\/p>\n<\/li>\n<li><p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category, and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1424-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1424-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A<\/script> be an object in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>. We will define a category <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.134ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2711.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-286-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-286-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-286-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-286-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-286-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-286-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(554,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-286-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1183.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-286-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1572.1,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-286-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2322.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-286-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">k_\\mathscr{C}(A)<\/script> whose objects are all morphisms from <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1424-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1424-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A<\/script>. Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"35.515ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 15697.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-282-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-282-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-282-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-282-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-282-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-282-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 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data-mml-node=\"mo\" transform=\"translate(6033.2,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-282-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6477.9,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-282-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7236.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-282-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7625.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-282-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8070.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-282-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-282-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9274.9,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-282-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(10219.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-282-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-282-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-282-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-282-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12964.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-282-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13353.8,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-282-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14103.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-282-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(14548.5,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-282-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15308.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-282-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f_1\\in \\operatorname{Hom}_\\mathscr C(A,B),f_2\\in \\operatorname{Hom}_\\mathscr C(A,C)<\/script>. Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.196ex\" height=\"2.497ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7158.8 1103.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.8ex;\"><defs><path id=\"MJX-283-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-283-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 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data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-283-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-283-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-283-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-176.7) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-283-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(554,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-283-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1183.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-283-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1572.1,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-283-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2322.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-283-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4083.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-283-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4472.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-283-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-283-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5398.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-283-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5843.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-283-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-283-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6769.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-283-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}_{k_\\mathscr{C}(A)}(f_1,f_2)<\/script> be the collection of morphisms <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.259ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7186.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-284-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 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61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-284-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-284-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-284-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-284-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-284-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-284-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-284-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-284-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-284-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(754.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-284-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1699.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-284-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-284-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-284-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-284-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4444.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-284-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4833.7,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-284-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5592.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-284-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6037.3,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-284-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6797.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-284-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">g\\in \\operatorname{Hom}_{\\mathscr C}(B,C)<\/script> with <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.289ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 3663.7 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-285-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-285-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-285-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-285-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-285-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-285-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(477,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-285-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-285-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1681.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-285-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2737.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-285-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-285-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">gf_1=f_2<\/script>. We also denote <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.134ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2711.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-286-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-286-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-286-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-286-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-286-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-286-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(554,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-286-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1183.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-286-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1572.1,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-286-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2322.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-286-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">k_\\mathscr{C}(A)<\/script> as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.434ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5495.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-287-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-287-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-287-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-287-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-287-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-287-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-287-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-287-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-287-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-287-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-287-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-287-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-287-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2745.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-287-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3134.1,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-287-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3884.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-287-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4328.8,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-287-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5106.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-287-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}_{\\mathscr C}(A,-)<\/script>.<\/p>\n<p>Like the last example, we can treat the objects of this category as diagrams like<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n93\" cid=\"n93\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container math-jax-postprocess\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"19.744ex\" height=\"12.088ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -958 8727 5343\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -9.921ex;\"><defs><path id=\"MJX-1385-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-1385-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-1385-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1385-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-1385-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-1385-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"xypic\" data-cmd-id=\"0\" data-text-mml-ids=\"[0,1,2,3,4,5,6,7]\"><g fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-width=\"0.64\" transform=\"translate(779.5,250) scale(62.5, -62.5)\"><g transform=\"translate(0,0) \"><\/g><line x1=\"0\" y1=\"11.33\" x2=\"0\" y2=\"50.08\" stroke-dasharray=\"\"><\/line><g transform=\"translate(0,50.08) rotate(90) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(11.5,0) \"><\/g><g transform=\"translate(52.47,0) \"><\/g><g transform=\"translate(104.95,0) \"><\/g><line x1=\"104.95\" y1=\"11.33\" x2=\"104.95\" y2=\"49.73\" stroke-dasharray=\"\"><\/line><g transform=\"translate(104.95,49.73) rotate(90) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(11.5,0) \"><\/g><g transform=\"translate(0,61.14) \"><\/g><g transform=\"translate(52.47,61.14) \"><\/g><g transform=\"translate(104.95,61.14) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.37499999999999994\" data-y=\"0.358\" data-xypic-id=\"0\" transform=\"translate(404.5,-108)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1385-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"0.3914213562373095\" data-y=\"2.0875266952966367\" data-xypic-id=\"1\" transform=\"translate(1170.9,-1837.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1385-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1385-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"6.1845\" data-y=\"0.358\" data-xypic-id=\"3\" transform=\"translate(6964,-108)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1385-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"6.950921356237309\" data-y=\"2.0875266952966367\" data-xypic-id=\"4\" transform=\"translate(7730.4,-1837.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1385-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1385-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.3795\" data-y=\"4.162999999999999\" data-xypic-id=\"5\" transform=\"translate(400,-3913)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1385-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"6.1795\" data-y=\"4.162999999999999\" data-xypic-id=\"7\" transform=\"translate(6959,-3913)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-1385-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>And morphisms are commutative diagrams like<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n95\" cid=\"n95\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container math-jax-postprocess\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.637ex\" height=\"13.054ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -958 6027.5 5769.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -10.887ex;\"><defs><path id=\"MJX-1386-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-1386-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-1386-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1386-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-1386-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-1386-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-1386-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"xypic\" data-cmd-id=\"0\" data-text-mml-ids=\"[0,1,2,3,4,5,6]\"><g fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-width=\"0.64\" transform=\"translate(1388,250) scale(62.5, -62.5)\"><g transform=\"translate(0,0) \"><\/g><line x1=\"0\" y1=\"11.33\" x2=\"0\" y2=\"50.08\" stroke-dasharray=\"\"><\/line><g transform=\"translate(0,50.08) rotate(90) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(-11.5,0) \"><\/g><line x1=\"11.44\" y1=\"11.33\" x2=\"50.22\" y2=\"49.73\" stroke-dasharray=\"\"><\/line><g transform=\"translate(50.22,49.73) rotate(44.72) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(10.23,-10.34) \"><\/g><g transform=\"translate(61.75,0) \"><\/g><g transform=\"translate(0,61.14) \"><\/g><line x1=\"11.67\" y1=\"61.14\" x2=\"50.07\" y2=\"61.14\" stroke-dasharray=\"\"><\/line><g transform=\"translate(50.07,61.14) rotate(0) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(0,9.92) \"><\/g><g transform=\"translate(61.75,61.14) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.37499999999999994\" data-y=\"0.358\" data-xypic-id=\"0\" transform=\"translate(1013,-108)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1386-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-1.0465935418572012\" data-y=\"2.0875266952966367\" data-xypic-id=\"1\" transform=\"translate(341.4,-1837.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1386-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1386-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"2.2418134962251237\" data-y=\"1.441516597460486\" data-xypic-id=\"2\" transform=\"translate(3629.8,-1191.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1386-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1386-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.3795\" data-y=\"4.162999999999999\" data-xypic-id=\"4\" transform=\"translate(1008.5,-3913)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1386-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.7611050326870084\" data-y=\"4.525462553521763\" data-xypic-id=\"5\" transform=\"translate(3149.1,-4275.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-1386-TEX-I-1D454\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.4795000000000003\" data-y=\"4.162999999999999\" data-xypic-id=\"6\" transform=\"translate(4867.5,-3913)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-1386-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<\/li>\n<li><p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category, and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.42ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 1953.7 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-289-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-289-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-289-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-289-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(750,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-289-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1194.7,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-289-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A,B<\/script> be objects in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>. We will define a category <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.553ex\" height=\"2.303ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 2012.6 1017.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.656ex;\"><defs><path id=\"MJX-291-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-291-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-291-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-291-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-291-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(699,-152.7) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-291-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(750,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-291-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1028,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-291-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C_{A,B}<\/script> like last two examples.<\/p>\n<p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.024ex\" height=\"2.353ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4430.6 1039.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.656ex;\"><defs><path id=\"MJX-292-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-292-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-292-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 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data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-292-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(699,-152.7) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-292-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(750,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-292-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1028,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-292-TEX-I-1D435\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4041.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-292-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr C_{A,B})<\/script> consists of diagrams<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n99\" cid=\"n99\" 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data-x=\"10.618776695296637\" data-y=\"3.8745\" data-xypic-id=\"9\" transform=\"translate(11578.6,-3624.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"13.718776695296636\" data-y=\"4.141\" data-xypic-id=\"10\" transform=\"translate(14678.6,-3891)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1388-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(716,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1388-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.6597766952966375\" data-y=\"8.074\" data-xypic-id=\"13\" transform=\"translate(4619.6,-7824)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1388-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"17.938330085889913\" data-y=\"8.074\" data-xypic-id=\"17\" transform=\"translate(18898.1,-7824)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1388-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>are commutative diagrams<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n104\" cid=\"n104\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container math-jax-postprocess\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"22.62ex\" height=\"20.887ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -958 9998.1 9232\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -18.719ex;\"><defs><path id=\"MJX-1389-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-1389-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-1389-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1389-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-1389-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-1389-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"xypic\" data-cmd-id=\"0\" data-text-mml-ids=\"[0,1,2,3,4,5,6,7,8,9]\"><g fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-width=\"0.64\" transform=\"translate(959.8,250) scale(62.5, -62.5)\"><g transform=\"translate(0,0) \"><\/g><g 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y2=\"11.13\" stroke-dasharray=\"\"><\/line><g transform=\"translate(120.54,11.13) rotate(-43.81) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><line x1=\"80.35\" y1=\"74.26\" x2=\"120.56\" y2=\"112.66\" stroke-dasharray=\"\"><\/line><g transform=\"translate(120.56,112.66) rotate(43.69) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(132.14,61.99) \"><\/g><g transform=\"translate(0,123.72) \"><\/g><g transform=\"translate(67.51,123.72) \"><\/g><g transform=\"translate(132.14,123.72) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"7.88383008588991\" data-y=\"0.358\" data-xypic-id=\"2\" transform=\"translate(8843.6,-108)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1389-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.559776695296637\" data-y=\"4.141\" data-xypic-id=\"3\" transform=\"translate(400,-3891)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1389-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(716,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1389-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.9078977092678775\" data-y=\"3.475300469169638\" data-xypic-id=\"4\" transform=\"translate(2867.7,-3225.3)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-1389-TEX-I-1D70E\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.659776695296637\" data-y=\"4.141\" data-xypic-id=\"5\" transform=\"translate(4619.6,-3891)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1389-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(716,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1389-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"7.87933008588991\" data-y=\"8.074\" data-xypic-id=\"9\" transform=\"translate(8839.1,-7824)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1389-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>.<\/p>\n<\/li>\n<li><p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category, and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"21.398ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 9457.9 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-296-TEX-I-1D6FC\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 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637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-296-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(917.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-296-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1473.6,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-296-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2501.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-296-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3779.1,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-296-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4539.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-296-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4983.8,0)\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-296-TEX-I-1D6FD\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5827.6,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-296-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6383.3,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-296-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7420.1,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-296-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8697.9,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-296-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\alpha:A\\to C,\\beta:B\\to C<\/script> be two morphisms in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>. We will define a category <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.069ex\" height=\"2.297ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 1798.3 1015.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-298-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 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-126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-298-TEX-I-1D6FD\" d=\"M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-298-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(699,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-298-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(640,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-298-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(918,0)\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-298-TEX-I-1D6FD\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C_{\\alpha,\\beta}<\/script>\u200b like examples above. <\/p>\n<p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.539ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4216.3 1037.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path 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648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-299-TEX-I-1D6FC\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z\"><\/path><path id=\"MJX-299-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-299-TEX-I-1D6FD\" d=\"M29 -194Q23 -188 23 -186Q23 -183 102 134T186 465Q208 533 243 584T309 658Q365 705 429 705H431Q493 705 533 667T573 570Q573 465 469 396L482 383Q533 332 533 252Q533 139 448 65T257 -10Q227 -10 203 -2T165 17T143 40T131 59T126 65L62 -188Q60 -194 42 -194H29ZM353 431Q392 431 427 419L432 422Q436 426 439 429T449 439T461 453T472 471T484 495T493 524T501 560Q503 569 503 593Q503 611 502 616Q487 667 426 667Q384 667 347 643T286 582T247 514T224 455Q219 439 186 308T152 168Q151 163 151 147Q151 99 173 68Q204 26 260 26Q302 26 349 51T425 137Q441 171 449 214T457 279Q457 337 422 372Q380 358 347 358H337Q258 358 258 389Q258 396 261 403Q275 431 353 431Z\"><\/path><path id=\"MJX-299-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-299-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-299-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-299-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-299-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2029,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-299-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(699,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-299-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(640,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-299-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(918,0)\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-299-TEX-I-1D6FD\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3827.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-299-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr C_{\\alpha,\\beta})<\/script> consists of commutative diagrams<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n109\" cid=\"n109\" mdtype=\"math_block\" 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xlink:href=\"#MJX-1392-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"11.73833008588991\" data-y=\"4.216\" data-xypic-id=\"9\" transform=\"translate(12698.1,-3966)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-1392-TEX-I-1D436\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"7.87933008588991\" data-y=\"8.074\" data-xypic-id=\"12\" transform=\"translate(8839.1,-7824)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1392-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"10.565544015888962\" data-y=\"6.561465299921925\" data-xypic-id=\"13\" transform=\"translate(11525.3,-6311.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D6FD\" xlink:href=\"#MJX-1392-TEX-I-1D6FD\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>\u200b.<\/p>\n<\/li>\n<li><p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category. We will define a category <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.751ex\" height=\"1.744ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -721 1216 771\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.113ex;\"><defs><path id=\"MJX-432-TEX-S-2133\" d=\"M112 -7Q86 -7 58 6T30 48T54 103T113 130Q129 130 141 121T153 94Q153 71 132 59T90 47H80Q95 30 133 30Q180 30 228 63T311 137T402 249T500 361Q566 425 703 529T910 693Q942 721 945 721T958 716T970 709Q974 704 964 691Q961 688 905 622T847 554L595 181Q553 121 527 77T496 19L492 5Q497 5 531 46Q579 98 685 224T850 409L972 524Q994 543 1004 556Q1012 567 1097 643T1186 720Q1194 720 1206 715T1215 703Q1215 701 1191 671T1133 599T1080 530Q1036 461 983 357T862 152Q802 64 799 17Q799 7 800 5T811 2Q836 2 882 37T969 126Q972 130 974 134T978 138T983 139T996 140H1012Q1018 134 1018 132Q1018 122 981 83T889 4T795 -35Q761 -35 745 -12T728 48Q728 122 781 190Q833 269 890 370L927 434L914 422Q848 360 752 245Q643 117 582 51T498 -33T461 -50Q424 -48 424 -4Q424 84 481 172L714 495Q591 406 523 333Q507 316 430 226T313 95Q263 48 221 24T162 -4T120 -7H112Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"2133\" xlink:href=\"#MJX-432-TEX-S-2133\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr M<\/script> whose objects are morphisms in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>; that is <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"18.945ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8373.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-499-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-499-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-499-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-499-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-499-TEX-S-2133\" d=\"M112 -7Q86 -7 58 6T30 48T54 103T113 130Q129 130 141 121T153 94Q153 71 132 59T90 47H80Q95 30 133 30Q180 30 228 63T311 137T402 249T500 361Q566 425 703 529T910 693Q942 721 945 721T958 716T970 709Q974 704 964 691Q961 688 905 622T847 554L595 181Q553 121 527 77T496 19L492 5Q497 5 531 46Q579 98 685 224T850 409L972 524Q994 543 1004 556Q1012 567 1097 643T1186 720Q1194 720 1206 715T1215 703Q1215 701 1191 671T1133 599T1080 530Q1036 461 983 357T862 152Q802 64 799 17Q799 7 800 5T811 2Q836 2 882 37T969 126Q972 130 974 134T978 138T983 139T996 140H1012Q1018 134 1018 132Q1018 122 981 83T889 4T795 -35Q761 -35 745 -12T728 48Q728 122 781 190Q833 269 890 370L927 434L914 422Q848 360 752 245Q643 117 582 51T498 -33T461 -50Q424 -48 424 -4Q424 84 481 172L714 495Q591 406 523 333Q507 316 430 226T313 95Q263 48 221 24T162 -4T120 -7H112Z\"><\/path><path id=\"MJX-499-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-499-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-499-TEX-N-4D\" d=\"M132 622Q125 629 121 631T105 634T62 637H29V683H135Q221 683 232 682T249 675Q250 674 354 398L458 124L562 398Q666 674 668 675Q671 681 683 682T781 683H887V637H854Q814 636 803 634T785 622V61Q791 51 802 49T854 46H887V0H876Q855 3 736 3Q605 3 596 0H585V46H618Q660 47 669 49T688 61V347Q688 424 688 461T688 546T688 613L687 632Q454 14 450 7Q446 1 430 1T410 7Q409 9 292 316L176 624V606Q175 588 175 543T175 463T175 356L176 86Q187 50 261 46H278V0H269Q254 3 154 3Q52 3 37 0H29V46H46Q78 48 98 56T122 69T132 86V622Z\"><\/path><path id=\"MJX-499-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-499-TEX-N-72\" d=\"M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z\"><\/path><path id=\"MJX-499-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-499-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-499-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-499-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-499-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2029,0)\"><g data-mml-node=\"mi\"><use data-c=\"2133\" xlink:href=\"#MJX-499-TEX-S-2133\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3245,0)\"><use data-c=\"29\" xlink:href=\"#MJX-499-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3911.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-499-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4967.6,0)\"><use data-c=\"4D\" xlink:href=\"#MJX-499-TEX-N-4D\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-499-TEX-N-6F\" transform=\"translate(917,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-499-TEX-N-72\" transform=\"translate(1417,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6776.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-499-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(7165.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-499-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7984.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-499-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr M)=\\operatorname{Mor}(\\mathscr C)<\/script>. Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"23.016ex\" height=\"2.084ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 10173 921\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-524-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-524-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-524-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-524-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-524-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-524-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><path id=\"MJX-524-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-524-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-524-TEX-I-1D436\" d=\"M50 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46Q499 46 581 128Q617 164 640 212T683 339T703 469Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-524-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-524-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1204.3,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-524-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1760.1,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-524-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2787.9,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-524-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4065.7,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-524-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4824.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-524-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5269.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-524-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-524-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6473.7,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-524-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7029.4,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-524-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8067.2,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-524-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9345,0)\"><use data-c=\"1D437\" xlink:href=\"#MJX-524-TEX-I-1D437\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f_1:A\\to B,f_2:C\\to D<\/script> are morphisms in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>. Then the morphisms from <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.096ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 926.6 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-517-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-517-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-517-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-517-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f_1<\/script> to <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.096ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 926.6 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-525-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-525-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-525-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-525-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f_2<\/script> are commutative diagrams<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n170\" 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stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.28492135623731\" data-y=\"2.079276695296637\" data-xypic-id=\"4\" transform=\"translate(5672.9,-1829.3)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1393-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1393-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.3795\" data-y=\"4.1465000000000005\" data-xypic-id=\"5\" transform=\"translate(1008.5,-3896.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1393-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.7431032470182743\" data-y=\"4.687153462380773\" data-xypic-id=\"6\" transform=\"translate(3131.1,-4437.2)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-1393-TEX-I-210E\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.4795000000000007\" data-y=\"4.1465000000000005\" data-xypic-id=\"7\" transform=\"translate(4867.5,-3896.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D437\" xlink:href=\"#MJX-1393-TEX-I-1D437\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>.<\/p>\n<\/li>\n\n<\/ul>\n<h3 id='17-definition'>1.7 Definition<\/h3>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category. A category <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.934ex\" height=\"1.672ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -708 855 739\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.07ex;\"><defs><path id=\"MJX-314-TEX-S-1D49F\" d=\"M251 272Q199 272 168 298T136 374Q136 432 175 496T267 603Q321 645 395 676T552 708Q610 708 655 694T724 659T763 618T784 584L790 569Q792 569 800 572T819 576T840 578Q855 578 855 571Q855 566 846 554T829 541Q824 541 817 539T804 533T799 529Q802 517 802 483Q802 376 742 256T585 53T392 -31Q353 -31 300 -7L289 -2L277 -6Q242 -17 192 -17Q141 -17 113 -13T77 -3T68 14Q68 20 73 28T93 45T131 58Q152 62 197 62Q241 62 272 48L282 44Q308 65 334 93Q380 145 467 266T622 454Q644 476 664 493T694 517T720 534T740 547Q699 671 540 671Q461 671 385 625T276 534Q265 520 251 499T219 438T202 368Q202 309 267 309Q325 309 396 383T491 545Q492 548 493 552Q494 558 496 559T511 564Q513 565 514 565Q523 568 527 568Q534 568 534 560Q534 554 529 537Q507 442 420 357T251 272ZM332 20Q332 19 341 15T368 6T400 2Q425 2 457 13T531 49T614 125T690 248Q750 376 750 476V490L738 479Q698 436 646 366T554 239T455 121T332 20ZM226 20Q226 25 186 25Q181 25 174 24H166Q161 24 158 23H152Q170 21 197 21Q205 21 213 21T223 20H226Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49F\" xlink:href=\"#MJX-314-TEX-S-1D49F\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr D<\/script> is called a <strong>subcategory<\/strong> of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> if<\/p>\n<ul>\n<li><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.746ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7843.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-305-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-305-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-305-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-305-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-305-TEX-S-1D49F\" d=\"M251 272Q199 272 168 298T136 374Q136 432 175 496T267 603Q321 645 395 676T552 708Q610 708 655 694T724 659T763 618T784 584L790 569Q792 569 800 572T819 576T840 578Q855 578 855 571Q855 566 846 554T829 541Q824 541 817 539T804 533T799 529Q802 517 802 483Q802 376 742 256T585 53T392 -31Q353 -31 300 -7L289 -2L277 -6Q242 -17 192 -17Q141 -17 113 -13T77 -3T68 14Q68 20 73 28T93 45T131 58Q152 62 197 62Q241 62 272 48L282 44Q308 65 334 93Q380 145 467 266T622 454Q644 476 664 493T694 517T720 534T740 547Q699 671 540 671Q461 671 385 625T276 534Q265 520 251 499T219 438T202 368Q202 309 267 309Q325 309 396 383T491 545Q492 548 493 552Q494 558 496 559T511 564Q513 565 514 565Q523 568 527 568Q534 568 534 560Q534 554 529 537Q507 442 420 357T251 272ZM332 20Q332 19 341 15T368 6T400 2Q425 2 457 13T531 49T614 125T690 248Q750 376 750 476V490L738 479Q698 436 646 366T554 239T455 121T332 20ZM226 20Q226 25 186 25Q181 25 174 24H166Q161 24 158 23H152Q170 21 197 21Q205 21 213 21T223 20H226Z\"><\/path><path id=\"MJX-305-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-305-TEX-N-2282\" d=\"M84 250Q84 372 166 450T360 539Q361 539 370 539T395 539T430 540T475 540T524 540H679Q694 532 694 520Q694 511 681 501L522 500H470H441Q366 500 338 496T266 472Q244 461 224 446T179 404T139 337T124 250V245Q124 157 185 89Q244 25 328 7Q348 2 366 2T522 0H681Q694 -10 694 -20Q694 -32 679 -40H526Q510 -40 480 -40T434 -41Q350 -41 289 -25T172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-305-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-305-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-305-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-305-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-305-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2029,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49F\" xlink:href=\"#MJX-305-TEX-S-1D49F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2884,0)\"><use data-c=\"29\" xlink:href=\"#MJX-305-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3550.8,0)\"><use data-c=\"2282\" xlink:href=\"#MJX-305-TEX-N-2282\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4606.6,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-305-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-305-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-305-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6246.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-305-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(6635.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-305-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7454.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-305-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr D)\\subset \\operatorname{Obj}(\\mathscr C)<\/script>;<\/li>\n<li><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" 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data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-306-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(17030.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-306-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17419.8,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-306-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18169.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-306-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(18614.5,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-306-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19373.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-306-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\forall A,B\\in \\operatorname{Obj}(\\mathscr D),\\operatorname{Hom}_{\\mathscr D}(A,B)\\subset \\operatorname{Hom}_{\\mathscr C}(A,B)<\/script>\u200b;<\/li>\n<li>the composition in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.934ex\" height=\"1.672ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -708 855 739\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.07ex;\"><defs><path id=\"MJX-314-TEX-S-1D49F\" d=\"M251 272Q199 272 168 298T136 374Q136 432 175 496T267 603Q321 645 395 676T552 708Q610 708 655 694T724 659T763 618T784 584L790 569Q792 569 800 572T819 576T840 578Q855 578 855 571Q855 566 846 554T829 541Q824 541 817 539T804 533T799 529Q802 517 802 483Q802 376 742 256T585 53T392 -31Q353 -31 300 -7L289 -2L277 -6Q242 -17 192 -17Q141 -17 113 -13T77 -3T68 14Q68 20 73 28T93 45T131 58Q152 62 197 62Q241 62 272 48L282 44Q308 65 334 93Q380 145 467 266T622 454Q644 476 664 493T694 517T720 534T740 547Q699 671 540 671Q461 671 385 625T276 534Q265 520 251 499T219 438T202 368Q202 309 267 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data-c=\"62\" xlink:href=\"#MJX-309-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-309-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4168.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-309-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4557.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49F\" xlink:href=\"#MJX-309-TEX-S-1D49F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5412.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-309-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5801.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-309-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6246.2,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-309-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,-152.7) scale(0.707)\"><use data-c=\"1D434\" xlink:href=\"#MJX-309-TEX-I-1D434\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7637.3,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-309-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8582.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-309-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-309-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-309-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49F\" xlink:href=\"#MJX-309-TEX-S-1D49F\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11352.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-309-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11741.7,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-309-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12491.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-309-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12936.4,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-309-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13686.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-309-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\forall A\\in \\operatorname{Obj}(\\mathscr D),1_A\\in \\operatorname{Hom}_{\\mathscr D}(A,A)<\/script>, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.519ex\" height=\"1.852ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 1113.3 818.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.345ex;\"><defs><path id=\"MJX-310-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-310-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-310-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(533,-152.7) scale(0.707)\"><use data-c=\"1D434\" xlink:href=\"#MJX-310-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">1_A<\/script> is the identity morphism of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1424-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1424-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A<\/script> in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 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xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>.<\/li>\n\n<\/ul>\n<p>And if <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"44.711ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 19762.5 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-313-TEX-N-2200\" d=\"M0 673Q0 684 7 689T20 694Q32 694 38 680T82 567L126 451H430L473 566Q483 593 494 622T512 668T519 685Q524 694 538 694Q556 692 556 674Q556 670 426 329T293 -15Q288 -22 278 -22T263 -15Q260 -11 131 328T0 673ZM414 410Q414 411 278 411T142 410L278 55L414 410Z\"><\/path><path id=\"MJX-313-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 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transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-313-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(17030.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-313-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17419.8,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-313-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18169.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-313-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(18614.5,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-313-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(19373.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-313-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\forall A,B\\in \\operatorname{Obj}(\\mathscr D),\\operatorname{Hom}_{\\mathscr D}(A,B)= \\operatorname{Hom}_{\\mathscr C}(A,B)<\/script>, we call <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.934ex\" height=\"1.672ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -708 855 739\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.07ex;\"><defs><path id=\"MJX-314-TEX-S-1D49F\" d=\"M251 272Q199 272 168 298T136 374Q136 432 175 496T267 603Q321 645 395 676T552 708Q610 708 655 694T724 659T763 618T784 584L790 569Q792 569 800 572T819 576T840 578Q855 578 855 571Q855 566 846 554T829 541Q824 541 817 539T804 533T799 529Q802 517 802 483Q802 376 742 256T585 53T392 -31Q353 -31 300 -7L289 -2L277 -6Q242 -17 192 -17Q141 -17 113 -13T77 -3T68 14Q68 20 73 28T93 45T131 58Q152 62 197 62Q241 62 272 48L282 44Q308 65 334 93Q380 145 467 266T622 454Q644 476 664 493T694 517T720 534T740 547Q699 671 540 671Q461 671 385 625T276 534Q265 520 251 499T219 438T202 368Q202 309 267 309Q325 309 396 383T491 545Q492 548 493 552Q494 558 496 559T511 564Q513 565 514 565Q523 568 527 568Q534 568 534 560Q534 554 529 537Q507 442 420 357T251 272ZM332 20Q332 19 341 15T368 6T400 2Q425 2 457 13T531 49T614 125T690 248Q750 376 750 476V490L738 479Q698 436 646 366T554 239T455 121T332 20ZM226 20Q226 25 186 25Q181 25 174 24H166Q161 24 158 23H152Q170 21 197 21Q205 21 213 21T223 20H226Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49F\" xlink:href=\"#MJX-314-TEX-S-1D49F\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr D<\/script> is a <strong>full subcategory<\/strong> of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 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xlink:href=\"#MJX-316-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1478.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-316-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"30\" xlink:href=\"#MJX-316-TEX-N-30\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2487.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-316-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(S,x_0)<\/script> in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.637ex\" height=\"1.728ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -752.9 2049.6 763.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-319-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-319-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-319-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-319-TEX-N-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-319-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-319-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-319-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1646,424.1) scale(0.707)\"><use data-c=\"2217\" xlink:href=\"#MJX-319-TEX-N-2217\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}^*<\/script> as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.77ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3876.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-318-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-318-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 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314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-318-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-318-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-318-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><path id=\"MJX-318-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-318-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-318-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1034,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-318-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1478.7,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-318-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1978.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-318-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"30\" xlink:href=\"#MJX-318-TEX-N-30\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2987.2,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-318-TEX-N-7D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3487.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-318-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(S,\\{x_0\\})<\/script>, then <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.637ex\" height=\"1.728ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -752.9 2049.6 763.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-319-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-319-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-319-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-319-TEX-N-2217\" d=\"M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-319-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-319-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-319-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1646,424.1) scale(0.707)\"><use data-c=\"2217\" xlink:href=\"#MJX-319-TEX-N-2217\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}^*<\/script> is a full subcategory of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.637ex\" height=\"2.05ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -895 2049.6 906\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-320-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-320-TEX-B-1D41E\" d=\"M32 225Q32 332 102 392T272 452H283Q382 452 436 401Q494 343 494 243Q494 226 486 222T440 217Q431 217 394 217T327 218H175V209Q175 177 179 154T196 107T236 69T306 50Q312 49 323 49Q376 49 410 85Q421 99 427 111T434 127T442 133T463 135H468Q494 135 494 117Q494 110 489 97T468 66T431 32T373 5T292 -6Q181 -6 107 55T32 225ZM383 276Q377 346 348 374T280 402Q253 402 230 390T195 357Q179 331 176 279V266H383V276Z\"><\/path><path id=\"MJX-320-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-320-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-320-TEX-B-1D412\"><\/use><use data-c=\"1D41E\" xlink:href=\"#MJX-320-TEX-B-1D41E\" transform=\"translate(639,0)\"><\/use><use data-c=\"1D42D\" xlink:href=\"#MJX-320-TEX-B-1D42D\" transform=\"translate(1166,0)\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(1646,424.1) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-320-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathbf{Set}^2<\/script>\u200b.<\/p>\n<h3 id='18-definition'>1.8 Definition<\/h3>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category. We will define a category <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.861ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 1706.4 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-334-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-334-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-334-TEX-N-70\" d=\"M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 -194H20V-148H36ZM424 218Q424 292 390 347T305 402Q234 402 182 337V98Q222 26 294 26Q345 26 384 80T424 218Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-334-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(909.7,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"6F\" xlink:href=\"#MJX-334-TEX-N-6F\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-334-TEX-N-70\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C^{\\mathrm{op}}<\/script> called the <strong>opposite category<\/strong> of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>.<\/p>\n<ul>\n<li><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"19.672ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 8694.9 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-365-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-365-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-365-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-365-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-365-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-365-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-365-TEX-N-70\" d=\"M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 -194H20V-148H36ZM424 218Q424 292 390 347T305 402Q234 402 182 337V98Q222 26 294 26Q345 26 384 80T424 218Z\"><\/path><path id=\"MJX-365-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-365-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"4F\" xlink:href=\"#MJX-365-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-365-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-365-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1640,0)\"><use data-c=\"28\" xlink:href=\"#MJX-365-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2029,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-365-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(909.7,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"6F\" xlink:href=\"#MJX-365-TEX-N-6F\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-365-TEX-N-70\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3735.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-365-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4402.1,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-365-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5457.9,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-365-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-365-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-365-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7097.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-365-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(7486.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-365-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8305.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-365-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Obj}(\\mathscr C^{\\mathrm{op}})=\\operatorname{Obj}(\\mathscr C)<\/script>;<\/li>\n<li><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"29.219ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 12914.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-415-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-415-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-415-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 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249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-415-TEX-N-70\" d=\"M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 -194H20V-148H36ZM424 218Q424 292 390 347T305 402Q234 402 182 337V98Q222 26 294 26Q345 26 384 80T424 218Z\"><\/path><path id=\"MJX-415-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 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336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-415-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-415-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-415-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-415-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(909.7,289) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"6F\" xlink:href=\"#MJX-415-TEX-N-6F\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-415-TEX-N-70\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3372.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-415-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3761.6,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-415-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4511.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-415-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4956.2,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-415-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5715.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-415-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6382,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-415-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7437.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-415-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-415-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-415-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-415-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10182.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-415-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10571.9,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-415-TEX-I-1D435\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11330.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-415-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11775.6,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-415-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12525.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-415-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}_{\\mathscr C^{\\mathrm{op}}}(A,B)=\\operatorname{Hom}_{\\mathscr C}(B,A)<\/script>;<\/li>\n<li><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.815ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 6106.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-411-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-411-TEX-N-2218\" d=\"M55 251Q55 328 112 386T249 444T386 388T444 249Q444 171 388 113T250 55Q170 55 113 112T55 251ZM245 403Q188 403 142 361T96 250Q96 183 141 140T250 96Q284 96 313 109T354 135T375 160Q403 197 403 250Q403 313 360 358T245 403Z\"><\/path><path id=\"MJX-411-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-411-TEX-N-70\" d=\"M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 -194H20V-148H36ZM424 218Q424 292 390 347T305 402Q234 402 182 337V98Q222 26 294 26Q345 26 384 80T424 218Z\"><\/path><path id=\"MJX-411-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-411-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-411-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(772.2,0)\"><g data-mml-node=\"mo\"><use data-c=\"2218\" xlink:href=\"#MJX-411-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(533,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"6F\" xlink:href=\"#MJX-411-TEX-N-6F\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-411-TEX-N-70\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(2324.1,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-411-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3078.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-411-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4134.7,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-411-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4833.9,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-411-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5556.1,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-411-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f \\circ^{\\mathrm{op}}g=g\\circ f<\/script>\u200b, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.008ex\" height=\"1.538ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -679.8 1329.7 679.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-422-TEX-N-2218\" d=\"M55 251Q55 328 112 386T249 444T386 388T444 249Q444 171 388 113T250 55Q170 55 113 112T55 251ZM245 403Q188 403 142 361T96 250Q96 183 141 140T250 96Q284 96 313 109T354 135T375 160Q403 197 403 250Q403 313 360 358T245 403Z\"><\/path><path id=\"MJX-422-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-422-TEX-N-70\" d=\"M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 -194H20V-148H36ZM424 218Q424 292 390 347T305 402Q234 402 182 337V98Q222 26 294 26Q345 26 384 80T424 218Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"mo\"><use data-c=\"2218\" xlink:href=\"#MJX-422-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(533,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"6F\" xlink:href=\"#MJX-422-TEX-N-6F\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-422-TEX-N-70\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\circ^{\\mathrm{op}}<\/script> denotes the composition in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.861ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 1706.4 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-334-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-334-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-334-TEX-N-70\" d=\"M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 -194H20V-148H36ZM424 218Q424 292 390 347T305 402Q234 402 182 337V98Q222 26 294 26Q345 26 384 80T424 218Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-334-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(909.7,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"6F\" xlink:href=\"#MJX-334-TEX-N-6F\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-334-TEX-N-70\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C^{\\mathrm{op}}<\/script>\u200b.<\/li>\n\n<\/ul>\n<h2 id='2-universal-properties'>2. Universal Properties<\/h2>\n<p>Some math books use the concept, universal properties, but with few explanations. Actually, universal properties are initial or final objects in a certain category.<\/p>\n<h3 id='21-definition'>2.1 Definition<\/h3>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category. If we call a object <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.786ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5209.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-586-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-586-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-586-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-586-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-586-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-586-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-586-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-586-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-586-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1027.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-586-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.6,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-586-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-586-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-586-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3612.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-586-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4001.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-586-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4820.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-586-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A\\in \\operatorname{Obj}(\\mathscr C)<\/script> an <strong>initial object<\/strong>, if for all <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.725ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5182.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-602-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-602-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-602-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-602-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-602-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-602-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-602-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-602-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-602-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1000.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-602-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1945.6,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-602-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-602-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-602-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3585.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-602-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3974.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-602-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4793.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-602-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">Z\\in\\operatorname{Obj}(\\mathscr C)<\/script>, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.309ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5440.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-604-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-604-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-604-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-604-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-604-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-604-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-604-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-604-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-604-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-604-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-604-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-604-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-604-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2745.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-604-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3134.1,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-604-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3884.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-604-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4328.8,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-604-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5051.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-604-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}_{\\mathscr C}(A,Z)<\/script> is a one-point set. If we call a object <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.786ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5209.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-586-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-586-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-586-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-586-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-586-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 200 652T217 609Q217 579 200 564T158 549Q133 549 116 564T98 609ZM28 -163Q58 -168 64 -168Q124 -168 135 -77Q137 -65 137 141T136 353Q132 371 120 377T72 385H52V408Q52 431 54 431L58 432Q62 432 70 432T87 433T108 434T133 436Q151 437 171 438T202 441T214 442H218V184Q217 -36 217 -59T211 -98Q195 -145 153 -175T58 -205Q9 -205 -23 -179T-55 -117Q-55 -94 -40 -79T-2 -64T36 -79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-586-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-586-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-586-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-586-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1027.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-586-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.6,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-586-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-586-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-586-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3612.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-586-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4001.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-586-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4820.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-586-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A\\in \\operatorname{Obj}(\\mathscr C)<\/script> a <strong>final object<\/strong>, if for all <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.725ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5182.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-602-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-602-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-602-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 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-79T52 -118Q52 -143 28 -163Z\"><\/path><path id=\"MJX-602-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-602-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-602-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-602-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1000.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-602-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1945.6,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-602-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-602-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-602-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3585.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-602-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3974.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-602-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4793.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-602-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">Z\\in\\operatorname{Obj}(\\mathscr C)<\/script>, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.309ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5440.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-607-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-607-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-607-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-607-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(2116,-161.2) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-607-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2745.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-607-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3134.1,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-607-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3857.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-607-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4301.8,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-607-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5051.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-607-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}_{\\mathscr C}(Z,A)<\/script> is a one-point set.<\/p>\n<h3 id='22-theorem'>2.2 Theorem<\/h3>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category. If there is an initial object <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.14ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 504 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-615-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-615-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">I<\/script> in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>, then <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.14ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 504 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-615-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-615-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">I<\/script> is the unique initial object in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> up to isomorphism.<\/p>\n<p>Similarly, if there is an final object <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.14ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 504 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-615-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-615-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">I<\/script> in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>, then <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.14ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 504 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-615-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-615-TEX-I-1D43C\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">I<\/script> is the unique final object in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> up to isomorphism.<\/p>\n<p><strong>Proof.<\/strong><\/p>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.972ex\" height=\"1.984ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2197.8 877\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-625-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-625-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-625-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-625-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-625-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-625-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(876.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-625-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1321.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-625-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-625-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">I_1,I_2<\/script> be initial objects in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>. Then we have <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"36.348ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 16065.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-650-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-650-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-650-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-650-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-650-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-650-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-650-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-650-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-650-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-650-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-650-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-650-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2083,0)\"><use data-c=\"28\" xlink:href=\"#MJX-650-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2472,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-650-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-650-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3348.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3793.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-650-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-650-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4669.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-650-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5058.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5503.4,0)\"><use data-c=\"48\" xlink:href=\"#MJX-650-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-650-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-650-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7586.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-650-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7975.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-650-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-650-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8852,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(9296.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-650-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-650-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10173.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-650-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10562.2,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11006.9,0)\"><use data-c=\"48\" xlink:href=\"#MJX-650-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-650-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-650-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13089.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-650-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(13478.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-650-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-650-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(14355.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(14800.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-650-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-650-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(15676.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-650-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}(I_1,I_1),\\operatorname{Hom}(I_1,I_2),\\operatorname{Hom}(I_2,I_1)<\/script> are one-point sets.<\/p>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"31.752ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 14034.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-674-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-674-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-674-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-674-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-674-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-674-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-674-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-674-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-674-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-674-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-674-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-674-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-674-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-674-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1772.6,0)\"><use data-c=\"48\" xlink:href=\"#MJX-674-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-674-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-674-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3855.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-674-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4244.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-674-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-674-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5121.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-674-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5565.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-674-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-674-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6442.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-674-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6831.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-674-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7276,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-674-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8030.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-674-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8975.6,0)\"><use data-c=\"48\" xlink:href=\"#MJX-674-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-674-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-674-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11058.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-674-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(11447.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-674-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-674-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12324.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-674-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(12768.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-674-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-674-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13645.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-674-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f\\in\\operatorname{Hom}(I_1,I_2),g\\in\\operatorname{Hom}(I_2,I_1)<\/script>, then we have <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.535ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7308.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-686-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-686-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-686-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-686-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-686-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-686-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-686-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-686-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-686-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-686-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-686-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-686-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(477,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-686-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1304.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-686-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2249.6,0)\"><use data-c=\"48\" xlink:href=\"#MJX-686-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-686-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-686-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4332.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-686-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4721.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-686-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-686-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5598.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-686-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6042.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-686-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-686-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6919.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-686-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">gf\\in \\operatorname{Hom}(I_1,I_1)<\/script>.<\/p>\n<p>Since <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"16.932ex\" height=\"2.276ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7484.1 1006.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.579ex;\"><defs><path id=\"MJX-707-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-707-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-707-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-707-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-707-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-707-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-707-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-707-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-707-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-707-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(533,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-707-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-707-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1480.6,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-707-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2425.4,0)\"><use data-c=\"48\" xlink:href=\"#MJX-707-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-707-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-707-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4508.4,0)\"><use data-c=\"28\" xlink:href=\"#MJX-707-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4897.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-707-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-707-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5773.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-707-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6218.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-707-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-707-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7095.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-707-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">1_{I_1}\\in \\operatorname{Hom}(I_1,I_1)<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.445ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5058.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-709-TEX-N-48\" d=\"M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 500V378H517V622Q510 629 506 631T490 634T447 637H414V683H425Q446 680 569 680Q704 680 713 683H724V637H691Q651 636 640 634T622 622V61Q628 51 639 49T691 46H724V0H713Q692 3 569 3Q434 3 425 0H414V46H447Q489 47 498 49T517 61V332H232V197L233 61Q239 51 250 49T302 46H335V0H324Q303 3 180 3Q45 3 36 0H25V46H58Q100 47 109 49T128 61V622Z\"><\/path><path id=\"MJX-709-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-709-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-709-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-709-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-709-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-709-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-709-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-709-TEX-N-48\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-709-TEX-N-6F\" transform=\"translate(750,0)\"><\/use><use data-c=\"6D\" xlink:href=\"#MJX-709-TEX-N-6D\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2083,0)\"><use data-c=\"28\" xlink:href=\"#MJX-709-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2472,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-709-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-709-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3348.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-709-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3793.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-709-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-709-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4669.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-709-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\operatorname{Hom}(I_1,I_1)<\/script> has only one element, we have <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.062ex\" height=\"2.174ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 3563.4 961.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.579ex;\"><defs><path id=\"MJX-716-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-716-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-716-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-716-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-716-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-716-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(477,0)\"><use data-c=\"1D453\" xlink:href=\"#MJX-716-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1304.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-716-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2360.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-716-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(533,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-716-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-716-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">gf=1_{I_1}<\/script>\u200b.<\/p>\n<p>Similarly, we have <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.062ex\" height=\"2.174ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 3563.4 961.1\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.579ex;\"><defs><path id=\"MJX-724-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-724-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><path id=\"MJX-724-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-724-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-724-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-724-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-724-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(550,0)\"><use data-c=\"1D454\" xlink:href=\"#MJX-724-TEX-I-1D454\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1304.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-724-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2360.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-724-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(533,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-724-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-724-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">fg=1_{I_2}<\/script>. Therefore <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.244ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 550 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-1183-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1183-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.079ex\" height=\"1.464ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 477 647\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-728-TEX-I-1D454\" d=\"M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D454\" xlink:href=\"#MJX-728-TEX-I-1D454\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">g<\/script> are isomorphisms. <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.76ex\" height=\"1.559ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -689 778 689\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-729-TEX-I-25FB\" d=\"M71 0Q59 4 55 16V346L56 676Q64 686 70 689H709Q719 681 722 674V15Q719 10 709 1L390 0H71ZM682 40V649H95V40H682Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"25FB\" xlink:href=\"#MJX-729-TEX-I-25FB\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\square<\/script><\/p>\n<p>The proof for final objects is similar to this.<\/p>\n<h3 id='23-example'>2.3 Example<\/h3>\n<ul>\n<li><p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1424-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1424-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A<\/script> be a set and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.695ex\" height=\"1.538ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -680 749 680\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-737-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D439\" xlink:href=\"#MJX-737-TEX-I-1D439\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">F<\/script> is the free abelian group on <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1424-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1424-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A<\/script>. Then we say <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.695ex\" height=\"1.538ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -680 749 680\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-737-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D439\" xlink:href=\"#MJX-737-TEX-I-1D439\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">F<\/script> has a universal property that for all abelian group <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.778ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 786 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-850-TEX-I-1D43A\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-850-TEX-I-1D43A\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">G<\/script> and map <mjx-container class=\"MathJax\" jax=\"SVG\" 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data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-747-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-747-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1383.6,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-747-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2411.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-747-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3689.1,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-747-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:A\\to B<\/script>, there is a unique group homomorphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.48ex\" height=\"1.493ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 654 660\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.493ex;\"><defs><path id=\"MJX-1350-TEX-I-1D711\" d=\"M92 210Q92 176 106 149T142 108T185 85T220 72L235 70L237 71L250 112Q268 170 283 211T322 299T370 375T429 423T502 442Q547 442 582 410T618 302Q618 224 575 152T457 35T299 -10Q273 -10 273 -12L266 -48Q260 -83 252 -125T241 -179Q236 -203 215 -212Q204 -218 190 -218Q159 -215 159 -185Q159 -175 214 -2L209 0Q204 2 195 5T173 14T147 28T120 46T94 71T71 103T56 142T50 190Q50 238 76 311T149 431H162Q183 431 183 423Q183 417 175 409Q134 361 114 300T92 210ZM574 278Q574 320 550 344T486 369Q437 369 394 329T323 218Q309 184 295 109L286 64Q304 62 306 62Q423 62 498 131T574 278Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D711\" xlink:href=\"#MJX-1350-TEX-I-1D711\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\varphi<\/script> making<\/p>\n<div contenteditable=\"false\" 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-7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(9.96,0) \"><\/g><g transform=\"translate(0,61.14) \"><\/g><g transform=\"translate(61.89,61.14) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.37499999999999994\" data-y=\"0.358\" data-xypic-id=\"0\" transform=\"translate(400,-108)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1394-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.1663737986093012\" data-y=\"2.6676728849685167\" data-xypic-id=\"1\" transform=\"translate(1941.4,-2417.7)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1394-TEX-I-1D453\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.8120240802453205\" data-y=\"-0.3991995308303615\" data-xypic-id=\"2\" transform=\"translate(2587,649.2)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-1394-TEX-I-1D456\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.4935\" data-y=\"0.3400000000000001\" data-xypic-id=\"3\" transform=\"translate(4268.5,-90)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D439\" xlink:href=\"#MJX-1394-TEX-I-1D439\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.259421356237309\" data-y=\"1.989945959492893\" data-xypic-id=\"4\" transform=\"translate(5034.4,-1739.9)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D711\" xlink:href=\"#MJX-1394-TEX-I-1D711\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.475\" data-y=\"4.162999999999999\" data-xypic-id=\"6\" transform=\"translate(4250,-3913)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-1394-TEX-I-1D43A\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>commute, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.781ex\" height=\"1.52ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -661 345 672\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-876-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-876-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">i<\/script>\u200b is the inclusion.<\/p>\n<p>In fact, we can regard this as an initial object in a category.<\/p>\n<p>Let&#39;s define a category <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>, whose objects are maps <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.994ex\" height=\"1.67ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 3091.6 738\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-847-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 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679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-847-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1027.8,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-847-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2305.6,0)\"><use data-c=\"1D43A\" xlink:href=\"#MJX-847-TEX-I-1D43A\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A\\to G<\/script>, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.778ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 786 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-850-TEX-I-1D43A\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-850-TEX-I-1D43A\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">G<\/script> is an abelian group.<\/p>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"35.024ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 15480.7 1000\" 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89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><path id=\"MJX-854-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-854-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-854-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1204.3,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-854-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1760.1,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-854-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2787.9,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-854-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4065.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-854-TEX-I-1D43A\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(819,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-854-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5288.2,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-854-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5732.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-854-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-854-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6937.2,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-854-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7493,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-854-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8520.8,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-854-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(9798.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-854-TEX-I-1D43A\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(819,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-854-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11298.9,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-854-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12243.7,0)\"><use data-c=\"4F\" xlink:href=\"#MJX-854-TEX-N-4F\"><\/use><use data-c=\"62\" xlink:href=\"#MJX-854-TEX-N-62\" transform=\"translate(778,0)\"><\/use><use data-c=\"6A\" xlink:href=\"#MJX-854-TEX-N-6A\" transform=\"translate(1334,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13883.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-854-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(14272.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-854-TEX-S-1D49E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(15091.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-854-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f_1:A\\to G_1,f_2:A\\to G_2\\in \\operatorname{Obj}(\\mathscr C)<\/script>. The morphisms <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.712ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 3408.7 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-817-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-817-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-817-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-817-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-817-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-817-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1204.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-817-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2482.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-817-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-817-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f_1\\to f_2<\/script> are commutative diagram<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n225\" cid=\"n225\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container math-jax-postprocess\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.57ex\" height=\"21.326ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1027.5 5997.9 9426\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -19.001ex;\"><defs><path id=\"MJX-1395-TEX-I-1D43A\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z\"><\/path><path id=\"MJX-1395-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1395-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-1395-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 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357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"xypic\" data-cmd-id=\"0\" data-text-mml-ids=\"[0,1,2,3,4,5,6,7,8]\"><g fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-width=\"0.64\" transform=\"translate(775,250) scale(62.5, -62.5)\"><g transform=\"translate(0,0) \"><\/g><g transform=\"translate(65.38,0) \"><\/g><g transform=\"translate(0,62.17) \"><\/g><line x1=\"11.6\" y1=\"51.14\" x2=\"52.3\" y2=\"12.44\" stroke-dasharray=\"\"><\/line><g transform=\"translate(52.3,12.44) rotate(-43.56) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(-10.02,-10.54) \"><\/g><line x1=\"11.6\" y1=\"73.2\" x2=\"52.3\" y2=\"111.9\" stroke-dasharray=\"\"><\/line><g transform=\"translate(52.3,111.9) rotate(43.56) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(-10.02,10.54) \"><\/g><g transform=\"translate(65.38,62.17) \"><\/g><g transform=\"translate(0,124.34) \"><\/g><g transform=\"translate(65.38,124.34) \"><\/g><line x1=\"65.38\" y1=\"111.9\" x2=\"65.38\" y2=\"12.44\" stroke-dasharray=\"\"><\/line><g transform=\"translate(65.38,12.44) rotate(-90) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(9.49,0) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.475\" data-y=\"0.2775\" data-xypic-id=\"1\" transform=\"translate(4250,-27.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-1395-TEX-I-1D43A\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(819,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1395-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.37499999999999994\" data-y=\"4.243499999999999\" data-xypic-id=\"2\" transform=\"translate(400,-3993.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1395-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.0892997941297444\" data-y=\"1.4609136891028804\" data-xypic-id=\"3\" transform=\"translate(1864.3,-1210.9)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1395-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1395-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.0892997941297442\" data-y=\"6.663639701490392\" data-xypic-id=\"4\" transform=\"translate(1864.3,-6413.6)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1395-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1395-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.475\" data-y=\"8.048499999999999\" data-xypic-id=\"7\" transform=\"translate(4250,-7798.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43A\" xlink:href=\"#MJX-1395-TEX-I-1D43A\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(819,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1395-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.477698051533946\" data-y=\"4.033992424049174\" data-xypic-id=\"8\" transform=\"translate(5252.7,-3784)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-1395-TEX-I-1D70E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.292ex\" height=\"1ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -431 571 442\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1181-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-1181-TEX-I-1D70E\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\sigma<\/script>\u200b is a group homomorphism.<\/p>\n<p>Then we can see, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.577ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 4233.1 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-865-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-865-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-865-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-865-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-865-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-865-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(622.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-865-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1178.6,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-865-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2206.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-865-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3484.1,0)\"><use data-c=\"1D439\" xlink:href=\"#MJX-865-TEX-I-1D439\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">i:A\\to F<\/script> is the initial object in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>.<\/p>\n<\/li>\n<li><p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-1064-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1064-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R<\/script> be a commutative ring with units, and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.143ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -470 500 505\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.079ex;\"><defs><path id=\"MJX-887-TEX-F-1D51E\" d=\"M86 363Q86 365 128 391T212 444L255 470Q256 470 263 465T283 453T308 442Q338 431 346 431Q350 431 358 436L414 465L421 459L418 434Q404 339 404 262Q404 147 408 119L425 97Q443 74 444 74L467 86L491 97Q492 97 497 83L436 28Q377 -26 374 -26L331 46L329 91L155 -35Q152 -35 114 10T68 65L67 86Q66 100 66 156Q66 177 66 198T68 236T71 268T74 296T77 320T80 338T83 351T86 360V363ZM314 365Q285 367 255 374T206 386L187 392Q181 392 172 376T154 311T145 197Q145 124 151 118Q156 110 187 78L208 56H214Q221 57 253 76L308 109L329 123V136Q328 143 328 257V365H314Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D51E\" xlink:href=\"#MJX-887-TEX-F-1D51E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathfrak a<\/script> is an proper ideal in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-1064-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1064-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R<\/script>. Then we say <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.143ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -470 500 505\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.079ex;\"><defs><path id=\"MJX-887-TEX-F-1D51E\" d=\"M86 363Q86 365 128 391T212 444L255 470Q256 470 263 465T283 453T308 442Q338 431 346 431Q350 431 358 436L414 465L421 459L418 434Q404 339 404 262Q404 147 408 119L425 97Q443 74 444 74L467 86L491 97Q492 97 497 83L436 28Q377 -26 374 -26L331 46L329 91L155 -35Q152 -35 114 10T68 65L67 86Q66 100 66 156Q66 177 66 198T68 236T71 268T74 296T77 320T80 338T83 351T86 360V363ZM314 365Q285 367 255 374T206 386L187 392Q181 392 172 376T154 311T145 197Q145 124 151 118Q156 110 187 78L208 56H214Q221 57 253 76L308 109L329 123V136Q328 143 328 257V365H314Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D51E\" xlink:href=\"#MJX-887-TEX-F-1D51E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathfrak a<\/script> has a universal property that for all ring homomorphisms <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.759ex\" height=\"1.765ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 4755.6 780\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-900-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-900-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-900-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-900-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-900-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-900-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(848.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-900-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1404.6,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-900-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2441.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-900-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3719.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-900-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(792,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-900-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\sigma:R\\to R'<\/script> with <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.092ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4460.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-910-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-910-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-910-TEX-F-1D51E\" d=\"M86 363Q86 365 128 391T212 444L255 470Q256 470 263 465T283 453T308 442Q338 431 346 431Q350 431 358 436L414 465L421 459L418 434Q404 339 404 262Q404 147 408 119L425 97Q443 74 444 74L467 86L491 97Q492 97 497 83L436 28Q377 -26 374 -26L331 46L329 91L155 -35Q152 -35 114 10T68 65L67 86Q66 100 66 156Q66 177 66 198T68 236T71 268T74 296T77 320T80 338T83 351T86 360V363ZM314 365Q285 367 255 374T206 386L187 392Q181 392 172 376T154 311T145 197Q145 124 151 118Q156 110 187 78L208 56H214Q221 57 253 76L308 109L329 123V136Q328 143 328 257V365H314Z\"><\/path><path id=\"MJX-910-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-910-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-910-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-910-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(571,0)\"><use data-c=\"28\" xlink:href=\"#MJX-910-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(960,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D51E\" xlink:href=\"#MJX-910-TEX-F-1D51E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1460,0)\"><use data-c=\"29\" xlink:href=\"#MJX-910-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2126.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-910-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3182.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-910-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3571.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-910-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4071.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-910-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\sigma(\\mathfrak a)=(0)<\/script>, there is a unique ring homomorphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.209ex\" height=\"2.283ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 5838.6 1009\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-926-TEX-I-1D711\" d=\"M92 210Q92 176 106 149T142 108T185 85T220 72L235 70L237 71L250 112Q268 170 283 211T322 299T370 375T429 423T502 442Q547 442 582 410T618 302Q618 224 575 152T457 35T299 -10Q273 -10 273 -12L266 -48Q260 -83 252 -125T241 -179Q236 -203 215 -212Q204 -218 190 -218Q159 -215 159 -185Q159 -175 214 -2L209 0Q204 2 195 5T173 14T147 28T120 46T94 71T71 103T56 142T50 190Q50 238 76 311T149 431H162Q183 431 183 423Q183 417 175 409Q134 361 114 300T92 210ZM574 278Q574 320 550 344T486 369Q437 369 394 329T323 218Q309 184 295 109L286 64Q304 62 306 62Q423 62 498 131T574 278Z\"><\/path><path id=\"MJX-926-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-926-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-926-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-926-TEX-F-1D51E\" d=\"M86 363Q86 365 128 391T212 444L255 470Q256 470 263 465T283 453T308 442Q338 431 346 431Q350 431 358 436L414 465L421 459L418 434Q404 339 404 262Q404 147 408 119L425 97Q443 74 444 74L467 86L491 97Q492 97 497 83L436 28Q377 -26 374 -26L331 46L329 91L155 -35Q152 -35 114 10T68 65L67 86Q66 100 66 156Q66 177 66 198T68 236T71 268T74 296T77 320T80 338T83 351T86 360V363ZM314 365Q285 367 255 374T206 386L187 392Q181 392 172 376T154 311T145 197Q145 124 151 118Q156 110 187 78L208 56H214Q221 57 253 76L308 109L329 123V136Q328 143 328 257V365H314Z\"><\/path><path id=\"MJX-926-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-926-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D711\" xlink:href=\"#MJX-926-TEX-I-1D711\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(931.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-926-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1487.6,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-926-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2246.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-926-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2746.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D51E\" xlink:href=\"#MJX-926-TEX-F-1D51E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3524.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-926-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(4802.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-926-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(792,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-926-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\varphi:R\/\\mathfrak a \\to R'<\/script> making<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n244\" cid=\"n244\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container math-jax-postprocess\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.52ex\" height=\"13.49ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1382.6 6418 5962.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -10.362ex;\"><defs><path id=\"MJX-1396-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-1396-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-1396-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-1396-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-1396-TEX-F-1D51E\" d=\"M86 363Q86 365 128 391T212 444L255 470Q256 470 263 465T283 453T308 442Q338 431 346 431Q350 431 358 436L414 465L421 459L418 434Q404 339 404 262Q404 147 408 119L425 97Q443 74 444 74L467 86L491 97Q492 97 497 83L436 28Q377 -26 374 -26L331 46L329 91L155 -35Q152 -35 114 10T68 65L67 86Q66 100 66 156Q66 177 66 198T68 236T71 268T74 296T77 320T80 338T83 351T86 360V363ZM314 365Q285 367 255 374T206 386L187 392Q181 392 172 376T154 311T145 197Q145 124 151 118Q156 110 187 78L208 56H214Q221 57 253 76L308 109L329 123V136Q328 143 328 257V365H314Z\"><\/path><path id=\"MJX-1396-TEX-I-1D711\" d=\"M92 210Q92 176 106 149T142 108T185 85T220 72L235 70L237 71L250 112Q268 170 283 211T322 299T370 375T429 423T502 442Q547 442 582 410T618 302Q618 224 575 152T457 35T299 -10Q273 -10 273 -12L266 -48Q260 -83 252 -125T241 -179Q236 -203 215 -212Q204 -218 190 -218Q159 -215 159 -185Q159 -175 214 -2L209 0Q204 2 195 5T173 14T147 28T120 46T94 71T71 103T56 142T50 190Q50 238 76 311T149 431H162Q183 431 183 423Q183 417 175 409Q134 361 114 300T92 210ZM574 278Q574 320 550 344T486 369Q437 369 394 329T323 218Q309 184 295 109L286 64Q304 62 306 62Q423 62 498 131T574 278Z\"><\/path><path id=\"MJX-1396-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"xypic\" data-cmd-id=\"0\" data-text-mml-ids=\"[0,1,2,3,4,5,6]\"><g fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-width=\"0.64\" transform=\"translate(779.5,250) scale(62.5, -62.5)\"><g transform=\"translate(0,0) \"><\/g><line x1=\"11.67\" y1=\"0\" x2=\"50.07\" y2=\"0\" stroke-dasharray=\"\"><\/line><g transform=\"translate(50.07,0) rotate(0) \"><path d=\"M-4.8,0 Q-8.35,0.32 -12.620000000000001,2.35\"><\/path><path d=\"M-4.8,0 Q-8.35,-0.32 -12.620000000000001,-2.35\"><\/path><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(0,-9.86) \"><\/g><line x1=\"11.67\" y1=\"10.68\" x2=\"56.81\" y2=\"52\" stroke-dasharray=\"\"><\/line><g transform=\"translate(56.81,52) rotate(42.47) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(-7.58,8.28) \"><\/g><g transform=\"translate(69.74,0) \"><\/g><line x1=\"69.74\" y1=\"14.8\" x2=\"69.74\" y2=\"52\" stroke-dasharray=\"7.2 7.2\"><\/line><g transform=\"translate(69.74,52) rotate(90) \"><path d=\"M0,0 Q-3.55,0.32 -7.82,2.35\"><\/path><path d=\"M0,0 Q-3.55,-0.32 -7.82,-2.35\"><\/path><\/g><g transform=\"translate(9.96,0) \"><\/g><g transform=\"translate(0,63.84) \"><\/g><g transform=\"translate(69.74,63.84) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.3795\" data-y=\"0.331\" data-xypic-id=\"0\" transform=\"translate(400,-81)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1396-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"2.0016626445315833\" data-y=\"-0.5286000717874997\" data-xypic-id=\"1\" transform=\"translate(2781.2,778.6)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1396-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.5040502351496088\" data-y=\"2.6608558573654375\" data-xypic-id=\"2\" transform=\"translate(2283.6,-2410.9)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-1396-TEX-I-1D70E\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.4795000000000003\" data-y=\"0.24999999999999997\" data-xypic-id=\"3\" transform=\"translate(4259,0)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1396-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(759,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-1396-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1259,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D51E\" xlink:href=\"#MJX-1396-TEX-F-1D51E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.750421356237309\" data-y=\"2.07419090885901\" data-xypic-id=\"4\" transform=\"translate(5529.9,-1824.2)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D711\" xlink:href=\"#MJX-1396-TEX-I-1D711\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.84077281758685\" data-y=\"4.358979797464467\" data-xypic-id=\"6\" transform=\"translate(4620.3,-4109)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msup\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1396-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(792,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-1396-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>commute, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.138ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 503 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-982-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-982-TEX-I-1D45D\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">p<\/script> is the quotient map.<\/p>\n<p>We can also see this as an initial object in a category like last example.<\/p>\n<\/li>\n<li><p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-1064-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1064-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R<\/script> be a commutative ring with units, and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.898ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6142.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1070-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-1070-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-1070-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-1070-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1070-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1070-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-1070-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1070-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1070-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(759,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-1070-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1037,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-1070-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1070-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2301.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1070-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2746.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-1070-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4084.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1070-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4529.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-1070-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1070-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5864.8,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-1070-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R[X_1,\\dots,X_n]<\/script> is a polynomial ring on <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-1064-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1064-TEX-I-1D445\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R<\/script>. Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.651ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 7801.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1052-TEX-C-58\" d=\"M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 413 551 442T648 509T705 561Q707 565 707 578Q707 610 682 614Q667 614 667 626Q667 641 695 662T755 683Q765 683 775 680T796 662T807 623Q807 596 792 572T713 499T530 376L505 361V356Q508 346 511 278T524 148T557 75Q569 69 580 69Q585 69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z\"><\/path><path id=\"MJX-1052-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-1052-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-1052-TEX-N-7B\" d=\"M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z\"><\/path><path id=\"MJX-1052-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-1052-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1052-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1052-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-1052-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1052-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"58\" xlink:href=\"#MJX-1052-TEX-C-58\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1084.8,0)\"><g data-mml-node=\"text\"><use data-c=\"3A\" xlink:href=\"#MJX-1052-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"text\" transform=\"translate(278,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1052-TEX-N-3D\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2418.6,0)\"><use data-c=\"7B\" xlink:href=\"#MJX-1052-TEX-N-7B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2918.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-1052-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1052-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4183.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1052-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4627.8,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-1052-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5966.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-1052-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1052-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7301.7,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-1052-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathcal X:=\\{X_1,\\dots X_n\\}<\/script> be the set of intermediates. We say <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.898ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6142.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1070-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-1070-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-1070-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-1070-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1070-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1070-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-1070-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1070-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1070-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(759,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-1070-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1037,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-1070-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1070-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2301.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1070-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2746.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-1070-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4084.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1070-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4529.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-1070-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-1070-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5864.8,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-1070-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">R[X_1,\\dots,X_n]<\/script> has a universal property that for all ring homomorphisms <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.759ex\" height=\"1.765ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 4755.6 780\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.048ex;\"><defs><path id=\"MJX-900-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-900-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-900-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-900-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-900-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-900-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(848.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-900-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1404.6,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-900-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2441.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-900-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3719.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-900-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(792,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-900-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\sigma:R\\to R'<\/script> and maps <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.82ex\" height=\"2.181ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 4782.6 964\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-1067-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-1067-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-1067-TEX-C-58\" d=\"M324 614Q291 576 250 573Q231 573 231 584Q231 589 232 592Q235 601 244 614T271 643T324 671T400 683H403Q462 683 481 610Q485 594 490 545T498 454L501 413Q504 413 551 442T648 509T705 561Q707 565 707 578Q707 610 682 614Q667 614 667 626Q667 641 695 662T755 683Q765 683 775 680T796 662T807 623Q807 596 792 572T713 499T530 376L505 361V356Q508 346 511 278T524 148T557 75Q569 69 580 69Q585 69 593 77Q624 108 660 110Q667 110 670 110T676 106T678 94Q668 59 624 30T510 0Q487 0 471 9T445 32T430 71T422 117T417 173Q416 183 416 188Q413 214 411 244T407 286T405 299Q403 299 344 263T223 182T154 122Q152 118 152 105Q152 69 180 69Q183 69 187 66T191 60L192 58V56Q192 41 163 21T105 0Q94 0 84 3T63 21T52 60Q52 77 56 90T85 131T155 191Q197 223 259 263T362 327T402 352L391 489Q391 492 390 505T387 526T384 547T379 568T372 586T361 602T348 611Q346 612 341 613T333 614H324Z\"><\/path><path id=\"MJX-1067-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-1067-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-1067-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1067-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(827.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-1067-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1383.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"58\" xlink:href=\"#MJX-1067-TEX-C-58\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2468.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-1067-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3746.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1067-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(792,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-1067-TEX-V-2032\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f:\\mathcal X\\to R'<\/script>, there is a unique ring homomorphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"23.128ex\" height=\"2.283ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 10222.4 1009\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1105-TEX-I-1D711\" d=\"M92 210Q92 176 106 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xlink:href=\"#MJX-1397-TEX-C-58\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.327384214863124\" data-y=\"6.81148895576408\" data-xypic-id=\"12\" transform=\"translate(4130.9,-6561.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-1397-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(378,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1397-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"7.879081983503071\" data-y=\"7.438482115168623\" data-xypic-id=\"13\" transform=\"translate(8682.6,-7188.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1397-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>commute, where <mjx-container 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xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 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data-x=\"-0.4034999999999999\" data-y=\"8.014979797464466\" data-xypic-id=\"5\" transform=\"translate(400,-7765)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"58\" xlink:href=\"#MJX-1398-TEX-C-58\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"2.379965258379284\" data-y=\"6.526048692808651\" data-xypic-id=\"6\" transform=\"translate(3183.5,-6276)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1398-TEX-I-1D453\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>, where <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.292ex\" height=\"1ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -431 571 442\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-1181-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-1181-TEX-I-1D70E\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\sigma<\/script> is a ring homomorphism and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: 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transform=\"translate(14802.6,-8001.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"58\" xlink:href=\"#MJX-1399-TEX-C-58\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"16.756118088123472\" data-y=\"6.766607760459868\" data-xypic-id=\"20\" transform=\"translate(17559.6,-6516.6)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1399-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1399-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>are commutative diagrams<\/p>\n<div contenteditable=\"false\" 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data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1400-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(792,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-1400-TEX-V-2032\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(792,-257.5) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1400-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"6.047527957955108\" data-y=\"3.561054130241481\" data-xypic-id=\"7\" transform=\"translate(6851,-3311.1)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D70B\" xlink:href=\"#MJX-1400-TEX-I-1D70B\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"7.799053390593275\" data-y=\"4.210979797464466\" data-xypic-id=\"8\" transform=\"translate(8602.6,-3961)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D445\" xlink:href=\"#MJX-1400-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(792,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-1400-TEX-V-2032\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(792,-257.5) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1400-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.4034999999999999\" data-y=\"8.251507322143686\" data-xypic-id=\"9\" transform=\"translate(400,-8001.5)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"58\" xlink:href=\"#MJX-1400-TEX-C-58\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"2.353564697530199\" data-y=\"6.76660776045987\" data-xypic-id=\"10\" transform=\"translate(3157.1,-6516.6)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1400-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1400-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.49599249000596\" data-y=\"7.440950076826124\" data-xypic-id=\"11\" transform=\"translate(5299.5,-7191)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1400-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1400-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>. As we can see, <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.303ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2785.8 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1263-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-1263-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 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95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-1263-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-1263-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(389,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-1263-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(378,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1263-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1170.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1263-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1615.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-1263-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(378,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1263-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2396.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1263-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(i_1,i_2)<\/script> is the initial object in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>.<\/p>\n<\/li>\n\n<\/ul>\n<h3 id='24-definition'>2.4 Definition<\/h3>\n<p>Let <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script> be a category, and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.42ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 1953.7 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-289-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-289-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-289-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-289-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(750,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-289-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1194.7,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-289-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A,B<\/script> are objects in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>. There are a object <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.636ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 723 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1423-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1423-TEX-I-1D44D\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">Z<\/script> and morphisms <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"22.753ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 10057 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-1303-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-1303-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1303-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-1303-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><path id=\"MJX-1303-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-1303-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-1303-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-1303-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-1303-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1303-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1303-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1217.3,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-1303-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1773.1,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1303-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2773.9,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-1303-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4051.7,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-1303-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4801.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1303-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5246.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1303-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1303-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6463.7,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-1303-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7019.4,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1303-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8020.2,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-1303-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9298,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-1303-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">p_1:Z\\to A,p_2:Z\\to B<\/script>, such that for all <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.63ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5140.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1332-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-1332-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-1332-TEX-N-4F\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM467 647Q426 665 388 665Q360 665 331 654T269 620T213 549T179 439Q174 411 174 354Q174 144 277 61Q327 20 385 20H389H391Q474 20 537 99Q603 188 603 354Q603 411 598 439Q577 592 467 647Z\"><\/path><path id=\"MJX-1332-TEX-N-62\" d=\"M307 -11Q234 -11 168 55L158 37Q156 34 153 28T147 17T143 10L138 1L118 0H98V298Q98 599 97 603Q94 622 83 628T38 637H20V660Q20 683 22 683L32 684Q42 685 61 686T98 688Q115 689 135 690T165 693T176 694H179V543Q179 391 180 391L183 394Q186 397 192 401T207 411T228 421T254 431T286 439T323 442Q401 442 461 379T522 216Q522 115 458 52T307 -11ZM182 98Q182 97 187 90T196 79T206 67T218 55T233 44T250 35T271 29T295 26Q330 26 363 46T412 113Q424 148 424 212Q424 287 412 323Q385 405 300 405Q270 405 239 390T188 347L182 339V98Z\"><\/path><path id=\"MJX-1332-TEX-N-6A\" d=\"M98 609Q98 637 116 653T160 669Q183 667 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xlink:href=\"#MJX-1332-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">L\\in \\operatorname{Obj}(\\mathscr C)<\/script> and morphisms <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"22.505ex\" height=\"2.084ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 9947 921\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-1346-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 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transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1346-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1204.3,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-1346-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1760.1,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-1346-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2718.9,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-1346-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3996.7,0)\"><use data-c=\"1D434\" xlink:href=\"#MJX-1346-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4746.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1346-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5191.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-1346-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1346-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6395.7,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-1346-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6951.4,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-1346-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7910.2,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-1346-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9188,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-1346-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">f_1:L\\to A,f_2:L\\to B<\/script>, there exists a unique morphism <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.48ex\" height=\"1.493ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 654 660\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.493ex;\"><defs><path id=\"MJX-1350-TEX-I-1D711\" d=\"M92 210Q92 176 106 149T142 108T185 85T220 72L235 70L237 71L250 112Q268 170 283 211T322 299T370 375T429 423T502 442Q547 442 582 410T618 302Q618 224 575 152T457 35T299 -10Q273 -10 273 -12L266 -48Q260 -83 252 -125T241 -179Q236 -203 215 -212Q204 -218 190 -218Q159 -215 159 -185Q159 -175 214 -2L209 0Q204 2 195 5T173 14T147 28T120 46T94 71T71 103T56 142T50 190Q50 238 76 311T149 431H162Q183 431 183 423Q183 417 175 409Q134 361 114 300T92 210ZM574 278Q574 320 550 344T486 369Q437 369 394 329T323 218Q309 184 295 109L286 64Q304 62 306 62Q423 62 498 131T574 278Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D711\" xlink:href=\"#MJX-1350-TEX-I-1D711\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\varphi<\/script>\u200b making<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" 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transform=\"translate(122.29,60.79) \"><\/g><g transform=\"translate(0,121.32) \"><\/g><g transform=\"translate(60.83,121.32) \"><\/g><g transform=\"translate(122.29,121.32) \"><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"7.268000000000001\" data-y=\"0.358\" data-xypic-id=\"2\" transform=\"translate(8008.5,-108)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1446-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"-0.34049999999999997\" data-y=\"4.141\" data-xypic-id=\"3\" transform=\"translate(400,-3891)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" 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xlink:href=\"#MJX-1446-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"1.669776082551999\" data-y=\"3.253929365464023\" data-xypic-id=\"6\" transform=\"translate(2410.3,-3003.9)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"1D711\" xlink:href=\"#MJX-1446-TEX-I-1D711\"><\/use><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"3.4405\" data-y=\"4.141\" data-xypic-id=\"7\" transform=\"translate(4181,-3891)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1446-TEX-I-1D44D\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.797144457143319\" data-y=\"1.3877790560593903\" data-xypic-id=\"8\" transform=\"translate(5537.6,-1137.8)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1446-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1446-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"4.798572409265445\" data-y=\"6.379499034172953\" data-xypic-id=\"9\" transform=\"translate(5539.1,-6129.5)\"><g data-mml-node=\"mstyle\" transform=\"scale(0.707)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1446-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1446-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><g stroke=\"currentColor\" fill=\"currentColor\" data-x=\"7.2635000000000005\" data-y=\"7.9239999999999995\" data-xypic-id=\"13\" transform=\"translate(8004,-7674)\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1446-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><\/div><\/div>\n<p>commute. Then we call <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.659ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4269.4 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-1415-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-1415-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 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id=\"MJX-1415-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-1415-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1415-TEX-I-1D44D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1112,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1415-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1556.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1415-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1415-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2496.2,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-1415-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2940.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1415-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1415-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3880.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-1415-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">(Z,p_1,p_2)<\/script> a <strong>product<\/strong> of <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.697ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 750 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1424-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1424-TEX-I-1D434\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.717ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 759 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1418-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D435\" xlink:href=\"#MJX-1418-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">B<\/script> in <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.853ex\" height=\"1.706ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -728 819 754\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.059ex;\"><defs><path id=\"MJX-513-TEX-S-1D49E\" d=\"M367 89Q367 84 353 77T334 70Q325 70 312 83T298 120Q298 169 364 233T496 298Q538 298 563 275T588 220V213Q588 132 501 53T306 -26Q251 -26 211 6T170 114Q171 148 181 184T205 248T232 298T255 334T265 349T246 350Q127 350 77 390T26 480Q26 533 71 581T178 656T295 683Q312 683 312 676Q312 674 311 672L302 660Q294 648 292 647L286 646Q280 646 276 646Q197 641 145 583T93 476Q93 387 265 387Q271 387 277 387T287 388T292 388T313 414T373 483T451 562Q618 712 732 727Q733 727 740 727T753 728Q790 727 804 708T819 665Q819 643 810 617T773 553T699 481T572 414T385 361Q381 361 367 341Q247 172 247 86Q247 11 325 11Q404 11 465 95Q518 165 528 208Q529 212 529 220Q529 237 518 249T480 261Q431 261 387 209T343 126Q343 111 355 101T367 89ZM777 658Q777 691 738 691Q704 691 658 662T570 590T491 504T432 432T410 400H411Q416 400 440 405T505 423T589 455T675 506T743 576Q777 627 777 658Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D49E\" xlink:href=\"#MJX-513-TEX-S-1D49E\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">\\mathscr C<\/script>, and denote <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.636ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 723 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1423-TEX-I-1D44D\" d=\"M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 100T579 247Q587 272 590 277T603 282H607Q628 282 628 271Q547 5 541 2Q538 0 300 0H124Q58 0 58 8Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44D\" xlink:href=\"#MJX-1423-TEX-I-1D44D\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">Z<\/script> as <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.18ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 2731.4 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1427-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-1427-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-1427-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1427-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(972.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-1427-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.4,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-1427-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A\\times B<\/script>. We call <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.126ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 939.6 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-1429-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-1429-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1429-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1429-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">p_1<\/script> and <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.126ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 939.6 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-1433-TEX-I-1D45D\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path id=\"MJX-1433-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45D\" xlink:href=\"#MJX-1433-TEX-I-1D45D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(536,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1433-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">p_2<\/script> are <strong>projections<\/strong>.<\/p>\n<p>As we talked above, if <mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.18ex\" height=\"1.62ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -716 2731.4 716\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-1427-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-1427-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-1427-TEX-I-1D435\" d=\"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-1427-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(972.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-1427-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1972.4,0)\"><use data-c=\"1D435\" xlink:href=\"#MJX-1427-TEX-I-1D435\"><\/use><\/g><\/g><\/g><\/svg><\/mjx-container><script type=\"math\/tex\">A\\times B<\/script> exists, it is unique up to isomorphism.<\/p>\n<p>&nbsp;<\/p>\n<h2 id='3-functors-and-nature-transformations'>3. Functors and Nature Transformations<\/h2>\n","protected":false},"excerpt":{"rendered":"<p>Here is my note of the category theory, which is the ba [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[11,8],"class_list":["post-455","post","type-post","status-publish","format-standard","hentry","category-6","tag-11","tag-8"],"_links":{"self":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/455","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=455"}],"version-history":[{"count":4,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/455\/revisions"}],"predecessor-version":[{"id":463,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/455\/revisions\/463"}],"wp:attachment":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=455"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=455"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=455"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}