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, "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 .
Categories of monoids.
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
In other words, the following diagrams
commute.
The category of monoids in C and their monoid morphisms is written MonC.