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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 Homotopy category of chain complexes ： ウィキペディア英語版 Homotopy category of chain complexes In homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes ''Kom(A)'' of ''A'' and the derived category ''D(A)'' of ''A'' when ''A'' is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that ''A'' is abelian. Philosophically, while ''D(A)'' makes isomorphisms of any maps of complexes that are quasi-isomorphisms in ''Kom(A)'', ''K(A)'' does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, ''K(A)'' is more understandable than ''D(A)''. == Definitions == Let ''A'' be an additive category. The homotopy category ''K(A)'' is based on the following definition: if we have complexes ''A'', ''B'' and maps ''f'', ''g'' from ''A'' to ''B'', a chain homotopy from ''f'' to ''g'' is a collection of maps $h^n\; \backslash colon\; A^n\; \backslash to\; B^$ (''not'' a map of complexes) such that :$f^n\; -\; g^n\; =\; d\_B^\; h^n\; +\; h^\; d\_A^n,$ or simply $f\; -\; g\; =\; d\_B\; h\; +\; h\; d\_A.$ This can be depicted as: : We also say that ''f'' and ''g'' are chain homotopic, or that $f\; -\; g$ is null-homotopic or homotopic to 0. It is clear from the definition that the maps of complexes which are null-homotopic form a group under addition. The homotopy category of chain complexes ''K(A)'' is then defined as follows: its objects are the same as the objects of ''Kom(A)'', namely chain complexes. Its morphisms are "maps of complexes modulo homotopy": that is, we define an equivalence relation :$f\; \backslash sim\; g\backslash$ if ''f'' is homotopic to ''g'' and define :$\backslash operatorname\_(A,\; B)\; =\; \backslash operatorname\_(A,B)/\backslash sim$ to be the quotient by this relation. It is clearer that this results in an additive category if one notes that this is the same as taking the quotient by the subgroup of null-homotopic maps. The following variants of the definition are also widely used: if one takes only ''bounded-below'' (''An=0 for n<<0''), ''bounded-above'' (''An=0 for n>>0''), or ''bounded'' (''An=0 for |n|>>0'') complexes instead of unbounded ones, one speaks of the ''bounded-below homotopy category'' etc. They are denoted by ''K+(A)'', ''K−(A)'' and ''Kb(A)'', respectively. A morphism $f\; :\; A\; \backslash rightarrow\; B$ which is an isomorphism in ''K(A)'' is called a homotopy equivalence. In detail, this means there is another map $g\; :\; B\; \backslash rightarrow\; A$, such that the two compositions are homotopic to the identities: $f\; \backslash circ\; g\; \backslash sim\; Id\_B$ and $g\; \backslash circ\; f\; \backslash sim\; Id\_A$. The name "homotopy" comes from the fact that homotopic maps of topological spaces induce homotopic (in the above sense) maps of singular chains. 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「Homotopy category of chain complexes」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.