Here is my note of the category theory, which is the basic of modern mathematics. While it's very useful, it's too hard and abstract to learn. I want to record my understanding of category theory, and so I write this note.

I'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't be more appreciated to you.

References:

• Advanced Modern Algebra, Joseph J. Rotman
• Algebra: Chapter 0, Paolo Aluffi
• 代数学方法 卷一：基础架构, 李文威
• A Term of Commutative Algebra, Allen Altman and Steven Kleiman
• An Introduction to Algebraic Topology, Joseph J. Rotman
• An Introduction to Homological Topology, Joseph J. Rotman
• nlab
• 香蕉空间

1. Categories

Before we begin, let's discuss why category theory is worth learning.

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.

1.1 Definition

A category $\mathscr{C}$ consists of:

• $\operatorname{Obj}(\mathscr{C})$: 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);
• $\operatorname{Mor}(\mathscr C)$: the class of morphisms. For every morphism $f\in \operatorname{Mor}(\mathscr C)$, it has a source $s(f)$ and a target $t(f)$, where both $s(f)$ and $t(f)$ are elements of $\operatorname{Obj}(\mathscr{C})$. Let $X$ be the source of $f$ and $Y$ be the target of $f$ we can denote $f$ as $f:X\to Y$. And we define $\operatorname{Hom}_\mathscr{C}(X,Y)$ (also $\operatorname{Hom}(X,Y)$ for short) as the class of morphisms with source $X$ and target $Y$; that is $\operatorname{Hom}_\mathscr{C}(X,Y):=s^{-1}(X)\cap t^{-1}(Y)$.

Additionally, objects and morphisms should satisfy these properties:

• $\forall X,Y,Z\in\operatorname{Obj}(\mathscr C)$, there is a composition $\circ:\operatorname{Hom}(Y,Z)\times \operatorname{Hom}(X,Y)\to \operatorname{Hom}(X,Z)$, $(g,f)\mapsto g\circ f$. We also abbreviate $g\circ f$ as $gf$​​.

  And we can use a commutative diagram

  to describe.

  You can see the definition of commutative diagram in wikipedia or 香蕉空间.

• The composition we defined above satisfies the association law; that is $\forall A,B,C,D\in\operatorname{Obj}(\mathscr C)$ and $f:A\to B,g:B\to C,h:C\to D$, we have

  And we can use a commutative diagram

  to describe.

• $\forall X\in\operatorname{Obj}(\mathscr C)$, there is a identity $1_X\in \operatorname{Hom}(X,X)$. For all $Y,Z\in\operatorname{Obj}(\mathscr C)$ and $f:X\to Y$ ,$g: Z\to X$, we have

The definition of category is simple but abstract. Let's understand it with some examples.

1.2 Example

• Let's begin with a simple example, the category of sets, denoted as $\mathbf{Set}$. $\operatorname{Obj}(\mathbf{Set})$ is the class of all sets (as we all know, it can't be a set because of Russell's paradox). $\operatorname{Hom}_{\mathbf{Set}}(A,B)$ are all maps from $A$ to $B$. It's easy to check $\mathbf{Set}$ satisfies the concept of category.
• The category of topological spaces, denoted as $\mathbf {Top}$. The objects of $\mathbf {Top}$ are all topological spaces, and the morphisms are continuous functions.
• The category of groups, denoted as $\mathbf{Grp}$, in which objects are all groups and the morphisms are group homomorphisms; similarly, the category of abelian groups, denoted as $\mathbf{Ab}$​, in which objects are all abelian groups and the morphisms are group homomorphisms.
• Let $k$ be a field. The category of the vector spaces on $k$ is denoted as $\mathbf{Vect}_k$, in which objects are all vector spaces on $k$ and morphisms are linear maps.
• Let $R$ be a ring with units. The category of the left modules on $R$ is denoted as ${}_R\mathbf{Mod}$, in which objects are all modules on $R$ and morphisms are $R$-module homomorphisms. Similarly, we have the category of right modules $\mathbf{Mod}_R$​.

1.3 Example

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.

Let $(X,\le)$ be a poset. We can code the partial order relation of $X$ into a category $\mathscr C$.

Let $\operatorname{Obj}(\mathscr{C})=X$, and $\operatorname{Hom}(a,b)=\begin{cases}\{(a,b)\}, & a\le b,\\\emptyset &\mathrm{otherwise.}\end{cases}$

And if $a\le b\le c$ define $(b,c)\circ(a,b)=(a,c)$. Then it's easy to check $\mathscr C$​ is a category.

Similarly, we can code the equivalence relation into a category.

1.4 Definition

A morphism $f:X\to Y$ is called a monomorphism if $\forall \alpha,\beta:Z\to X,f\alpha=f\beta\Rightarrow\alpha=\beta$. And we can use $f:X\hookrightarrow Y$ to describe a monomorphism.

A morphism $f:X\to Y$ is called an epimorphism if $\forall \alpha,\beta:Y\to Z,\alpha f=\beta f\Rightarrow\alpha=\beta$. And we can use $f:X\twoheadrightarrow Y$ to describe an epimorphism.

A morphism $f:X\to Y$ is called an isomorphism if $\exists g:Y\to X,gf=1_X,fg=1_Y$. We call $g$ is the inverse of $f$ and denote it as $f^{-1}$.

Monomorphism is a concept like an injection, and epimorphism is a concept like a surjection. Actually, in many categories such as $\mathbf{Set}$, $\mathbf {Top}$, $\mathbf{Grp}$, $\mathbf{Ab}$, $\mathbf{Vect}_k$ and ${}_R\mathbf{Mod}$, a monomorphism is equivalence to an injection as well as an epimorphism is equivalence to a surjection.

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 $\mathbf {Top}$, $f$​ is bijective may not have a continuous inverse.

1.5 Example

Let $\mathscr C$ be a category. We call $\mathscr C$ is a groupoid if all morphisms in $\mathscr C$ are isomorphisms.

For example, give a topological space $X$. Let $\operatorname{Obj}(\mathscr C)=X$ and the morphisms from $x_0$ to $x_1$ are the homotopy classes of path from $x_0$ to $x_1$. We call $\mathscr C$ is the fundamental groupoid of $X$.

And we can regard a group as a groupoid with only one object.

1.6 Example

In this part, let me show you some special examples of categories.

• $\mathbf{Set}^*$ is the category of sets with based point. $\operatorname{Obj}(\mathbf{Set}^*)$ consists of all ordered pairs like $(S,x)$, where $S$ is a set and $x\in S$. The morphism $f:(S_1,x_1)\to (S_2,x_2)$ is a map from $S_1$ to $S_2$ satisfied $f(x_1)=x_2$.

• $\mathbf{Set}^2$. The Objects of $\mathbf{Set}^2$ consists of all ordered pairs like $(S,T)$, where $S,T$ are sets and $T\subset S$. The morphism $f:(S_1,T_1)\to (S_2,T_2)$ is a map from $S_1$ to $S_2$ satisfied $f(T_1)\subset T_2$. We can describe this use a commutative diagram

  , where $i_1,i_2$​ are inclusions.

  We can also consider the objects of this category are some diagrams like

  and morphisms are commutative diagrams like the one above.

• Let $\mathscr C$ be a category, and $A$ be an object in $\mathscr C$. We will define a category $k_\mathscr{C}(A)$ whose objects are all morphisms from $A$. Let $f_1\in \operatorname{Hom}_\mathscr C(A,B),f_2\in \operatorname{Hom}_\mathscr C(A,C)$. Let $\operatorname{Hom}_{k_\mathscr{C}(A)}(f_1,f_2)$ be the collection of morphisms $g\in \operatorname{Hom}_{\mathscr C}(B,C)$ with $gf_1=f_2$. We also denote $k_\mathscr{C}(A)$ as $\operatorname{Hom}_{\mathscr C}(A,-)$.

  Like the last example, we can treat the objects of this category as diagrams like

  And morphisms are commutative diagrams like

• Let $\mathscr C$ be a category, and $A,B$ be objects in $\mathscr C$. We will define a category $\mathscr C_{A,B}$ like last two examples.

  $\operatorname{Obj}(\mathscr C_{A,B})$ consists of diagrams

  in $\mathscr C$.

  And morphisms

  are commutative diagrams

  in $\mathscr C$.

• Let $\mathscr C$ be a category, and $\alpha:A\to C,\beta:B\to C$ be two morphisms in $\mathscr C$. We will define a category $\mathscr C_{\alpha,\beta}$​ like examples above.

  $\operatorname{Obj}(\mathscr C_{\alpha,\beta})$ consists of commutative diagrams

  in $\mathscr C$.

  And morphisms

  are commutative diagrams

  in $\mathscr C$​.

• Let $\mathscr C$ be a category. We will define a category $\mathscr M$ whose objects are morphisms in $\mathscr C$; that is $\operatorname{Obj}(\mathscr M)=\operatorname{Mor}(\mathscr C)$. Let $f_1:A\to B,f_2:C\to D$ are morphisms in $\mathscr C$. Then the morphisms from $f_1$ to $f_2$ are commutative diagrams

  in $\mathscr C$.

1.7 Definition

Let $\mathscr C$ be a category. A category $\mathscr D$ is called a subcategory of $\mathscr C$ if

• $\operatorname{Obj}(\mathscr D)\subset \operatorname{Obj}(\mathscr C)$;
• $\forall A,B\in \operatorname{Obj}(\mathscr D),\operatorname{Hom}_{\mathscr D}(A,B)\subset \operatorname{Hom}_{\mathscr C}(A,B)$​;
• the composition in $\mathscr D$ is the same as $\mathscr C$;
• $\forall A\in \operatorname{Obj}(\mathscr D),1_A\in \operatorname{Hom}_{\mathscr D}(A,A)$, where $1_A$ is the identity morphism of $A$ in $\mathscr C$.

And if $\forall A,B\in \operatorname{Obj}(\mathscr D),\operatorname{Hom}_{\mathscr D}(A,B)= \operatorname{Hom}_{\mathscr C}(A,B)$, we call $\mathscr D$ is a full subcategory of $\mathscr C$.

For example, if we regard $(S,x_0)$ in $\mathbf{Set}^*$ as $(S,\{x_0\})$, then $\mathbf{Set}^*$ is a full subcategory of $\mathbf{Set}^2$​.

1.8 Definition

Let $\mathscr C$ be a category. We will define a category $\mathscr C^{\mathrm{op}}$ called the opposite category of $\mathscr C$.

• $\operatorname{Obj}(\mathscr C^{\mathrm{op}})=\operatorname{Obj}(\mathscr C)$;
• $\operatorname{Hom}_{\mathscr C^{\mathrm{op}}}(A,B)=\operatorname{Hom}_{\mathscr C}(B,A)$;
• $f \circ^{\mathrm{op}}g=g\circ f$​, where $\circ^{\mathrm{op}}$ denotes the composition in $\mathscr C^{\mathrm{op}}$​.

2. Universal Properties

Some math books use the concept, universal properties, but with few explanations. Actually, universal properties are initial or final objects in a certain category.

2.1 Definition

Let $\mathscr C$ be a category. If we call a object $A\in \operatorname{Obj}(\mathscr C)$ an initial object, if for all $Z\in\operatorname{Obj}(\mathscr C)$, $\operatorname{Hom}_{\mathscr C}(A,Z)$ is a one-point set. If we call a object $A\in \operatorname{Obj}(\mathscr C)$ a final object, if for all $Z\in\operatorname{Obj}(\mathscr C)$, $\operatorname{Hom}_{\mathscr C}(Z,A)$ is a one-point set.

2.2 Theorem

Let $\mathscr C$ be a category. If there is an initial object $I$ in $\mathscr C$, then $I$ is the unique initial object in $\mathscr C$ up to isomorphism.

Similarly, if there is an final object $I$ in $\mathscr C$, then $I$ is the unique final object in $\mathscr C$ up to isomorphism.

Proof.

Let $I_1,I_2$ be initial objects in $\mathscr C$. Then we have $\operatorname{Hom}(I_1,I_1),\operatorname{Hom}(I_1,I_2),\operatorname{Hom}(I_2,I_1)$ are one-point sets.

Let $f\in\operatorname{Hom}(I_1,I_2),g\in\operatorname{Hom}(I_2,I_1)$, then we have $gf\in \operatorname{Hom}(I_1,I_1)$.

Since $1_{I_1}\in \operatorname{Hom}(I_1,I_1)$ and $\operatorname{Hom}(I_1,I_1)$ has only one element, we have $gf=1_{I_1}$​.

Similarly, we have $fg=1_{I_2}$. Therefore $f$ and $g$ are isomorphisms. $\square$

The proof for final objects is similar to this.

2.3 Example

• Let $A$ be a set and $F$ is the free abelian group on $A$. Then we say $F$ has a universal property that for all abelian group $G$ and map $f:A\to B$, there is a unique group homomorphism $\varphi$ making

  commute, where $i$​ is the inclusion.

  In fact, we can regard this as an initial object in a category.

  Let's define a category $\mathscr C$, whose objects are maps $A\to G$, where $G$ is an abelian group.

  Let $f_1:A\to G_1,f_2:A\to G_2\in \operatorname{Obj}(\mathscr C)$. The morphisms $f_1\to f_2$ are commutative diagram

  , where $\sigma$​ is a group homomorphism.

  Then we can see, $i:A\to F$ is the initial object in $\mathscr C$.

• Let $R$ be a commutative ring with units, and $\mathfrak a$ is an proper ideal in $R$. Then we say $\mathfrak a$ has a universal property that for all ring homomorphisms $\sigma:R\to R'$ with $\sigma(\mathfrak a)=(0)$, there is a unique ring homomorphism $\varphi:R/\mathfrak a \to R'$ making

  commute, where $p$ is the quotient map.

  We can also see this as an initial object in a category like last example.

• Let $R$ be a commutative ring with units, and $R[X_1,\dots,X_n]$ is a polynomial ring on $R$. Let $\mathcal X:=\{X_1,\dots X_n\}$ be the set of intermediates. We say $R[X_1,\dots,X_n]$ has a universal property that for all ring homomorphisms $\sigma:R\to R'$ and maps $f:\mathcal X\to R'$, there is a unique ring homomorphism $\varphi:R[X_1,\dots,X_n]\to R'$ making

  commute, where $i_1,i_2$​ are inclusions.

  Let's define a category $\mathscr C$, whose objects are diagrams

  , where $\sigma$ is a ring homomorphism and $f$ is a map; and morphisms

  are commutative diagrams

  . As we can see, $(i_1,i_2)$ is the initial object in $\mathscr C$.

2.4 Definition

Let $\mathscr C$ be a category, and $A,B$ are objects in $\mathscr C$. There are a object $Z$ and morphisms $p_1:Z\to A,p_2:Z\to B$, such that for all $L\in \operatorname{Obj}(\mathscr C)$ and morphisms $f_1:L\to A,f_2:L\to B$, there exists a unique morphism $\varphi$​ making

commute. Then we call $(Z,p_1,p_2)$ a product of $A$ and $B$ in $\mathscr C$, and denote $Z$ as $A\times B$. We call $p_1$ and $p_2$ are projections.

As we talked above, if $A\times B$ exists, it is unique up to isomorphism.

 

3. Functors and Nature Transformations