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associative unital algebra

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Definition

Over ordinary rings

For RR a commutative ring, an associative unital RR-algebra is equivalently

If there is no danger for confusion, one often says simply ‘associative algebra’, or even only ‘algebra’.

More generally, a (merely) associative algebra need not have i:RVi: R \to V; that is, it is a semigroup instead of a monoid.

Less generally, a commutative algebra (where associative and unital are usually assumed) is an abelian monoid in VectVect.

Over monoids in a monoidal category

Definition

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

  1. an object A𝒞A \in \mathcal{C};

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

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

such that

  1. (associativity) the following diagram commutes

    (AA)A a A,A,A A(AA) Aμ AA μA μ AA 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 𝒞\mathcal{C};

  2. (unitality) the following diagram commutes:

    1A eid AA ide A1 μ 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 \ell and rr are the left and right unitor isomorphisms of 𝒞\mathcal{C}.

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

  • (commutativity) the following diagram commutes

    AA τ A,A AA μ μ 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 1A 2 f \;\colon\; A_1 \longrightarrow A_2

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

A 1A 1 ff A 2A 2 μ 1 μ 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 2 f A 2. \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.

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

Definition

Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1), and given (A,μ,e)(A,\mu,e) a monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 1), then a left module object in (𝒞,,1)(\mathcal{C}, \otimes, 1) over (A,μ,e)(A,\mu,e) is

  1. an object N𝒞N \in \mathcal{C};

  2. a morphism ρ:ANN\rho \;\colon\; A \otimes N \longrightarrow N (called the action);

such that

  1. (unitality) the following diagram commutes:

    1N eid AN ρ A, \array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && A } \,,

    where \ell is the left unitor isomorphism of 𝒞\mathcal{C}.

  2. (action property) the following diagram commutes

    (AA)N a A,A,N A(AN) Aρ AN μN ρ AN ρ N, \array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,

A homomorphism of left AA-module objects

(N 1,ρ 1)(N 2,ρ 2) (N_1, \rho_1) \longrightarrow (N_2, \rho_2)

is a morphism

f:N 1N 2 f\;\colon\; N_1 \longrightarrow N_2

in 𝒞\mathcal{C}, such that the following diagram commutes:

AN 1 Af AN 2 ρ 1 ρ 2 N 1 f N 2. \array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,.

For the resulting category of modules of left AA-modules in 𝒞\mathcal{C} with AA-module homomorphisms between them, we write

AMod(𝒞). A Mod(\mathcal{C}) \,.

This is naturally a (pointed) topologically enriched category itself.

Definition

Given a (pointed) topological symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1), given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 1), and given (N 1,ρ 1)(N_1, \rho_1) and (N 2,ρ 2)(N_2, \rho_2) two left AA-module objects (def.1), then the tensor product of modules N 1 AN 2N_1 \otimes_A N_2 is, if it exists, the coequalizer

N 1AN 2AAAAρ 1(τ N 1,AN 2)N 1ρ 2N 1N 1coequN 1 AN 2 N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coequ}{\longrightarrow} N_1 \otimes_A N_2
Proposition

Given a symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. \ref{SymmetricMonoidalCategory}), and given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. 1). If all coequalizers exist in 𝒞\mathcal{C}, then the tensor product of modules A\otimes_A from def. 3 makes the category of modules AMod(𝒞)A Mod(\mathcal{C}) into a symmetric monoidal category, (AMod, A,A)(A Mod, \otimes_A, A) with tensor unit the object AA itself.

Definition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) as in prop. 1, then a monoid (E,μ,e)(E, \mu, e) in (AMod, A,A)(A Mod , \otimes_A , A) (def. 1) is called an AA-algebra.

Propposition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) in a monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) as in prop. 1, and an AA-algebra (E,μ,e)(E,\mu,e) (def. 4), then there is an equivalence of categories

AAlg comm(𝒞)CMon(AMod)CMon(𝒞) A/ A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/}

between the category of commutative monoids in AModA Mod and the coslice category of commutative monoids in 𝒞\mathcal{C} under AA, hence between commutative AA-algebras in 𝒞\mathcal{C} and commutative monoids EE in 𝒞\mathcal{C} that are equipped with a homomorphism of monoids AEA \longrightarrow E.

(e.g. EKMM 97, VII lemma 1.3)

Proof

In one direction, consider a AA-algebra EE with unit e E:AEe_E \;\colon\; A \longrightarrow E and product μ E/A:E AEE\mu_{E/A} \colon E \otimes_A E \longrightarrow E. There is the underlying product μ E\mu_E

EAE AAA EE coeq E AE μ E μ E/A E. \array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,.

By considering a diagram of such coequalizer diagrams with middle vertical morphism e Ee Ae_E\circ e_A, one find that this is a unit for μ E\mu_E and that (E,μ E,e Ee A)(E, \mu_E, e_E \circ e_A) is a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1).

Then consider the two conditions on the unit e E:AEe_E \colon A \longrightarrow E. First of all this is an AA-module homomorphism, which means that

()AA ide E AE μ A ρ A e E E (\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E }

commutes. Moreover it satisfies the unit property

A AE e Aid E AE μ E/A E. \array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,.

By forgetting the tensor product over AA, the latter gives

AE eid EE A AE e Eid E AE μ E/A E = EAE e Eid EE ρ μ E E id E, \array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,,

where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be pasted to the square ()(\star) above, to yield a commuting square

AA ide E AE e Eid EE μ A ρ μ E A e E E id E=AA e Ee E EE μ A μ E A e E E. \array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,.

This shows that the unit e Ae_A is a homomorphism of monoids (A,μ A,e A)(E,μ E,e Ee A)(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A).

Now for the converse direction, assume that (A,μ A,e A)(A,\mu_A, e_A) and (E,μ E,e E)(E, \mu_E, e'_E) are two commutative monoids in (𝒞,,1)(\mathcal{C}, \otimes, 1) with e E:AEe_E \;\colon\; A \to E a monoid homomorphism. Then EE inherits a left AA-module structure by

ρ:AEe AidEEμ EE. \rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,.

By commutativity and associativity it follows that μ E\mu_E coequalizes the two induced morphisms EAEAAEEE \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E. Hence the universal property of the coequalizer gives a factorization through some μ E/A:E AEE\mu_{E/A}\colon E \otimes_A E \longrightarrow E. This shows that (E,μ E/A,e E)(E, \mu_{E/A}, e_E) is a commutative AA-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

Variants

Examples

Properties

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

References

Revised on June 9, 2016 10:58:24 by Urs Schreiber (131.220.184.222)