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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 compact closed category ： ウィキペディア英語版 compact closed category In category theory, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the category with finite-dimensional vector spaces as objects and linear maps as morphisms. == Symmetric compact closed category == A symmetric monoidal category $(\backslash mathbf,\backslash otimes,I)$ is compact closed if every object $A\; \backslash in\; C$ has a dual object. If this holds, the dual object is unique up to canonical isomorphism, and it is denoted $A^$ *. In a bit more detail, an object $A^$ * is called the dual of A if it is equipped with two morphisms called the unit $\backslash eta\_A:I\backslash to\; A^$ *\otimes A and the counit $\backslash varepsilon\_A:A\backslash otimes\; A^$ *\to I, satisfying the equations :$\backslash lambda\_A\backslash circ(\backslash varepsilon\_A\backslash otimes\; A)\backslash circ\backslash alpha\_^\backslash circ(A\backslash otimes\backslash eta\_A)\backslash circ\backslash rho\_A^=\backslash mathrm\_A$ and :$\backslash rho\_\backslash circ(A^$ *\otimes\varepsilon_A)\circ\alpha_\circ(\eta_A\otimes A^ *)\circ\lambda_^=\mathrm_, where $\backslash lambda,\backslash rho$ are the introduction of the unit on the left and right, respectively. For clarity, we rewrite the above compositions diagramatically. In order for $(\backslash mathbf,\backslash otimes,I)$ to be compact closed, we need the following composites to equal $\backslash mathrm\_A$: :$A\backslash xrightarrow\; A\backslash otimes\; I\backslash xrightarrowA\backslash otimes\; (A^$ *\otimes A)\xrightarrow (A\otimes A^ *)\otimes A\xrightarrow I\otimes A\xrightarrow A and $\backslash mathrm\_$: :$A^$ *\xrightarrow I\otimes A^ *\xrightarrow(A^ *\otimes A)\otimes A^ *\xrightarrow A^ *\otimes (A\otimes A^ *)\xrightarrow A^ *\otimes I\xrightarrow A^ * 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「compact closed 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.