Monoid (category theory)

In , a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a (C, ⊗, I) is an M together with two morphisms μ: M ⊗ M → M called multiplication, η: I → M called unit, such that the pentagon and the unitor diagram commute. In the above notation, 1 is the identity morphism of M, 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 Cop. Suppose that the monoidal category C has a γ. A monoid M in C is commutative when μ o γ = μ. A monoid object in , the (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense. A monoid object in , the (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 argument. 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 (, ⊗Z, Z), the , is a ring. For a commutative ring R, a monoid object in (, ⊗R, R), the over R, is a R-algebra. the category of graded modules is a graded R-algebra. the of R-modules is a differential graded algebra. A monoid object in K-Vect, the (again, with the tensor product), is a unital associative K-algebra, and a comonoid object is a K-coalgebra. For any category C, the category [C,C] of its endofunctors has a monoidal structure induced by the composition and the identity functor IC. A monoid object in [C,C] is a on C. For any category with a terminal object and , every object becomes a comonoid object via the diagonal morphism . Dually in a category with an initial object and finite coproducts every object becomes a monoid object via . Given two monoids (M, μ, η) and (M', μ', η') in a monoidal category C, a morphism f : M → M ' is a morphism of monoids when f o μ = μ''' o (f ⊗ f), f o η = η.