Abelian category: Difference between revisions

In [[mathematics]], an '''abelian category''' is a [[Category (mathematics)|category]] in which [[morphism]]s and [[Object (category theory)|objects]] can be added and in which [[Kernel (category theory)|kernel]]s and [[cokernel]]s exist and have desirable properties. The motivating prototypical example of an abelian category is the [[category of abelian groups]], '''Ab'''. The theory originated in an effort to unify several [[Cohomology theory|cohomology theories]] by [[Alexander Grothendieck]] and independently in the slightly earlier work of [[David Buchsbaum]]. Abelian categories are very ''stable'' categories; for example they are [[regular category|regular]] and they satisfy the [[snake lemma]]. The [[Class (set theory)|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]].