nLab commutative monoid in a symmetric monoidal category

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

(differential graded-commutative algebras)

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

(differential graded-commutative superalgebras)

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)

	$\phantom{A}$Deligne’s convention$\phantom{A}$	$\phantom{A}$Bernstein’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-D’Auria-Fré 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 $CommMon$.

Related concepts

• monoid, commutative monoid
• monoid in a monoidal category
• category of monoids
• module over a monoid
• semilattice object
• commutative monoid in a symmetric monoidal (infinity,1)-category

References

General

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

• Mark Hovey, Brooke Shipley, Jeff Smith, pp. 15 in: Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149-208 [arXiv:math/9801077, doi:10.1090/S0894-0347-99-00320-3]

See also:

• MO/180673.

• Marty 2009, 1.2 and 1.3

• Martin Brandenburg, Section 4.1 Tensor categorical foundations of algebraic geometry, arXiv:1410.1716.

Lecture notes:

• Urs Schreiber: Algebras and Modules, in Introduction to Stable Homotopy Theory 1.2: Structured Spectra

Opposite categories

Discussion of the opposite categories of commutative monoid objects and regarded as categories of generalized affine schemes:

• Bertrand Toën, Michel Vaquié, Au-dessous de $Spec \mathbb{Z}$, Journal of K-Theory 3 3 (2009) 437-500 [doi:10.1017/is008004027jkt048]

• Florian Marty, Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Toulouse (2009) [theses:2009TOU30071, pdf]

• Florian Marty, Relative Zariski Open Objects, Journal of K-Theory 10 1 (2012) 9-39 [arXiv:0712.3676, doi:10.1017/is011012004jkt176]

• Abhishek Banerjee, The relative Picard functor on schemes over a symmetric monoidal category, Bulletin des Sciences Mathématiques 135 4 (2011) 400-419 [doi:10.1016/j.bulsci.2011.02.001]

• Abhishek Banerjee, On integral schemes over symmetric monoidal categories [arXiv:1506.04890]

• Abhishek Banerjee, Noetherian Schemes over abelian symmetric monoidal categories, International Journal of Mathematics 28 07 (2017) 1750051 [doi:1410.3212, doi:10.1142/S0129167X17500513]