# Category

The definition of a category in category theory evolved over time, according to the author's chosen goals and framework. Eilenberg & Mac Lane gave a purely abstract definition of a category. Others, starting with Grothendieck (1957) and Freyd (1964), elected for reasons of practicality to define categories in set-theoretic terms. I will define it as follows.

# Idea

A category consists of a collection of things and binary relationships (or transitions) between them, such that these relationships can be combined and include the "identity" relationship: "is the same as".
A category is a directed graph with multiple edges with a rule saying how to compose two edges that fit together to get a new edge. Furthermore, each vertex, which acts as an identity for this composition.
A category consists of a collection of objects and a collection of morphisms. Every morphism has a source and a target object. If $f$ is a morphism with $x$ as its source and $y$ as its target, we write:

f : x \rightarrow y

and we say that $f$ is a morphism from $x$ to $y$ . In a category, we can compose a morphism $g: x \rightarrow y$ and a morphism $f: y \rightarrow z$ to get a morphism $f \circ g : x \rightarrow z$ . Composition is associative and satisfies the left and right unit laws.