Two Kinds of Categorification And Other Stuff

I’d just like to post something about a conceptual clarification that came up recently. Last week I gave the first of a couple of talks in the Algebra seminar in our department, about the ideas of structure types and stuff types, more or less as outlined in this paper which I put out a couple of years ago. It summarizes and traipses a little way beyond the matter of the 2003/2004 quantum gravity seminar at UCR, whence on this paper by John Baez and Jim Dolan, and even further back on work by André Joyal, particularly in the paper “Foncteurs analytiques et espèces de structures“, which regrettably doesn’t seem to be available either online. (I gave a blackboard version of the talk, but it was an expanded form of this one hour version.)

(Semantic side note: these espèces de structures are often referred to as “combinatorial species” in English. This is the more common translation than “structure type”, but unfortunately, it doesn’t capture the modifier “de structures“, instead choosing the more generic “combinatorial”, which makes it hard to distinguish “structure types” from “stuff types” in the Baez-Dolan sense. Also, “species” is probably over-specific as a translation of “espèces” in a way that “type” isn’t. The generic sense of “species” as “a kind of” in English is a bit recherché.)

In any case, what I’m interested in this post is the sense in which stuff types give a “categorification” of a vector space. In a nutshell, a stuff type is a groupoid over (the groupoid whose objects are finite sets, and whose morphisms are bijections). That is, it’s really a functor , which we call the “underlying set” functor. For example, consider the groupoid of all binary trees, where the underlying set is the set of nodes (or, a different example, the set of leaves). Any isomorphism between two such trees gives a bijection between the underlying sets, so this actually is a functor. Or one could take the functor , where the “underlying set” of a pair of sets is just , and likewise for morphisms. (Notice that different bijections “up above” in the bundle may give the same bijection “below” - in cases where this doesn’t happen, we have one of Joyal’s “structure types”). In some ways, it’s better to think of it as a bundle of groupoids - one fibre over each object in

The thing is, that map gives an invariant for objects in the category of groupoids, but not a complete invariant. Unlike, say, finite sets and the natural numbers. Natural numbers correspond exactly to isomorphism classes of sets - not so with groupoid cardinalities. So there’s an equivalence relation, and reducing the object set modulo that equivalence relation gives a structure - but it’s not the minimal throwing-away of information about objects that taking isomorphism classes would be.

But in any case, it’s the whole category of groupoids (over ) which gets “decategorified” down to a vector space, in that world. There is a concept of groupoid cardinality, which is given by Baez and Dolan in the paper above, and which is also linked to Tom Leinster’s definition of the Euler characteristic of a category. This adds up, over all the isomorphism classes of objects, , the reciprocals of the sizes of automorphism groups. Reasons why this is the nicest concept of cardinality are described in some of those references, but all that really matters here is that groupoid cardinality gets along with disjoint unions of groupoids (corresponding to sums of cardinalitys), and products of groupoids (which get the product of the two cardinalities). That is, the categorical coproduct and product, respectively, define operations on the set of cardinalities!

In particular, taking stuff types - groupoids over , we can take the cardinalities of the fibres over sets of each size giving the coordinate in a vector. So then is, the slice category has this “cardinality” on objects into a set, and the structure of the category gives well-defined operations on this set, turning it into a vector space. In fact, there’s an operation (weak pullback) which makes it an inner product space. (To make this work in complex cardinalities takes some fudging with phases in , but it can be done.)

The details are interesting, and I’m coming back to looking at some of this again, but what I want to point out at the moment is a more fundamental point, which has to do with the offhanded use of the handy, but imprecise, term “categorify”. With the category of (-) stuff types, we have a category with a “decategorification” map that compresses it into a vector space. This sure sounds like a “categorified vector space”. In fact, this seems to be what people who hear the term “categorification” often want it to mean: I look for a categorification of mathematical object X by finding a category which, secretly, looks like X.

The problem is, there’s another concept attached to the phrase “categorified vector space”, namely that of 2-vector space in the sense of Kapranov and Voevodski, as discussed, say, here. There’s a different level of abstraction at work here. The specific category of stuff types provides a categorification (if that indeed is the right word to use) of a specific vector space. The concept of a KV 2-vector space categorifies the concept of a regular vector space in a particular way: putting “additive” structure on objects, and “C-linear” structure on morphisms. (The Baez-Crans version does the same job in a different way).

You don’t think of a specific KV 2-vector space “decategorifying to” a specific vector space. Indeed, just taking the “minimal” equivalence relation - isomorphism classes of objects - what we get from a KV 2-vector space is more like an -module (over a rig, not a ring). Basically, 2-vectors have components which are vector spaces, and therefore classified by their dimension. The relationship between THIS kind of 2-vector space and the non-categorified concept is that real vector spaces show up as the -sets in a KV 2-vector space.

Elucidating exactly what’s going on with these two forms of categorification would be nice - perhaps somebody’s done it, but if so, I don’t know who. I also don’t know any nice conditions that tell you when you have a “category that can be mistaken for a vector space”, like stuff types: a good characterization of these things would be nice. Or again: both versions of “categorification” of vector space have special relationships to groupoids - but of two very different natures (in one, the groupoids can be interpreted as 2-vectors - in the other, there are whole 2-vector spaces associated to groupoids). Just a coincidence?

Another possibility that comes to mind would be to form some kind of hybrid structure - where the “vector spaces” which show up in the -sets in a KV 2-v.s. are secretly this fake-vector space type of category. Since both types seem to have physics-y ambitions, such a setup that combines both approaches is appealing, rather than a muddled and confusing competition for the term “categorification”.

I don’t have a good ending to this story, which is why this is a blog, not a book.

 

n-group Representation Theory - part 3: Rep(Poinc)

I’m going to be giving a talk on extended TQFT stuff and quantum gravity at Perimeter Institute next thursday, and then in mid-March I’ll be heading to UC Davis to give the same/similar talk for the String Theory and Quantum Gravity seminar being run by Derek Wise. So I have a bunch of things on my mind right now. However, before heading to Davis, I wanted to go back and look at some of the stuff Derek has done having to do with Cartan geometry, which I was following somewhat at the time, and blog about it a bit here. Before that, I’d like to wrap up this presentation of the talks I gave here about representation theory of the Poincaré 2-group, .

As a side note, thanks to Dan for pointing out these notes on representations of the (normal, uncategorified) Poincaré group, including some general comments on representations of semidirect products. It’s interesting to consider how this relates to the more general picture of 2-group representations - but I won’t do so here and now.

…

In Part 1 I talked about what representations 2-categories of 2-groups are like in general, and in Part 2 a fairly concrete description of . Here I’ll wrap up by summarizing the results of Crane and Sheppeard about what looks like concretely.

It has three parts: the objects are representations (also known as functors from as a 2-category with one object, into ); the morphisms are 1-intertwiners (a.k.a. natural transformations) between reps; and the 2-morphisms are 2-intertwiners (a.k.a. modifications) between 1-intertwiners.

1) Representations: A functor

will pick out some measurable space for the lone object of the 2-group - or rather, , the 2-vector space of all measurable fields of Hilbert spaces on . (This is a matter of taste since to know the one is to know the other.) Then for the morphisms and 2-morphisms of we get, respectively, 2-linear maps from to itself, and natural transformations between them.

The morphisms of are just the group in the crossed-module picture I described in Part 2. For the usual Poincaré 2-group, this is . For each such element, we’re supposed to get an invertible 2-linear map from to itself - that is, a measurable field of Hilbert spaces on (together with measures to do “matrix multiplication” with by direct integrals). This can only be invertible if the only Hilbert spaces which appear are 1-dimensional (since these maps compose by a “matrix multiplication” involving direct sums of tensor products of the components - and the discreteness of dimensions means that if any dimension is higher than 1, you’ll never get back the identity).

So any representation turns out to give what amounts to an action of on - the component is if and 0 otherwise. An irreducible representation gives an with a transitive action (otherwise, you can decompose it into orbits, each of which corresponds to a subrepresentation). Crane and Sheppeard classify several kinds of these, associated to various subgroups of , but an easy example would be a mass shell in Minkowski space - a sphere or hyperboloid (depending on ) that is the full orbit of some point under rotations and boosts (a “mass shell” because it gives all the possible momenta for a particle of a given mass, as seen by an observer in some inertial frame).

The 2-morphism part of gives a homomorphism from at each of these points. Now, one-by-one matrices of complex numbers are just complex numbers, so what we have here is a character of - at each point on . To be functorial, this has to be done in an equivariant way (so that acting on the point by affects the character by acting on by the same ).

2) 1-Intertwiners:

If representations and correspond to actions of on spaces and respectively, with characters , then what is a 1-intertwiner ? Remember from Part 1 that it’s a natural transformation: to the object of it assigns a specific 2-linear map

To each (object of it gives a transformation

This is a specified map which replaces the naturality square in the old definition of an intertwiner. It has to make a certain “pillow” diagram commute (Part 1).

Now, back in the posts on 2-Hilbert spaces, I explained that a 2-linear map is given by some field of Hilbert spaces on (a “matrix” of Hilbert spaces, though of course needn’t be finite), along with a family of measures on indexed by (which allow us to do integration when doing the sum in “matrix multiplication”). The transformations also can be written in components, so that

(Note this uses the two actions given by on - one forward, and one backward. This is the current form of what, in uncategorified representation theory, would be a naturality condition.)

What does this all amount to? One way to think of it is as a representation of itself! In particular, it’s a representation on the direct sum of all the Hilbert spaces which appear as components of . This is since the maps given by the have to satisfy a condition which says that composition is preserved (as long as you’re careful about indexing things):

To get a representation of the group, we can say that elements shuffle vector spaces over points in by the action of and then act within vector spaces by . So then has both intertwiner-like and representation-like properties.

The “intertwiner-ness” of has to do with how it interpolates between two actions on by turning them into an action on the product - but it also has some “representation-ness”, by giving this action of a (semidirect product) group on a big vector space.

3) 2-intertwiners

If a 1-intertwiner can be thought of as a representation of , it shouldn’t be too surprising that a 2-intertwiner between 1-intertwiners ends up being an intertwiner between the associated representations. If 1-intertwiners have some qualities of both reps and intertwiners, the 2-intertwiners are more single-minded.

In particular, a 2-intertwiner assigns to the only object of a 2-morphism in (that is, a field of linear maps between the vector spaces which are the components of ), which satisfies some “pillow” diagram. When we form the big rep. by taking a direct integral of all those spaces, the field of linear maps turns into one big linear map, and the diagram it satisfies just collapses into the condition that it be an intertwiner.

So the representation theory of this interesting 2-group looks a lot like the representation theory of the group of 2-morphisms. The extra structure involving actions on measurable spaces by would be mostly invisible if you just thought about irreducible reps of the group, since the space would be just a single point.

This phenomenon where a lower-order structure turns up in some form at the top level of morphisms of its categorified version has cropped up before in this blog - namely, when extended TQFT’s turn out to contain normal TQFT’s in individual components. In these examples, categorification is less a matter of building more floors “on top” of structures we already know, as “higher morphisms” suggests, but excavating additional floors of subbasement - interpreting what were objects as morphisms.

 

n-group Representation Theory - (part 2: the Poincaré 2-group)

It’s been a while since I wrote the last entry, on representation theory of n-groups, partly because I’ve been polishing up a draft of a paper on a different subject. Now that I have it at a plateau where other people are looking at it, I’ll carry on with a more or less concrete description of the situation of a 2-group. For higher values of , describing things concretely would get very elaborate quite quickly, but interesting things already happen for . In particular, the case that I gave the talk about, a while back, was mostly the Poincaré 2-group, since this is the one Crane, Sheppeard, and Yetter talk about, and probably the one most interesting to physicists.  It was first described by John Baez.

So what’s the Poincaré 2-group? To begin with, what’s a 2-group again?

I already said that a 2-group is a 2-category with only one object, and all morphisms and 2-morphisms invertible. That’s all very good for summing up the representation theory of as I described last time, but it’s sometimes more informative to describe the structure of concretely. A good tool for doing this is a crossed module. (A lot more on 2-groups can be found in Baez and Lauda’s HDA V, and there are some more references and information in this page by Ronald Brown, who’s done a lot to popularize crossed modules).

A crossed module has two layers, which correspond to the morphisms and 2-morphisms of . These can be represented as , where is the group of morphisms in , consists of the 2-morphisms ending at the identity of (a group under horizontal composition).

There has to be an action of on (morphisms can be composed “horizontally” with 2-morphisms), and a map (which picks out the source of the 2-morphism). The data have to fit together a certain way, which amounts to giving the axioms for a 2-category.

A handy way to remember the conditions is to realize that the action and the injection give ways for elements of to act on each other and for elements of to act on each other. These amount to doing first and then or vice versa, and both of these must amount to conjugation. That is:

and

Both of these are simplified in the case that maps everything in to the identity of - in this case, can be interpreted as the group of 2-automorphisms of the identity 1-morphism of the sole object of . In this case, by the Eckmann-Hilton argument (the clearest explanation of which that I know being the one in TWF Week 100) it turns out that has to be commutative, so the first condition is trivial since , and the second is trivial since it follows from commutativity. This simpler situation is known as an automorphic 2-group.

In any case, given a 2-group represented as a crossed module, automorphic or not, the collection of all morphisms can be seen as a group in itself - namely the semidirect product , which is to say with the multiplication . “What?” you may ask, or maybe “Why?”

Maybe a concrete example would help, since we’d like one anyway: the Poincaré 2-group, which is an automorphic 2-group. There are versions of various signatures , in which case , and .

The group , then, consists of metric-preserving transformations of Minkowski space with the metric of signature - rotations and boosts (if any). The (abelian) group consists of translations of this space - in fact, being a vector space, it’s just a copy of it. Between them, they cover the basic types of transformation. Thinking of the translations as having a “projection” down to the identity rotation/boost may seem a bit artificial, except insofar as translations “don’t rotate” anything. More obvious is that rotations or boosts act on translations: the same translation can look differently in rotated/boosted coordinate systems - that is, to different observers.

So where does the Poincaré group come in? It’s the group of all metric-preserving transformations of Minkowski space, and is built from these two types: but how?

Well, the vector space is the group of transformations of the identity Lorentz transformation , since the map is trivial. But suppose that there is another copy of over each point in . Then we have the set of points , but notice that to talk about this as a group, we’d want a way to act on an element of one copy of over by another over . The obvious way is to just treat the whole set as a product of groups, but this misses the fundamental relation between and , which is that can act on , just as morphisms can act on 2-morphisms by “whiskering with the identity”. (Via Google books, here is the description of this in MacLane’s Categories for the Working Mathematician).

Concretely, this is the fact that there is a sensible way for both parts of to affect the , so we can say (using additive notation for translations, since they’re abelian). The point is that the first rotation we do, , changes coordinates, and therefore the definition of the translation .

So that’s the construction of the Poincaré group from the Poincaré 2-group. What would be nice would be to have some clear description of some higher analog of Minkowski space where it makes sense to say the Poincaré 2-group acts as a 2-group. I don’t quite know how to set this up, but if anyone has thoughts, it would be interesting to hear them.

One reason is that, when describing representations of the 2-group, there’s an important role for spaces (or at least sets) with an action of the group - which raises questions like whether there’s a role for 2-spaces with 2-group actions in representation theory of higher -groups. Again - I don’t really know the answer to this. However, in Part 3 I’ll describe concretely how this works for 2-groups, and particularly the Poincaré 2-group.

 

n-Group Representation Theory - Part 1

Recently I finished up my series of talks on 2-Hilbert spaces with a description of the basics of 2-group representation theory, and a little about the special case of the Poincaré 2-group. The main sources were a paper by Crane and Yetter describing 2-group representations in general, and another by Crane and Sheppeard. The Poincaré 2-group, so far as I know, was first explicitly mentioned by John Baez in the context of higher gauge theory. It’s an example of a kind of 2-group which can be cooked up from any group and abelian group , and which is related to the semidirect product .

One reason people are starting to take an interest in the representation theory of the Poincaré 2-group is that representations of the Poincaré group (among others) and intertwiners between them play a role in spin foam models for field theories such as BF theory, various models of quantum gravity, and so on. Some of these, turn up naturally when looking at TQFT’s, and generalizations of these, which is how I got here. Extending this to 2-groups gives a richer structure to work with. (Whether the extra richness is useful is another matter).

Before getting into more detail, I first would like to take a look at representation theory for groups from a categorical point of view, and then see what happens when we move to -groups - that is, when we categorify.

To begin with, we can think of a representation of a group as a functor. The group can be thought of as a category with one object and all morphisms invertible - so that the group elements are morphisms, and the group operation is composition. In this case, a representation of the group is just any functor:

since this assigns some one vector space (the representation space, ) to the one object of , and a linear map to each morphism of (i.e. to each group element) in a way consistent with composition. The nice thing about this point of view is that knowing a little category theory is enough to suggest one of the fundamental ideas of representation theory, namely intertwining operators (”intertwiners”). These are natural transformations between functors. This is the idea to categorify.

The point is that functors can be organized into a structure , and this is most naturally seen as a category, not just a set. The category of representations of is usually called , but seen as a category of functors, it is a general case of a category $hom(C,D)$ of functors from category to category . Let’s look at how this is structured, then consider what happens with higher dimensional categories. There seems to be a general pattern which one can just begin to see with 1-categories:

• a functor is a map between categories, assigning
  • to each -object a corresponding -object
  • to each -morphism a corresponding -morphism

  in a way compatible with composition and identities

• a natural transformation between functors assigns
  • to each -object a -morphism

  making a naturality square commute for any morphism in :

(In the case where the functors are representations of a group, this is an intertwiner - a linear map which commutes with the action of the group on .)

The pattern is a little more obvious for 2-categories:

• a 2-functor is a map between 2-categories, assigning
  • to each -object a corresponding -object
  • to each -morphism a corresponding -morphism
  • to each -2-morphism a corresponding -2-morphism

  in a way compatible with composition and identities

• a natural transformation between 2-functors assigns
  • to each -object a -morphism
  • to each -morphism a -2-morphism

  making a generalized naturality square commute for any 2-morphism in (where ):

• a modification (what I might have named a “2-natural transformation” or similar) between natural transformations assigns
  • to each -object a -2-morphism

  making a similar diagram commute (OK, well, it appears on p11 of John Baez’ Introduction to n-Categories, but I don’t have a web-ified version of it - I haven’t learned how to turn LaTeX diagrams into handy web format).

…

In the case where is a 2-group - a 2-category with one object and all -morphisms invertible, and , then we have here the (quite abstract!) definition of a representation, an 1-intertwiner between representations, and a 2-intertwiner between 1-intertwiners.

It’s not too hard to see the pattern suggested here - a “-natural transformation” assigns a -morphism in to an object in the -category , and a -morphism in to each -morphism in . This morphism fits into a diagram filling a commutative diagram which was the coherence law for the top dimensional transformation for -categories. (I might point out that if I were to come up with terminology for these things from scratch, I’d try to build in some flexibility from the start. Instead of “functor”, “natural transformation”, and “modification”, I’d have used terms more analogous to the terminology for morphisms. Probably I’d have used, respectively, “1-functor”, “2-functor”, “3-functor”, and so on. This is already a problem, since these terms are in use with a different meaning! Instead, I’ve used “natural -transformation”.) It’s less easy to say what, explicitly, the various coherence laws should be at each stage, except that there should be an equation between the composites of (a) the -morphisms in an -natural transformation with (b) the two possible images of any chosen lower dimensional morphisms.

There is a lot of useful information out there about various forms of -categories, such as the Illustrated Guidebook by Cheng and Lauda, and Tom Leinster’s “Higher Operads, Higher Categories” (also in print). They’re a little less packed with information on functors, natural transformations, and their higher generalizations. I don’t know a reference that explains the generalization thoroughly, though. If anyone does know a good source on this, I’d like to hear about it. Probably this is somewhere in the work of Street, Kelly, maybe Batanin (whose definition of -category is the one implicitly used here) or others, but I’m not familiar enough with the literature to know where this is done.

These generalizations of functors and natural transformations to higher -categories describe what functor -categories are like. When written down and decoded, these definitions can be turned into a concrete definition of representations and the various -intertwiners involved in the representation theory of -groups.

However, next time I’ll take a look at some of what is known in the slightly more down to earth world where .

 

2-Hilbert Spaces - pt 2

So I gave a little talk shortly before leaving London for Christmas. I had mostly written it up, but then I’ve been on the road for a while in Montreal, Ottawa, and Calgary, without consistent net access. However, now I have a moment to put this up.

The talk carried on from the previous one I described last post. It began to move in the direction of representation theory of 2-groups on 2-vector spaces and 2-Hilbert spaces, but didn’t get that far. This was partly because I had to finish describing what 2-linear maps and 2-maps look like for such spaces, and then because I had to explain about 2-groups and give some examples. I’ll say more about the representation-theory stuff in January. But here I’ll just summarize at least the rest of the description of the category , and also by describing 2-linear maps and so forth. Then I’ll comment a little more philosophically about what these are about.

So I explain how there’s a 2-vector space (in some suitable sense, not the KV sense) of measurable fields of Hilbert spaces on a space , analogous to the vector space of complex functions on a space. Also similarly, given a measure on , we get an inner product. Then there’s a (2-)Hilbert space where this inner product is always well-defined (as a complex scalar, or a genuine Hilbert space - which is the equivalent of a scalar at the next level up).

Well, then Crane and Yetter’s paper describes constructively how to get 2-linear maps (additive, linear functors) between such 2-vector spaces. They don’t as far as I can see, show that all functors arise this way, but it seems likely. The way is to say you get a functor from:

1) A measurable field of Hilbert spaces (this is similar to the linear maps between KV 2-vector spaces, which are like matrices of vector spaces)

2) A -indexed family of measures on - these give you the measures you need to do the “inner product” involved in “matrix multiplication” at each (note that this stuff is only well-defined up to sets of measure 0, as usual). So we have, on objects:

and a related expression for morphisms, using the identity on .

It’s probably worth pointing out that the measures on are used in the direct integral here, and so their only real role is to define the inner product on - the underlying vector space at each point in the new field would be the same no matter what these measures were (up to the fact that if the resulting inner product is degenerate, we need a quotient space where it’s not).

So this gives 2-linear maps, which are functors. Natural transformation between these functors come from the fact that is itself a category, and in fact a 2-vector space in the sense we’re using here ( is “enriched over itself”). So morphisms between these fields of Hilbert spaces basically amount to fields of bounded operators as usual. This is actually not quite right, because we need to account for the different measures: basically, you use a measure which is the geometric mean of those associated to source and target - check out Crane and Yetter’s paper if you want the details.

That finishes up a summary of how 2-Hilbert spaces work. The next thing I’ll be talking to our group about is how to use these for a categorified form of representation theory.

…

But first, what is the point of all this stuff? Not yet asking about representation theory in this setting - why is it interesting enough to bother? It’s worth thinking about what a categorification of a Hilbert space is supposed to be. In particular, let’s try locating them in the world of quantum mechanics.

A quantum system is usally portrayed as having states represented by vectors in a Hilbert space. The only things you can “do” to states involve applying operators to the whole space: project them into subspaces, “rotating” them by some unitary evolution operator, and so on. In a 2-Hilbert space, states, or “2-vectors” are objects in a category, which means there are not only these “macro” operations on the whole space, but also morphisms between any two states you pick. In fact, this is the source of the inner product on a 2-Hilbert space - there is a Hilbert space (in the usual sense) of morphisms between any two states, and in the world of 2-Hilbert spaces, this is the equivalent of a scalar.

In QM, the inner product is telling you an amplitude to observe a system in state if it was set up in state - this is saying something about “how related” is to . The categorifed picture saying this is just makes more explicit what kind of relationship this is.

Now, if you happen to pick the same vector to start and end with, considering , what this is saying is that there’s some bunch of “symmetry operations” on a state. (Taking just the invertible ones gives an actual symmetry group for a given state.) This is saying that “state 2-vectors” have some internal degrees of freedom. Their amplitudes give a measure of how many such degrees of freedom there are.

The fact that a 2-Hilbert space is described as an enriched category means that the usual picture of a quantum system returns when you look in individual components of a state 2-vector. In particular, the coefficients of a 2-state vector can be thought of as Hilbert spaces representing a system in that particular component. So, for instance, part of the big project I’m describing in these notes is to depict quantum gravity (at least in 3 dimensions) as an extended TQFT, which represents a physical system with these 2-Hilbert spaces. A 2-state vector here describes the situation on a boundary of space - matrix elements of a 2-linear map are Hilbert spaces of connections on a given manifold interpolating between chosen boundary states. Natural transformations between 2-linear maps are what give amplitudes for spacetimes joining such slices of space.

So what is a state 2-vector? All these properties should fit together into some nice scheme: classical configurations can exist in a “2-state” in some kind of superposition, where each configuration gets its own internal degrees of freedom. The inner product emerges naturally from this, considering morphisms between 2-states. Every morphism between 2-states has to respect the classical configurations, giving for each one a map between the internal spaces associated to it in the two 2-states. Is there a more elegant way to sum this up? Probably so, but at the moment I don’t quite see how to put it.

However: next time, I’ll carry on with some representation theory.

 

2-Hilbert Spaces - pt 1

So one of the missing pieces in some of what I’ve been posting about recently is a discussion of 2-Hilbert spaces, and particularly the kind that categorify infinite dimensional Hilbert spaces. Part of the issue with these is that there are a number of ways of looking at them, and how these all fit together isn’t quite as clearly developed as with mere finite-dimensional 2-vector spaces.

I gave a little talk about this to our group at UWO, leading up to representation theory on 2-Hilbert spaces, which touches - potentially - on some of the stuff with spin foams that Wade and Igor are working on especially. It’s also a part of the project of trying to work out how the approach to extended TQFT’s I’ve described a bit should work for an infinite gauge group - in particular, a Lie group. The descriptions of these theories which I’ve given describe functors valued in , the 2-category of Kapranov-Voevodsky 2-vector spaces. These had a basis indexed by conjugacy classes in , and representations of their stabilizer subgroups - and for, say, , this is infinite.

Furthermore, a good categorification of a quantum field theory should use something deserving the name 2-Hilbert space (unless you prefer the -algebra approach to quantum theory which doesn’t pick a specific representation on a Hilbert space - this would also be interesting to categorify). So it needs some structure analogous to an inner product, complex conjugation, and so no. There are a number of concepts that stab in this direction, so in my talk I tried to summarize the main points.

An early reference on this is HDA II by John Baez, which defines 2-Hilbert spaces in a nice axiomatic way - abelian categories, enriched over , with a *-structure, satisfying some properties, etc. There are a few provisos: the version of things are presumed to be enriched over includes only finite dimensional Hilbert spaces. On the other hand, there’s no assumption - as there is for Kapranov-Voevodsky 2-vector spaces - that the category itself is finitely generated by simple objects. In other words, there’s no assumption of finite dimensionality for the 2-Hilbert space itself, but there is for its component Hilbert spaces. Then there’s a classification theorem which says what 2-Hilbert spaces in this sense are like. If they are finitely generated, then in particular they happen to be KV 2-vector spaces. But there is more structure, corresponding to two features of Hilbert spaces: the inner product, and complex conjugation.

The interesting thing about the inner product is that every KV 2-vector space is automatically equipped with one. Since it needs to be a map , the obvious choice is , which takes a pair of 2-vectors and gives a vector space - namely, the one containing all morphisms between these two objects. In , the components of a 2-vector are themselves inner-product spaces, so we have a little extra structure. It turns out this has all the properties it needs to be a categorified inner product. As for the equivalent of complex conjugation, the categorified version is just adjunction - it leaves objects as they are, but turns morphisms into their (componentwise, vector-space) adjoints. This process has some important properties, such as being an involution (like conjugation) and so on. This makes into a *-category.

There’s another possible approach to the subject, or a closely related subject, is described by David Yetter in this paper on measurable categories, and which Crane and Yetter use to support representations of 2-groups, the way Hilbert spaces can support representations of groups. This is a more concrete, constructive approach - like describing spaces of complex functions on a topological space, rather than giving an axiomatic definition of a Hilbert space. Actually, what they describe is more like the space of measurable functions on a space. These are measurable fields of Hilbert spaces on a measurable space - such a field defines (a) a Hilbert space at each point, and (b) a space of “measurable sections”, namely ways of picking a vector in the space at each point which are considered measurable. (There are some properties, like the fact that the function giving the local norm of these vectors at each point is measurable, plus some closure-type properties.)

Well, that’s measurable. Given a measure, so you can do integration, you can define something like spaces. Integration works by means of the direct integal, which produces not a scalar, but a Hilbert space; in this kind of categorification, takes the role of . The way this works is that the direct integral

as a vector space is just the whole space of measurable sections. The inner product of sections is

So integrability of a measurable field means not finiteness, per se (which we think of as saying that an inner product gives a well defined map ), but that this direct integral gives an object of (so the inner product integral should be finite, for instance, but also the space of measurable sections needs to be complete in the norm from this inner product). There is clearly a relationship between this way of describing an inner product and the way of describing it as a “hom-space”.

Some things are less clear… This gives a construction for how to get an “infinite dimensional 2-Hilbert space”. There doesn’t seem to be a known classification theorem here analogous to the one for KV 2-vector space, saying that this construction describes all “2-Hilbert spaces”. In fact, a general abstract definition of this concept seems to be a bit trickier than in the finite-dimensional case, and Crane and Yetter don’t really address it in their papers. One would hope that given a nice infinite-dimensional version of the usual definition of a 2-Hilbert space, this type would turn out to be generic.

Another question I’d like the answer to is - can one get one of these 2-Hilbert spaces from a (smooth, let’s say compact, probably) infinite groupoid, the way one can get 2-vector spaces (and, in particular, ones which can be made easily into finite-dimensional 2-Hilbert spaces) from an essentially finite groupoid? I think so - but there are some analysis issues to work out.  Assuming it works, this would be the right setting to support extended TQFT based on topological gauge theory with a Lie group like as gauge group.  (For some analysis reasons I may talk about later on, I only see reason to think this works with a compact group - but happily, that’s one right there!)

However, the question I’ll actually address in the second talk, which I’m giving on Friday, is how these are used for representation theory of 2-groups, since I’ve thought about that some, and some work has already been done with it - by, e.g. Crane, Yetter, Sheppeard, and also in some discussions I had the chance to participate in with John Baez, Laurent Freidel, Derek Wise, and Aristide Baratin (they are putting a paper together on the subject - as far as I know, not released yet).

 

A Problem Arises

In “The Fabric of Reality”, David Deutch gives a refutation of solipsism. I’m not entirely sure it works - all he really tries to do is to show that the difference between solipsism and realism is more nearly a mere semantic distinction than is generally assumed. But in any case, along the way, there’s an anecdote about a solipsist professor lecturing his (imaginary?) class merely to help him clarify his ideas. The idea being that, even if the imaginary students don’t really exist, it helps to clarify the professor’s own ideas by lecturing to them, answering questions, and so forth. In this view, you don’t really understand your own opinions - let alone justifiably believe in them - unless you’ve argued for them against a variety of possible criticisms. (J.S. Mill gave a defense of full-fledged freedom of speech, even for grossly offensive and even “dangerous” opinion, on this ground.)

I mention this because, when I told Dan about the blog, he seemed dubious about blogging as a way of communicating math. It’s certainly more solipsistic than a usenet newsgroup, or a mailing list. Those are channels devoted to a particular subject, with many participants. A blog, comments notwithstanding, is mainly a channel devoted to one voice, on many particular subjects. It’s true that half the point of communicating ideas is to get feedback on them from other people. You make your thinking part of one of those great processes like cathedral-building - ad-hoc, gradual, and (significantly) collective. Even so, relatively solipsistic channels are not entirely pointless.

To wit: by working through my theorems about transporting 2-vectors through spans - both for this blog, and for my talk at Groupoidfest, I discovered some problems. Nobody pointed them out, but discovering them was a consequence of approaching the material again from a new angle, with an audience in mind.

The problem is a conceptually important one - mistaking an n-dimensional space for a 1-dimensional space. I’m fairly sure, for various reasons, that the theorem that there is a 2-functor is still true, but the proof I have in my thesis (in the special case where the groupoids are flat connection groupoids on spaces) has a problem. Since that affects the Part 4 of “Spans and Vector Spaces” which I was going to post, I’ll put that off for a while as I get the proof straightened out.

Here is the issue in a nutshell, however:

The proof I have involves a construction of a functor by a particular method, which I’ve been describing in the last three posts. The final step I was going to describe involved what the contstruction does for 2-morphisms - spans between spans. (There is more to the proof, but the remainder is technical enough to be fairly unenlightening - basically, to be a 2-functor, there need to be specified natural isomorphisms replacing the equations for preserving identities and composition in the definition of a functor, and these have to obey some equations which need to be checked.)

The construction given in my thesis is supposed to give a way to take a span of spans of groupoids, and give a natural transformation between a pair of 2-linear maps. But a 2-linear map can be written as a matrix of vector spaces, and a natural transformation is then written as a matrix of linear operators which act componentwise. So one way to look at the problem is to construct a linear map between vector spaces from a span of groupoids.

That is, we have spans and . Picking basis objects for and (namely, objects and , plus representations of their automorphism groups) gives a subgroupoid of of , consisting of those objects which are sent to and under the maps in the span. It also gives a vector space which is built as a colimit of some vector spaces associated to these objects. Assuming is skeletal, this works out (as I described before) to for each of the in question. The same holds for .

Now suppose we have a span-of-spans making the obvious diagram commute. Then because of that commutation, we also have a span of groupoids over each of the choices of objects, and so then the question becomes, partly, how to get a linear map between the vector spaces we just constructed. If you have bases for all the vector spaces here, it’s not too bad: vectors can be seen as complex-valued functions on the basis. We can push these through the span just as we’ve been talking about in the last few posts here: first pull back a function along one leg by composition, then push forward along the other leg. The push-forward will involve a sum over some objects, and some normalizing factors having to do with the groupoid cardinalities of the groupoids in the span.

However, I won’t go too far into detail about this, because the construction I actually outlined doesn’t adequately specify the basis to use. In fact, it will really only work if all the vector spaces is one-dimensional. Then there is a basis for the combined space which just consists of all the objects . I’d hoped that Schur’s lemma (that intertwiners from to itself, or from to itself, have to be multiples of the identity) would get out of this problem, but I’m not sure it does. So there is a problem with the construction I was trying to use.

As I say, I’m fairly sure the theorem remains true - it’s just the proof needs fixing, which I don’t expect to be too hard. However, I’ll refrain from getting sidetracked until I know I have it worked out.

Instead, next time I’ll describe some of the things I learned at Groupoidfest 07 when I presented a talk on this stuff. (At first I was nervous, having discovered this flaw while preparing the talk - but then, a lot of people were talking about work-in-progress, so I don’t feel too bad now. Plus, the meeting was a lot of fun.)

 

Spans and Vector Spaces - pt 3

Well, I was out of town for a weekend, and then had a miserable cold that went away but only after sleeping about 4 extra hours per day for a few days. So it’s been a while since I continued the story here.

To recap: I first explained how to turn a span of sets into a linear operator between the free vector spaces on those sets. Then I described the “free” 2-vector space on a groupoid - namely, the category of functors from to . So now the problem is to describe how to turn a span of groupoids into a 2-linear map. Here’s a span of groupoids:

Here we have a span , of groupoids. In fact, they’re skeletal groupoids: there’s only one object in each isomorphism class, so they’re completely described, up to isomorphism, by the automorphism groups of each object. The object , for instance, has automorphism group , and the object has automorphism group . This diagram shows the object maps of the “source” and “target” functors and explicitly, but note that with each arrow indicated in the diagram, there is a group homomorphism. So, since the object map for sends to , that strand must be labelled with a group homomorphism . (We’re leaving these out of the diagram for clarity).

So, we want to know how to transport a -valued functor - along this span. We know that such a functor attaches to each a representation of on some vector space . As with spans of sets, the first stage is easy: we have the composable pair of functors , so “pulling back” to gives .

What about the other leg of the span? Remember back in Part 1 what happened when we pushed down a function (not a functor) along the second leg of a span. To find the value of the pushed-forward function on an element , we took a sum of the complex values on every element of the preimage . For vector-space-valued functors, we expect to use a direct sum of some terms. Since we’re dealing with functors, things are a little more complex than before, but there should still be a contribution from each object in the preimage (or, if we’re not talking about skeletal groupoids, the essential preimage) of the object we look at.

However, we have to deal with the fact that there are morphisms. Instead of adding scalars, we have to combine vector spaces using the fact that they are given as representation spaces for some particular groups.

To see what needs to be done, consider the situation of groupoids with just one object, so the only important information is the homomorphism of groups. These can be seen as one-object groupoids, which we can just call and . A functor between them is given by the single group homomorphism .

Now suppose we have a representation of the group on (so that and ). Then somehow we need to get a representation of which is “induced” by the homomorphism , :

This diagram shows “the answer” - but how does it work? Essentially, we use the fact that there’s a nice, convenient representation of any group , namely the regular representation of on the group algebra . Elements of are just complex linear combinations of elemenst of , which are acted on by by left multiplication. The group also has regular representation, on . These are the most easily available building blocks with which to build the “push-forward” of onto .

To see how, we use the fact that has a right-action of , and hence , by way of . An element acts on by right-multiplication by - and this extends linearly to . So we can combine this with the left action of on (also extended linearly from ) by taking a tensor product of with over . This lets us “mod out” by the actions of which are not detected in . The result, called the induced representation of , in turn gives us back a left-action of on . I’ll call this .

(Note that usually this name refers to the situation where is a subgroup of , but in fact this can be defined for any homomorphism.)

This tells us what to do for single-object groupoids. As we remarked earlier, if more than one object is sent to the same , we should get a direct sum of all their contributions. So I want to describe the 2-linear map, which I’ll now call which we get from the span above, thought of as in . Here and (where I’m now being more explicit that this whole process is a functor in some reasonable sense).

I have to say what does to a given 2-vector (what it does to morphisms between 2-vectors is straightforward to work out, since every operation we do is a tensor product or direct sum). Suppose we have is one. Then . We can now say what this works out to. At some object , we get (still assuming everything is skeletal for simplicity):

And this is a direct sum of a bunch of such expressions where is a basis 2-vector - i.e. assigns an irreducible representation to some one object, and the trivial rep on the zero vector space to every other. That allows this to be written as a matrix with vector-space components, just like any 2-linear map.

So the 2-linear map has a matrix representation. The indices of the matrix are the simple objects in and , which consist of a choice of (a) object in