The Basic of Category Theory

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 consists of:

  • : 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);
  • : the class of morphisms. For every morphism , it has a source and a target , where both and are elements of . Let be the source of and be the target of we can denote as . And we define (also for short) as the class of morphisms with source and target ; that is .

Additionally, objects and morphisms should satisfy these properties:

  • , there is a composition , . We also abbreviate as ​​.

    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 and , we have

    And we can use a commutative diagram

    to describe.

  • , there is a identity . For all and ,, 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 . is the class of all sets (as we all know, it can't be a set because of Russell's paradox). are all maps from to . It's easy to check satisfies the concept of category.
  • The category of topological spaces, denoted as . The objects of are all topological spaces, and the morphisms are continuous functions.
  • The category of groups, denoted as , in which objects are all groups and the morphisms are group homomorphisms; similarly, the category of abelian groups, denoted as ​, in which objects are all abelian groups and the morphisms are group homomorphisms.
  • Let be a field. The category of the vector spaces on is denoted as , in which objects are all vector spaces on and morphisms are linear maps.
  • Let be a ring with units. The category of the left modules on is denoted as , in which objects are all modules on and morphisms are -module homomorphisms. Similarly, we have the category of right modules ​.

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 be a poset. We can code the partial order relation of into a category .

Let , and

And if define . Then it's easy to check ​ is a category.

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

1.4 Definition

A morphism is called a monomorphism if . And we can use to describe a monomorphism.

A morphism is called an epimorphism if . And we can use to describe an epimorphism.

A morphism is called an isomorphism if . We call is the inverse of and denote it as .

Monomorphism is a concept like an injection, and epimorphism is a concept like a surjection. Actually, in many categories such as , , , , and , 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 , ​ is bijective may not have a continuous inverse.

1.5 Example

Let be a category. We call is a groupoid if all morphisms in are isomorphisms.

For example, give a topological space . Let and the morphisms from to are the homotopy classes of path from to . We call is the fundamental groupoid of .

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.

  • is the category of sets with based point. consists of all ordered pairs like , where is a set and . The morphism is a map from to satisfied .

  • . The Objects of consists of all ordered pairs like , where are sets and . The morphism is a map from to satisfied . We can describe this use a commutative diagram

    , where ​ 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 be a category, and be an object in . We will define a category whose objects are all morphisms from . Let . Let be the collection of morphisms with . We also denote as .

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

    And morphisms are commutative diagrams like

  • Let be a category, and be objects in . We will define a category like last two examples.

    consists of diagrams

    in .

    And morphisms

    are commutative diagrams

    in .

  • Let be a category, and be two morphisms in . We will define a category ​ like examples above.

    consists of commutative diagrams

    in .

    And morphisms

    are commutative diagrams

    in ​.

  • Let be a category. We will define a category whose objects are morphisms in ; that is . Let are morphisms in . Then the morphisms from to are commutative diagrams

    in .

1.7 Definition

Let be a category. A category is called a subcategory of if

  • ;
  • ​;
  • the composition in is the same as ;
  • , where is the identity morphism of in .

And if , we call is a full subcategory of .

For example, if we regard in as , then is a full subcategory of ​.

1.8 Definition

Let be a category. We will define a category called the opposite category of .

  • ;
  • ;
  • ​, where denotes the composition in ​.

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 be a category. If we call a object an initial object, if for all , is a one-point set. If we call a object a final object, if for all , is a one-point set.

2.2 Theorem

Let be a category. If there is an initial object in , then is the unique initial object in up to isomorphism.

Similarly, if there is an final object in , then is the unique final object in up to isomorphism.

Proof.

Let be initial objects in . Then we have are one-point sets.

Let , then we have .

Since and has only one element, we have ​.

Similarly, we have . Therefore and are isomorphisms.

The proof for final objects is similar to this.

2.3 Example

  • Let be a set and is the free abelian group on . Then we say has a universal property that for all abelian group and map , there is a unique group homomorphism making

    commute, where ​ is the inclusion.

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

    Let's define a category , whose objects are maps , where is an abelian group.

    Let . The morphisms are commutative diagram

    , where ​ is a group homomorphism.

    Then we can see, is the initial object in .

  • Let be a commutative ring with units, and is an proper ideal in . Then we say has a universal property that for all ring homomorphisms with , there is a unique ring homomorphism making

    commute, where is the quotient map.

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

  • Let be a commutative ring with units, and is a polynomial ring on . Let be the set of intermediates. We say has a universal property that for all ring homomorphisms and maps , there is a unique ring homomorphism making

    commute, where ​ are inclusions.

    Let's define a category , whose objects are diagrams

    , where is a ring homomorphism and is a map; and morphisms

    are commutative diagrams

    . As we can see, is the initial object in .

2.4 Definition

Let be a category, and are objects in . There are a object and morphisms , such that for all and morphisms