- Monoid (category theory)
-
In category theory, a monoid (or monoid object) (M,μ,η) in a monoidal category
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
.
Contents
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 Top (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 theorem.
- 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) is a ring.
- For a commutative ring R, a monoid object in (R-Mod, ⊗R, R) is an R-algebra.
- A monoid object in K-Vect (again, with the tensor product) is a K-algebra, 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. A monoid object in [C,C] is a monad on C.
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 monoids in C and their monoid morphisms is written
.
See also
- monoid (non-categorical definition)
- Act-S, the category of monoids acting on sets
References
- Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, Monoids, Acts and Categories (2000), Walter de Gruyter, Berlin ISBN 3-11-015248-7
Categories:- Monoidal categories
- Objects (category theory)
- Category-theoretic categories
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