Editing Homological algebra (section)

===Abelian categories===
{{Main|Abelian category}}
In [[mathematics]], an '''abelian category''' is a [[category (category theory)|category]] in which [[morphism]]s and objects can be added and in which [[kernel (category theory)|kernel]]s and [[cokernel]]s exist and have desirable properties. The motivating prototype example of an abelian category is the [[category of abelian groups]], '''Ab'''. The theory originated in a tentative attempt to unify several [[cohomology theory|cohomology theories]] by [[Alexander Grothendieck]]. Abelian categories are very ''stable'' categories, for example they are [[regular category|regular]] and they satisfy the [[snake lemma]]. The class of Abelian categories is closed under several categorical constructions, for example, the category of [[chain complex]]es of an Abelian category, or the category of [[functor]]s from a [[small category]] to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in [[algebraic geometry]], [[cohomology]] and pure [[category theory]]. Abelian categories are named after [[Niels Henrik Abel]].

More concretely, a category is '''abelian''' if
*it has a [[zero object]],
*it has all binary [[Product (category theory)|products]] and binary [[coproduct]]s, and
*it has all [[kernel (category theory)|kernels]] and [[cokernel]]s.
*all [[monomorphism]]s and [[epimorphism]]s are [[normal morphism|normal]].