Monoid (category theory)

From formulasearchengine
Revision as of 15:36, 11 April 2013 by en>Yobot (→‎Examples: WP:CHECKWIKI error fixes using AWB (9075))
Jump to navigation Jump to search

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

such that the pentagon diagram

Monoid mult.png

and the unitor diagram

Monoid unit.png

commute. In the above notations, I is the unit element and , and 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 . A monoid in C is symmetric when

.

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 and 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