# nLab monoid in a monoidal category

Contents

### Context

#### Categorical algebra

internalization and categorical algebra

universal algebra

categorical semantics

#### Monoidal categories

monoidal categories

## In higher category theory

#### Monoid theory

monoid theory in algebra:

# 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:

$\array{ & (M \otimes M) \otimes M & \stackrel{\alpha}{\longrightarrow} & M \otimes (M \otimes M) & \stackrel{1 \otimes \mu}{\longrightarrow} & M \otimes M \\ & {}_{\mu \otimes 1}\searrow && && \swarrow_{\mu} & \\ && M \otimes M & \stackrel{\mu}{\longrightarrow} M && }$

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.

## Properties

### Preservation by lax monoidal functors

Monoid structure is preserved by lax monoidal functors. Comonoid structure by oplax monoidal functors. See lax monoidal functor for more details.

### Category of monoids

For special properties of categories of monoids, see category of monoids.

## Examples

• A monoid object in Ab (with the usual tensor product of $\mathbb{Z}$-modules as the tensor product) is a ring. A monoid object in the category of vector spaces over a field $k$ (with the usual tensor product of vector spaces) is an algebra over $k$.
• A monoid in a category of modules is an associative unital algebra.
• A monoid object in Top (with cartesian product as the tensor product) is a topological monoid.
• A monoid object in Ho(Top) is an H-monoid?.
• A monoid object in the category of monoids (with cartesian product as the tensor product) is a commutative monoid. This is a version of the Eckmann-Hilton argument.
• A monoid object in the category of complete join-semilattices (with its tensor product that represents maps preserving joins in each variable separately) is a unital quantale.
• The category of pointed sets has a monoidal structure given by the smash product. A monoid object in this monoidal category is an absorption monoid.
• Given any monoidal category $C$, a monoid in the monoidal category $C^{op}$ is called a comonoid in $C$.
• In a cocartesian monoidal category, every object is a monoid object in a unique way.
• For any category $C$, the endofunctor category $C^C$ has a monoidal structure induced by composition of endofunctors, and a monoid object in $C^C$ is a monad on $C$.

These are examples of monoids internal to monoidal categories. More generally, given any bicategory $B$ and a chosen object $a$, the hom-category $B(a,a)$ has the structure of a monoidal category. So, the concept of monoid makes sense in any bicategory $B$: we define a monoid in $B$ to be a monoid in $B(a,a)$ for some object $a \in B$. This often called a monad in $B$. The reason is that a monad in Cat is the same as monad on a category.

A monoid in a bicategory $B$ may also be described as the hom-object of a $B$-enriched category with a single object.

Original reference (including the case of a commutative monoid in a symmetric monoidal category):