翻訳と辞書 Words near each other ・ Closet flange ・ Closet Land ・ Closet Monster ・ Closet Monster (film) ・ Closet screenplay ・ Closet Space ・ Closet? You'll Love It! ・ ClosetCon '13 ・ Closeted ・ Closed list ・ Closed loop ・ Closed loop communication ・ Closed loop manufacturing ・ Closed manifold ・ Closed Mondays ・ Closed monoidal category ・ Closed on Account of Rabies ・ Closed platform ・ Closed position ・ Closed railway stations in Western Australia ・ Closed range theorem ・ Closed schools in the Northland Region ・ Closed session of the United States Congress ・ Closed sessions of the United States House of Representatives ・ Closed sessions of the United States Senate ・ Closed set ・ Closed shop ・ Closed subgroup theorem ・ Closed subscriber group ・ Closed system Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Closed monoidal category ： ウィキペディア英語版 Closed monoidal category In mathematics, especially in category theory, a closed monoidal category is a context where it is possible both to form tensor products of objects and to form 'mapping objects'. A classic example is the category of sets, Set, where the tensor product of sets $A$ and $B$ is the usual cartesian product $A\; \backslash times\; B$, and the mapping object $B^A$ is the set of functions from $A$ to $B$. Another example is the category FdVect, consisting of finite-dimensional vector spaces and linear maps. Here the tensor product is the usual tensor product of vector spaces, and the mapping object is the vector space of linear maps from one vector space to another. The 'mapping object' referred to above is also called the 'internal Hom'. The internal language of closed symmetric monoidal categories is the linear type system. ==Definition== A closed monoidal category is a monoidal category $\backslash mathcal$ such that for every object $B$ the functor given by right tensoring with $B$ :$A\backslash mapsto\; A\backslash otimes\; B$ has a right adjoint, written :$A\backslash mapsto\; (B\; \backslash Rightarrow\; A).$ This means that there exists a bijection, called 'currying', between the Hom-sets :$\backslash text\_\backslash mathcal(A\backslash otimes\; B,\; C)\backslash cong\backslash text\_\backslash mathcal(A,B\backslash Rightarrow\; C)$ that is natural in both ''A'' and ''C''. In a different, but common notation, one would say that the functor :$-\backslash otimes\; B:\backslash mathcal\backslash to\backslash mathcal$ has a right adjoint :$(-):\backslash mathcal\backslash to\backslash mathcal$ Equivalently, a closed monoidal category $\backslash mathcal$ is a category equipped, for every two objects ''A'' and ''B'', with * an object $A\backslash Rightarrow\; B$, * a morphism $\backslash mathrm\_\; :\; (A\backslash Rightarrow\; B)\; \backslash otimes\; A\; \backslash to\; B$, satisfying the following universal property: for every morphism :$f\; :\; X\backslash otimes\; A\backslash to\; B$ there exists a unique morphism :$h\; :\; X\; \backslash to\; A\backslash Rightarrow\; B$ such that :$f\; =\; \backslash mathrm\_\backslash circ(h\; \backslash otimes\; \backslash mathrm\_A).$ It can be shown that this construction defines a functor $\backslash Rightarrow\; :\; \backslash mathcal^\; \backslash otimes\; \backslash mathcal\; \backslash to\; \backslash mathcal$. This functor is called the internal Hom functor, and the object $A\; \backslash Rightarrow\; B$ is called the internal Hom of $A$ and $B$. Many other notations are in common use for the internal Hom. When the tensor product on $\backslash mathcal$ is the cartesian product, the usual notation is $B^A$ and this object is called the exponential object. 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「Closed monoidal 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.