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geometry of physics -- modules

Contents

This entry contains one chapter of geometry of physics. See there for background and context.

previous chapters: representations and associated bundles, stable homotopy types

next chapters: geometric quantization with KU-coefficients


Contents

Modules

In the previous section we have seen ∞-actions of ∞-groups on objects in the ambient (∞,1)-topos. Here we specialize this to an important class of cases and then generalize from ∞-actions to (,n)(\infty,n)-actions for nn \in \mathbb{N}. That special case is the one where all objects involved are required to be equipped with additive structure (in generalization of the sense in which abelian groups are additive) and where all actions are linear (in generalization of the sense of abelian group homomorphisms, hence linear functions). Moreover, we equip groups with additional (linear) monoidal structure such as to become higher analogs of rings. Linear actions of a ring are called modules and hence here we discuss modules and their higher analogs.

Notice the different role of the “(r,n)(r,n)” index:

(r,n)(r,n)-ringpresentation
ring = 1-ring = (1,1)(1,1)-ringcompatibly monoidal abelian group
2-ring = (2,2)(2,2)-ringcompatibly monoidal category
commutative (,1)(\infty,1)-ringE-∞ ring = compatibly monoidal ∞-group / spectrum
(,2)(\infty,2)-ringcompatibly monoidal (∞,1)-category

A special case of modules are vector spaces, which are the modules over a ring that is a field. For them the theory of modules is the theory of linear algebra. But as opposed to the notion of ring, the more specialized notion of field has in general no natural or useful lift to higher category theory. Therefore we generally speak here of modules over rings and their higher analogs. But the reader who feels more at home with vector spaces may find it helpful to think of higher modules as being higher analogs of vector spaces.

1-Modules

We briefly disucss here some basics of ordinary linear algebra with an emphasis on the generalization from vector spaces over fields to modules over rings.

Physics motivation: Superposition of states and fields

One key aspect of quantum mechanics, that distinguishes it from classical mechanics, is that there is linear structure on quantum states, as follows (the following applies to quantum states in the Schrödinger picture only, not to states in the sense of state on an operator algebra, which are really the expectation values of observables in the states as discussed here):

  • The superposition principle in quantum mechanics says that for ψ 1\psi_1 and ψ 2\psi_2 two quantum states one can form their sum ψ 1+ψ 2\psi_1 + \psi_2 as well as their difference ψ 1ψ 2\psi_1 - \psi_2 such that this is again a quantum state. Moreover there is a 0-state such that ψ+0=ψ\psi + 0 = \psi for all ψ\psi. Together this means there is the structure of an additive abelian group on the set of quantum states.

  • Quantum states have complex phases. This means that for ψ\psi a quantum state and for cc \in \mathbb{C} a complex number, there is a new quantum state cψc \psi and this is such that for two such complex numbers we have c 1(c 2ψ)=(c 1c 2)ψc_1(c_2 \psi) = (c_1 c_2) \psi and for two states we have c(ψ 1+ψ 2)=cψ 1+cψ 2c(\psi_1 + \psi_2) = c \psi_1 + c \psi_2. This means that multiplying by phases is a linear action of the ring of complex numbers on quantum states.

Together this says equivalently that the additive group of quantum states is a module over the complex numbers. Since the ring of complex numbers is special in that it is a field, such a module is equivalently called a complex vector space.

This linear structure is a crucial aspect of quanum theory. It is at the heart of phenomena such quantum interference and entanglement

In macroscopic physics similar behaviour is known in wave mechanics for freely propagating waves. Not unrelated to this is the term wave function for a quantum state. However, apart from wave mechanics, linearity is not manifest in macoscopic physical phenomena, even though it is fundamentally present; this is called decoherence.

It is this linear structure of quantum theory that, since shortly after its conception, led to a close relaton to group representation theory which was unknown in classical physics. (And this came unexpected to physicist at the beginning of the 20th century and was not embraced by all theoreticians at first, see at Gruppenpest.) If a symmetry group acts on field configurations of a physical system and is preserved by quantization, then it still acts on the quantum states and does so in a linear way; hence then the space of quantum states forms a linear representation space of the symmetry group. This relation between quantum mechanical systems with symmetry and linear representation theory goes so far that under natural assumptions every representation of a group arises as the action of a group of symmetries on a quantum system – a relation known as the “orbit method” in geometric quantization. Deep results about abstract linear representation theory have been proven by considering systems quantum mechanics this way.

But the relation between quantum physics and linear representation theory goes deeper still. Not only do the quantum states form modules over the complex numbers, but already the matter fields in fundamental physics form modules over the algebra of functions on spacetime. This means in turn that matter fields form an additive abelian group where for ϕ 1\phi_1 and ϕ 2\phi_2 two field configurations also ϕ 1+ϕ 2\phi_1 + \phi_2 is a field configuration, such that with ϕ\phi a matter field for each smooth function ff on spacetime there is also a matter field denoted fϕf \phi and such that f 1(f 2ϕ)=(f 1f 2)ϕf_1 (f_2 \phi) = (f_1 f_2) \phi and f(ϕ 1+ϕ 2)=fϕ 1+ϕ 2f(\phi_1 + \phi_2) = f \phi_1 + \phi_2.

There is a more geometric expression of this: in fact the matter fields form a module with special properties (they are what is called projective and finitely generated) and the Serre-Swan theorem asserty that modules with these special properties consist of sections of a fiber bundle whose typical fiber is a module over the complex numbers, hence a complex vector space: a complex vector bundle. Moreover, these vector bundles are associated bundles via linear representations of the gauge group of the force fields under which the given matter fields are charged. So in addition to being a linear module over the algebra of functions on spacetime, matter fields also form a linear representation of the relevant gauge group. Finally all this combines with the symmetry group actions as for the quantum states themselves: is a group acts by symmetries on spacetime – notably for instance the action of the Poincaré group on Minkowski spacetime – then this translates into a linear representation of the group on the module of matter fields, via pullback of functions. This goes as far as that under suitable conditions one can classify/identify particle species with unitary irreducible representations of the spacetime symmetry group, a relation known as Wigner classification, discussed in more detail at unitary representation of the Poincaré group. A closely related relation is the Doplicher-Roberts reconstruction theorem which shows that superselection sectors of a quantum field theory form the representation category of the global gauge group of a local net of observables of a quanutm field theory on spacetime – a special case of the deep principle of Tannaka duality in representation theory.

In summary this means that the relation between quantum theory and module theory/linear algebra/representation theory is intimate.

Modules

Vector bundles

(,n)(\infty,n)-Modules

We turn now to the discussion of the generalization of the notion of module in higher category theory. After a motivation from physics

we first discuss how the 2-category (i.e. (2,2)-category) of 2-modules and their 2-linear maps is presented by the classical 2-category of associative algebras with bimodules between them and intertwiners between those.

This serves as an instructive blueprint for all the following generalizations and is in itself a useful theory that neatly subsumes and organizes classical constructions and results such as Tannaka duality for associative algebras, the Eilenberg-Watts theorem, and the basic Morita invariant theory of commutative rings given by Brauer group, Picard group and group of units.

Next we pass to homotopy theory proper and discuss the construction of the (∞,2)-category of A-∞ algebras and ∞-bimodules internal to any compatibly cocomplete monoidal (∞,1)-category.

By iterating this construction in analogy to the iterative construction of (∞,n)-categories as nn-fold categories in an (∞,1)-category, we obtain a notion of (∞,n)-modules

Physics motivation: Higher order states in extended QFT

There two key classes of examples of (∞,n)-categories of relevance in physics. One is clearly the (∞,n)-categories of cobordisms Bord n SBord_n^S (with some specified extra structure SS) which by the theory of extended quantum field theory are the domain of monoidal (∞,n)-functors

Z:Bord n S𝒞. Z \;\colon\; Bord_n^S \to \mathcal{C} \,.

In the abstract mathematical discussion the codomain 𝒞\mathcal{C} here is usually allowed to be any symmetric monoidal (∞,n)-category. But we expect that for those quantum field theories of actual relevance in physics (notably those obtained by quantization from extended prequantum field theories) the codomain here is special. Notably in codimension 1 for Σ n1\Sigma_{n-1} a closed manifold of dimension n1n-1 the value Z(Σ n1)Z(\Sigma_{n-1}) should be the space of quantum states over Σ n1\Sigma_{n-1} and thus be a \mathbb{C}-module (a complex vector space) possibly with some extra properties and structure (e.g. a topological vector space, a Hilbert space, etc., depending on the precise details).

Therefore we expect that for relevant theories of physics in dimension nn, 𝒞\mathcal{C} is something that deserves to be called an (∞,n)-category of (∞,n)-modules (maybe: (∞,n)-vector spaces)

𝒞nMod R \mathcal{C} \simeq n Mod_{R}

over some base E-∞ ring RR.

Here we discuss such nn-modules.

2-Abelian groups

An abelian group semigroup is a set equipped with a commutative and hence “additive” and unital pairing. A natural categorification of addition is the coproduct operation in a category, hence more generally the operation of taking colimits. Hence a sensible categorification of the notion of additive semigroup is that of category with all colimits. But in the spirit of category theory we should not try just to categorify abelian semigroups, but rather the category that these form. The category Ab of abelian groups is notably a closed symmetric monoidal category and accordingly we demand that the 2-category of 2-abelian semigroups to replace it is similarly a closed symmetric monoidal 2-category. This is achived by restricting attention among all categories with colimits to the presentable categories. Moreover, since the homomorphisms of abelian groups are linear functions, hence addition-respecting functions, so the 2-linear maps between these 2-abelian semigroups should be colimit-preserving functors. This way we arrive at the following definition.

Definition

Write

2Ab2Cat 2 Ab \in 2Cat

for the 2-category of presentable categories and colimit-preserving functors between them.

(CJF, def. 2.1.8)

Remark

By the adjoint functor theorem this is equivalently the 2-category of presentable categories and left adjoint functors between them.

Example

Given a small category 𝒞\mathcal{C}, the presheaf category Set 𝒞Set^{\mathcal{C}} is a presentable category.

Remark

Forming presheaves on 𝒞\mathcal{C} is the free cocompletion of 𝒞\mathcal{C}. Under the interpretation of colimits as the categorification of additive sums, this means that Set 𝒞Set^{\mathcal{C}} is the 2-free abelian group generated by the “2-set” 𝒞\mathcal{C}.

Moreover, every presentable category (as discussed there) is a reflective subcategory hence a left localization of a category of presheaves. This means that not every 2-abelian group is 2-free, but is possibly a localization of a 2-free abelian group.

Example

Given an ordinary ring RR, its category of modules Mod RMod_R is presentable, hence may be regarded as a 2-abelian group. This example we discuss in more detail below in Bases for 2-modules: Tannaka duality for associative algebras.

(CJF, example 2.1.5)

Remark

These two examples are directly analogous from the perspective of enriched category theory. For AA an RR-algebra write BA\mathbf{B}A for the corresponding 1-object Mod RMod_R-enriched category with AA as its RR-module of morphisms. Then

Mod A[BA,Mod R] Mod_A \simeq [\mathbf{B}A, Mod_R]

is the enriched functor category.

Compare oth situation to how an ordinary free module NN (of finite rank) over a commutative ring is equivalently an algebra of functions

N[B,R] N \simeq [B,R]

for BB a basis set.

Proposition

The 2-category 2Ab=PrCat2Ab = PrCat of def. is a closed? symmetric monoidal 2-category with respect to the tensor product :2Ab×2Ab2Ab\boxtimes \colon 2Ab \times 2Ab \to 2Ab which corepresents “bi-2-linear” functors; in that for A,B,C2AbA,B, C \in 2Ab the hom-category Hom 2Ab(AB,C)Hom_{2Ab}(A \boxtimes B, C) is equivalently the full subcategory of the functor category Hom Cat(A×B,C)Hom_{Cat}(A \times B, C) on those that preserve colimits in each argument separately.

See also at Pr(∞,1)Cat for more on this.

Remark

This is analous to the Deligne tensor product of abelian categories.

Example

For 𝒞\mathcal{C} a small category, the category of presheaves Set 𝒞Set^{\mathcal{C}} is presentable and

Set 𝒞 1Set 𝒞 2Set 𝒞 1×𝒞 2. Set^{\mathcal{C}_1} \boxtimes Set^{\mathcal{C}_2} \simeq Set^{\mathcal{C}_1 \times \mathcal{C}_2} \,.
Proposition

For RR a ring the category of modules Mod RMod_R is presentable and

Mod R 1Mod R 2Mod R 1R 2, Mod_{R_1} \boxtimes Mod_{R_2} \simeq Mod_{R_1 \otimes R_2} \,,

(CJF, example 2.2.7)

Remark

As we discuss below in Bases for 2-modules: Tannaka duality for associative algebras, a category of modules over an RR-algebra is a presheaf category in Mod RMod_R-enriched category theory. Notice that in this language a plain presheaf over a locally small category is a presheaf in Set-enriched category theory.

Therefore comparison of example with example shows that in the context of 2-abelian semigroups plain sets play a role of a “nonlinear generalization” of linear algebra. We see this in a more pronounced way once we have introduced the notion of 2-modules below. (It has become fashionable to speak of sets regarded as non-linear module categories as being modules over the “field with one element”.)

2-Rings

With the ambient monoidal 2-category 2Ab2Ab chosen, it is now straightforward to define 2-rings as monoids internal to that.

Definition

Write

2Ring2Cat 2Ring \in 2Cat

for the 2-category of monoid objects internal to 2Ab2 Ab. An object of this 2-category we call a 2-ring.

Equivalently, a 2-ring in this sense is a presentable category equipped with the structure of a monoidal category where the tensor product preserves colimits.

(CJF, def. 2.1.8)

Example

The category Set with its cartesian product is a 2-ring and it is the initial object in 2Ring2Ring.

(CJF, example 2.3.4)

Example

The category Ab of abelian groups with its standard tensor product of abelian groups is a 2-ring.

Example

For RR an ordinary commutative ring, Mod RMod_R equipped with its usual tensor product of modules is a commutative 2-ring.

Proposition

For RR an ordinary commutative ring and Mod RMod_R its ordinary category of modules, regarded as a 2-abelian group by example , the structure of a 2-ring on Mod RMod_R is equivalently the structure of a sesquiunital sesquialgebra on RR.

Proof

By prop. , a 2-linear map

Mod AMod AMod A Mod_A \boxtimes Mod_A \to Mod_A

is equivalently a 2-linear map

Mod AAMod A. Mod_{A \otimes A} \to Mod_A \,.

By the Eilenberg-Watts theorem (which we discuss in more detail as theorem below) this is equivalently an AAA \otimes A-AA-bimodule. Similarly the unit map

Mod RMod A Mod_R \to Mod_A

is equivalently an RR-AA-bimodule.

Proposition

For RR a commutative ring, Mod RMod_R is a commutative 2-ring and is canonically an 2-algebra over the 2-ring AbAb in that we have a canonical 2-ring homomorphism

AbMod Mod R. Ab \simeq Mod_{\mathbb{Z}} \to Mod_R \,.

(CJF, example 2.3.7)

2-Modules

Definition

For \mathcal{R} a 2-ring, def. , write

2Mod 2Cat 2Mod_{\mathcal{R}} \in 2Cat

for the 2-category of module objects over AA in 2Ab2Ab.

This means that a 2-module over AA is a presentable category NN equipped with a functor

ANN A \boxtimes N \to N

which satisfies the evident action property.

(CJF, def. 2.3.3)

Example

Let RR be an ordinary commutative ring and AA an ordinary RR-algebra. Then by example Mod AMod_A is a 2-abelian group and by example Mod RMod_R is a commutative ring. By example Mod RMod_R-2-module structures on Mod AMod_A

Mod RMod AMod A Mod_R \boxtimes \Mod_A \to Mod_A

correspond to colimit-preserving functors

Mod R AMod A Mod_{R \otimes_{\mathbb{Z}} A} \to Mod_{A}

that satisfy the action property. Such as presented under the Eilenberg-Watts theorem, prop. , by R AR \otimes_{\mathbb{Z}} A-AA bimodules. AA itself is canonically such a bimodule and it exhibits a Mod RMod_R-2-module structure on Mod AMod_A.

Remark

Coming back to remark we observe that in 2-module theory plain presentable categories which do not “look linear” in the ordinary sense are naturally regarded as Set-2-modules, hence as being “Set-linear”, whereas the categories of modules over an algebra that are traditionally regarded as being “RR-linear categories” are Mod RMod_R-2-modules. This unification of sets as “nonlinear linear structure” has become fashionable as “linear algebra over the field with one element”.

Bases for 2-modules: Tannaka duality for associative algebras

The construction in example of 2-modules over the 2-ring induced by an ordinary commutative ring RR as categories of 1-modules over some RR-algebra may be understood as presenting a 2-free module by a 2-basis.

To see this, write

𝒱Mod R \mathcal{V} \coloneqq Mod_R

for the closed symmetric monoidal category of modules and regard it as an enriching context for 𝒱\mathcal{V}-enriched category theory.

So a 𝒱\mathcal{V}-enriched category 𝒜\mathcal{A} serves as a 2-basis for a free 𝒱\mathcal{V}-2-module. Notice that an RR-algebra AA is equivalently a (pointed) 𝒱\mathcal{V}-enriched category 𝒜=BA\mathcal{A} = \mathbf{B}A with a single object (the delooping of AA). Generally, 𝒜\mathcal{A} may be called RR-algebroid.

Since colimits in the category underlying a 2-abelian group? categorify the sum operation in an abelian group the 2-analog of the free module on a basis set is the free cocompletion of 𝒜\mathcal{A} as a 𝒱\mathcal{V}-enriched category. This is the 𝒱\mathcal{V}-presheaf category, hence the enriched functor category

[𝒜,𝒱]𝒱Cat. [\mathcal{A}, \mathcal{V}] \in \mathcal{V}Cat \,.

Compare this to how for a (finite dimensional) vector space a presentation by a basis is a set BB such that the vector space is

Maps(B,k)Vect k Maps(B,k) \in Vect_k

Indeed, for 𝒜=BA\mathcal{A} = \mathbf{B}A this is the ordinary category of modules over AA:

[BA,𝒱]Mod A. [\mathbf{B}A, \mathcal{V}] \simeq Mod_A \,.

In fact in enriched category theory it is customary generally to call 𝒱\mathcal{V}-enriched functors into 𝒱\mathcal{V}modules” (see there).

Without further assumptions on 𝒜\mathcal{A}, the category [𝒜,𝒱][\mathcal{A}, \mathcal{V}] is just a 𝒱\mathcal{V}-2-module. But if 𝒜\mathcal{A} is equipped with further extra structure and/or property, also [𝒜,𝒱][\mathcal{A}, \mathcal{V}] inherits further property. This relation between types of algebras and types of monoidal categories is known as Tannaka duality.

The following table lists the main items of the disctionary of Tannaka duality for associtive algebras that are used in the literature.

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
Matrix calculus for 2-modules: Bimodules and Eilenberg-Watts theorem

If a 𝒱Mod R\mathcal{V} \coloneqq Mod_R-enriched category serves as a 2-basis for a Mod RMod_R-2-module by the above discussion, there should be an analog of matrix calculus to present 2-linear maps between 2-modules equipped with such a 2-basis.

Such a higher matrix is what in enriched category theory is called a 𝒱\mathcal{V}-profunctor (or “distributor”): a profunctor between 𝒱\mathcal{V}-enriched categories 𝒜 1,𝒜 2𝒱Cat\mathcal{A}_1, \mathcal{A}_2 \in \mathcal{V}Cat is a 𝒱\mathcal{V}-enriched functor

K:𝒜 1 op×𝒜 2𝒱. K \;\colon\; \mathcal{A}_1^{op} \times \mathcal{A}_2 \to \mathcal{V} \,.

Compare this to how a matrix between two vector spaces equipped with basis sets B 1,B 2B_1, B_2 is a function

K:B 1×B 2k K \colon B_1 \times B_2 \to k

to the ground field. Notice that if the vector space is given by [B 1,k][B_1,k], then the action of this matrix on a vector v:B 1kv \colon B_1 \to k is given by the vector Kv:B 2kK v \colon B_2 \to k

b 2 b 1B 1v(b 1)K(b 1,b 2). b_2 \mapsto \sum_{b_1 \in B_1} v(b_1) \cdot K(b_1,b_2) \,.

If we replace in this formula the sum by a coend, then we get the action of the profunctor KK on a 𝒱\mathcal{V}-functor v:𝒜 1𝒱v \colon \mathcal{A}_1 \to \mathcal{V} to yield Kv:𝒜 2𝒱K \otimes v \colon \mathcal{A}_2 \to \mathcal{V} with

a 2 a 1A 1v(a 1)K(a 1,a 2). a_2 \mapsto \int^{a_1 \in A_1} v(a_1) \otimes K(a_1, a_2) \,.

For the case at hand where 𝒱=Mod R\mathcal{V} = Mod_R and in the special case that 𝒜 1=BA 1\mathcal{A}_1 = \mathbf{B}A_1 and 𝒜 2=BA 2\mathcal{A}_2 = \mathbf{B}A_2, such a profunctor BA 1×BA 2𝒱\mathbf{B}A_1 \times \mathbf{B}A_2 \to \mathcal{V} is equivalently an A 1A_1-A2A-2-bimodule in the traditional sense of associative algebra. Moreover, the coend-action above is equivalently the traditional tensor product of modules over the given algebra. Motivated by this example one also generally calls profunctors “bimodules”.

Now one should ask to which extent these “2-matrices” given by profunctors capture all the 2-linear maps between 2-modules. This is what the classical Eilenberg-Watts theorem solves:

Theorem

There is a natural equivalence of categories

() A 1(): A 1Mod A 2Hom 2Ab(Mod A 1,Mod A 2) (-) \otimes_{A_1} (-) \;\colon\; {}_{A_1} Mod_{A_2} \stackrel{\simeq}{\to} Hom_{2Ab}(Mod_{A_1}, Mod_{A_2})

between colimit-preserving functors Mod A 1Mod A 2Mod_{A_1} \to Mod_{A_2} between categories of modules and A 1A_1-A2A-2-bimodules, given by sending a bimodule NN to the tensor product functor

() A 1N:Mod A 1Mod A 2. (-) \otimes_{A_1} N \;\colon\; Mod_{A_1} \to Mod_{A_2} \,.

In summary we this find that for RR a commutative ring, the 2-category Mod Mod RMod_{Mod_R} of Mod RMod_R-2-modules with 2-basis is equivalent to the 2-category Prof(Mod R)Prof(Mod_R) of Mod RMod_R-enriched profunctors. Indeed, another common notation for Prof is Mod, and so we have the unambiguous notation

2Mod RMod(Mod R). 2 Mod_R \simeq Mod(Mod_R) \,.

Notice how this is analogous to the identification

(,2)CatCat(Cat (,1)) (\infty,2)Cat \simeq Cat(Cat_{(\infty,1)})

discussed at internal (∞,1)-category.

2-Module bundle

2-Lines

Definition

For \mathcal{R} a commutative 2-ring, def. , a 2-line over \mathcal{R} is an \mathbb{R}-2-module, def. which is invertible under the tensor product of \mathbb{R}-2-modules.

Definition

Write

Line Mod Line^\simeq_\mathbb{R} \to Mod_{\mathbb{R}}

for the maximal 2-groupoid of \mathbb{R} 2-lines with 2-basis inside all \mathbb{R}-2-modules. This is the Picard 2-groupoid of \mathcal{R}. Moreover, by construction it in inherits the structure of a 3-group from the tensor products of 2-modules: as such it is the Picard 3-group.

Classification: Azumaya algebras, Brauer group, Picard group
Proposition

For RR an ordinary commutative ring and Mod R\mathcal{R} \coloneqq Mod_R the homotopy groups of the Picard 2-groupoid are

  • π 0(2Line R )Br(R)\pi_0(2Line^\simeq_R) \simeq Br(R) the Brauer group of RR;

  • π 1(2Line R )Pic(R)\pi_1(2Line^\simeq_R) \simeq Pic(R) the Picard group of RR;

  • π 2(2Line R )R ×\pi_2(2Line^\simeq_R) \simeq R^\times the group of units of RR.

Proof

By the Eilenberg-Watts theorem, , an equivalence in 2Mod R2Mod_R between 2-modules with 2-basis given by RR-algebras is a Morita equivalence of RR-algebras.

By the prop. and the Eilenberg-Watts theorem, the invertible 2-modules with 2-basis are therefore precisely the Azumaya algebras. This shows that the connected components of 2Line R 2Line^\simeq_R form the Brauer group of RR.

Next, an isomorphism class of an automorphism of the canonical base point Mod R2Line R Mod_R \in 2Line^\simeq_R is by Eilenberg-Watts equivalently an isomorphism class of an RR-RR-bimodule which is invertible under horizontal composition of bimodules. Since RR is commutative this just means that it is an isomorphism class of an RR-module which is invertible under the ordinary tensor product of modules. But this just means that it is the isomorphism class of an ordinary RR-line. This shows that the fundamental group of 2Line R 2Line^\simeq_R is the ordinary Picard group of RR.

Finally an automorphism of the RR regarded as the identity RR-RR-bimodule is an invertible RR-linear function RRR \to R, hence is given by multiplication with an invertible element of RR. This shows that π 2(2Line R )\pi_2(2Line^\simeq_R) is the group of units of RR.

Remark

Below in 2-line bundles and their sections we discuss how forming a fiber 2-bundle of 2-lines as above produces a structure whose sections are twisted bundles with typical fiber Mod RMod_R, hence for instance twisted unitary bundles if RR \simeq \mathbb{C}. This means that the characteristic classes of Mod RMod_R-2-line bundles are the twists for the twisted cohomology classifying these twisted bundles. After stabilization the latter are the cocycles of twisted K-theory, hence the Brauer group, Picard group and group of units of RR are the degree 0, 1, 2-twists of twisted K-theory over RR-resectively. Below we discuss two examples of this phenomenon.

2-Line bundles and their sections: twisted K-theory

(,2)(\infty,2)-Abelian groups

(…)

(,2)(\infty,2)-rings

  • (∞,2)-ring?

(,2)(\infty,2)-modules

Bases for (,2)(\infty,2)-modules: A A_\infty-algebras
Matrix calculus for (,2)(\infty,2)-modules: \infty-Bimodules

(,n)(\infty,n)-Abelian groups

(…)

(,n)(\infty,n)-rings

  • (∞,n)-ring?

(,n)(\infty,n)-modules

Bases for (,n)(\infty,n)-modules

(…)

Matrix calculus for (,n)(\infty,n)-modules

(…)

References

The notions of 2-rings and 2-modules are nicely set up in

The (,2)(\infty,2)-category of A A_\infty-algebras and \infty-bimodules between them is constructed in section 3.4 of

For further references see behind the relevant links.

Last revised on April 14, 2019 at 14:34:48. See the history of this page for a list of all contributions to it.