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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 Quotient category ： ウィキペディア英語版 Quotient category In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. The notion is similar to that of a quotient group or quotient space, but in the categorical setting. ==Definition== Let ''C'' be a category. A ''congruence relation'' ''R'' on ''C'' is given by: for each pair of objects ''X'', ''Y'' in ''C'', an equivalence relation ''R''''X'',''Y'' on Hom(''X'',''Y''), such that the equivalence relations respect composition of morphisms. That is, if :$f\_1,f\_2\; :\; X\; \backslash to\; Y\backslash ,$ are related in Hom(''X'', ''Y'') and :$g\_1,g\_2\; :\; Y\; \backslash to\; Z\backslash ,$ are related in Hom(''Y'', ''Z'') then ''g''1''f''1, ''g''1''f''2, ''g''2''f''1 and ''g''2''f''2 are related in Hom(''X'', ''Z''). Given a congruence relation ''R'' on ''C'' we can define the quotient category ''C''/''R'' as the category whose objects are those of ''C'' and whose morphisms are equivalence classes of morphisms in ''C''. That is, :$\backslash mathrm\_\}(X,Y)\; =\; \backslash mathrm\_.$ Composition of morphisms in ''C''/''R'' is well-defined since ''R'' is a congruence relation. There is also a notion of taking the quotient of an Abelian category ''A'' by a Serre subcategory ''B''. This is done as follows. The objects of ''A/B'' are the objects of ''A''. Given two objects ''X'' and ''Y'' of ''A'', we define the set of morphisms from ''X'' to ''Y'' in ''A/B'' to be $\backslash varinjlim\; \backslash mathrm\_A(X\text{'},\; Y/Y\text{'})$ where the limit is over subobjects $X\text{'}\; \backslash subseteq\; X$ and $Y\text{'}\; \backslash subseteq\; Y$ such that $X/X\text{'},\; Y\text{'}\; \backslash in\; B$. Then ''A/B'' is an Abelian category, and there is a canonical functor $Q\; \backslash colon\; A\; \backslash to\; A/B$. This Abelian quotient satisfies the universal property that if ''C'' is any other Abelian category, and $F\; \backslash colon\; A\; \backslash to\; C$ is an exact functor such that ''F(b)'' is a zero object of ''C'' for each $b\; \backslash in\; B$, then there is a unique exact functor $\backslash overline\; \backslash colon\; A/B\; \backslash to\; C$ such that $F\; =\; \backslash overline\; \backslash circ\; Q$. (See ().) 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「Quotient 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.