---
tag: A2DE66937D
kind: concept
name: Category
msc: [18A05, 18A15]
sources:
  - url: https://en.wikipedia.org/wiki/Category_(mathematics)
same_as:
  wikidata: Q719395
  wikipedia: Category (mathematics)
---

## Summary

A category is made of objects, of morphisms (arrows) between pairs of objects, and of a composition of morphisms that is associative and has identities.
It is the basic notion of [[category theory|Q1RZT7DV0F]].
Sets with [[functions|0EPAT51DXQ]], [[groups|G5S1M9]] with [[group homomorphisms|0JYTG5K9Y4]], and [[topological spaces|0CQHFPGJ45]] with [[continuous maps|0VGFHTM75Y]] are categories.
A category can also be a single structure: a [[monoid|0N6R902FG1]] is a category with one object.
A [[preorder|0VABYWJWH9]] is a category with at most one morphism between any two objects.

## Explanations

### L1 human

Picture dots for the objects and arrows between them for the morphisms.
Whenever one arrow ends where another starts, the two can be chained into one arrow.
Chaining is associative, and each dot carries an identity arrow that changes nothing when chained.
For example, take as dots the numbers $1, 2, 3$, and draw one arrow from $a$ to $b$ when $a \le b$.
Arrows chain because $a \le b$ and $b \le c$ give $a \le c$, and the identity arrows come from $a \le a$.