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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 Rigid category ： ウィキペディア英語版 Rigid category In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual ''X'' * (the internal Hom ''(1 )'') and a morphism 1 → ''X'' ⊗ ''X'' * satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined (following Alexandre Grothendieck) by Neantro Saavedra-Rivano in his thesis on Tannakian categories.〔 * N. Saavedra Rivano, ''Catégories Tannakiennes'', Springer LNM 265, 1972〕 == Definition == There are at least two equivalent definitions of a rigidity. *An object ''X'' of a monoidal category is called left rigid if there is an object ''Y'' and morphisms $\backslash eta\_X\; :\; \backslash mathbf\; \backslash to\; X\; \backslash otimes\; Y$ and $\backslash epsilon\_X\; :\; Y\; \backslash otimes\; X\; \backslash to\; \backslash mathbf$ such that both compositions $X\; ~\; \backslash xrightarrow\; ~\; (X\; \backslash otimes\; Y)\; \backslash otimes\; X\; ~\; \backslash xrightarrow\; \}\; ~\; X\; \backslash otimes\; (Y\; \backslash otimes\; X)\; ~\; \backslash xrightarrow\; ~\; X$ $Y\; ~\; \backslash xrightarrow\; ~\; Y\; \backslash otimes\; (X\; \backslash otimes\; Y\; )\; ~\; \backslash xrightarrow\; ~\; (Y\; \backslash otimes\; X)\; \backslash otimes\; Y\; ~\; \backslash xrightarrow\; ~\; Y$ are identities. A right rigid object is defined similarly. An inverse is an object ''X−1'' such that both ''X'' ⊗ ''X−1'' and ''X−1'' ⊗ ''X'' are isomorphic to 1, the one object of the monoidal category. If an object ''X'' has a left (resp. right) inverse ''X''−1 with respect to the tensor product then it is left (resp. right) rigid, and ''X'' * = ''X''−1. The operation of taking duals gives a contravariant functor on a rigid category. 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「Rigid 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.