Abelian category

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In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels 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 theories by Alexander Grothendieck. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors 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.

Contents

Definitions

A category is abelian if

By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition:

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