representation theory


I’ve written here before about building topological quantum field theories using groupoidification, but I haven’t yet gotten around to discussing a refinement of this idea, which is in the most recent version of my paper on the subject.  I also gave a talk about this last year in Erlangen. The main point of the paper is to pull apart some constructions which are already fairly well known into two parts, as part of setting up a category which is nice for supporting models of fairly general physical systems, using an extension of the  concept of groupoidification. So here’s a somewhat lengthy post which tries to unpack this stuff a bit.

Factoring TQFT

The older version of this paper talked about the untwisted version of the Dijkgraaf-Witten (DW for short) model, which is a certain kind of TQFT based on a gauge theory with a finite gauge group.  (Freed and Quinn put it as: “Chern-Simons theory with finite gauge group”).  The new version gets the general – that is, the twisted – form in the same way: factoring the theory into two parts. So, the DW model, which was originally described by Dijkgraaf and Witten in terms of a state-sum, is a functor

Z : 3Cob \rightarrow Vect

The “twisting” is the point of their paper, “Topological Gauge Theories and Group Cohomology”.  The twisting has to do with the action for some physical theory. Now, for a gauge theory involving flat connections, the kind of gauge-theory actions which involve the curvature of a connection make no sense: the curvature is zero.  So one wants an action which reflects purely global features of connections.  The cohomology of the gauge group is where this comes from.

Now, the machinery I describe is based on a point of view which has been described in a famous paper by Freed, Hopkins, Lurie and Teleman (FHLT for short – see further discussion here) in terms in which the two stages are called the “classical field theory” (which has values in groupoids), and the “quantization functor”, which takes one into Hilbert spaces.

Actually, we really want to have an “extended” TQFT: a TQFT gives a Hilbert space for each 2D manifold (“space”), and a linear map for a 3D cobordism (“spacetime”) between them. An extended TQFT will assign (higher) algebraic data to lower-dimension boundaries still.  My paper talks only about the case where we’ve extended down to codimension 2, whereas FHLT talk about extending “down to a point”. The point of this first stopping point is to unpack explicitly and computationally what the factorization into two parts looks like at the first level beyond the usual TQFT.

In the terminology I use, the classical field theory is:

A^{\omega} : nCob_2 \rightarrow Span_2(Gpd)^{U(1)}

This depends on a cohomology class [\omega] \in H^3(G,U(1)). The “quantization functor” (which in this case I call “2-linearization”):

\Lambda^{U(1)} : Span_2(Gpd)^{U(1)} \rightarrow 2Vect

The middle stage involves the monoidal 2-category I call Span_2(Gpd)^{U(1)}.  (In FHLT, they use different terminology, for instance “families” rather than “spans”, but the principle is the same.)

Freed and Quinn looked at the quantization of the “extended” DW model, and got a nice geometric picture. In it, the action is understood as a section of some particular line-bundle over a moduli space. This geometric picture is very elegant once you see how it works, which I found was a little easier in light of a factorization through Span_2(Gpd).

This factorization isolates the geometry of this particular situation in the “classical field theory” – and reveals which of the features of their setup (the line bundle over a moduli space) are really part of some more universal construction.

In particular, this means laying out an explicit definition of both Span_2(Gpd)^{U(1)} and \Lambda^{U(1)}.

2-Linearization Recalled

While I’ve talked about it before, it’s worth a brief recap of how 2-linearization works with a view to what happens when you twist it via groupoid cohomology. Here we have a 2-category Span(Gpd), whose objects are groupoids (A, B, etc.), whose morphisms are spans of groupoids:

A \stackrel{s}{\leftarrow} X \stackrel{t}{\rightarrow} B

and whose 2-morphisms are spans of span-maps (taken up to isomorphism), which look like so:

span of span maps

(And, by the by: how annoying that WordPress doesn’t appear to support xypic figures…)

These form a (symmetric monoidal) 2-category, where composition of spans works by taking weak pullbacks.  Physically, the idea is that a groupoid has objects which are configurations (in the cause of gauge theory, connections on a manifold), and morphisms which are symmetries (gauge transformations, in this case).  Then a span is a groupoid of histories (connections on a cobordism, thought of as spacetime), and the maps s,t pick out its starting and ending configuration.  That is, A = A_G(S) is the groupoid of flat G-connections on a manifold S, and X = A_G(\Sigma) is the groupoid of flat G-connections on some cobordism \Sigma, of which S is part of the boundary.  So any such connection can be restricted to the boundary, and this restriction is s.

Now 2-linearization is a 2-functor:

\Lambda : Span_2(Gpd)^{U(1)} \rightarrow 2Vect

It gives a 2-vector space (a nice kind of category) for each groupoid G.  Specifically, the category of its representations, Rep(G).  Then a span turns into a functor which comes from “pulling” back along s (the restricted representation where X acts by first applying s then the representation), then “pushing” forward along t (to the induced representation).

What happens to the 2-morphisms is conceptually more complicated, but it depends on the fact that “pulling” and “pushing” are two-sided adjoints. Concretely, it ends up being described as a kind of “sum over histories” (where “histories” are the objects of Y), which turns out to be exactly the path integral that occurs in the TQFT.

Or at least, it’s the path integral when the action is trivial! That is, if S=0, so that what’s integrated over paths (“histories”) is just e^{iS}=1. So one question is: is there a way to factor things in this way if there’s a nontrivial action?

Cohomological Twisting

The answer is by twisting via cohomology. First, let’s remember what that means…

We’re talking about groupoid cohomology for some groupoid G (which you can take to be a group, if you like).  “Cochains” will measure how much some nice algebraic fact, such as being a homomorphism, or being associative, “fails to occur”.  “Twisting by a cocycle” is a controlled way to force some such failure to happen.

So, an n-cocycle is some function of n composable morphisms of G (or, if there’s only one object, “group elements”, which amounts to the same thing).  It takes values in some group of coefficients, which for us is always U(1)

The trivial case where n=0 is actually slightly subtle: a 0-cocycle is an invariant function on the objects of a groupoid. (That is, it takes the same value on any two objects related by an (iso)morphism. (Think of the object as a sequence of zero composable morphisms: it tells you where to start, but nothing else.)

The case n=1 is maybe a little more obvious. A 1-cochain f \in Z^1_{gpd}(G,U(1)) can measure how a function h on objects might fail to be a 0-cocycle. It is a U(1)-valued function of morphisms (or, if you like, group elements).  The natural condition to ask for is that it be a homomorphism:

f(g_1 \circ g_2) = f(g_1) f(g_2)

This condition means that a cochain f is a cocycle. They form an abelian group, because functions satisfying the cocycle condition are closed under pointwise multiplication in U(1). It will automatically by satisfied for a coboundary (i.e. if f comes from a function h on objects as f(g) = \delta h (g) = h(t(g)) - h(s(g))). But not every cocycle is a coboundary: the first cohomology H^1(G,U(1)) is the quotient of cocycles by coboundaries. This pattern repeats.

It’s handy to think of this condition in terms of a triangle with edges g_1, g_2, and g_1 \circ g_2.  It says that if we go from the source to the target of the sequence (g_1, g_2) with or without composing, and accumulate f-values, our f gives the same result.  Generally, a cocycle is a cochain satisfying a “coboundary” condition, which can be described in terms of an n-simplex, like this triangle. What about a 2-cocycle? This describes how composition might fail to be respected.

So, for instance, a twisted representation R of a group is not a representation in the strict sense. That would be a map into End(V), such that R(g_1) \circ R(g_2) = R(g_1 \circ g_2).  That is, the group composition rule gets taken directly to the corresponding rule for composition of endomorphisms of the vector space V.  A twisted representation \rho only satisfies this up to a phase:

\rho(g_1) \circ \rho(g_2) = \theta(g_1,g_2) \rho(g_1 \circ g_2)

where \theta : G^2 \rightarrow U(1) is a function that captures the way this “representation” fails to respect composition.  Still, we want some nice properties: \theta is a “cocycle” exactly when this twisting still makes \rho respect the associative law:

\rho(g_1) \rho( g_2 \circ g_3) = \rho( g_1 \circ g_2) \circ \rho( g_3)

Working out what this says in terms of \theta, the cocycle condition says that for any composable triple (g_1, g_2, g_3) we have:

\theta( g_1, g_2 \circ g_3) \theta (g_2,g_3) = \theta(g_1,g_2) \theta(g_1 \circ g_2, g_3)

So H^2_{grp}(G,U(1)) – the second group-cohomology group of G – consists of exactly these \theta which satisfy this condition, which ensures we have associativity.

Given one of these \theta maps, we get a category Rep^{\theta}(G) of all the \theta-twisted representations of G. It behaves just like an ordinary representation category… because in fact it is one! It’s the category of representations of a twisted version of the group algebra of G, called C^{\theta}(G). The point is, we can use \theta to twist the convolution product for functions on G, and this is still an associative algebra just because \theta satisfies the cocycle condition.

The pattern continues: a 3-cocycle captures how some function of 2 variable may fail to be associative: it specifies an associator map (a function of three variables), which has to satisfy some conditions for any four composable morphisms. A 4-cocycle captures how a map might fail to satisfy this condition, and so on. At each stage, the cocycle condition is automatically satisfied by coboundaries. Cohomology classes are elements of the quotient of cocycles by coboundaries.

So the idea of “twisted 2-linearization” is that we use this sort of data to change 2-linearization.

Twisted 2-Linearization

The idea behind the 2-category Span(Gpd)^{U(1)} is that it contains Span(Gpd), but that objects and morphisms also carry information about how to “twist” when applying the 2-linearization \Lambda.  So in particular, what we have is a (symmetric monoidal) 2-category where:

  • Objects consist of (A, \theta), where A is a groupoid and $\theta \in Z^2(A,U(1))$
  • Morphisms from A to B consist of a span (X,s,t) from A to B, together with \alpha \in Z^1(X,U(1))
  • 2-Morphisms from X_1 to X_2 consist of a span (Y,\sigma,\tau) from X, together with \beta \in Z^0(Y,U(1))

The cocycles have to satisfy some compatibility conditions (essentially, pullbacks of the cocycles from the source and target of a span should land in the same cohomology class).  One way to see the point of this requirement is to make twisted 2-linearization well-defined.

One can extend the monoidal structure and composition rules to objects with cocycles without too much trouble so that Span(Gpd) is a subcategory of Span(Gpd)^{U(1)}. The 2-linearization functor extends to \Lambda^{U(1)} : Span(Gpd)^{U(1)} \rightarrow 2Vect:

  • On Objects: \Lambda^{U(1)} (A, \theta) = Rep^{\theta}(A), the category of \theta-twisted representation of A
  • On Morphisms: \Lambda^{U(1)} ( (X,s,t) , \alpha ) comes by pulling back a twisted representation in Rep^{\theta_A}(A) to one in Rep^{s^{\ast}\theta_A}(X), pulling it through the algebra map “multiplication by \alpha“, and pushing forward to Rep^{\theta_B}(B)
  • On 2-Morphisms: For a span of span maps, one uses the usual formula (see the paper for details), but a sum over the objects y \in Y picks up a weight of \beta(y) at each object

When the cocycles are trivial (evaluate to 1 always), we get back the 2-linearization we had before. Now the main point here is that the “sum over histories” that appears in the 2-morphisms now carries a weight.

So the twisted form of 2-linearization uses the same “pull-push” ideas as 2-linearization, but applied now to twisted representations. This twisting (at the object level) uses a 2-cocycle. At the morphism level, we have a “twist” between “pull” and “push” in constructing . What the “twist” actually means depends on which cohomology degree we’re in – in other words, whether it’s applied to objects, morphisms, or 2-morphisms.

The “twisting” by a 0-cocycle just means having a weight for each object – in other words, for each “history”, or connection on spacetime, in a big sum over histories. Physically, the 0-cocycle is playing the role of the Lagrangian functional for the DW model. Part of the point in the FHLT program can be expressed by saying that what Freed and Quinn are doing is showing how the other cocycles are also the Lagrangian – as it’s seen at higher codimension in the more “local” theory.

For a TQFT, the 1-cocycles associated to morphisms describe how to glue together values for the Lagrangian that are associated to histories that live on different parts of spacetime: the action isn’t just a number. It is a number only “locally”, and when we compose 2-morphisms, the 0-cocycle on the composite picks up a factor from the 1-morphism (or 0-morphism, for a horizontal composite) where they’re composed.

This has to do with the fact that connections on bits of spacetime can be glued by particular gauge transformations – that is, morphisms of the groupoid of connections. Just as the gauge transformations tell how to glue connections, the cocycles associated to them tell how to glue the actions. This is how the cohomological twisting captures the geometric insight that the action is a section of a line bundle – not just a function, which is a section of a trivial bundle – over the moduli space of histories.

So this explains how these cocycles can all be seen as parts of the Lagrangian when we quantize: they explain how to glue actions together before using them in a sum-over histories. Gluing them this way is essential to make sure that \Lambda^{U(1)} is actually a functor. But if we’re really going to see all the cocycles as aspects of “the action”, then what is the action really? Where do they come from, that they’re all slices of this bigger thing?

Twisting as Lagrangian

Now the DW model is a 3D theory, whose action is specified by a group-cohomology class [\omega] \in H^3_{grp}(G,U(1)). But this is the same thing as a class in the cohomology of the classifying space: [\omega] \in H^3(BG,U(1)). This takes a little unpacking, but certainly it’s helpful to understand that what cohomology classes actually classify are… gerbes. So another way to put a key idea of the FHLT paper, as Urs Schreiber put it to me a while ago, is that “the action is a gerbe on the classifying space for fields“.

What does this mean?

This map is given as a path integral over all connections on the space(-time) S, which is actually just a sum, since the gauge group is finite and so all the connections are flat.  The point is that they’re described by assigning group elements to loops in S:

A : \pi_1(M) \rightarrow G

But this amounts to the same thing as a map into the classifying space of G:

f_A : M \rightarrow BG

This is essentially the definition of BG, and it implies various things, such as the fact that BG is a space whose fundamental group is G, and has all other homotopy groups trivial. That is, BG is the Eilenberg-MacLane space K(G,1). But the point is that the groupoid of connections and gauge transformations on S just corresponds to the mapping space Maps(S,BG). So the groupoid cohomology classes we get amount to the same thing as cohomology classes on this space. If we’re given [\omega] \in H^3(BG,U(1)), then we can get at these by “transgression” – which is very nicely explained in a paper by Simon Willerton.

The essential idea is that a 3-cocycle \omega (representing the class [\omega]) amounts to a nice 3-form on BG which we can integrate over a 3-dimentional submanifold to get a number. For a d-dimensional S, we get such a 3-manifold from a (3-d)-dimensional submanifold of Maps(S,BG): each point gives a copy of S in BG. Then we get a (3-d)-cocycle on Maps(S,BG) whose values come from integrating \omega over this image. Here’s a picture I used to illustrate this in my talk:

Now, it turns out that this gives 2-cocycles for 1-manifolds (the objects of 3Cob_2, 1-cocycles on 2D cobordisms between them, and 0-cocycles on 3D cobordisms between these cobordisms. The cocycles are for the groupoid of connections and gauge transformations in each case. In fact, because of Stokes’ theorem in BG, these have to satisfy all the conditions that make them into objects, morphisms, and 2-morphisms of Span^{U(1)}(Gpd). This is the geometric content of the Lagrangian: all the cocycles are really “reflections” of \omega as seen by transgression: pulling back along the evaluation map ev from the picture. Then the way you use it in the quantization is described exactly by \Lambda^{U(1)}.

What I like about this is that \Lambda^{U(1)} is a fairly universal sort of thing – so while this example gets its cocycles from the nice geometry of BG which Freed and Quinn talk about, the insight that an action is a section of a (twisted) line bundle, that actions can be glued together in particular ways, and so on… These presumably can be moved to other contexts.

Well, as promised in the previous post, I’d like to give a summary of some of what was discussed at the conference I attended (quite a while ago now, late last year) in Erlangen, Germany.  I was there also to visit Derek Wise, talking about a project we’ve been working on for some time.

(I’ve also significantly revised this paper about Extended TQFT since then, and it now includes some stuff which was the basis of my talk at Erlangen on cohomological twisting of the category Span(Gpd).  I’ll get to that in the next post.  Also coming up, I’ll be describing some new things I’ve given some talks about recently which relate the Baez-Dolan groupoidification program to Khovanov-Lauda categorification of algebras – at least in one example, hopefully in a way which will generalize nicely.)

In the meantime, there were a few themes at the conference which bear on the Extended TQFT project in various ways, so in this post I’ll describe some of them.  (This isn’t an exhaustive description of all the talks: just of a selection of illustrative ones.)


Categories with Structures

A few talks were mainly about facts regarding the sorts of categories which get used in field theory contexts.  One important type, for instance, are fusion categories is a monoidal category which is enriched in vector spaces, generated by simple objects, and some other properties: essentially, monoidal 2-vector spaces.  The basic example would be categories of representations (of groups, quantum groups, algebras, etc.), but fusion categories are an abstraction of (some of) their properties.  Many of the standard properties are described and proved in this paper by Etingof, Nikshych, and Ostrik, which also poses one of the basic conjectures, the “ENO Conjecture”, which was referred to repeatedly in various talks.  This is the guess that every fusion category can be given a “pivotal” structure: an isomorphism from Id to **.  It generalizes the theorem that there’s always such an isomorphism into ****.  More on this below.

Hendryk Pfeiffer talked about a combinatorial way to classify fusion categories in terms of certain graphs (see this paper here).  One way I understand this idea is to ask how much this sort of category really does generalize categories of representations, or actually comodules.  One starting point for this is the theorem that there’s a pair of functors between certain monoidal categories and weak Hopf algebras.  Specifically, the monoidal categories are (Cat \downarrow Vect)^{\otimes}, which consists of monoidal categories equipped with a forgetful functor into Vect.  Then from this one can get (via a coend), a weak Hopf algebra over the base field k(in the category WHA_k).  From a weak Hopf algebra H, one can get back such a category by taking all the modules of H.  These two processes form an adjunction: they’re not inverses, but we have maps between the two composites and the identity functors.

The new result Hendryk gave is that if we restrict our categories over Vect to be abelian, and the functors between them to be linear, faithful, and exact (that is, roughly, that we’re talking about concrete monoidal 2-vector spaces), then this adjunction is actually an equivalence: so essentially, all such categories C may as well be module categories for weak Hopf algebras.  Then he gave a characterization of these in terms of the “dimension graph” (in fact a quiver) for (C,M), where M is one of the monoidal generators of C.  The vertices of \mathcal{G} = \mathcal{G}_{(C,M)} are labelled by the irreducible representations v_i (i.e. set of generators of the category), and there’s a set of edges j \rightarrow l labelled by a basis of Hom(v_j, v_l \otimes M).  Then one can carry on and build a big graded algebra H[\mathcal{G}] whose m-graded part consists of length-m paths in \mathcal{G}.  Then the point is that the weak Hopf algebra of which C is (up to isomorphism) the module category will be a certain quotient of H[\mathcal{G}] (after imposing some natural relations in a systematic way).

The point, then, is that the sort of categories mostly used in this area can be taken to be representation categories, but in general only of these weak Hopf algebras: groups and ordinary algebras are special cases, but they show up naturally for certain kinds of field theory.

Tensor Categories and Field Theories

There were several talks about the relationship between tensor categories of various sorts and particular field theories.  The idea is that local field theories can be broken down in terms of some kind of n-category: n-dimensional regions get labelled by categories, (n-1)-D boundaries between regions, or “defects”, are labelled by functors between the categories (with the idea that this shows how two different kinds of field can couple together at the defect), and so on (I think the highest-dimension that was discussed explicitly involved 3-categories, so one has junctions between defects, and junctions between junctions, which get assigned some higher-morphism data).  Alteratively, there’s the dual picture where categories are assigned to points, functors to 1-manifolds, and so on.  (This is just Poincaré duality in the case where the manifolds come with a decomposition into cells, which they often are if only for convenience).

Victor Ostrik gave a pair of talks giving an overview role tensor categories play in conformal field theory.  There’s too much material here to easily summarize, but the basics go like this: CFTs are field theories defined on cobordisms that have some conformal structure (i.e. notion of angles, but not distance), and on the algebraic side they are associated with vertex algebras (some useful discussion appears on mathoverflow, but in this context they can be understood as vector spaces equipped with exactly the algebraic operations needed to model cobordisms with some local holomorphic structure).

In particular, the irreducible representations of these VOA’s determine the “conformal blocks” of the theory, which tell us about possible correlations between observables (self-adjoint operators).  A VOA V is “rational” if the category Rep(V) is semisimple (i.e. generated as finite direct sums of these conformal blocks).  For good VOA’s, Rep(V) will be a modular tensor category (MTC), which is a fusion category with a duality, braiding, and some other strucutre (see this for more).   So describing these gives us a lot of information about what CFT’s are possible.

The full data of a rational CFT are given by a vertex algebra, and a module category M: that is, a fusion category is a sort of categorified ring, so it can act on M as an ring acts on a module.  It turns out that choosing an M is equivalent to finding a certain algebra (i.e. algebra object) \mathcal{L}, a “Lagrangian algebra” inside the centre of Rep(V).  The Drinfel’d centre Z(C) of a monoidal category C is a sort of free way to turn a monoidal category into a braided one: but concretely in this case it just looks like Rep(V) \otimes Rep(V)^{\ast}.  Knowing the isomorphism class \mathcal{L} determines a “modular invariant”.  It gets “physics” meaning from how it’s equipped with an algebra structure (which can happen in more than one way), but in any case \mathcal{L} has an underlying vector space, which becomes the Hilbert space of states for the conformal field theory, which the VOA acts on in the natural way.

Now, that was all conformal field theory.  Christopher Douglas described some work with Chris Schommer-Pries and Noah Snyder about fusion categories and structured topological field theories.  These are functors out of cobordism categories, the most important of which are n-categories, where the objects are points, morphisms are 1D cobordisms, and so on up to n-morphisms which are n-dimensional cobordisms.  To keep things under control, Chris Douglas talked about the case Bord_0^3, which is where n=3, and a “local” field theory is a 3-functor Bord_0^3 \rightarrow \mathcal{C} for some 3-category \mathcal{C}.  Now, the (Baez-Dolan) Cobordism Hypothesis, which was proved by Jacob Lurie, says that Bord_0^3 is, in a suitable sense, the free symmetric monoidal 3-category with duals.  What this amounts to is that a local field theory whose target 3-category is \mathcal{C} is “just” a dualizable object of \mathcal{C}.

The handy example which links this up to the above is when \mathcal{C} has objects which are tensor categories, morphisms which are bimodule categories (i.e. categories acted), 2-morphisms which are functors, and 3-morphisms which are natural transformations.  Then the issue is to classify what kind of tensor categories these objects can be.

The story is trickier if we’re talking about, not just topological cobordisms, but ones equipped with some kind of structure regulated by a structure group G(for instance, orientation by G=SO(n), spin structure by its universal cover G= Spin(n), and so on).  This means the cobordisms come equipped with a map into BG.  They take O(n) as the starting point, and then consider groups G with a map to O(n), and require that the map into BG is a lift of the map to BO(n).  Then one gets that a structured local field theory amounts to a dualizable objects of \mathcal{C} with a homotopy-fixed point for some G-action – and this describes what gets assigned to the point by such a field theory.  What they then show is a correspondence between G and classes of categories.  For instance, fusion categories are what one gets by imposing that the cobordisms be oriented.

Liang Kong talked about “Topological Orders and Tensor Categories”, which used the Levin-Wen models, from condensed matter phyiscs.  (Benjamin Balsam also gave a nice talk describing these models and showing how they’re equivalent to the Turaev-Viro and Kitaev models in appropriate cases.  Ingo Runkel gave a related talk about topological field theories with “domain walls”.).  Here, the idea of a “defect” (and topological order) can be understood very graphically: we imagine a 2-dimensional crystal lattice (of atoms, say), and the defect is a 1-dimensional place where the two lattices join together, with the internal symmetry of each breaking down at the boundary.  (For example, a square lattice glued where the edges on one side are offset and meet the squares on the other side in the middle of a face, as you typically see in a row of bricks – the slides linked above have some pictures).  The Levin-Wen models are built using a hexagonal lattice, starting with a tensor category with several properties: spherical (there are dualities satisfying some relations), fusion, and unitary: in fact, historically, these defining properties were rediscovered independently here as the requirement for there to be excitations on the boundary which satisfy physically-inspired consistency conditions.

These abstract the properties of a category of representations.  A generalization of this to “topological orders” in 3D or higher is an extended TFT in the sense mentioned just above: they have a target 3-category of tensor categories, bimodule categories, functors and natural transformations.  The tensor categories (say, \mathcal{C}, \mathcal{D}, etc.) get assigned to the bulk regions; to “domain walls” between different regions, namely defects between lattices, we assign bimodule categories (but, for instance, to a line within a region, we get \mathcal{C} understood as a \mathcal{C}-\mathcal{C}-bimodule); then to codimension 2 and 3 defects we attach functors and natural transformations.  The algebra for how these combine expresses the ways these topological defects can go together.  On a lattice, this is an abstraction of a spin network model, where typically we have just one tensor category \mathcal{C} applied to the whole bulk, namely the representations of a Lie group (say, a unitary group).  Then we do calculations by breaking down into bases: on codimension-1 faces, these are simple objects of \mathcal{C}; to vertices we assign a Hom space (and label by a basis for intertwiners in the special case); and so on.

Thomas Nickolaus spoke about the same kind of G-equivariant Dijkgraaf-Witten models as at our workshop in Lisbon, so I’ll refer you back to my earlier post on that.  However, speaking of equivariance and group actions:

Michael Müger  spoke about “Orbifolds of Rational CFT’s and Braided Crossed G-Categories” (see this paper for details).  This starts with that correspondence between rational CFT’s (strictly, rational chiral CFT’s) and modular categories Rep(F).  (He takes F to be the name of the CFT).  Then we consider what happens if some finite group G acts on F (if we understand F as a functor, this is an action by natural transformations; if as an algebra, then ).  This produces an “orbifold theory” F^G (just like a finite group action on a manifold produces an orbifold), which is the “G-fixed subtheory” of F, by taking G-fixed points for every object, and is also a rational CFT.  But that means it corresponds to some other modular category Rep(F^G), so one would like to know what category this is.

A natural guess might be that it’s Rep(F)^G, where C^G is a “weak fixed-point” category that comes from a weak group action on a category C.  Objects of C^G are pairs (c,f_g) where c \in C and f_g : g(c) \rightarrow c is a specified isomorphism.  (This is a weak analog of S^G, the set of fixed points for a group acting on a set).  But this guess is wrong – indeed, it turns out these categories have the wrong dimension (which is defined because the modular category has a trace, which we can sum over generating objects).  Instead, the right answer, denoted by Rep(F^G) = G-Rep(F)^G, is the G-fixed part of some other category.  It’s a braided crossed G-category: one with a grading by G, and a G-action that gets along with it.  The identity-graded part of Rep(F^G) is just the original Rep(F).

State Sum Models

This ties in somewhat with at least some of the models in the previous section.  Some of these were somewhat introductory, since many of the people at the conference were coming from a different background.  So, for instance, to begin the workshop, John Barrett gave a talk about categories and quantum gravity, which started by outlining the historical background, and the development of state-sum models.  He gave a second talk where he began to relate this to diagrams in Gray-categories (something he also talked about here in Lisbon in February, which I wrote about then).  He finished up with some discussion of spherical categories (and in particular the fact that there is a Gray-category of spherical categories, with a bunch of duals in the suitable sense).  This relates back to the kind of structures Chris Douglas spoke about (described above, but chronologically right after John).  Likewise, Winston Fairbairn gave a talk about state sum models in 3D quantum gravity – the Ponzano Regge model and Turaev-Viro model being the focal point, describing how these work and how they’re constructed.  Part of the point is that one would like to see that these fit into the sort of framework described in the section above, which for PR and TV models makes sense, but for the fancier state-sum models in higher dimensions, this becomes more complicated.

Higher Gauge Theory

There wasn’t as much on this topic as at our own workshop in Lisbon (though I have more remarks on higher gauge theory in one post about it), but there were a few entries.  Roger Picken talked about some work with Joao Martins about a cubical formalism for parallel transport based on crossed modules, which consist of a group G and abelian group H, with a map \partial : H \rightarrow G and an action of G on H satisfying some axioms.  They can represent categorical groups, namely group objects in Cat (equivalently, categories internal to Grp), and are “higher” analogs of groups with a set of elements.  Roger’s talk was about how to understand holonomies and parallel transports in this context.  So, a “connection” lets on transport things with G-symmetries along paths, and with H-symmetries along surfaces.  It’s natural to describe this with squares whose edges are labelled by G-elements, and faces labelled by H-elements (which are the holonomies).  Then the “cubical approach” means that we can describe gauge transformations, and higher gauge transformations (which in one sense are the point of higher gauge theory) in just the same way: a gauge transformation which assigns H-values to edges and G-values to vertices can be drawn via the holonomies of a connection on a cube which extends the original square into 3D (so the edges become squares, and so get H-values, and so on).  The higher gauge transformations work in a similar way.  This cubical picture gives a good way to understand the algebra of how gauge transformations etc. work: so for instance, gauge transformations look like “conjugation” of a square by four other squares – namely, relating the front and back faces of a cube by means of the remaining faces.  Higher gauge transformations can be described by means of a 4D hypercube in an analogous way, and their algebraic properties have to do with the 2D faces of the hypercube.

Derek Wise gave a short talk outlining his recent paper with John Baez in which they show that it’s possible to construct a higher gauge theory based on the Poincare 2-group which turns out to have fields, and dynamics, which are equivalent to teleparallel gravity, a slightly unusal theory which nevertheless looks in practice just like General Relativity.  I discussed this in a previous post.

So next time I’ll talk about the new additions to my paper on ETQFT which were the basis of my talk, which illustrates a few of the themes above.

So I’ve been travelling a lot in the last month, spending more than half of it outside Portugal. I was in Ottawa, Canada for a Fields Institute workshop, “Categorical Methods in Representation Theory“. Then a little later I was in Erlangen, Germany for one called “Categorical and Representation-Theoretic Methods in Quantum Geometry and CFT“. Despite the similar-sounding titles, these were on fairly different themes, though Marco Mackaay was at both, talking about categorifying the q-Schur algebra by diagrams.  I’ll describe the meetings, but for now I’ll start with the first.  Next post will be a summary of the second.

The Ottawa meeting was organized by Alistair Savage, and Alex Hoffnung (like me, a former student of John Baez). Alistair gave a talk here at IST over the summer about a q-deformation of Khovanov’s categorification of the Heisenberg Algebra I discussed in an earlier entry. A lot of the discussion at the workshop was based on the Khovanov-Lauda program, which began with categorifying quantum version of the classical Lie groups, and is now making lots of progress in the categorification of algebras, representation theory, and so on.

The point of this program is to describe “categorifications” of particular algebras. This means finding monoidal categories with the property that when you take the Grothendieck ring (the ring of isomorphism classes, with a multiplication given by the monoidal structure), you get back the integral form of some algebra. (And then recover the original by taking the tensor over \mathbb{Z} with \mathbb{C}). The key thing is how to represent the algebra by generators and relations. Since free monoidal categories with various sorts of structures can be presented as categories of string diagrams, it shouldn’t be surprising that the categories used tend to have objects that are sequences (i.e. monoidal products) of dots with various sorts of labelling data, and morphisms which are string diagrams that carry those labels on strands (actually, usually they’re linear combinations of such diagrams, so everything is enriched in vector spaces). Then one imposes relations on the “free” data given this way, by saying that the diagrams are considered the same morphism if they agree up to some local moves. The whole problem then is to find the right generators (labelling data) and relations (local moves). The result will be a categorification of a given presentation of the algebra you want.

So for instance, I was interested in Sabin Cautis and Anthony Licata‘s talks connected with this paper, “Heisenberg Categorification And Hilbert Schemes”. This is connected with a generalization of Khovanov’s categorification linked above, to include a variety of other algebras which are given a similar name. The point is that there’s such a “Heisenberg algebra” associated to different subgroups \Gamma \subset SL(2,\mathbf{k}), which in turn are classified by Dynkin diagrams. The vertices of these Dynkin diagrams correspond to some generators of the Heisenberg algebra, and one can modify Khovanov’s categorification by having strands in the diagram calculus be labelled by these vertices. Rules for local moves involving strands with different labels will be governed by the edges of the Dynkin diagram. Their paper goes on to describe how to represent these categorifications on certain categories of Hilbert schemes.

Along the same lines, Aaron Lauda gave a talk on the categorification of the NilHecke algebra. This is defined as a subalgebra of endomorphisms of P_a = \mathbb{Z}[x_1,\dots,x_a], generated by multiplications (by the x_i) and the divided difference operators \partial_i. There are different from the usual derivative operators: in place of the differences between values of a single variable, they measure how a function behaves under the operation s_i which switches variables x_i and x_{i+1} (that is, the reflection in the hyperplane where x_i = x_{i+1}). The point is that just like differentiation, this operator – together with multiplication – generates an algebra in End(\mathbb{Z}[x_1,\dots,x_a]. Aaron described how to categorify this presentation of the NilHecke algebra with a string-diagram calculus.

So anyway, there were a number of talks about the explosion of work within this general program – for instance, Marco Mackaay’s which I mentioned, as well as that of Pedro Vaz about the same project. One aspect of this program is that the relatively free “string diagram categories” are sometimes replaced with categories where the objects are bimodules and morphisms are bimodule homomorphisms. Making the relationship precise is then a matter of proving these satisfy exactly the relations on a “free” category which one wants, but sometimes they’re a good setting to prove one has a nice categorification. Thus, Ben Elias and Geordie Williamson gave two parts of one talk about “Soergel Bimodules and Kazhdan-Lusztig Theory” (see a blog post by Ben Webster which gives a brief intro to this notion, including pointing out that Soergel bimodules give a categorification of the Hecke algebra).

One of the reasons for doing this sort of thing is that one gets invariants for manifolds from algebras – in particular, things like the Jones polynomial, which is related to the Temperley-Lieb algebra. A categorification of it is Khovanov homology (which gives, for a manifold, a complex, with the property that the graded Euler characteristic of the complex is the Jones polynomial). The point here is that categorifying the algebra lets you raise the dimension of the kind of manifold your invariants are defined on.

So, for instance, Scott Morrison described “Invariants of 4-Manifolds from Khonanov Homology“.  This was based on a generalization of the relationship between TQFT’s and planar algebras.  The point is, planar algebras are described by the composition of diagrams of the following form: a big circle, containing some number of small circles.  The boundaries of each circle are labelled by some number of marked points, and the space between carries curves which connect these marked points in some way.  One composes these diagrams by gluing big circles into smaller circles (there’s some further discussion here including a picture, and much more in this book here).  Scott Morrison described these diagrams as “spaghetti and meatball” diagrams.  Planar algebras show up by associating a vector spaces to “the” circle with n marked points, and linear maps to each way (up to isotopy) of filling in edges between such circles.  One can think of the circles and marked-disks as a marked-cobordism category, and so a functorial way of making these assignments is something like a TQFT.  It also gives lots of vector spaces and lots of linear maps that fit together in a particular way described by this category of marked cobordisms, which is what a “planar algebra” actually consists of.  Clearly, these planar algebras can be used to get some manifold invariants – namely the “TQFT” that corresponds to them.

Scott Morrison’s talk described how to get invariants of 4-dimensional manifolds in a similar way by boosting (almost) everything in this story by 2 dimensions.  You start with a 4-ball, whose boundary is a 3-sphere, and excise some number of 4-balls (with 3-sphere boundaries) from the interior.  Then let these 3D boundaries be “marked” with 1-D embedded links (think “knots” if you like).  These 3-spheres with embedded links are the objects in a category.  The morphisms are 4-balls which connect them, containing 2D knotted surfaces which happen to intersect the boundaries exactly at their embedded links.  By analogy with the image of “spaghetti and meatballs”, where the spaghetti is a collection of 1D marked curves, Morrison calls these 4-manifolds with embedded 2D surfaces “lasagna diagrams” (which generalizes to the less evocative case of “(n,k) pasta diagrams”, where we’ve just mentioned the (2,1) and (4,2) cases, with k-dimensional “pasta” embedded in n-dimensional balls).  Then the point is that one can compose these pasta diagrams by gluing the 4-balls along these marked boundaries.  One then gets manifold invariants from these sorts of diagrams by using Khovanov homology, which assigns to

Ben Webster talked about categorification of Lie algebra representations, in a talk called “Categorification, Lie Algebras and Topology“. This is also part of categorifying manifold invariants, since the Reshitikhin-Turaev Invariants are based on some monoidal category, which in this case is the category of representations of some algebra.  Categorifying this to a 2-category gives higher-dimensional equivalents of the RT invariants.  The idea (which you can check out in those slides) is that this comes down to describing the analog of the “highest-weight” representations for some Lie algebra you’ve already categorified.

The Lie theory point here, you might remember, is that representations of Lie algebras \mathfrak{g} can be analyzed by decomposing them into “weight spaces” V_{\lambda}, associated to weights \lambda : \mathfrak{g} \rightarrow \mathbf{k} (where \mathbf{k} is the base field, which we can generally assume is \mathbb{C}).  Weights turn Lie algebra elements into scalars, then.  So weight spaces generalize eigenspaces, in that acting by any element g \in \mathfrak{g} on a “weight vector” v \in V_{\lambda} amounts to multiplying by \lambda{g}.  (So that v is an eigenvector for each g, but the eigenvalue depends on g, and is given by the weight.)  A weight can be the “highest” with respect to a natural order that can be put on weights (\lambda \geq \mu if the difference is a nonnegative combination of simple weights).  Then a “highest weight representation” is one which is generated under the action of \mathfrak{g} by a single weight vector v, the “highest weight vector”.

The point of the categorification is to describe the representation in the same terms.  First, we introduce a special strand (which Ben Webster draws as a red strand) which represents the highest weight vector.  Then we say that the category that stands in for the highest weight representation is just what we get by starting with this red strand, and putting all the various string diagrams of the categorification of \mathfrak{g} next to it.  One can then go on to talk about tensor products of these representations, where objects are found by amalgamating several such diagrams (with several red strands) together.  And so on.  These categorified representations are then supposed to be usable to give higher-dimensional manifold invariants.

Now, the flip side of higher categories that reproduce ordinary representation theory would be the representation theory of higher categories in their natural habitat, so to speak. Presumably there should be a fairly uniform picture where categorifications of normal representation theory will be special cases of this. Vlodymyr Mazorchuk gave an interesting talk called 2-representations of finitary 2-categories.  He gave an example of one of the 2-categories that shows up a lot in these Khovanov-Lauda categorifications, the 2-category of Soergel Bimodules mentioned above.  This has one object, which we can think of as a category of modules over the algebra \mathbb{C}[x_1, \dots, x_n]/I (where I  is some ideal of homogeneous symmetric polynomials).  The morphisms are endofunctors of this category, which all amount to tensoring with certain bimodules – the irreducible ones being the Soergel bimodules.  The point of the talk was to explain the representations of 2-categories \mathcal{C} – that is, 2-functors from \mathcal{C} into some “classical” 2-category.  Examples would be 2-categories like “2-vector spaces”, or variants on it.  The examples he gave: (1) [small fully additive \mathbf{k}-linear categories], (2) the full subcategory of it with finitely many indecomposible elements, (3) [categories equivalent to module categories of finite dimensional associative \mathbf{k}-algebras].  All of these have some claim to be a 2-categorical analog of [vector spaces].  In general, Mazorchuk allowed representations of “FIAT” categories: Finitary (Two-)categories with Involutions and Adjunctions.

Part of the process involved getting a “multisemigroup” from such categories: a set S with an operation which takes pairs of elements, and returns a subset of S, satisfying some natural associativity condition.  (Semigroups are the case where the subset contains just one element – groups are the case where furthermore the operation is invertible).  The idea is that FIAT categories have some set of generators – indecomposable 1-morphisms – and that the multisemigroup describes which indecomposables show up in a composite.  (If we think of the 2-category as a monoidal category, this is like talking about a decomposition of a tensor product of objects).  So, for instance, for the 2-category that comes from the monoidal category of \mathfrak{sl}(2) modules, we get the semigroup of nonnegative integers.  For the Soergel bimodule 2-category, we get the symmetric group.  This sort of thing helps characterize when two objects are equivalent, and in turn helps describe 2-representations up to some equivalence.  (You can find much more detail behind the link above.)

On the more classical representation-theoretic side of things, Joel Kamnitzer gave a talk called “Spiders and Buildings”, which was concerned with some geometric and combinatorial constructions in representation theory.  These involved certain trivalent planar graphs, called “webs”, whose edges carry labels between 1 and (n-1).  They’re embedded in a disk, and the outgoing edges, with labels (k_1, \dots, k_m) determine a representation space for a group G, say G = SL_n, namely the tensor product of a bunch of wedge products, \otimes_j \wedge^{k_j} \mathbb{C}^n, where SL_n acts on \mathbb{C}^n as usual.  Then a web determines an invariant vector in this space.  This comes about by having invariant vectors for each vertex (the basic case where m =3), and tensoring them together.  But the point is to interpret this construction geometrically.  This was a bit outside my grasp, since it involves the Langlands program and the geometric Satake correspondence, neither of which I know much of anything about, but which give geometric/topological ways of constructing representation categories.  One thing I did pick up is that it uses the “Langlands dual group” \check{G} of G to get a certain metric space called Gn_{\check{G}}.  Then there’s a correspondence between the category of representations of G and the category of (perverse, constructible) sheaves on this space.  This correspondence can be used to describe the vectors that come out of these webs.

Jim Dolan gave a couple of talks while I was there, which actually fit together as two parts of a bigger picture – one was during the workshop itself, and one at the logic seminar on the following Monday. It helped a lot to see both in order to appreciate the overall point, so I’ll mix them a bit indiscriminately. The first was called “Dimensional Analysis is Algebraic Geometry”, and the second “Toposes of Quasicoherent Sheaves on Toric Varieties”. For the purposes of the logic seminar, he gave the slogan of the second talk as “Algebraic Geometry is a branch of Categorical Logic”. Jim’s basic idea was inspired by Bill Lawvere’s concept of a “theory”, which is supposed to extend both “algebraic theories” (such as the “theory of groups”) and theories in the sense of physics.  Any given theory is some structured category, and “models” of the theory are functors into some other category to represent it – it thus has a functor category called its “moduli stack of models”.  A physical theory (essentially, models which depict some contents of the universe) has some parameters.  The “theory of elastic scattering”, for instance, has the masses, and initial and final momenta, of two objects which collide and “scatter” off each other.  The moduli space for this theory amounts to assignments of values to these parameters, which must satisfy some algebraic equations – conservation of energy and momentum (for example, \sum_i m_i v_i^{in} = \sum_i m_i v_i^{out}, where i \in 1, 2).  So the moduli space is some projective algebraic variety.  Jim explained how “dimensional analysis” in physics is the study of line bundles over such varieties (“dimensions” are just such line bundles, since a “dimension” is a 1-dimensional sort of thing, and “quantities” in those dimensions are sections of the line bundles).  Then there’s a category of such bundles, which are organized into a special sort of symmetric monoidal category – in fact, it’s contrained so much it’s just a graded commutative algebra.

In his second talk, he generalized this to talk about categories of sheaves on some varieties – and, since he was talking in the categorical logic seminar, he proposed a point of view for looking at algebraic geometry in the context of logic.  This view could be summarized as: Every (generalized) space studied by algebraic geometry “is” the moduli space of models for some theory in some doctrine.  The term “doctrine” is Bill Lawvere’s, and specifies what kind of structured category the theory and the target of its models are supposed to be (and of course what kind of functors are allowed as models).  Thus, for instance, toposes (as generalized spaces) are supposed to be thought of as “geometric theories”.  He explained that his “dimensional analysis doctrine” is a special case of this.  As usual when talking to Jim, I came away with the sense that there’s a very large program of ideas lurking behind everything he said, of which only the tip of the iceberg actually made it into the talks.

Next post, when I have time, will talk about the meeting at Erlangen…

Marco Mackaay recently pointed me at a paper by Mikhail Khovanov, which describes a categorification of the Heisenberg algebra H (or anyway its integral form H_{\mathbb{Z}}) in terms of a diagrammatic calculus.  This is very much in the spirit of the Khovanov-Lauda program of categorifying Lie algebras, quantum groups, and the like.  (There’s also another one by Sabin Cautis and Anthony Licata, following up on it, which I fully intend to read but haven’t done so yet. I may post about it later.)

Now, as alluded to in some of the slides I’ve from recent talks, Jamie Vicary and I have been looking at a slightly different way to answer this question, so before I talk about the Khovanov paper, I’ll say a tiny bit about why I was interested.

Groupoidification

The Weyl algebra (or the Heisenberg algebra – the difference being whether the commutation relations that define it give real or imaginary values) is interesting for physics-related reasons, being the algebra of operators associated to the quantum harmonic oscillator.  The particular approach to categorifying it that I’ve worked with goes back to something that I wrote up here, and as far as I know, originally was suggested by Baez and Dolan here.  This categorification is based on “stuff types” (Jim Dolan’s term, based on “structure types”, a.k.a. Joyal’s “species”).  It’s an example of the groupoidification program, the point of which is to categorify parts of linear algebra using the category Span(Gpd).  This has objects which are groupoids, and morphisms which are spans of groupoids: pairs of maps G_1 \leftarrow X \rightarrow G_2.  Since I’ve already discussed the backgroup here before (e.g. here and to a lesser extent here), and the papers I just mentioned give plenty more detail (as does “Groupoidification Made Easy“, by Baez, Hoffnung and Walker), I’ll just mention that this is actually more naturally a 2-category (maps between spans are maps X \rightarrow X' making everything commute).  It’s got a monoidal structure, is additive in a fairly natural way, has duals for morphisms (by reversing the orientation of spans), and more.  Jamie Vicary and I are both interested in the quantum harmonic oscillator – he did this paper a while ago describing how to construct one in a general symmetric dagger-monoidal category.  We’ve been interested in how the stuff type picture fits into that framework, and also in trying to examine it in more detail using 2-linearization (which I explain here).

Anyway, stuff types provide a possible categorification of the Weyl/Heisenberg algebra in terms of spans and groupoids.  They aren’t the only way to approach the question, though – Khovanov’s paper gives a different (though, unsurprisingly, related) point of view.  There are some nice aspects to the groupoidification approach: for one thing, it gives a nice set of pictures for the morphisms in its categorified algebra (they look like groupoids whose objects are Feynman diagrams).  Two great features of this Khovanov-Lauda program: the diagrammatic calculus gives a great visual representation of the 2-morphisms; and by dealing with generators and relations directly, it describes, in some sense1, the universal answer to the question “What is a categorification of the algebra with these generators and relations”.  Here’s how it works…

Heisenberg Algebra

One way to represent the Weyl/Heisenberg algebra (the two terms refer to different presentations of isomorphic algebras) uses a polynomial algebra P_n = \mathbb{C}[x_1,\dots,x_n].  In fact, there’s a version of this algebra for each natural number n (the stuff-type references above only treat n=1, though extending it to “n-sorted stuff types” isn’t particularly hard).  In particular, it’s the algebra of operators on P_n generated by the “raising” operators a_k(p) = x_k \cdot p and the “lowering” operators b_k(p) = \frac{\partial p}{\partial x_k}.  The point is that this is characterized by some commutation relations.  For j \neq k, we have:

[a_j,a_k] = [b_j,b_k] = [a_j,b_k] = 0

but on the other hand

[a_k,b_k] = 1

So the algebra could be seen as just a free thing generated by symbols \{a_j,b_k\} with these relations.  These can be understood to be the “raising and lowering” operators for an n-dimensional harmonic oscillator.  This isn’t the only presentation of this algebra.  There’s another one where [p_k,q_k] = i (as in i = \sqrt{-1}) has a slightly different interpretation, where the p and q operators are the position and momentum operators for the same system.  Finally, a third one – which is the one that Khovanov actually categorifies – is skewed a bit, in that it replaces the a_j with a different set of \hat{a}_j so that the commutation relation actually looks like

[\hat{a}_j,b_k] = b_{k-1}\hat{a}_{j-1}

It’s not instantly obvious that this produces the same result – but the \hat{a}_j can be rewritten in terms of the a_j, and they generate the same algebra.  (Note that for the one-dimensional version, these are in any case the same, taking a_0 = b_0 = 1.)

Diagrammatic Calculus

To categorify this, in Khovanov’s sense (though see note below1), means to find a category \mathcal{H} whose isomorphism classes of objects correspond to (integer-) linear combinations of products of the generators of H.  Now, in the Span(Gpd) setup, we can say that the groupoid FinSet_0, or equvialently \mathcal{S} = \coprod_n  \mathcal{S}_n, represents Fock space.  Groupoidification turns this into the free vector space on the set of isomorphism classes of objects.  This has some extra structure which we don’t need right now, so it makes the most sense to describe it as \mathbb{C}[[t]], the space of power series (where t^n corresponds to the object [n]).  The algebra itself is an algebra of endomorphisms of this space.  It’s this algebra Khovanov is looking at, so the monoidal category in question could really be considered a bicategory with one object, where the monoidal product comes from composition, and the object stands in formally for the space it acts on.  But this space doesn’t enter into the description, so we’ll just think of \mathcal{H} as a monoidal category.  We’ll build it in two steps: the first is to define a category \mathcal{H}'.

The objects of \mathcal{H}' are defined by two generators, called Q_+ and Q_-, and the fact that it’s monoidal (these objects will be the categorifications of a and b).  Thus, there are objects Q_+ \otimes Q_- \otimes Q_+ and so forth.  In general, if \epsilon is some word on the alphabet \{+,-\}, there’s an object Q_{\epsilon} = Q_{\epsilon_1} \otimes \dots \otimes Q_{\epsilon_m}.

As in other categorifications in the Khovanov-Lauda vein, we define the morphisms of \mathcal{H}' to be linear combinations of certain planar diagrams, modulo some local relations.  (This type of formalism comes out of knot theory – see e.g. this intro by Louis Kauffman).  In particular, we draw the objects as sequences of dots labelled + or -, and connect two such sequences by a bunch of oriented strands (embeddings of the interval, or circle, in the plane).  Each + dot is the endpoint of a strand oriented up, and each - dot is the endpoint of a strand oriented down.  The local relations mean that we can take these diagrams up to isotopy (moving the strands around), as well as various other relations that define changes you can make to a diagram and still represent the same morphism.  These relations include things like:

which seems visually obvious (imagine tugging hard on the ends on the left hand side to straighten the strands), and the less-obvious:

and a bunch of others.  The main ingredients are cups, caps, and crossings, with various orientations.  Other diagrams can be made by pasting these together.  The point, then, is that any morphism is some \mathbf{k}-linear combination of these.  (I prefer to assume \mathbf{k} = \mathbb{C} most of the time, since I’m interested in quantum mechanics, but this isn’t strictly necessary.)

The second diagram, by the way, are an important part of categorifying the commutation relations.  This would say that Q_- \otimes Q_+ \cong Q_+ \otimes Q_- \oplus 1 (the commutation relation has become a decomposition of a certain tensor product).  The point is that the left hand sides show the composition of two crossings Q_- \otimes Q_+ \rightarrow Q_+ \otimes Q_- and Q_+ \otimes Q_- \rightarrow Q_- \otimes Q_+ in two different orders.  One can use this, plus isotopy, to show the decomposition.

That diagrams are invariant under isotopy means, among other things, that the yanking rule holds:

(and similar rules for up-oriented strands, and zig zags on the other side).  These conditions amount to saying that the functors - \otimes Q_+ and - \otimes Q_- are two-sided adjoints.  The two cups and caps (with each possible orientation) give the units and counits for the two adjunctions.  So, for instance, in the zig-zag diagram above, there’s a cup which gives a unit map \mathbf{k} \rightarrow Q_- \otimes Q_+ (reading upward), all tensored on the right by Q_-.  This is followed by a cap giving a counit map Q_+ \otimes Q_- \rightarrow \mathbf{k} (all tensored on the left by Q_-).  So the yanking rule essentially just gives one of the identities required for an adjunction.  There are four of them, so in fact there are two adjunctions: one where Q_+ is the left adjoint, and one where it’s the right adjoint.

Karoubi Envelope

Now, so far this has explained where a category \mathcal{H}' comes from – the one with the objects Q_{\epsilon} described above.  This isn’t quite enough to get a categorification of H_{\mathbb{Z}}: it would be enough to get the version with just one a and one b element, and their powers, but not all the a_j and b_k.  To get all the elements of the (integral form of) the Heisenberg algebras, and in particular to get generators that satisfy the right commutation relations, we need to introduce some new objects.  There’s a convenient way to do this, though, which is to take the Karoubi envelope of \mathcal{H}'.

The Karoubi envelope of any category \mathcal{C} is a universal way to find a category Kar(\mathcal{C}) that contains \mathcal{C} and for which all idempotents split (i.e. have corresponding subobjects).  Think of vector spaces, for example: a map p \in End(V) such that p^2 = p is a projection.  That projection corresponds to a subspace W \subset V, and W is actually another object in Vect, so that p splits (factors) as V \rightarrow W subset V.  This might not happen in any general \mathcal{C}, but it will in Kar(\mathcal{C}).  This has, for objects, all the pairs (C,p) where p : C \rightarrow C is idempotent (so \mathcal{C} is contained in Kar(\mathcal{C}) as the cases where p=1).  The morphisms f : (C,p) \rightarrow (C',p') are just maps f : C \rightarrow C' with the compatibility condition that p' f = p f = f (essentially, maps between the new subobjects).

So which new subobjects are the relevant ones?  They’ll be subobjects of tensor powers of our Q_{\pm}.  First, consider Q_{+^n} = Q_+^{\otimes n}.  Obviously, there’s an action of the symmetric group \mathcal{S}_n on this, so in fact (since we want a \mathbf{k}-linear category), its endomorphisms contain a copy of \mathbf{k}[\mathcal{S}_n], the corresponding group algebra.  This has a number of different projections, but the relevant ones here are the symmetrizer,:

e_n = \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} \sigma

which wants to be a “projection onto the symmetric subspace” and the antisymmetrizer:

e'_n = \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} sign(\sigma) \sigma

which wants to be a “projection onto the antisymmetric subspace” (if it were in a category with the right sub-objects). The diagrammatic way to depict this is with horizontal bars: so the new object S^n_+ = (Q_{+^n}, e) (the symmetrized subobject of Q_+^{\oplus n}) is a hollow rectangle, labelled by n.  The projection from Q_+^{\otimes n} is drawn with n arrows heading into that box:

The antisymmetrized subobject \Lambda^n_+ = (Q_{+^n},e') is drawn with a black box instead.  There are also S^n_- and \Lambda^n_- defined in the same way (and drawn with downward-pointing arrows).

The basic fact – which can be shown by various diagram manipulations, is that S^n_- \otimes \Lambda^m_+ \cong (\Lambda^m_+ \otimes S^n_-) \oplus (\Lambda_+^{m-1} \otimes S^{n-1}_-).  The key thing is that there are maps from the left hand side into each of the terms on the right, and the sum can be shown to be an isomorphism using all the previous relations.  The map into the second term involves a cap that uses up one of the strands from each term on the left.

There are other idempotents as well – for every partition \lambda of n, there’s a notion of \lambda-symmetric things – but ultimately these boil down to symmetrizing the various parts of the partition.  The main point is that we now have objects in \mathcal{H} = Kar(\mathcal{H}') corresponding to all the elements of H_{\mathbb{Z}}.  The right choice is that the \hat{a}_j  (the new generators in this presentation that came from the lowering operators) correspond to the S^j_- (symmetrized products of “lowering” strands), and the b_k correspond to the \Lambda^k_+ (antisymmetrized products of “raising” strands).  We also have isomorphisms (i.e. diagrams that are invertible, using the local moves we’re allowed) for all the relations.  This is a categorification of H_{\mathbb{Z}}.

Some Generalities

This diagrammatic calculus is universal enough to be applied to all sorts of settings where there are functors which are two-sided adjoints of one another (by labelling strands with functors, and the regions of the plane with categories they go between).  I like this a lot, since biadjointness of certain functors is essential to the 2-linearization functor \Lambda (see my link above).  In particular, \Lambda uses biadjointness of restriction and induction functors between representation categories of groupoids associated to a groupoid homomorphism (and uses these unit and counit maps to deal with 2-morphisms).  That example comes from the fact that a (finite-dimensional) representation of a finite group(oid) is a functor into Vect, and a group(oid) homomorphism is also just a functor F : H \rightarrow G.  Given such an F, there’s an easy “restriction” F^* : Fun(G,Vect) \rightarrow Fun(H,Vect), that just works by composing with F.  Then in principle there might be two different adjoints Fun(H,Vect) \rightarrow Fun(G,Vect), given by the left and right Kan extension along F.  But these are defined by colimits and limits, which are the same for (finite-dimensional) vector spaces.  So in fact the adjoint is two-sided.

Khovanov’s paper describes and uses exactly this example of biadjointness in a very nice way, albeit in the classical case where we’re just talking about inclusions of finite groups.  That is, given a subgroup H < G, we get a functors Res_G^H : Rep(G) \rightarrow Rep(H), which just considers the obvious action of H act on any representation space of G.  It has a biadjoint Ind^G_H : Rep(H) \rightarrow Rep(G), which takes a representation V of H to \mathbf{k}[G] \otimes_{\mathbf{k}[H]} V, which is a special case of the formula for a Kan extension.  (This formula suggests why it’s also natural to see these as functors between module categories \mathbf{k}[G]-mod and \mathbf{k}[H]-mod).  To talk about the Heisenberg algebra in particular, Khovanov considers these functors for all the symmetric group inclusions \mathcal{S}_n < \mathcal{S}_{n+1}.

Except for having to break apart the symmetric groupoid as S = \coprod_n \mathcal{S}_n, this is all you need to categorify the Heisenberg algebra.  In the Span(Gpd) categorification, we pick out the interesting operators as those generated by the - \sqcup \{\star\} map from FinSet_0 to itself, but “really” (i.e. up to equivalence) this is just all the inclusions \mathcal{S}_n < \mathcal{S}_{n+1} taken at once.  However, Khovanov’s approach is nice, because it separates out a lot of what’s going on abstractly and uses a general diagrammatic way to depict all these 2-morphisms (this is explained in the first few pages of Aaron Lauda’s paper on ambidextrous adjoints, too).  The case of restriction and induction is just one example where this calculus applies.

There’s a fair bit more in the paper, but this is probably sufficient to say here.


1 There are two distinct but related senses of “categorification” of an algebra A here, by the way.  To simplify the point, say we’re talking about a ring R.  The first sense of a categorification of R is a (monoidal, additive) category C with a “valuation” in R that takes \otimes to \times and \oplus to +.  This is described, with plenty of examples, in this paper by Rafael Diaz and Eddy Pariguan.  The other, typical of the Khovanov program, says it is a (monoidal, additive) category C whose Grothendieck ring is K_0(C) = R.  Of course, the second definition implies the first, but not conversely.  The objects of the Grothendieck ring are isomorphism classes in C.  A valuation may identify objects which aren’t isomorphic (or, as in groupoidification, morphisms which aren’t 2-isomorphic).

So a categorification of the first sort could be factored into two steps: first take the Grothendieck ring, then take a quotient to further identify things with the same valuation.  If we’re lucky, there’s a commutative square here: we could first take the category C, find some surjection C \rightarrow C', and then find that K_0(C') = R.  This seems to be the relation between Khovanov’s categorification of H_{\mathbb{Z}} and the one in Span(Gpd). This is the sense in which it seems to be the “universal” answer to the problem.

On a tangential note, let me point out John Baez’ most recent “This Week’s Finds”, which has an accessible but fairly in-depth discussion of climate modelling.  There have been many years of very loud public discussion of this which, for reasons of politics, seems to involve putting the “Mathematical models are inherently elitist gibberish” and “Science knows everything so shut up, moron” positions on display and letting viewer decide.  This is known in the journalism trade as “balance”.  Obviously, within the research community working on them, there’s a mountain of literature on what the models model, how detailed they are, how they work, etc., but it mostly goes over my head, so John’s post strikes a nice balance for me.

Like most computer simulation models, they’re basically discrete approximations to big systems of differential equations – but exactly which systems, how they’re developed, how accurately they model the real thing, and the relative merits of simple vs. complex models is the main point.  The use of Monte Carlo methods and Bayesian analysis to tune the various free parameters is a key part of the matter of how accurate they should be.  Anyway – check it out.

Meanwhile, the TQFT club at IST recently started up its series of seminars.  The first few speakers were Rui Carpentier, Anne-Laure Thiel, and Marco Mackaay.  Rui is faculty here at IST, and a former student of Roger Picken (his thesis was on a topic closely related to what he was talking about).  Anne-Laure is a post-doc here at IST, mainly working with Marco, who, however, is actually at the University of the Algarve in Faro, Portugal, and had to come up to Lisbon specially for the seminar.  Anne-Laure and Marco were both speaking mainly about some of the Soergel bimodule stuff which came up at the Oporto meeting on categorification, which I posted about previously, so I’ll go over that in a bit more detail here.

First, though, Rui Carpentier’s talk:

3-colourings of Cubic Graphs and Operators

All these talks involve algebraic representations of categories that can be represented by some graphical calculus, but in this case, one starts with a category whose morphisms are precisely graphs with loose ends.  (The objects are non-negative integers, or, if you like, finite sets of dots which act as the vertices of the loose ends).  The graphs are trivalent (except at the input and output vertices, which are 1-valent), hence “cubic graphs”.  This category is therefore called \mathbf{CG}, and it has a small number of generators, which happen to be quite similar to those which generate the category of 2D-cobordisms (one of the connections to TQFT), though the relations are slightly different.

Roughly, and without drawing the pictures: the generators are cup and cap (the shapes \cup and \cap), two different trivalent vertices (a Y, and the same upside-down), the swap (an X where the strands cross without a vertex), and the identity (just a vertical line).  There are a number of relations, including Reidemeister moves, on these generating pictures, which ensure that they’re enough to identify graphs up to isotopy of the pictures.

Then the point is to describe graphs using operators – that is, construct a representation F :\mathbf{CG} \rightarrow \mathbf{Vect}.   Given any such representation, these generators provide all the structure maps of a bialgebra – chiefly, unit, counit, multiplication and co-multiplication – and the relations imposed by isotopy make this work (though unlike some other situations, it’s neither commutative nor cocommutative).  The representation F he constructs is based on 3-colourings of the edges of the graphs.  At the object level, it assigns to a dot the 3-dimensional vector space V= span(e_1,e_2,e_3).  Being monoidal, F takes the object n to V^{\otimes n} – the tensor product of the spaces at each vertex.

The idea is that choosing a basis vector in this space amounts to picking a colouring of the incoming and outgoing edges.  For morphisms, we should note that the rule that says when a colouring is admissible is that all the edges incident to a given vertex must have different colours.  Then, given a morphism (graph) G : m \rightarrow n, we can describe the linear map F(G) most easily by saying that the component in the matrix, given an incoming and outgoing basis vector, just counts the number of admissible graphs that agree with the chosen colourings on the in-edges and out-edges.

There’s another functor, \hat{F}, which counts these graphs with a sign, which marks whether the graph contains an odd or an even number of crossings of differently-coloured edges – negative for odd, positive for even.  This  is the “Penrose evaluation” of the graph.

So these maps give the “operators” of the title, and the rest of the point is to use them to study graphs and their colourings.  One can, in this setup, rewrite some graphs as linear combinations of others – so-called “Skein relations” hold, for example, so that, after applying F, the composite of multiplication and comultiplication (taking two points to two points, through one cut-edge) is the same as the identity minus the swap.  This sort of thing appears in formal knot theory all the time, and is a key tool for recoupling in spin networks, and so on…

Given this “recoupling” idea, there are some important facts: first, any graph can be rewritten as a linear combination of planar graphs, and any planar graph with cycles can be reduced to a sum of planar graphs without cycles.  (Rui gave the example of decomposing a pentagonal cycle as a linear combination of four other graphs, three of which are disconnected).  So in fact any graph decomposes as a linear combination of forests (cycle-free graphs, the connected components of which are called “trees”, hence the name).  Another essential fact is that, due to the Euler characteristic of the plane, any planar graph can be split into two parts with at most five edges between them (the basis of the solution to the three utilities puzzle).  Then it so happens that the space of graphs connecting zero in-edges to five out-edges is a 6-dimensional space, \mathcal{V}^o_5, generated by just six forests (including one lonesome tree).

So one theorem which Rui told us about, which can be shown using the so-called Penrose relations (provable using the representations F and \hat{F}), is that there’s just one such graph (which he described in the particular basis above) that evaluates to zero when composed with some other graph.  The proof of this uses the Four Colour Theorem (3-colouring of graph edges being related to 4-colouring of planar regions); in fact, the two theorems are equivalent so if anyone can find an alternative proof of this one, the bonus is another proof of the FCT.

Finally, he gave a conjecture that, if true, would help recognize planar graphs just by the operators produced by the representation \hat{F} (at least it proposes a necessary condition).  This conjecture says that if a planar graph with five output edges (the maximum, remember) is written in the basis mentioned above, then the sum of the coefficients of the five disconnected trees is nonnegative.  (Thus, the connected tree doesn’t contribute to this measure).  This is still just a conjecture – Rui said that to date neither proof nor counterexample has been found.

Soergel Bimodules, Singular and Virtual Braids

As I mentioned up top, I previously posted a bit about work on Soergel bimodules when describing Catharina Stroppel’s talk at the meeting in Faro in July.  To recap: they are associated with categories of modules over rings – specifically, rings of certain classes of symmetric functions.  Even more specifically, given a partition \lambda of an integer n, there is a subgroup of the symmetric group S_{\lambda} \subset S_n which fixes the partition.  All such groups act on the ring of n-variable polynomial functions R =\mathbb{Q}[x_1, \dots, x_n], and the ones fixed by S_{\lambda} form the ring R^{\lambda}.

Now, these groups are all related to each other in a web of containments, hence so are the rings.  So the module categories R^{\lambda} are connected by various functors.  Given a containment R^{\lambda '} \subset R^{\lambda}, modules over R^{\lambda} can be restricted to ones over R^{\lambda '}, and modules over R^{\lambda '} can be induced up to ones over R^{\lambda}.  The restriction and induction functors can be represented as “tensor with a bimodule” (this is much the same classification as that for 2-linear maps which I’ve said a bunch about here, except that those must be free).  Applying induction functors repeatedly gives abitrarily large bimodules, but they are built as direct sums of simple parts.  Those simple parts, and any direct sums of them, are Soergel bimodules.  The point is that such bimodules describe morphisms.

So in the TQFT club, Marco Mackaay gave the first of a series of survey talks on this topic, and Anne-Laure Thiel gave a talk about the “Categorification of Singular Braid Monoids and Virtual Braid Groups”.  Since Marco’s talk was the first in a series of surveys, and a lot of what it surveyed was work described in my post on the Faro meeting, I’ll just mention that it dealt with the original motivation of a lot of this work in categorifying representation theory of Lie algebras (c.f. the discussion of the Khovanov-Lauda categorification of quantum groups in the previous post), and also got a bit into some of the different diagrammatic calculi created for that purpose, along the lines of the talks by Ben Webster and Geordie Williamson at that meeting.  Maybe when Marco has given more of these talks, I’ll return to this one here as well.

Now, the starting point of Anne-Laure’s talk was that the setup above lets one define a category with a presentation like that of the Hecke algebra (a quotient of the group algebra of the braid group), where exact relations become isomorphisms.  That is, we go from a category where morphisms are braids (up to isotopy and Reidemeister moves and so forth as usual) to a 2-category where the morphisms are bimodules, which happen to satisfy the same relations.  (The 2-morphisms, bimodule maps, are what allow relations to hold weakly…)

Specifically, the generators of the braid group are \sigma_i, the braids taking the i^{th} strand over the (i+1)^{st}.  The parallel thing is B_i = R \otimes_{R^{\sigma_i}} R, where here we’re talking about the subgroup generated by the transposition of i and i+1.  In the language of partitions, this corresponds to a \lambda with one part of size two, (i,i+1), and the rest of size one.  Now, since this bimodule is actually built from polynomials in R, it naturally has a grading – this corresponds to the degree of q, since the Hecke algebra involves a quotient giving q-deformed relations – so there is a degree-shift operation that categorifies multiplication by q.  This much is due to Soergel.

Anne-Laure’s talk was about extending this to talk about a categorification, first of the braid group in terms of complexes of these bimodules (due actually to Rouquier), then virtual and singular braids.  These, again, are basically creatures of formal knot theory (see link above).  They can be described by a presentation similar to that for braids – just as the braid group has a generators-and-relations presentation in terms of over-crossings of adjacent strands, these incorporate other kinds of crossings.  Singular braids allow a sort of “through” crossing, where the i^{th} strand goes neither over nor under the (i+1)^{st}.  Virtual braids (the braid variant on virtual knots) have a special type of marked crossing called the “virtual crossing”, drawn with a little circle around it.  These are included as new generators in describing the virtual braid group, and of course some new relations are added to show how they relate to the original generators – variations on the Reidemeister moves, for example.

To categorify this, Anne-Laure explained that these new generators can also be represented by bimodules, but these ones need to be twisted.  In particular, twisting the bimodule R by the action of a permutation \omega \in S_n gives R_{\omega}, which is the same as R as a left R-module, but is acted on by an element a \in R on the right through multiplication by \omega(a), so that b \cdot p \cdot a = bp(\omega(a)).  Then the new generators, beyond the B_i = R \otimes_{R^{\sigma_i}} R, are of the form R_{\omega} \otimes_{R^{\omega '}} R.  These then satsify the right relations for this to categorify the virtual braid group.

Whatever ultimately becomes of some aspects of the Standard Model – the Higgs boson, for example – here is a report (based on an experiment described here) that some of the fundamentals hold up well to experimental test. Specifically, the Spin-Statistics Theorem – the relationship between quantum numbers of elementary particles and the representation theory of the Poincare group. It would have been very surprising if things had been otherwise, but as usual, the more you rely on an idea, the more important it is to be sure it fits the facts. The association between physics and representation theory is one of those things.

So the fact that it all seems to work correctly is a bit of a relief for me. See below.


Since the paperwork is now well on its way, I may as well now mention here that I’ve taken a job as a postdoctoral researcher at CAMGSD, a centre at IST in Lisbon, starting in September. In a week or so I will be heading off to visit there – there are quite a few people there doing things I find quite interesting, so it should be an interesting trip. After that, I’ll be heading down to the south of the country for the Oporto meeting on Geometry, Topology and Physics, which is held this year in Faro. This year the subject is “categorification”, so my talk will be mainly about my paper on ETQFT. There are a bunch of interesting speakers – two I happen to know personally are Aaron Lauda and Joel Kamnitzer, but many others look quite promising.

In particular, one of the main invited speakers is Mikhail Khovanov, whose name is famously (for some values of “famous”) attached to Khovanov Homology, which is a categorification of the Jones Polynomial. Instead of a polynomial, it associates a graded complex of vector spaces to a knot. (Dror Bar-Natan wrote an intro, with many pictures and computations). Khovanov’s more recent work, with Aaron Lauda, has been on categorifying quantum groups (starting with this).

Now, as for me, since my talk in Faro will only be about 20 minutes, I’m glad of the opportunity to give some more background during the visit at IST. In particular, a bunch of the background to the ETQFT paper really depends on this paper on 2-linearization. I’ve given some previous talks on the subject, but this time I’m going to try to get a little further into how this fits into a more general picture. To repeat a bit of what’s in this post, 2-linearization describes a (weak) 2-functor:

\Lambda : Span(Gpd) \rightarrow 2Vect

where Span(Gpd) has groupoids as its objects, spans of groupoid homomorphisms as its arrows, and spans-of-span-maps as 2-morphisms. 2Vect is the 2-category of 2-vector spaces, which I’ve explained before. This 2-functor is supposed to be a sort of “linearization”, which is a very simple functor

L : Span(FinSet) \rightarrow Vect

It takes a set X to the free vector space L(X) = \mathbb{C}^X, and a span X \stackrel{s}{\leftarrow} S \stackrel{t}{\rightarrow} Y to a linear map L(S) : L(X) \rightarrow L(Y). This can be described in two stages, starting with a vector in L(S), namely, a function \psi : X \rightarrow \mathbb{C}. The two stages are:

  • First, “pull” \psi up along s to \mathbb{C}^S (note: I’m conflating the set S with the span (S,s,t)), to get the function s^*\psi = \psi \circ s : S \rightarrow \mathbb{C}.
  • Then “push” this along t to get t_*(s^*\psi). The “push” operation f_* along any map f : X \rightarrow Y is determined by the fact that it takes the basis vector \delta_x \in \mathbb{C}^X to the basis vector \delta_{f(x)} \in \mathbb{C}^Y (these are the delta functions which are 1 on the given element and 0 elsewhere)

It’s helpful to note that, for a given map f : X \rightarrow Y, are linear adjoints (using the standard inner product where the delta functions are orthonormal). Combining them together – it’s easy to see – gives a linear map which can be described in the basis of delta functions by a matrix. The (x,y)-entry of the matrix counts the elements of S which map to (x,y) under (s,t) : S \rightarrow X \times Y. We interpret this by saying the matrix “counts histories” connecting x to y.

In groupoidification, a-la Baez and Dolan (see the various references beyond the link), one replaces FinSet with FinGpd, the 2-category of (essentially) finite groupoids, but we still have a functor into Vect. In fact, into FinHilb: the vector space D(G) is the free one on isomorphism classes in G, but the linear maps (and the inner product) are tweaked using the groupoid cardinality, which can be any positive rational number. Then we say the matrix does a “sum over histories” of certain weights. In this paper, I extend this to “U(1)-groupoids”, which are labelled by phases – which represent the exponentiated action in quantum mechanics – and end up with complex matrices. So far so good.

The 2-linearization process is really “just” a categorification of what happens for sets, where we treat “groupoid” as the right categorification of “set”, and “Kapranov-Voevodsky 2-vector space” as the right categorification of “vector space”. (To treat “category” as the right categorification of “set”, one would have to use Elgueta’s “generalized 2-vector space“, which is probably morally the right thing to do, but here I won’t.) To a groupoid X, we assign the category of functors into Vect – that is, Rep(X) (in smooth cases, we might want to restrict what kind of representations we mean – see below).

To pull such a functor along a groupoid homomorphism f : X \rightarrow Y is again done by precomposition: f^*F = F \circ f. The push map in 2-linearization is the Kan extension of the functor \Psi along f. This is the universal way to push a functor forward, and is the (categorical!) adjoint to the pull map. (Kan extensions are supposed to come equipped with some natural transformations: these are the ones associated to the adjunction). Then composing “pull” and “push”, one categorifies “sum over histories”.

So here’s one thing this process is related to: in the case where our groupoids have just one object (i.e. are groups), and the homomorphism f : X \rightarrow Y is an inclusion (conventionally written H < G), this goes by a familiar name in representation theory: restriction and induction. So, given a representation \rho of G (that is, a functor from Y into Vect), there is an induced representation res_H^G \rho = f^*\rho, which is just the same representation space, acted on only by elements of H (that is, X). This is the easy one. The harder one is the induced representation of G from a representation \tau of H (i.e. \tau : X \rightarrow Vect, which is to say ind^G_H \tau = f_* \tau : Y \rightarrow Vect. The fact that these operations are adjoints goes in representation theory by the name “Frobenius reciprocity”.

These two operations were studied by George Mackey (in particular, though I’ve been implicitly talking about discrete groups, Mackey’s better known for looking at the case of unitary representations of compact Lie groups). The notion of a Mackey functor is supposed to abstract the formal properties of these operations. (A Mackey functor is really a pair of functors, one covariant and one contravariant – giving restriction and “transfer”/induction maps for – which have formal properties similar to the functor from groups into their representation rings – which it’s helpful to think of as the categories of representations, decategorificatied. In nice cases, a Mackey functor from a category C is the same as a functor out of Span(C)).

Anyway, by way of returning to groupoids: the induced representation for groups is found by \mathbb{C}[G] \otimes_{\mathbb{C}[H]} V, where V is the representation space of \tau. (For compact Lie groups, replace the group algebra \mathbb{C}[G] with L^2(G), and likewise for H). A similar formula shows up in the groupoid case, but with a contribution from each object (see the paper on 2-linearization for more details). This is also the formula for the Kan extension.

“Now wait a minute”, the categorically aware may ask, “do you mean the left Kan extension, or the right Kan extension?” That’s a good question! For one thing, they have different formulas: one involving limits, and the other involving colimits. Instead of answering it, I’ll talk about something not entirely unrelated – and a little more context for 2-linearization.

The setup here is actually a rather special case of Grothendieck’s six-operation framework, in the algebro-geometric context, for sheaves on (algebraic) spaces (there’s an overview in this talk by Joseph Lipman, the best I’ve been able to find online). Now, , these operations as extended to derived categories of sheaves (see this intro by R.P. Thomas). The derived category D(X) is described concretely in terms of chain complexes of sheaves in Sh(X), taken “up to homotopy” – it is a sort of categorification of cohomology. But of course, this contains Sh(X) as trivial complexes (i.e. concentrated at level zero). The fact that our sheaves come from functors into Vect, which form a 2-vector space, so that functors between these are exact, means that there’s no nontrivial homology – so in our special case, the machinery of derived categories is more than we need.

This framework has been extended to groupoids – so the sheaves are on the space of objects, and are equivariant – as described in a paper by Moerdijk called “Etale Groupoids, Derived Categories, and Operations” (the situation of sheaves that are equivariant under a group action is described in more detail by Bernstein and Lunts in the Springer lecture notes “Equivariant Sheaves and Functors”). Sheaves on groupoids are essentially just equivariant sheaves on the space of objects. Now, given a morphism $f : X \ra Y$, there are four induced operations:

  • f^* , f^! : D(Y) \rightarrow D(X)
  • f_*, f^! : D(X) \rightarrow D(Y) (in general right adjoint to f^* and f^!)

(The other operations of the “six” are hom and \otimes). The basic point here is that we can “pull” and “push” sheaves along the map f in various ways. For our purposes, it’s enough to consider f^* and f_*. The sheaves we want come from functors into Vect (we actually have a vector space at each point in the space of objects). These are equivariant “bundles”, albeit not necessarily locally trivial. The fact that we can think of these as sheaves – of sections – tends to stay in the background most of the time, but in particular, being functors automatically makes the resulting sheaves equivariant. In the discrete case, we can just think of these as sheaves of vector spaces: just take F(U) to be the direct sum of all the vector spaces at each object in any subset U – all subsets are open in the discrete topology… For the smooth situation, it’s better not to do this, and think of the space of sections as a module over the ring of suitable functions.

Now to return to your very good question about “left or right Kan extension”… the answer is both. since for Vect-valued functors (where Vect is the category of finite dimensional vector spaces), we have natural isomorphisms f^* \cong f^! and f_* \cong f_!: these functors are \textit{ambiadjoint} (ie. both left and right adjoint). We use this to define the effect of \Lambda on 2-morphisms in Span_2(Gpd).

This isomorphism is closely related to the fact that finite-dimensional vector spaces are canonically isomorphic to their double-dual: V \cong V^{**}. That’s because the functors f^* and f_* are 2-linear maps. These are naturally isomorphic to maps represented as matrices of vector spaces. Taking an adjoint – aside from transposing the matrix, naturally replaces the matrices with their duals. Doing this twice, we get the isomorphisms above. So the functors are both left and right adjoint to each other, and thus in particular we have what is both left and right Kan extension. (This is also connected with the fact that, in Vect, the direct sum is both product and coproduct – i.e. limit and colimit.)

It’s worth pointing out, then, that we wouldn’t generally expect this to happen for infinite-dimensional vector spaces, since these are generally not canonically isomorphic to their double-duals. Instead, for this case we would need to be looking at functors valued in Hilb, since Hilbert spaces do have that property. That’s why, in the case of smooth groupoids (say, Lie groupoids), we end up talking about “(measurable) equivariant Hilbert bundles”. (In particular, the ring of functions over which our sheaves are modules is: the measurable ones. Why this is the right choice would be a bit of a digression, but roughly it’s analogous to the fact that L^2(X) is a space of measurable functions. This is the limitation on which representations we want that I alluded to above.).

Now, \Lambda is supposed to be a 2-functor. In general, given a category C with all pullbacks, Span_2(C) is the universal 2-category faithfully containing C such that every morphism has an ambiadjoint. So the fact that the “pull” and “push” operations are ambiadjoint lets this 2-functor respect that property. It’s the unit and counits of the adjunctions which produce the effect of \Lambda on 2-morphisms: given a span of span-maps, we take the two maps in the middle, consider the adjoint pairs of functors that come from them, and get a natural transformation which is just the composite of the counit of one adjunction and the unit of the other.

Here’s where we understand how this fits into the groupoidification program – because the effect of \Lambda on 2-morphisms exactly reproduces the “degroupoidification” functor of Baez and Dolan, from spans of groupoids into Vect, when we think of such a span as a 2-morphism in Hom(1,1) – that is, a span of maps of spans from the terminal groupoid to itself. In other words, degroupoidification is an example something we can do between ANY pair of groupoids – but in the special case where the representation theory all becomes trivial. (This by no means makes it uninteresting: in fact, it’s a perfect setting to understand almost everything else about the subject).

Now, to actually get all the coefficients to work out to give the groupoid cardinality, one has to be a bit delicate – the exact isomorphism between the construction of the left and right adjoint has some flexibility when we’re working over the field of complex numbers. But there’s a general choice – the Nakayama isomorphism – which works even when we’re replace Vect by R-modules for some ring R. To make sure, for general R, that we have a true isomorphism, the map needs some constants. These happen to be, in our case, exactly the groupoid cardinalities to make the above statement true!

To me, this last part is a rather magical aspect of the whole thing, since the motivation I learned for groupoid cardinalities is quite remote from this – it’s just a valuation on groupoids which gets along with products and coproducts, and also with group actions (so that |X/G| = |X|/|G|, even when the action isn’t free). So one thing I’d like to know, but currently don’t is: how is it that this is “secretly” the same thing as the Nakayama isomorphism?

While I’d like to write up right away a description of the talk which Derek Wise gave recently at the Perimeter Institute (mostly about some work of mine which is preliminary to a collaboration we’re working on), I think I’ll take this next post as a chance to describe a couple of talks given in the seminar on stacks, groupoids, and algebras which I’m organizing, namely mine on representation theory of groupoids (focusing on Morita equivalence), and Peter Oman’s, called Toposes and Groupoids about how any topos can be compared to sheaves on a groupoid (sort of!). So here we go:

Representations of Groupoids and Morita Equivalence

The motivation here is to address directly what Morita equivalence means for groupoids, and particularly Lie groupoids. (One of the main references I used to prepare on this was this paper by Klaas Landsman, which gives Morita equivalence theorems for a variety of bicategories). The classic description of a Morita equivalence of rings R and L is often described in terms of the existence of an L-R-bimodule M having certain properties. But the point of this bimodule is that on can turn R-modules into L-modules by tensoring with it, and vice versa. Actually, it’s better than this, in that there are functors

- \otimes_L M : Mod(L) \rightarrow Mod(R)

and

M \otimes_R - : Mod(R) \rightarrow Mod(L)

And moreover, either composite of these is naturally isomorphic to the appropriate identity, so in particular one has M \otimes_L M \cong R and M \otimes_R M \cong L (since tensoring with the base ring is the identity for modules). But this just says that these two functors are actually giving an equivalence of the categories L-Mod and R-Mod.

So this is the point of Morita equivalence. Suppose, for a class of algebraic gadget (ring, algebra, groupoid, etc.), one has the notion of a representation of such a gadget R (as a module is the right idea of the representation of a ring), and all the representations of R form a category Rep(R). Then Morita equivalence is the equivalence relation induced by equivalence of the representation categories – gadgets R and L are Morita equivalent if there is an equivalence of the representation categories. For nice categories of gadgets – rings and von Neumann algebras, for instance, this occurs if and only if a condition like the existence of the bimodule M above is true. In other cases, this is only a sufficient condition for Morita equivalence, not a necessary one.

I’ll comment here that there are therefore several natural notions of Morita equivalence, which a priori might be different, since categories like Rep(R) carry quite a bit of structure. For example, there is a tensor product of representations that makes it a symmetric monoidal category; there is a direct sum of representations making it abelian. So we might want to ask that the equivalence between them be an equivalence of:

  • categories
  • abelian categories
  • monoidal abelian categories
  • symmetric monoidal abelian categories

(in principle we could also take the last two and drop “abelian”, for a total of six versions of the concept, but this progression is most natural in much the same way that “set – abelian group – ring” is a natural progression).

Reallly, what one wants is the strongest of these notions. Equivalence as abelian categories just means having the same number of irreducible representations (which are the generators). It’s less obvious that the “symmetric” qualifier is important, but there are examples where these are different.

So then one gets Morita equivalence for groupoids \mathbf{G} from the categories Rep(\mathbf{G}) in this standard way. One point here is that, whereas representations of groups are actions on vector spaces, representations of groupoids are actions on vector bundles E over the space of objects of \mathbf{G} (call this M). So for A morphism from x \in M to y \in M, the representation gives a linear map from the fibre E_x to the fibre E_y (which is necessarily iso).

The above paper by Landsman is nice in that it defines this concept for several different categories, and gives the corresponding versions of a theorem showing that this Morita equivalence is either the same as, or implied by (depending on the case) equivalence in a certain bicategory. For Lie groupoids, this bicategory \mathbf{LG} has Lie groupoids for objects, certain bibundles as morphisms, and bibundle maps as 2-morphisms – the others are roughly analogous. The bibundles in question are called “Hilsum-Skandalis maps” (on this, I found Janez Mrcun’s thesis a useful place to look). This \mathbf{LG} does in this context essentially what the bicategory of spans does for finite groupoids (many simplifying features about the finite case obscure what’s really going on, so in some ways it’s better to look at this case).

The general phenomenon here is the idea of “strong Morita equivalence” of rings/algebras/groupoids R and L. What, precisely, this means depends on the setting, but generally it means there is some sort of interpolating object between R and L. The paper by Landsman gives specifics in various cases – the interpolating object may be a bimodule, or bibundle of some sort (these Hilsum-Skandalis maps), and in the case of discrete groupoids one can derive this from a span. In any case, strong Morita equivalence appears to amount to an equivalence internal to a bicategory in which these are the morphisms (and the 2-morphisms are something natural, such as bimodule maps in the case of rings – just linear maps compatible with the left and right actions on two bimodules). In all cases, strong Morita equivalence implies Morita equivalence, but only in some cases (not including the case of Lie groupoids) is the converse true.

There are more details on this in my slides, and in the references above, but now I’d like to move on…

Toposes and Groupoids

Peter Oman gave the most recent talk in our seminar, the motivation for which is to explain how the idea of a topos as a generalization of space fits in with the idea of a groupoid as a generalization of space. As a motivation, he mentioned a theorem of Butz and Moerdijk, that any topos \mathcal{E} with “enough points” is equivalent to the topos of sheaves on some topological groupoid. The generalization drops the “enough points” condition, to say that any topos is equivalent to a topos of sheaves on a localic groupoid. Locales are a sort of point-free generalization of topological spaces – they are distributive lattices closed under meets and finite joins, just like the lattice of open sets in a topological space (the meet and join operations then are just unions and intersections). Actually, with the usual idea of a map of lattices (which are functors, since a lattice is a poset, hence a category), the morphisms point the wrong way, so one actually takes the opposite category, \mathbf{Loc}.

(Let me just add that as a generalization of space that isn’t essentially about points, this is nice, but in a rather “commutative” way. There is a noncommutative notion, namely quantales, which are related to locales in rather the same way lattices of subspaces of a Hilbert space H relate to those of open sets in a space. It would be great if an analogous theorem applied there, but neither I nor Peter happen to know if this is so.)

Anyway, the main theorem (due to Joyal and Tierney, in “An Extension of the Galois Theory of Grothendieck” – though see this, for instance) is that one can represent any topos as sheaves on a localic groupoid – ie. internal groupoids in \mathbf{Loc}.

The essential bit of these theorems is localic reflection. This refers to an adjoint pair of functors between \mathbf{Loc} and \mathbf{Top}. The functor pt : \mathbf{Loc} \rightarrow \mathbf{Top} gives the space of points of a locale (i.e. atomic elements of the lattice – those with no other elements between them and the minimal elements which corresponds to the empty set in a topology). The functor Loc : \mathbf{Top} \rightarrow \mathbf{Loc} gives, for any topological space, the locale which is its lattice of open sets. This adjunction turns out to give an equivalence when one restricts to “sober” spaces (for example, Hausdorff spaces are sober), and locales with “enough points” (having no other context for the term, I’ll take this to be a definition of “enough points” for the time being).

Now, part of the point is that locales are a generalization of topological space, and topoi generalize this somewhat further: any locale gives rise to a topos of sheaves on it (analogous to the sheaf of continuous functions on a space). A topos \mathcal{E} may or may not be equivalent to a topos of sheaves on a locale: i.e. \mathcal{E} \simeq Sh(X) might hold for some locale X. If so, the topos is “localic”. Localic reflection just says that Sh induces an equivalence between hom-categories in the 2-categories \mathbf{Loc} and \mathbf{Topoi}. Now, not every topos is localic, but, there is always some locale such that we can compare \mathcal{E} to Sh(X).

In particular, given a map of locales (or even more particularly, a continuous map of spaces) f : X \rightarrow Y, there’s an adjoint pair of inverse-image and direct-image maps f^*: Sh(Y) \rightarrow Sh(X) and f_* : Sh(X) \rightarrow Sh(Y) for passing sheaves back and forth. This gives the idea of a “geometric morphism” of topoi, which is just such an adjoint pair. The theorem is that given any topos \mathcal{E}, these is some “surjective” geometric morphism Sh(X) \rightarrow \mathcal{E} (surjectivity amounts to the claim that the inverse image functor f^* is faithful – i.e. ignores no part of \mathcal{E}). Of course, this f might not be an equivalence (so \mathbf{Topoi} is bigger than \mathbf{Loc}).

Now, the point, however, is that this comparison functor means that \mathcal{E} can’t be TOO much more general than
sheaves on a locale. The point is, given this geometric morphism f, one can form the pullback of f along itself, to get a “fibre product” of topoi: Sh(X) \times_{\mathcal{E}} Sh(X) with the obvious projection maps to Sh(X). Indeed, one can get Sh(X) \times_{\mathcal{E}} Sh(X) \times_{\mathcal{E}} Sh(X), and so on. It turns out these topoi, and these projection maps (thought of, via localic reflection, as locales, and maps between locales) can be treated as the objects and structure maps for a groupoid internal to \mathbf{Loc}. So in particular, we can think of Sh(X) \times_{\mathcal{E}} Sh(X) as the locale of morphisms in the groupoid, and Sh(X) \times_{\mathcal{E}} Sh(X) \times_{\mathcal{E}} Sh(X) as the locale of composable pairs of morphisms.

The theorem, then, is that \mathcal{E} is related to the topos of sheaves on this localic groupoid. More particularly, it is equivalent to the subcategory of objects which satisfy a descent condition. Descent, of course, is a huge issue – and one that’s likely to get much more play in future talks in this seminar, but for the moment, it’s probably sufficient to point to Peter’s slides, and observe that objects which satisfy descent are “global” in some sense (in the case of a sheaf of functions on a space, they correspond to sheaves in which locally defined functions which match on intersections of open sets can be “pasted” to form global functions).

So part of the point here is that locales generalize spaces, and toposes generalize locales, but only about as far as groupoids generalize spaces (by encoding local symmetry). There is also a more refined version (due to Moerdijk and Pronk) that has to do with ringed topoi (which generalize ringed spaces), giving a few conditions which amount to being equivalent to the topos of sheaves on an orbifold (which has some local manifold-like structure, and where the morphisms in the groupoid are fairly tame in that the automorphism groups at each point are finite).

Coming up in the seminar, Tom Prince will be talking about an approach to this whole subject due to Rick Jardine, involving simplicial presheaves.

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