Monoid (category theory)

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In category theory, a monoid (or monoid object) (M,\mu,\eta) in a monoidal category (\mathbf{C}, \otimes, I) is an object M together with two morphisms

  • \mu : M\otimes M\to M called multiplication,
  • and \eta : I\to M called unit,

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 \alpha, \lambda and \rho 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 \gamma. A monoid M in C is symmetric when



Categories of monoids[edit]

Given two monoids (M,\mu,\eta) and (M',\mu',\eta') 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 monoids in C and their monoid morphisms is written MonC.

See also[edit]

  • monoid (non-categorical definition)
  • Act-S, the category of monoids acting on sets


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