A short introduction to category theory

This short note originated from a short talk of basic category theory in 2022.

Category theory was first introduced by Samuel Eilenberg and Saunders Mac Lane in the 20th century. It rapidly became a strong tool in almost all domains of mathematics. Category theory studies the abstract structures of different mathematical objects, and it also has applications on computer science, including AI.

1. Categories

In this section, we will first lay out the general definition of categories, then we give some concrete and common examples of categories.

Definition 1. A category $\mathcal{C}$ consists of the following data:

1. A class of objects, denoted by ${\rm{Ob}}(\mathcal{C})$.
2. To each pair of objects $A, B$, a set ${\rm{Hom}}(A, B)$ of morphisms from $A$ to $B$.
3. To each triple of objects $A, B, C$, a composition law $${\rm{Hom}}(A,B)\times {\rm{Hom}}(B, C)\longrightarrow {\rm{Hom}}(A, C),\ (f,g)\longmapsto f\circ g.$$

Moreover, it subjects to the following axioms:

(1) Composition is associative, i.e. $(f\circ g)\circ h = f\circ (g\circ h)$ for morphisms $f,g,h$.

(2) For each object $A$, there is a unique identity morphism $1_{A}: A\rightarrow A$ such that $1_{A}\circ f=f\circ 1_{A}$ if the composition makes sense.

Examples 2. (1) The category $\textbf{Sets}$ of sets, whose objects are sets and morphisms are functions.

(2) The category $\textbf{Groups}$ of groups, whose objects are groups and morphisms are homomorphisms.

(3) The category $\textbf{Ab}$ of abelian groups, whose objects are abelian groups and morphisms are homomorphisms.

By Example 2 (3), one can observe that ${\rm{Ob}}(\textbf{Ab})\subset{\rm{Ob}}(\textbf{Groups})$, and for any $x,y\in{\rm{Ob}}(\textbf{Ab})$, ${\rm{Hom}}_{\textbf{Ab}}(x,y)={\rm{Hom}}_{\textbf{Groups}}(x,y)$. This leads to the definition of subcategories.

Definition 3. Let $\mathcal{C}$ be a category. A subcategory of $\mathcal{C}$ is a category $\mathcal{D}$ such that ${\rm{Ob}}(\mathcal{D})\subset{\rm{Ob}}(\mathcal{C})$ and ${\rm{Hom}}_{\textbf{Ab}}(x,y)\subset{\rm{Hom}}_{\textbf{Groups}}(x,y)$ for all $x,y\in{\rm{Ob}}(\mathcal{D})$.

The subcategory $\mathcal{D}$ is said to be full if we have ${\rm{Hom}}_{\mathcal{D}}(x,y)\cong{\rm{Hom}}_{\mathcal{C}}(x,y)$ for all $x,y\in{\rm{Ob}}(\mathcal{D})$.

An invertible morphism in a category is the so-called isomorphism.

Definition 4. Let $\mathcal{C}$ be a category and $f$ is a morphism in $\mathcal{C}$. We say that $f$ is an isomorphism if there is a morphism $g$ in $\mathcal{C}$ such that $f\circ g=1$ and $g\circ f=1$ when the composition makes sense. The morphism $g$ is called the inverse of $f$ and is denoted by $f^{-1}$.

2. Functors

Like objects in the category, we could define transformations between categories. In fact, one could view categories as objects of some bigger category.

Definition 5. Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A functor $F:\mathcal{C}\rightarrow\mathcal{D}$ is an assignment that

1. To each $x\in{\rm{Ob}}(\mathcal{C})$, it assigns $Fx\in{\rm{Ob}}(\mathcal{D})$.
2. For all $x,y\in{\rm{Ob}}(\mathcal{C})$ and $f\in{\rm{Hom}}_{\mathcal{C}}(x,y)$, it assigns $F(f)\in{\rm{Hom}}_{\mathcal{D}}(Fx,Fy)$ such that $F(1)=1$ and $F(fg)=F(f)F(g)$ when the composition makes sense.

A contravariant functor is a functor $G:\mathcal{C}^{opp}\rightarrow\mathcal{D}$.