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commutative monoid in a symmetric monoidal category

Context

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,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.

Definition

Definition

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,B)(\mathcal{C}, \otimes, 1, B) 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,B)(\mathcal{C}, \otimes, 1, B) 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 } \,.

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 }

and

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.

Examples

Example

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.

Example

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.

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 GΓ©omΓ©trie Relative, Ph.D. Thesis, 2009, web

A summary is in section 4.1 of

See also MO/180673.

Revised on June 9, 2016 09:39:15 by Urs Schreiber (131.220.184.222)