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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 . The objects of this category are called ''1-cells'' and its morphisms are called ''2-cells''; the composition in this category is usually written or 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 , 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 , 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 is often omitted, the horizontal composite of 2-cells and being written simply as . 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 : The interchange law follows from the fact that 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
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