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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 Karoubi envelope ： ウィキペディア英語版 Karoubi envelope In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician Max Karoubi. Given a category C, an idempotent of C is an endomorphism :$e:\; A\; \backslash rightarrow\; A$ with :$e\backslash circ\; e\; =\; e$. An idempotent ''e'': ''A'' → ''A'' is said to split if there is an object ''B'' and morphisms ''f'': ''A'' → ''B'', ''g'' : ''B'' → ''A'' such that ''e'' = ''g'' ''f'' and 1''B'' = ''f'' ''g''. The Karoubi envelope of C, sometimes written Split(C), is the category whose objects are pairs of the form (''A'', ''e'') where ''A'' is an object of C and $e\; :\; A\; \backslash rightarrow\; A$ is an idempotent of C, and whose morphisms are the triples : $(e,\; f,\; e^):\; (A,\; e)\; \backslash rightarrow\; (A^,\; e^)$ where $f:\; A\; \backslash rightarrow\; A^$ is a morphism of C satisfying $e^\; \backslash circ\; f\; =\; f\; =\; f\; \backslash circ\; e$ (or equivalently $f=e\text{'}\backslash circ\; f\backslash circ\; e$). Composition in Split(C) is as in C, but the identity morphism on $(A,e)$ in Split(C) is $(e,e,e)$, rather than the identity on $A$. The category C embeds fully and faithfully in Split(C). In Split(C) every idempotent splits, and Split(C) is the universal category with this property. The Karoubi envelope of a category C can therefore be considered as the "completion" of C which splits idempotents. The Karoubi envelope of a category C can equivalently be defined as the full subcategory of $\backslash hat\{\backslash mathbf\{C\}\}$ (the presheaves over C) of retracts of representable functors. The category of presheaves on C is equivalent to the category of presheaves on Split(C). == Automorphisms in the Karoubi envelope == An automorphism in Split(C) is of the form $(e,\; f,\; e):\; (A,\; e)\; \backslash rightarrow\; (A,\; e)$, with inverse $(e,\; g,\; e):\; (A,\; e)\; \backslash rightarrow\; (A,\; e)$ satisfying: : $g\; \backslash circ\; f\; =\; e\; =\; f\; \backslash circ\; g$ : $g\; \backslash circ\; f\; \backslash circ\; g\; =\; g$ : $f\; \backslash circ\; g\; \backslash circ\; f\; =\; f$ If the first equation is relaxed to just have $g\; \backslash circ\; f\; =\; f\; \backslash circ\; g$, then ''f'' is a partial automorphism (with inverse ''g''). A (partial) involution in Split(C) is a self-inverse (partial) automorphism. 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「Karoubi envelope」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.