In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category 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 notation, is the identity morphism of , 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 . 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 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 (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 over R, is an R-algebra. the category of graded modules is a graded R-algebra. the category of chain complexes of R-modules is a differential graded algebra. A monoid object in K-Vect, the category of K-vector spaces (again, with the tensor product), is a 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 monad on C. For any category with a terminal object and finite products, 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 . Categories of monoids Given two monoids (M, μ, η) and (M', μ', η') in a monoidal category C, a morphism f : M →