nLab commutative monoid in a symmetric monoidal category



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



Generalizing the classical notion of commutative monoid, one can define a commutative monoid (or commutative monoid object) internal to any symmetric monoidal category (C,βŠ—,I)(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.



Given a monoidal category (π’ž,βŠ—,1)(\mathcal{C}, \otimes, 1), then a monoid internal to (π’ž,βŠ—,1)(\mathcal{C}, \otimes, 1) is

  1. an object Aβˆˆπ’žA \in \mathcal{C};

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

  3. a morphism ΞΌ:AβŠ—A⟢A\mu \;\colon\; A \otimes A \longrightarrow A (called the product);

such that

  1. (associativity) the following diagram commutes

    (AβŠ—A)βŠ—A βŸΆβ‰ƒa A,A,A AβŠ—(AβŠ—A) ⟢AβŠ—ΞΌ AβŠ—A ΞΌβŠ—A↓ ↓ ΞΌ AβŠ—A ⟢ ⟢μ A, \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 aa is the associator isomorphism of π’ž\mathcal{C};

  2. (unitality) the following diagram commutes:

    1βŠ—A ⟢eβŠ—id AβŠ—A ⟡idβŠ—e AβŠ—1 β„“β†˜ ↓ ΞΌ ↙ r A, \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 β„“\ell and rr are the left and right unitor isomorphisms of π’ž\mathcal{C}.

Moreover, if (π’ž,βŠ—,1)(\mathcal{C}, \otimes , 1) has the structure of a symmetric monoidal category (π’ž,βŠ—,1,Ο„)(\mathcal{C}, \otimes, 1, \tau) with symmetric braiding Ο„\tau, then a monoid (A,ΞΌ,e)(A,\mu, e) as above is called a commutative monoid in (π’ž,βŠ—,1,Ο„)(\mathcal{C}, \otimes, 1, \tau) if in addition

  • (commutativity) the following diagram commutes

    AβŠ—A βŸΆβ‰ƒΟ„ A,A AβŠ—A ΞΌβ†˜ ↙ ΞΌ A. \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 Ο„\tau is not symmetric, as in a braided monoidal category. Such a monoid is also called a braided monoid in (π’ž,βŠ—,1,Ο„)(\mathcal{C}, \otimes, 1, \tau).

A homomorphism of monoids (A 1,μ 1,e 1)⟢(A 2,μ 2,f 2)(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2) is a morphism

f:A 1⟢A 2 f \;\colon\; A_1 \longrightarrow A_2

in π’ž\mathcal{C}, such that the following two diagrams commute

A 1βŠ—A 1 ⟢fβŠ—f A 2βŠ—A 2 ΞΌ 1↓ ↓ ΞΌ 2 A 1 ⟢f A 2 \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 }


1 𝒸 ⟢e 1 A 1 e 2β†˜ ↓ f A 2. \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.

Write Mon(π’ž,βŠ—,1)Mon(\mathcal{C}, \otimes,1) for the category of monoids in π’ž\mathcal{C}, and CMon(π’ž,βŠ—,1)CMon(\mathcal{C}, \otimes, 1) for its subcategory of commutative monoids.



(commutative rings)

Write (Ab,βŠ— β„€,β„€)(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,βŠ— β„€,β„€)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) is equivalently a commutative ring.


(differential graded-commutative algebras)

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

Ο„:;vβŠ—W↦(βˆ’1) n vn wwβŠ—v. \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.


(differential graded-commutative superalgebras)

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

  1. Ο„ Deligne:vβŠ—w↦(βˆ’1) (n vn w+Οƒ vΟƒ w)wβŠ—v\tau_{Deligne} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v n_w + \sigma_v \sigma_w) } w \otimes v;

  2. Ο„ Bernst:vβŠ—w↦(βˆ’1) (n v+Οƒ v)(n w+Οƒ w)wβŠ—v\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)

A\phantom{A}Deligne’s conventionA\phantom{A}A\phantom{A}Bernstein’s conventionA\phantom{A}
A\phantom{A}Ξ± iβ‹…Ξ± j= \alpha_i \cdot \alpha_j = A\phantom{A}A\phantom{A}(βˆ’1) (n iβ‹…n j+Οƒ iβ‹…Οƒ j)Ξ± jβ‹…Ξ± i(-1)^{ (n_i \cdot n_j + \sigma_i \cdot \sigma_j) } \alpha_j \cdot \alpha_iA\phantom{A}A\phantom{A}(βˆ’1) (n i+Οƒ i)β‹…(n j+Οƒ j)Ξ± jβ‹…Ξ± i (-1)^{ (n_i + \sigma_i) \cdot (n_j + \sigma_j) } \alpha_j \cdot \alpha_iA\phantom{A}
A\phantom{A}common inA\phantom{A}
A\phantom{A}discussion ofA\phantom{A}
A\phantom{A}supergravityA\phantom{A}A\phantom{A}AKSZ sigma-modelsA\phantom{A}
A\phantom{A}Bonora et. al 87,A\phantom{A}
A\phantom{A}Castellani-D’Auria-FrΓ© 91,A\phantom{A}
A\phantom{A}Deligne-Freed 99A\phantom{A}
A\phantom{A}AKSZ 95,A\phantom{A}
A\phantom{A}Carchedi-Roytenberg 12A\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(Ch(SuperVect),βŠ—,Ο„ Deligne)≃ComMon(Ch(SuperVect),βŠ—,Ο„ Bernst) ComMon\left( Ch(SuperVect), \otimes, \tau_{Deligne} \right) \;\simeq\; ComMon\left( Ch(SuperVect), \otimes, \tau_{Bernst} \right)
sdgcAlg Deligne≃sdgcAlg Bernst sdgcAlg_{Deligne} \;\simeq\; sdgcAlg_{Bernst}

(commutative ring spectra, E-infinity rings)

Write (SymSpec(Top cg),∧,π•Š sym)(SymSpec(Top_{cg}),\wedge, \mathbb{S}_{sym}) and (OrthSpec(Top cg),∧,π•Š orth)(OrthSpec(Top_{cg}),\wedge, \mathbb{S}_{orth}) and ([Top cg,fin */,Top cg */],∧,π•Š)([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.


(in a cocartesian monoidal category)

Every object AA in a cocartesian monoidal category CC becomes a commutative monoid in a unique way: the multiplication must be the fold map βˆ‡:A+Aβ†’A\nabla \colon A + A \to A, and the counit must be the unique map !:0β†’A! \colon 0 \to A. Similarly every morphism in CC becomes a morphism of commutative monoid objects, so the category of commutative monoid objects in CC is isomorphic to CC.


(in CommMonCommMon)

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



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

See also:

Lecture notes:

Opposite categories

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

Last revised on February 13, 2024 at 10:15:28. See the history of this page for a list of all contributions to it.