---
tag: Y9CZ3MTNF1
kind: concept
name: Abelian category
msc: ["18E10"]
specializes: [B2VP96NJSK]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_category
same_as:
  wikidata: Q318737
  wikipedia: Abelian category
---

## Summary

An abelian category is an [[additive category|B2VP96NJSK]] in which every morphism has a [[kernel|S417YNEJ6M]] and a [[cokernel|H8W38933NC]].
In it, monomorphisms and epimorphisms behave as in abelian groups.
In it, exact sequences, homology and the standard diagram lemmas make sense.
Modules over a ring and sheaves of abelian groups on a space are abelian categories.
Abelian categories were introduced by Buchsbaum and by Grothendieck to give a common framework for homological algebra.

## Explanations

### L1 human

Abelian groups have a rich toolkit: subgroups, quotients, images, and exact sequences that track how they fit together.
An abelian category is any category where the same toolkit works, described only through maps.
In such a category every injective-like map is the kernel of something, and every surjective-like map is the cokernel of something.
So a theorem of homological algebra proved once for abelian categories applies at once to modules, sheaves and representations.