In category theory, a monoid (or monoid object) $(M,\backslash mu,\backslash eta)$ in a monoidal category C is an object M together with two morphisms

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

such that the diagrams

and Monoid_unit.png

commute. In the above notations, I is the unit element and $\backslash alpha$, $\backslash lambda$ and $\backslash rho$ 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 $\backslash mathbf\{C\}^\{\backslash mathrm\{op\}\}$.

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

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

Categories of monoids

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Given two monoids $(M,\backslash mu,\backslash eta)$ and $(M\text{'},\backslash mu\text{'},\backslash eta\text{'})$ in a monoidal category C, a morphism $f:M\backslash to\; M\text{'}$ is a morphism of monoids when

• $f\backslash circ\backslash mu\; =\; \backslash mu\text{'}\backslash circ(f\backslash otimes\; f)$,
• $f\backslash circ\backslash eta\; =\; \backslash eta\text{'}$.

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

See also

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• monoid (non-categorical definition)

Category theory