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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 2-categories ： ウィキペディア英語版 2-category In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories). == Definition == A 2-category C consists of: * A class of ''0-cells'' (or ''objects'') , , .... * For all objects and , a category $\backslash mathbf(A,B)$. The objects $f,g:A\backslash to\; B$ of this category are called ''1-cells'' and its morphisms $\backslash alpha:f\backslash Rightarrow\; g$ are called ''2-cells''; the composition in this category is usually written $\backslash circ$ or $\backslash circ\_1$ and called ''vertical composition'' or ''composition along a 1-cell''. * For any object  there is a functor from the terminal category (with one object and one arrow) to $\backslash mathbf(A,A)$, that picks out the identity 1-cell  on and its identity 2-cell . In practice these two are often denoted simply by . * For all objects , and , there is a functor $\backslash circ\_0\; :\; \backslash mathbf(A,B)\backslash times\backslash mathbf(B,C)\backslash to\backslash mathbf(A,C)$, called ''horizontal composition'' or ''composition along a 0-cell'', which is associative and admits the identity 1 and 2-cells of as identities. The composition symbol $\backslash circ\_0$ is often omitted, the horizontal composite of 2-cells $\backslash alpha:f\backslash Rightarrow\; g:A\backslash to\; B$ and $\backslash beta:f\text{'}\backslash Rightarrow\; g\text{'}:B\backslash to\; C$ being written simply as $\backslash beta\backslash alpha:f\text{'}f\backslash Rightarrow\; g\text{'}g:A\backslash to\; C$. The notion of 2-category differs from the more general notion of a bicategory in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in a bicategory it needs only be associative up to a 2-isomorphism. The axioms of a 2-category are consequences of their definition as Cat-enriched categories: * Vertical composition is associative and unital, the units being the identity 2-cells . * Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells on the identity 1-cells .. * The interchange law holds; i.e. it is true that for composable 2-cells $\backslash alpha,\backslash beta,\backslash gamma,\backslash delta$ :$(\backslash alpha\backslash circ\_0\backslash beta)\backslash circ\_1(\backslash gamma\backslash circ\_0\backslash delta)\; =\; (\backslash alpha\backslash circ\_1\backslash gamma)\backslash circ\_0(\backslash beta\backslash circ\_1\backslash delta)$ The interchange law follows from the fact that $\backslash circ\_0$ is a functor between hom categories. It can be drawn as a pasting diagram as follows: Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both. 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「2-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.