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
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 $(\infty,n)$-actions for $n \in \mathbb{N}$. 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 “$(r,n)$” index:
$(r,n)$-ring | presentation |
---|---|
ring = 1-ring = $(1,1)$-ring | compatibly monoidal abelian group |
2-ring = $(2,2)$-ring | compatibly monoidal category |
commutative $(\infty,1)$-ring | E-∞ ring = compatibly monoidal ∞-group / spectrum |
$(\infty,2)$-ring | compatibly 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.
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.
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 $\psi_1$ and $\psi_2$ two quantum states one can form their sum $\psi_1 + \psi_2$ as well as their difference $\psi_1 - \psi_2$ such that this is again a quantum state. Moreover there is a 0-state such that $\psi + 0 = \psi$ for all $\psi$. 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 $\psi$ a quantum state and for $c \in \mathbb{C}$ a complex number, there is a new quantum state $c \psi$ and this is such that for two such complex numbers we have $c_1(c_2 \psi) = (c_1 c_2) \psi$ and for two states we have $c(\psi_1 + \psi_2) = c \psi_1 + c \psi_2$. 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 $\phi_1$ and $\phi_2$ two field configurations also $\phi_1 + \phi_2$ is a field configuration, such that with $\phi$ a matter field for each smooth function $f$ on spacetime there is also a matter field denoted $f \phi$ and such that $f_1 (f_2 \phi) = (f_1 f_2) \phi$ and $f(\phi_1 + \phi_2) = f \phi_1 + \phi_2$.
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.
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 $n$-fold categories in an (∞,1)-category, we obtain a notion of (∞,n)-modules
There two key classes of examples of (∞,n)-categories of relevance in physics. One is clearly the (∞,n)-categories of cobordisms $Bord_n^S$ (with some specified extra structure $S$) which by the theory of extended quantum field theory are the domain of monoidal (∞,n)-functors
In the abstract mathematical discussion the codomain $\mathcal{C}$ 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 $\Sigma_{n-1}$ a closed manifold of dimension $n-1$ the value $Z(\Sigma_{n-1})$ should be the space of quantum states over $\Sigma_{n-1}$ 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 $n$, $\mathcal{C}$ is something that deserves to be called an (∞,n)-category of (∞,n)-modules (maybe: (∞,n)-vector spaces)
over some base E-∞ ring $R$.
Here we discuss such $n$-modules.
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.
Write
for the 2-category of presentable categories and colimit-preserving functors between them.
By the adjoint functor theorem this is equivalently the 2-category of presentable categories and left adjoint functors between them.
Given a small category $\mathcal{C}$, the presheaf category $Set^{\mathcal{C}}$ is a presentable category.
Forming presheaves on $\mathcal{C}$ is the free cocompletion of $\mathcal{C}$. Under the interpretation of colimits as the categorification of additive sums, this means that $Set^{\mathcal{C}}$ is the 2-free abelian group generated by the “2-set” $\mathcal{C}$.
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.
Given an ordinary ring $R$, its category of modules $Mod_R$ 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.
These two examples are directly analogous from the perspective of enriched category theory. For $A$ an $R$-algebra write $\mathbf{B}A$ for the corresponding 1-object $Mod_R$-enriched category with $A$ as its $R$-module of morphisms. Then
is the enriched functor category.
Compare oth situation to how an ordinary free module $N$ (of finite rank) over a commutative ring is equivalently an algebra of functions
The 2-category $2Ab = PrCat$ of def. is a closed? symmetric monoidal 2-category with respect to the tensor product $\boxtimes \colon 2Ab \times 2Ab \to 2Ab$ which corepresents “bi-2-linear” functors; in that for $A,B, C \in 2Ab$ the hom-category $Hom_{2Ab}(A \boxtimes B, C)$ is equivalently the full subcategory of the functor category $Hom_{Cat}(A \times B, C)$ on those that preserve colimits in each argument separately.
See also at Pr(∞,1)Cat for more on this.
This is analous to the Deligne tensor product of abelian categories.
For $\mathcal{C}$ a small category, the category of presheaves $Set^{\mathcal{C}}$ is presentable and
For $R$ a ring the category of modules $Mod_R$ is presentable and
As we discuss below in Bases for 2-modules: Tannaka duality for associative algebras, a category of modules over an $R$-algebra is a presheaf category in $Mod_R$-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”.)
With the ambient monoidal 2-category $2Ab$ chosen, it is now straightforward to define 2-rings as monoids internal to that.
Write
for the 2-category of monoid objects internal to $2 Ab$. 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.
The category Set with its cartesian product is a 2-ring and it is the initial object in $2Ring$.
The category Ab of abelian groups with its standard tensor product of abelian groups is a 2-ring.
For $R$ an ordinary commutative ring, $Mod_R$ equipped with its usual tensor product of modules is a commutative 2-ring.
For $R$ an ordinary commutative ring and $Mod_R$ its ordinary category of modules, regarded as a 2-abelian group by example , the structure of a 2-ring on $Mod_R$ is equivalently the structure of a sesquiunital sesquialgebra on $R$.
is equivalently a 2-linear map
By the Eilenberg-Watts theorem (which we discuss in more detail as theorem below) this is equivalently an $A \otimes A$-$A$-bimodule. Similarly the unit map
is equivalently an $R$-$A$-bimodule.
For $R$ a commutative ring, $Mod_R$ is a commutative 2-ring and is canonically an 2-algebra over the 2-ring $Ab$ in that we have a canonical 2-ring homomorphism
For $\mathcal{R}$ a 2-ring, def. , write
for the 2-category of module objects over $A$ in $2Ab$.
This means that a 2-module over $A$ is a presentable category $N$ equipped with a functor
which satisfies the evident action property.
Let $R$ be an ordinary commutative ring and $A$ an ordinary $R$-algebra. Then by example $Mod_A$ is a 2-abelian group and by example $Mod_R$ is a commutative ring. By example $Mod_R$-2-module structures on $Mod_A$
correspond to colimit-preserving functors
that satisfy the action property. Such as presented under the Eilenberg-Watts theorem, prop. , by $R \otimes_{\mathbb{Z}} A$-$A$ bimodules. $A$ itself is canonically such a bimodule and it exhibits a $Mod_R$-2-module structure on $Mod_A$.
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 “$R$-linear categories” are $Mod_R$-2-modules. This unification of sets as “nonlinear linear structure” has become fashionable as “linear algebra over the field with one element”.
The construction in example of 2-modules over the 2-ring induced by an ordinary commutative ring $R$ as categories of 1-modules over some $R$-algebra may be understood as presenting a 2-free module by a 2-basis.
To see this, write
for the closed symmetric monoidal category of modules and regard it as an enriching context for $\mathcal{V}$-enriched category theory.
So a $\mathcal{V}$-enriched category $\mathcal{A}$ serves as a 2-basis for a free $\mathcal{V}$-2-module. Notice that an $R$-algebra $A$ is equivalently a (pointed) $\mathcal{V}$-enriched category $\mathcal{A} = \mathbf{B}A$ with a single object (the delooping of $A$). Generally, $\mathcal{A}$ may be called $R$-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 $\mathcal{A}$ as a $\mathcal{V}$-enriched category. This is the $\mathcal{V}$-presheaf category, hence the enriched functor category
Compare this to how for a (finite dimensional) vector space a presentation by a basis is a set $B$ such that the vector space is
Indeed, for $\mathcal{A} = \mathbf{B}A$ this is the ordinary category of modules over $A$:
In fact in enriched category theory it is customary generally to call $\mathcal{V}$-enriched functors into $\mathcal{V}$ “modules” (see there).
Without further assumptions on $\mathcal{A}$, the category $[\mathcal{A}, \mathcal{V}]$ is just a $\mathcal{V}$-2-module. But if $\mathcal{A}$ is equipped with further extra structure and/or property, also $[\mathcal{A}, \mathcal{V}]$ 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 algebra | category of modules |
---|---|
$A$ | $Mod_A$ |
$R$-algebra | $Mod_R$-2-module |
sesquialgebra | 2-ring = monoidal presentable category with colimit-preserving tensor product |
bialgebra | strict 2-ring: monoidal category with fiber functor |
Hopf algebra | rigid monoidal category with fiber functor |
hopfish algebra (correct version) | rigid monoidal category (without fiber functor) |
weak Hopf algebra | fusion category with generalized fiber functor |
quasitriangular bialgebra | braided monoidal category with fiber functor |
triangular bialgebra | symmetric monoidal category with fiber functor |
quasitriangular Hopf algebra (quantum group) | rigid braided monoidal category with fiber functor |
triangular Hopf algebra | rigid symmetric monoidal category with fiber functor |
supercommutative Hopf algebra (supergroup) | rigid symmetric monoidal category with fiber functor and Schur smallness |
form Drinfeld double | form Drinfeld center |
trialgebra | Hopf monoidal category |
2-Tannaka duality for module categories over monoidal categories
monoidal category | 2-category of module categories |
---|---|
$A$ | $Mod_A$ |
$R$-2-algebra | $Mod_R$-3-module |
Hopf monoidal category | monoidal 2-category (with some duality and strictness structure) |
3-Tannaka duality for module 2-categories over monoidal 2-categories
monoidal 2-category | 3-category of module 2-categories |
---|---|
$A$ | $Mod_A$ |
$R$-3-algebra | $Mod_R$-4-module |
If a $\mathcal{V} \coloneqq Mod_R$-enriched category serves as a 2-basis for a $Mod_R$-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 $\mathcal{V}$-profunctor (or “distributor”): a profunctor between $\mathcal{V}$-enriched categories $\mathcal{A}_1, \mathcal{A}_2 \in \mathcal{V}Cat$ is a $\mathcal{V}$-enriched functor
Compare this to how a matrix between two vector spaces equipped with basis sets $B_1, B_2$ is a function
to the ground field. Notice that if the vector space is given by $[B_1,k]$, then the action of this matrix on a vector $v \colon B_1 \to k$ is given by the vector $K v \colon B_2 \to k$
If we replace in this formula the sum by a coend, then we get the action of the profunctor $K$ on a $\mathcal{V}$-functor $v \colon \mathcal{A}_1 \to \mathcal{V}$ to yield $K \otimes v \colon \mathcal{A}_2 \to \mathcal{V}$ with
For the case at hand where $\mathcal{V} = Mod_R$ and in the special case that $\mathcal{A}_1 = \mathbf{B}A_1$ and $\mathcal{A}_2 = \mathbf{B}A_2$, such a profunctor $\mathbf{B}A_1 \times \mathbf{B}A_2 \to \mathcal{V}$ is equivalently an $A_1$-$A-2$-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:
There is a natural equivalence of categories
between colimit-preserving functors $Mod_{A_1} \to Mod_{A_2}$ between categories of modules and $A_1$-$A-2$-bimodules, given by sending a bimodule $N$ to the tensor product functor
In summary we this find that for $R$ a commutative ring, the 2-category $Mod_{Mod_R}$ of $Mod_R$-2-modules with 2-basis is equivalent to the 2-category $Prof(Mod_R)$ of $Mod_R$-enriched profunctors. Indeed, another common notation for Prof is Mod, and so we have the unambiguous notation
Notice how this is analogous to the identification
discussed at internal (∞,1)-category.
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.
Write
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.
For $R$ an ordinary commutative ring and $\mathcal{R} \coloneqq Mod_R$ the homotopy groups of the Picard 2-groupoid are
$\pi_0(2Line^\simeq_R) \simeq Br(R)$ the Brauer group of $R$;
$\pi_1(2Line^\simeq_R) \simeq Pic(R)$ the Picard group of $R$;
$\pi_2(2Line^\simeq_R) \simeq R^\times$ the group of units of $R$.
By the Eilenberg-Watts theorem, , an equivalence in $2Mod_R$ between 2-modules with 2-basis given by $R$-algebras is a Morita equivalence of $R$-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 $2Line^\simeq_R$ form the Brauer group of $R$.
Next, an isomorphism class of an automorphism of the canonical base point $Mod_R \in 2Line^\simeq_R$ is by Eilenberg-Watts equivalently an isomorphism class of an $R$-$R$-bimodule which is invertible under horizontal composition of bimodules. Since $R$ is commutative this just means that it is an isomorphism class of an $R$-module which is invertible under the ordinary tensor product of modules. But this just means that it is the isomorphism class of an ordinary $R$-line. This shows that the fundamental group of $2Line^\simeq_R$ is the ordinary Picard group of $R$.
Finally an automorphism of the $R$ regarded as the identity $R$-$R$-bimodule is an invertible $R$-linear function $R \to R$, hence is given by multiplication with an invertible element of $R$. This shows that $\pi_2(2Line^\simeq_R)$ is the group of units of $R$.
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 $Mod_R$, hence for instance twisted unitary bundles if $R \simeq \mathbb{C}$. This means that the characteristic classes of $Mod_R$-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 $R$ are the degree 0, 1, 2-twists of twisted K-theory over $R$-resectively. Below we discuss two examples of this phenomenon.
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The notions of 2-rings and 2-modules are nicely set up in
The $(\infty,2)$-category of $A_\infty$-algebras and $\infty$-bimodules between them is constructed in section 3.4 of
For further references see behind the relevant links.
Last revised on December 28, 2019 at 18:04:17. See the history of this page for a list of all contributions to it.