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In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. == Strict higher categories == An ordinary category has objects and morphisms. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n-1)-morphisms gives an n-category. Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) ''n''-categories is actually an ''n''+1-category. An ''n''-category is defined by induction on ''n'' by: * A 0-category is a set, * An (''n''+1)-category is a category enriched over the category ''n''-Cat. So a 1-category is just a (locally small) category. The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of ''n''-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too. While this concept is too strict for some purposes in for example, homotopy theory, where "weak" structures arise in the form of higher categories,〔Baez, p 6〕 strict cubical higher homotopy groupoids have also arisen as giving a new foundation for algebraic topology on the border between homology and homotopy theory, see the book "Nonabelian algebraic topology" referenced below. 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「Higher category theory」の詳細全文を読む スポンサード リンク
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