monoid

(category theory)

In category theory, a monoid (or monoid object) $(M,μ,η)$ in a monoidal category C is an object M together with two morphisms

$\mu : M\otimes M\to M$ called multiplication,
and $η: I \to M$ called unit,

such that the diagrams

Image:Monoid_mult.png and Image:Monoid_unit.png

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 $\mathbf{C}^{\mathrm{op}}$ .

Suppose that the monoidal category C has a symmetry $γ$ . A monoid $M$ in C is symmetric when

$\mu\circ\gamma=\mu$ .

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 the category of complete join-semilattices Sup (with the monoidal structure induced by the cartesian product) is a unital quantale.
A monoid object in Vect_K 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 $f : M \to M'$ is a morphism of monoids when

$f\circ\mu = \mu'\circ(f\otimes f)$ ,
$f\circ\eta = \eta'$ .

The category of whose objects are the monoids and monoid morphisms in C is written $\mathbf{Mon}_\mathbf{C}$ .

References

Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, Monoids, Acts and Categories (2000), Walter de Gruyter, Berlin ISBN 3-11-015248-7

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

monoid

Examples

Categories of monoids

See also

References

Tackle These

Keep Reading