Monoid (category theory)

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) ${\displaystyle (M,\mu ,\eta )}$ in a monoidal category ${\displaystyle ({\mathcal {C}},\otimes ,I)}$ is an object ${\displaystyle M}$ together with two morphisms

• ${\displaystyle \mu \colon M\otimes M\to M}$ called multiplication,
• ${\displaystyle \eta \colon I\to M}$ called unit,

such that the pentagon diagram

and the unitor diagram

commute. In the above notation, ${\displaystyle 1}$ is the identity morphism of ${\displaystyle M}$, ${\displaystyle I}$ is the unit element and ${\displaystyle \alpha ,\lambda }$ and ${\displaystyle \rho }$ are respectively the associator, the left unitor and the right unitor of the monoidal category ${\displaystyle {\mathcal {C}}}$.

Dually, a comonoid in a monoidal category ${\displaystyle {\mathcal {C}}}$ is a monoid in the dual category ${\displaystyle {\mathcal {C}}^{\mathrm {op} }}$.

Suppose that the monoidal category ${\displaystyle {\mathcal {C}}}$ has a braiding ${\displaystyle \gamma }$. A monoid ${\displaystyle M}$ in ${\displaystyle {\mathcal {C}}}$ is commutative when ${\displaystyle \mu \circ \gamma =\mu }$.