# nLab commutative monoid in a symmetric monoidal category

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Idea

Generalizing the classical notion of commutative monoid, one can define a commutative monoid (or commutative monoid object) internal to 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 $(\mathcal{C}, \otimes, 1, \tau)$ with symmetric braiding $\tau$, then a monoid $(A,\mu, e)$ as above is called a commutative monoid in $(\mathcal{C}, \otimes, 1, \tau)$ if in addition

• (commutativity) the following diagram commutes

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

Note that this condition makes sense even if the braiding $\tau$ is not symmetric, as in a braided monoidal category. Such a monoid is also called a braided monoid in $(\mathcal{C}, \otimes, 1, \tau)$.

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

(commutative rings)

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

The category of chain complexes $Ch(Vect)$ with its tensor product of chain complexes carries a symmetric monoidal braiding given on elements in definite degree $n \in \mathbb{Z}$ by

$\tau \;\colon; v \otimes W \mapsto (-1)^{ n_v n_w } w \otimes v \,.$

The corresponding commutative monoid objects are the differential graded-commutative algebras.

###### Example

The category of chain complexes of super vector spaces $Ch(SuperVect)$ with its tensor product of chain complexes carries two symmetric monoidal braidings given on elements in definite bidegree $(n,\sigma) \in \mathbb{Z} \times \mathbb{Z}/2$ by

1. $\tau_{Deligne} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v n_w + \sigma_v \sigma_w) } w \otimes v$;

2. $\tau_{Bernst} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v + \sigma_v) (n_w + \sigma_w) } w \otimes v$.

The corresponding commutative monoid objects are the differential graded-commutative superalgebras.

sign rule for differential graded-commutative superalgebras
(different but equivalent)

$\phantom{A}$Deligneβs convention$\phantom{A}$$\phantom{A}$Bernsteinβs convention$\phantom{A}$
$\phantom{A}$$\alpha_i \cdot \alpha_j =$$\phantom{A}$$\phantom{A}$$(-1)^{ (n_i \cdot n_j + \sigma_i \cdot \sigma_j) } \alpha_j \cdot \alpha_i$$\phantom{A}$$\phantom{A}$$(-1)^{ (n_i + \sigma_i) \cdot (n_j + \sigma_j) } \alpha_j \cdot \alpha_i$$\phantom{A}$
$\phantom{A}$common in$\phantom{A}$
$\phantom{A}$discussion of$\phantom{A}$
$\phantom{A}$supergravity$\phantom{A}$$\phantom{A}$AKSZ sigma-models$\phantom{A}$
$\phantom{A}$representative$\phantom{A}$
$\phantom{A}$references$\phantom{A}$
$\phantom{A}$Bonora et. al 87,$\phantom{A}$
$\phantom{A}$Castellani-DβAuria-FrΓ© 91,$\phantom{A}$
$\phantom{A}$Deligne-Freed 99$\phantom{A}$
$\phantom{A}$AKSZ 95,$\phantom{A}$
$\phantom{A}$Carchedi-Roytenberg 12$\phantom{A}$

Since the two braidings above are equivalent (this Prop) the corresponding two categories of differential graded-commutative superalgebras are also canonically equivalence of categories:

$ComMon\left( Ch(SuperVect), \otimes, \tau_{Deligne} \right) \;\simeq\; ComMon\left( Ch(SuperVect), \otimes, \tau_{Bernst} \right)$
$sdgcAlg_{Deligne} \;\simeq\; sdgcAlg_{Bernst}$
###### Example

(commutative ring spectra, E-infinity rings)

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.

###### Example

(in a cocartesian monoidal category)

Every object $A$ in a cocartesian monoidal category $C$ becomes a commutative monoid in a unique way: the multiplication must be the fold map $\nabla \colon A + A \to A$, and the counit must be the unique map $! \colon 0 \to A$. Similarly every morphism in $C$ becomes a morphism of commutative monoid objects, so the category of commutative monoid objects in $C$ is isomorphic to $C$.

###### Example

(in $CommMon$)

Since the category $CommMon$ of commutative monoids (in $Set$) is cocartesian, the category of commutative monoids in $(CommMon,+)$ is again $CommMon$. Finite coproducts of commutative monoids are also finite products, so the category of commutative monoids in $(CommMon,\times)$ is also $CommMon$.

## References

### General

Discussion including proof that/when the category of module objects is itself closed symmetric monoidal: