# nLab commutative monoid in a symmetric monoidal category

### Context

#### Monoidal categories

monoidal categories

# Contents

## Idea

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

## Definition

###### Definition

Given a monoidal category $(\mathcal{C}, \otimes, 1)$, then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is

1. an object $A \in \mathcal{C}$;

2. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the unit)

3. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the product);

such that

1. (associativity) the following diagram commutes

$\array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,$

where $a$ is the associator isomorphism of $\mathcal{C}$;

2. (unitality) the following diagram commutes:

$\array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,,$

where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.

Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a symmetric monoidal category $(\mathcal{C}, \otimes, 1, B)$ with symmetric braiding $\tau$, then a monoid $(A,\mu, e)$ as above is called a commutative monoid in $(\mathcal{C}, \otimes, 1, B)$ if in addition

• (commutativity) the following diagram commutes

$\array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,.$

A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

$f \;\colon\; A_1 \longrightarrow A_2$

in $\mathcal{C}$, such that the following two diagrams commute

$\array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }$

and

$\array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.$

Write $Mon(\mathcal{C}, \otimes,1)$ for the category of monoids in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.

## Examples

###### Example

Write $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ for the category Ab of abelian groups, equipped with the tensor product of abelian groups whose tensor unit is the additive group of integers. With the evident braiding this is a symmetric monoidal category.

A commutative monoid in $(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})$ is equivalently a commutative ring.

###### Example

Write $(SymSpec(Top_{cg}),\wedge, \mathbb{S}_{sym})$ and $(OrthSpec(Top_{cg}),\wedge, \mathbb{S}_{orth})$ and $([Top^{\ast/}_{cg,fin}, Top^{\ast/}_{cg}], \wedge, \mathbb{S} )$ for the categories, respectively of symmetric spectra, orthogonal spectra and pre-excisive functors, equipped with their symmetric monoidal smash product of spectra, whose tensor unit is the corresponding standard incarnation of the sphere spectrum.

A commutative monoid in any one of these three categories is equivalently a commutative ring spectrum in the strong sense: via the respective model structure on spectra it represents an E-infinity ring.

## References

Categorical properties of commutative monoid objects in symmetric 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