Monoid (category theory)

In category theory, monoids or monoid objects generalise the traditional concept of a monoid from abstract algebra — naturally, the algebraic monoid is a categorical monoid in $𝐒𝐞𝐭$.

Definition

Within a monoidal category $(𝐂,\otimes ,I)$, a monoid, monoid object, or internal monoid is an object $M\in 𝐂$ equipped with a morphism $\mu :M\otimes M\to M$ and an arrow $\eta :I\to M$ subject to the laws of associativity:[cite:@riehl2017category]

(quiver)\begin{tikzcd} {M \otimes (M \otimes M)} && {M \otimes M} \\ {(M \otimes M) \otimes M} \\ {M \otimes M} && M \arrow["{\id{M} \otimes \mu}", from=1-1, to=1-3] \arrow["\alpha"', from=1-1, to=2-1] \arrow["\mu\text{,}", from=1-3, to=3-3] \arrow["{\mu \otimes \id{M}}"', from=2-1, to=3-1] \arrow["\mu"', from=3-1, to=3-3] \end{tikzcd} ParseError: No such environment: tikzcd at position 7: \begin{̲t̲i̲k̲z̲c̲d̲}̲ {M \otimes (M…

and identity:

(quiver)\begin{tikzcd} {I \otimes M} & M & {M \otimes I} \\ & {M\otimes M} \arrow["\lambda", from=1-1, to=1-2] \arrow["{\eta \otimes \id{m}}"', from=1-1, to=2-2] \arrow["\rho"', from=1-3, to=1-2] \arrow["{\id{m} \otimes \eta}\text{,}", from=1-3, to=2-2] \arrow["\mu"', from=2-2, to=1-2] \end{tikzcd} ParseError: No such environment: tikzcd at position 7: \begin{̲t̲i̲k̲z̲c̲d̲}̲ {I \otimes …

where $\alpha$ is the associator and $\lambda$ and $\rho$ are the left and right unitors, all provided by the host category's monoidal structure.

See also

• Monoidal category

• Monoid

• Monoids as categories