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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 Waldhausen category ： ウィキペディア英語版 Waldhausen category In mathematics a Waldhausen category (after Friedhelm Waldhausen) is a category ''C'' with a zero object equipped with cofibrations co(''C'') and weak equivalences we(''C''), both containing all isomorphisms, both compatible with pushout, and co(''C'') containing the unique morphisms :$\backslash scriptstyle\; 0\backslash ,\backslash rightarrowtail\backslash ,\; A$ from the zero-object to any object ''A''. To be more precise about the pushouts, we require when :$\backslash scriptstyle\; A\backslash ,\; \backslash rightarrowtail\backslash ,\; B$ is a cofibration and :$\backslash scriptstyle\; A\backslash ,\backslash to\backslash ,\; C$ is any map, that we have a push-out :$\backslash scriptstyle\; B\backslash ,\; \backslash cup\_A\backslash ,\; C$ where the map :$\backslash scriptstyle\; C\backslash ,\; \backslash rightarrowtail\backslash ,\; B\backslash ,\backslash cup\_A\backslash ,\; C$ is a cofibration: A category ''C'' is equipped with bifibrations if it has cofibrations and its opposite category ''C''OP has so also. In that case, we denote the fibrations of ''C''OP by quot(''C''). In that case, ''C'' is a biWaldhausen category if ''C'' has bifibrations and weak equivalences such that both (''C'', co(''C''), we) and (''C''OP, quot(''C''), weOP) are Waldhausen categories. As examples one may think of exact categories, where the cofibrations are the admissible monomorphisms. Another example is the full subcategory of cofibrant objects in a pointed model categories, that is, the full subcategory consisting of those objects $X$ for which $0\; \backslash to\; X$ is a cofibration. (The bifibrant objects do not in general form a Waldhausen category, as a pushout of fibrant objects need not be fibrant. For more information on this second example see the paper by Sagave in the references) Waldhausen and biWaldhausen categories are linked with algebraic K-theory. There, many interesting categories are complicial biWaldhausen categories. For example: The category $\backslash scriptstyle\; C^b(\backslash mathcal)$ of bounded chaincomplexes on an exact category $\backslash scriptstyle\; \backslash mathcal$ The category $\backslash scriptstyle\; S\_n\; \backslash mathcal$ of functors $\backslash scriptstyle\; Ar(\backslash Delta\; ^n)\backslash ,\; \backslash to\backslash ,\; \backslash mathcal$ when $\backslash scriptstyle\backslash mathcal$ is so. And given a diagram $\backslash scriptstyle\; I$, then $\backslash scriptstyle\; \backslash mathcal^I$ is a nice complicial biWaldhausen category when $\backslash scriptstyle\; \backslash mathcal$ is. == References == * F. Waldhausen, ''Algebraic K-Theory of spaces'' — http://www.maths.ed.ac.uk/~aar/surgery/rutgers/wald.pdf * C. Weibel, ''The K-book, an introduction to algebraic K-theory'' — http://www.math.rutgers.edu/~weibel/Kbook.html * G. Garkusha, ''Systems of Diagram Categories and K-theory'' — http://front.math.ucdavis.edu/0401.5062 * 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「Waldhausen 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.