# nLab monoid in a monoidal category

Contents

### Context

#### Monoidal categories

monoidal categories

# Contents

## Idea

Generalizing the classical notion of monoid, one can define a monoid (or monoid object) in any monoidal category $(C,\otimes,I)$. Classical monoids are of course just monoids in Set with the cartesian product.

By the microcosm principle, in order to define monoid objects in $C$, $C$ itself must be a “categorified monoid” in some way. The natural requirement is that it be a monoidal category. In fact, it suffices if $C$ is a multicategory. Contrast this with a group object, which can only be defined in a cartesian monoidal category (or a cartesian multicategory).

## Definition

Namely, a monoid in $C$ is an object $M$ equipped with a multiplication $\mu: M \otimes M \to M$ and a unit $\eta: I \to M$ satisfying the associative law:

$\array{ & (M \otimes M) \otimes M & \stackrel{\alpha}{\longrightarrow} & M \otimes (M \otimes M) & \stackrel{\mu}{\longrightarrow} & M \otimes M \\ & {}_{\mu}\searrow && && \swarrow_{\mu} & \\ && M \otimes M & \stackrel{\mu}{\longrightarrow} M && }$

and the left and right unit laws:

$\array{ & I \otimes M & \stackrel{\eta \otimes 1}{\longrightarrow} & M \otimes M & \stackrel{1 \otimes \eta}{\longleftarrow} & M \otimes I \\ & & {}_{\lambda}\searrow & {}_{\mu}\downarrow & \swarrow_{\rho} & \\ & & & M & & }$

Here $\alpha$ is the associator in $C$, while $\lambda$ and $\rho$ are the left and right unitors.

## Morphism of monoids

The analogue of a monoid homomorphism, called a morphism of monoids, is a morphism, $\f: M \to M'$ between two monoid objects, satisfying the equations;

$f \circ \mu = \mu' \circ (f \otimes f)$

$f \circ \eta = \eta'$

corresponding to the commutative diagrams;

$\array{ & M \otimes M & \stackrel{f \otimes f}{\longrightarrow} & M' \otimes M' \\ & {}_{\mu}\downarrow & & \downarrow_{\mu'} \\ & M & \stackrel{f}{\longrightarrow} & M' }$
$\array{ & I & \stackrel{\eta}{\longrightarrow} & M \\ & & {}_{\eta'}\searrow & \downarrow_{f} \\ & & & M' }$

## As categories with one object

Just as the category of regular monoids is equivalent to the category of locally small (i.e. Set-enriched) categories with one object, the category of monoids in $C$ (with the obvious morphisms) is equivalent to the category of $C$-enriched categories with one object.

## Examples

A monoid in a category of modules is an associative unital algebra. A monoid in a category of endofunctors where tensor product is defined by composition, is a monad.

## References

Categorical properties of monoid objects in monoidal categories are spelled out in sections 1.2 and 1.3 of

• Florian Marty, Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Thesis, 2009, web

A summary is in section 4.1 of