# Monoid (category theory)

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

and the unitor diagram

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

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

## Categories of monoids

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.