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In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an opaque object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint (i.e., making the category symmetric monoidal or even cartesian closed, respectively). Enriched category theory thus encompasses within the same framework a wide variety of structures including * ordinary categories where the hom-set carries additional structure beyond being a set. That is, there are operations on, or properties of morphisms that need to be respected by composition (e.g., the existence of 2-cells between morphisms and horizontal composition thereof in a 2-category, or the addition operation on morphisms in an abelian category) * category-like entities that don't themselves have any notion of individual morphism but whose hom-objects have similar compositional aspects (e.g., preorders where the composition rule ensures transitivity, or Lawvere's metric spaces, where the hom-objects are numerical distances and the composition rule provides the triangle inequality). In the case where the hom-object category happens to be the category of sets with the usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory. An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or an enriched category in M, or simply an M-category. Due to Mac Lane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as V-categories. ==Definition== Let be a monoidal category. Then an ''enriched category'' C (alternatively, in situations where the choice of monoidal category needs to be explicit, a ''category enriched over M'', or ''M-category''), consists of * a class ''ob''(C) of ''objects'' of C, * an object ''C''(''a'', ''b'') of M for every pair of objects ''a'', ''b'' in C, * an arrow in M designating an ''identity'' for every object ''a'' in C, and * an arrow in M designating a ''composition'' for each triple of objects ''a'', ''b'', ''c'' in C, together with three commuting diagrams, discussed below. The first diagram expresses the associativity of composition: : That is, the associativity requirement is now taken over by the associator of the hom-category M. For the case that M is the category of sets and is the monoidal structure given by the cartesian product, the terminal single-point set, and the canonical isomorphisms they induce, then each ''C(a,b)'' is a set whose elements may be thought of as "individual morphisms" of ''C'', while °, now a function, defines how consecutive morphisms compose. In this case, each path leading to in the first diagram corresponds to one of the two ways of composing three consecutive individual morphisms from from ''C''(''a'', ''b''), ''C''(''b'', ''c'') and ''C''(''c'', ''d''). Commutativity of the diagram is then merely the statement that both orders of composition give the same result, exactly as required for ordinary categories. What is new here is that the above expresses the requirement for associativity without any explicit reference to individual morphisms in the enriched category C — again, these diagrams are for morphisms in hom-category M, and not in C — thus making the concept of associativity of composition meaningful in the general case where the hom-objects ''C''(''a'', ''b'') are abstract, and ''C'' itself need not even ''have'' any notion of individual morphism. The notion that an ordinary category must have identity morphisms is replaced by the second and third diagrams, which express identity in terms of left and right unitors: :File:Math-enriched category identity1.svg and :File:Math-enriched category identity2.svg Returning to the case where M is the category of sets with cartesian product, the morphisms become functions from the one-point set ''I'' and must then, for any given object ''a'', identify a particular element of each set ''C''(''a'', ''a''), something we can then think of as the "identity morphism for ''a'' in C". Commutativity of the latter two diagrams is then the statement that compositions (as defined by the functions °) involving these distinguished individual "identity morphisms in C" behave exactly as per the identity rules for ordinary categories. Note that there are several distinct notions of "identity" being referenced here: * the ''monoidal identity object'' of M, being an identity for ⊗ only in the monoid-theoretic sense, and even then only up to canonical isomorphism . * the ''identity morphism'' that M has for each of its objects by virtue of it being (at least) an ordinary category. * the enriched category ''identity'' for each object a in C, which is again a morphism of M which, even in the case where C ''is'' deemed to have individual morphisms of its own, is not necessarily identifying a specific one. 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「Enriched category」の詳細全文を読む スポンサード リンク
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