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In category theory, a monoid (or monoid object) (''M'', μ, η) in a monoidal category (C, ⊗, ''I'') is an object ''M'' together with two morphisms * μ: ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * η: ''I'' → ''M'' called ''unit'', such that the pentagon diagram : and the unitor diagram : commute. In the above notations, ''I'' is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the dual category Cop. Suppose that the monoidal category C has a symmetry γ. A monoid ''M'' in C is commutative when μ o γ = μ. == Examples == * A monoid object in Set, the category of sets, (with the monoidal structure induced by the Cartesian product) is a monoid in the usual sense. * A monoid object in Top, the category of topological spaces, (with the monoidal structure induced by the product topology) is a topological monoid. * A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton theorem. * A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale. * A monoid object in (Ab, ⊗Z, Z), the category of abelian groups, is a ring. * For a commutative ring ''R'', a monoid object in (''R''-Mod, ⊗''R'', ''R''), the category of modules, is an ''R''-algebra. * A monoid object in ''K''-Vect, the category of vector spaces, (again, with the tensor product) is a ''K''-algebra, a comonoid object is a ''K''-coalgebra. * For any category ''C'', the category () of its endofunctors has a monoidal structure induced by the composition. A monoid object in () is a monad on ''C''. 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「Monoid (category theory)」の詳細全文を読む スポンサード リンク
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