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**

: 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 isand , 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

Let

And if

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

### 1.4 Definition

A morphism **monomorphism** if

A morphism **epimorphism** if

A morphism **isomorphism** if

Monomorphism is a concept like an injection, and epimorphism is a concept like a surjection. Actually, in many categories such as

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

### 1.5 Example

Let **groupoid** if all morphisms in

For example, give a topological space

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 **subcategory** of

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

And if **full subcategory** of

For example, if we regard

### 1.8 Definition

Let **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 **initial object**, if for all **final object**, if for all

### 2.2 Theorem

Let

Similarly, if there is an final object

**Proof.**

Let

Let

Since

Similarly, we have

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