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In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor :⊗ : C × C → C which is associative up to a natural isomorphism, and an object ''I'' which is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions which ensure that all the relevant diagrams commute. In a monoidal category, analogs of usual monoids from abstract algebra can be defined using the same commutative diagrams. In fact, usual monoids are exactly the monoid objects in the monoidal category of sets with Cartesian product. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the category. They are also used in the definition of an enriched category. Monoidal categories have numerous applications outside of category theory proper. They are used to define models for the multiplicative fragment of intuitionistic linear logic. They also form the mathematical foundation for the topological order in condensed matter. Braided monoidal categories have applications in quantum field theory and string theory. ==Formal definition== A monoidal category is a category equipped with *a bifunctor called the ''tensor product'' or ''monoidal product'', *an object called the ''unit object'' or ''identity object'', *three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation * *is associative: there is a natural (in each of three arguments , , ) isomorphism , called ''associator'', with components , * *has as left and right identity: there are two natural isomorphisms and , respectively called ''left'' and ''right unitor'', with components and . : The coherence conditions for these natural transformations are: * for all , , and in , the pentagon diagram :: : commutes; * for all and in , the triangle diagram :: : commutes; It follows from these three conditions that ''a large class'' of such diagrams (i.e. diagrams whose morphisms are built using , , , identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated that ''all'' such diagrams commute. A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal category. 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「Monoidal category」の詳細全文を読む スポンサード リンク
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