---
tag: Q1RZT7DV0F
kind: concept
name: Category theory
msc: [18A05]
sources:
  - url: https://en.wikipedia.org/wiki/Category_theory
same_as:
  wikidata: Q217413
  wikipedia: Category theory
---

## Summary

Category theory is the part of mathematics that studies [[categories|A2DE66937D]], [[functors|28NQ0KRHD3]] between them and [[natural transformations|KYCPCC57FT]] between functors.
It describes mathematical structures through the maps between them rather than through their elements.
It was introduced by Samuel Eilenberg and Saunders Mac Lane in the 1940s, first to make precise the word "natural" in algebraic topology.
Today it is the common language of algebraic topology, algebraic geometry, homological algebra and parts of logic and computer science.

## Explanations

### L1 human

Many kinds of objects come with their own kind of maps: sets with functions, groups with homomorphisms, spaces with continuous maps.
Category theory looks only at the objects, the arrows between them, and how arrows compose.
From this one picture it defines notions that work in every field at once, such as products, isomorphisms and universal properties.
For example, the product of two sets, of two groups and of two spaces are all instances of one categorical notion of product.
A theorem proved at this level holds in every field that fits the picture.