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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Exact category ： ウィキペディア英語版 Exact category In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence. ==Definition== An exact category E is an additive category possessing a class ''E'' of "short exact sequences": triples of objects connected by arrows : $M\text{'}\; \backslash to\; M\; \backslash to\; M\text{'}\text{'}\backslash$ satisfying the following axioms inspired by the properties of short exact sequences in an abelian category: * ''E'' is closed under isomorphisms and contains the canonical ("split exact") sequences: ::$M\text{'}\; \backslash rightarrow\; M\text{'}\; \backslash oplus\; M\text{'}\text{'}\backslash rightarrow\; M\text{'}\text{'};$ * Suppose $M\; \backslash to\; M\text{'}\text{'}$ occurs as the second arrow of a sequence in ''E'' (it is an admissible epimorphism) and $N\; \backslash to\; M\text{'}\text{'}$ is any arrow in E. Then their pullback exists and its projection to $N$ is also an admissible epimorphism. Dually, if $M\text{'}\; \backslash to\; M$ occurs as the first arrow of a sequence in ''E'' (it is an admissible monomorphism) and $M\text{'}\; \backslash to\; N$ is any arrow, then their pushout exists and its coprojection from $N$ is also an admissible monomorphism. (We say that the admissible epimorphisms are "stable under pullback", resp. the admissible monomorphisms are "stable under pushout".); * Admissible monomorphisms are kernels of their corresponding admissible epimorphisms, and dually. The composition of two admissible monomorphisms is admissible (likewise admissible epimorphisms); * Suppose $M\; \backslash to\; M\text{'}\text{'}$ is a map in E which admits a kernel in E, and suppose $N\; \backslash to\; M$ is any map such that the composition $N\; \backslash to\; M\; \backslash to\; M\text{'}\text{'}$ is an admissible epimorphism. Then so is $M\; \backslash to\; M\text{'}\text{'}.$ Dually, if $M\text{'}\; \backslash to\; M$ admits a cokernel and $M\; \backslash to\; N$ is such that $M\text{'}\; \backslash to\; M\; \backslash to\; N$ is an admissible monomorphism, then so is $M\text{'}\; \backslash to\; M.$ Admissible monomorphisms are generally denoted $\backslash rightarrowtail$ and admissible epimorphisms are denoted $\backslash twoheadrightarrow.$ These axioms are not minimal; in fact, the last one has been shown by to be redundant. One can speak of an exact functor between exact categories exactly as in the case of exact functors of abelian categories: an exact functor $F$ from an exact category D to another one E is an additive functor such that if :$M\text{'}\; \backslash rightarrowtail\; M\; \backslash twoheadrightarrow\; M\text{'}\text{'}$ is exact in D, then :$F(M\text{'})\; \backslash rightarrowtail\; F(M)\; \backslash twoheadrightarrow\; F(M\text{'}\text{'})$ is exact in E. If D is a subcategory of E, it is an exact subcategory if the inclusion functor is fully faithful and exact. 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「Exact category」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.