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

Categories are usually drawn as diagrams with the objects represented by variables or points with (labeled) arrows between them representing morphisms. For this reason, morphisms are also referred to as arrows.

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

Definition

A category consists of a collection of objects and a collection of morphisms. 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 source or domain of $f$ and $Y$ is called the target or codomain of $f$ . This is written as $f : X \to Y$ .

These morphisms must satisfy three conditions:

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

Motivation

Many mathematical constructions (such as products) appear across different fields of mathematics, consisting of different ingredients but nevertheless ...