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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. Abelian categories are named for Niels Henrik Abel. ==Definitions== A category is abelian if *it has a zero object, *it has all binary products and binary coproducts, *it has all kernels and cokernels, and *all monomorphisms and epimorphisms are normal. This definition is equivalent〔Peter Freyd, (Abelian Categories )〕 to the following "piecemeal" definition: * A category is ''preadditive'' if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. * A preadditive category is ''additive'' if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products. In 〔Handbook of categorical algebra, vol. 2, F. Borceux〕 Def. 1.2.6, it is required that an additive category has a zero object (empty biproduct). * An additive category is ''preabelian'' if every morphism has both a kernel and a cokernel. * Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism. Note that the enriched structure on hom-sets is a ''consequence'' of the three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between Abelian categories. This ''exactness'' concept has been axiomatized in the theory of exact categories, forming a very special case of regular categories. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「abelian category」の詳細全文を読む スポンサード リンク
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