Monoid (category theory)
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 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 C. Suppose that the monoidal category C has a  symmetry "γ". A monoid "M" in C is commutative when . Categories of monoids. Given two monoids ("M", "μ", "η") and ("<nowiki>M'</nowiki>", "<nowiki>μ'</nowiki>", "<nowiki>η'</nowiki>") in a monoidal category C, a morphism "f" : "M" → "M" ' is a morphism of monoids when In other words, the following diagrams commute. The category of monoids in C and their monoid morphisms is written Mon.