Category theory

Introduction

Category theory studies the abstraction of mathematical objects (such as sets, groups, and topological spaces) in terms of the morphisms between them. These morphisms often represent functions. For example, in the category of sets, morphisms are functions, in the category of groups they are group homomorphism and in the category of topological spaces they are continuous maps.

Morphisms do not have to represent functions. For example, any partially ordered set

may be seen as a category where there is a (unique) morphism between two elements

x

and

y

if and only if

x \leq y

Categories are usually drawn as diagrams where the objects of interest in the category are usually represented by variables or points with (labelled) arrows between them representing morphisms. For this reason morphisms are also referred to as arrows.

Definition

A category consists of a collection of objects with morphisms between them. A morphism

f

goes from one object, say

X

, to another, say

Y

, and is drawn as an arrow from

X

Y

. Note that

X

may equal

Y

(in which case

f

is referred to as an endomorphism). The object

X

is called the domain of

f

and

Y

is called the codomain of

f

. This is written as

f : X \to Y

These morphisms must satisfy three conditions:

For any two morphisms

f : X \to Y

and

g : Y \to Z

, there exists a morphism

X \to Z

, written as

g \circ f

or simply

g f

. This is called composition.

For any morphisms

f : X \to Y

g : Y \to Z

and

h : Z \to W

composition is associative, i.e.,

h (g f) = (h g) f

For any object

X

, there is a (unique) morphism,

1_{X} : X \to X

which, when composed with another morphism, leaves it unchanged. I.e., given

f : W \to X

and

g : X \to Y

it holds that:

1_{X} f = f

and

g 1_{X} = g

Note that composition is written 'backwards' since given an element

x \in X

and two functions

f : X \to Y

and

g : Y \to Z

, the result of applying

f

then

g

g (f (x))

which equals

(g \circ f) (x)

Introduction

Morphisms do not have to represent functions. For example, any partially ordered set may be seen as a category where there is a (unique) morphism between two elements $x$ and $y$ if and only if $x \leq y$ .

Definition

A category consists of a collection of objects with morphisms between them. A morphism $f$ goes from one object, say $X$ , to another, say $Y$ , and is drawn as an arrow from $X$ to $Y$ . Note that $X$ may equal $Y$ (in which case $f$ is referred to as an endomorphism). The object $X$ is called the domain of $f$ and $Y$ is called the codomain of $f$ . This is written as $f : X \to Y$ .

These morphisms must satisfy three conditions:

For any two morphisms $f : X \to Y$ and $g : Y \to Z$ , there exists a morphism $X \to Z$ , written as $g \circ f$ or simply $g f$ . This is called composition.

For any morphisms $f : X \to Y$ , $g : Y \to Z$ and $h : Z \to W$ composition is associative, i.e., $h (g f) = (h g) f$ .

For any object $X$ , there is a (unique) morphism, $1_{X} : X \to X$ which, when composed with another morphism, leaves it unchanged. I.e., given $f : W \to X$ and $g : X \to Y$ it holds that: $1_{X} f = f$ and $g 1_{X} = g$ .

Note that composition is written 'backwards' since given an element $x \in X$ and two functions $f : X \to Y$ and $g : Y \to Z$ , the result of applying $f$ then $g$ is $g (f (x))$ which equals $(g \circ f) (x)$ .

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Introduction

Definition

Category theory

Introduction

Definition