Monoid (category theory)
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In category theory, a monoid (or monoid object) (M,μ,η) in a monoidal category C is an object M together with two morphisms
called multiplication,- and
called unit,
such that the diagrams
and 
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 
.
Suppose that the monoidal category C has a symmetry γ. A monoid M in C is symmetric when
.
[edit] Examples
- A monoid object in Set (with the monoidal structure induced by the cartesian product) is a monoid in the usual sense.
- A monoid object in VectK is a K-algebra.
- For any category C, the category [C,C] of its endofunctors has a monoidal structure induced by the composition. A monoid object in [C,C] is a monad on C.
[edit] Categories of monoids
Given two monoids (M,μ,η) and (M',μ',η') in a monoidal category C, a morphism 
is a morphism of monoids when
,
.
The category of whose objects are the monoids and monoid morphisms in C is written 
.
[edit] See also
- monoid (non-categorical definition)
