Due to the rapid-fire the nature of the blogosphere (or, in deference to John Armstrong, the “Blathysphere”, or maybe “blathyscape”), my blog (“blath”) has been discovered before I expected, and in particular before I’ve had the chance to put anything very interesting in it. So here I’ll just say something about “coming attractions” – a sort of mid-level executive summary of the next batch of things I expect to be working and commenting on. Also possibly later on I should have a math post or two about some talks I saw recently.

Since I graduated at UCR in June, I haven’t had much chance to do any actual work – partly because I broke my wrist in a bike accident, and lost the use of my writing hand for six weeks. Between that and the hassle of moving, I wasn’t able to do much but some reading. Now that the cast is off, I’ve been getting back to work. The first “real” research-related post I expect to make will be an announcement that a (slightly) polished version of my dissertation, “Extended TQFT’s and Quantum Gravity” has been released on the preprint archive – hopefully this week. That in turn should kick off some descriptions of what’s inside as I get more into the process of turning it into some smaller, more digestible papers.

These will fall, at first, into three parts:

1) A paper which has already been posted as math.CT/0611930, describing how to get a “double bicategory” of cobordisms with corners, and from that, a bicategory. Here I explain how cobordisms are cospans of manifolds with boundary, so the new structures are double cospans of manifolds with corners, and how that works.

This may end up being two parts. One is a decription of Dominic Verity‘s notion of a “double bicategory”, an aside on how to interpret it as a special case of bicategories internal to $\mathbf{Bicat}$, and how to get one from double spans (functors $DS:\Lambda^2 \rightarrow C$). Marco Grandis has a pretty thorough description of these in this paper and its sequels, although our approaches are slightly different.

The second part has to do with how to apply this to cobordisms with corners (cobordisms between cobordisms) – also something Grandis discusses in the second paper of that series. I also need to show how to collapse the more complicated structure to a mere bicategory, in order to do what I will want to do in part (3) below.

There’s an issue here I’ll want to think about at some point, related to a question Aaron Lauda raised. The question was this. The category whose objects are 1-D manifolds and whose morphisms are 2D cobordisms between them has a nice abstract description. It is the free symmetric monoidal category with a Frobenius object.

In Aaron’s work with Hendryk Pfeiffer, they likewise described a category of “open closed strings”, which can have either 1-D manifolds or 1-D manifolds with boundary (collections of circles and line segments, basically) as objects, and cobordisms between them as morphisms. They showed this has a similar characterization, but with “Knowledgeable Frobenius” replacing “Frobenius” in the above. These have a nice description in terms of adjunctions, so Aaron was asking me if the same could be done for the double bicategory I talk about. That would need a concept of adjunction in double categories (or cubical n-categories, more generally). I don’t know what the state of understanding is on this.

More generally, it’s strange that “cobordisms of cobordisms” really wants to be a cubical 2-category in some sense, whereas, to do what I want to do with them (see below), I have to convert them into a globular one, to take functors into $\mathbf{2Vect}$. I don’t know the best way to deal with this: is there a cubical version of $\mathbf{2Vect}$, for example?

2) One part will deal with building 2-vector spaces from groupoids using functors into the category $\mathbf{Vect}$; and 2-linear maps from spans of groupoids, using the pullback (composition) along an inclusion, and its (two-sided) adjoint. Along the way, it includes some proofs of well-known folklore theorems about 2-vector spaces which are hard to find anywhere. I plan to give a talk based on this at Groupoidfest ’07 in Iowa City in November.

Soon enough – certainly before the Groupoidfest, I’ll have a bigger post about this stuff (and most likely post slides). The basic idea is that the category of functors from an essentially finite groupoid $X$ into $\mathbf{Vect}$ is a Kapranov-Voevodsky 2-vector space – that is, a $\mathbbm{C}$-linear additive category which is generated by a finite number of simple objects. (The fact that this definition is equivalent to the one given by Kapranov and Voevodsky is one of those theorems which seems to be well known, but hard to track down). The finite number of simple objects correspond to the equivalence classes of $X$. From a span of groupoids, it is possible to build a linear map between the corresponding 2-vector spaces.

The motivation for building 2-vector spaces on groupoids in the new work is to categorify the quantization of a classical system, but the two ways I’ve looked at are a bit different in how they accomplish it. Ignoring complications like symplectic geometry for the moment, the configuration space of a classical system is described as a set $X$. Each element of the set is one possible state of the system. The corresponding quantum system will have states which live in $L^2(X)$ – in particular, they are complex-valued functions on the set $X$. And instead of being able to read off values like position, momentum, energy, and other features of the system by looking at the value these have at a single point, you need some algebra of operators on $L^2(X)$, whose eigenvalues are the values you can observe for the observable that corresponds to a given operator. In categorifying this, $X$ becomes a groupoid, in which the elements of the set can be related to each other – by “symmetries”. Instead of functions into the complex numbers, we take functors into $\mathbf{Vect}$, and obtain a 2-vector space of what I suppose should be called “2-states”. Given spans of groupoids, it becomes possible to get linear maps from one 2-vector space to another, using “pullback” and “pushforward” of these functors into $\mathbf{Vect}$.

I’ll say more about this later on, but one thing that I find perplexing about this is how (if at all), it relates to some earlier work I did in this paper on the categorified harmonic oscillator, which is heavily based on this paper by John Baez and Jim Dolan, which introduces “stuff types”. Both involve groupoids, and spans of groupoids giving rise to linear operators, as part of a categorification of some elementary quantum theory, but there are significant differences. At some point, I’d like to return to the question of whether they’re related, and if so, how.

3) One part uses the above to build an “extended TQFT”. A TQFT, or topological quantum field theory is a quantum field theory, in that it gives a Hilbert space of states for some field on a specifed “space” (i.e. manifold), and linear maps associated to “spacetimes” (cobordisms) joining them. It is topological, in that its states are topologically invariant – that is, they have no local degrees of freedom, only global ones. These started life in physics, but have fallen by the wayside there, and now mostly find life in the subject of quantum topology, where they give manifold invariants.

A TQFT can be described as a functor from a category of manifolds and cobordisms (see (1)) into $\mathbf{Vect}$. This way of putting it makes it relatively easy to see what to do if one wants to categorify – which we do, in order to get higher codimension (more on this later, I’m sure). The idea is to build a 2-functor from the bicategory of cobordisms with corners (see (1)) into $\mathbf{2Vect}$. This can be done using gauge theory. The main idea is to turn a cobordism, seen as a cospan of manifolds (with corners) into a span of groupoids – namely, the groupoids of flat connections on these spaces, with gauge transformations as morphisms, and then build 2-vector spaces and 2-linear maps, etc. as laid out in the program of (2) above. The main theorem proving that such a 2-functor exists and is given by this construction was the organizing theme of my dissertation defense talk. This part is the mathematical core of what I’ve been working on.

4) Finally, this is supposed to be related to quantum gravity somehow. I’ll put off talking about this until I actually put the thesis on the archive.

Until then, I may decide to post a little about some talks I’ve been to recently. UWO has a great department with lots of interesting talks. I recently attended a couple of these by graduate students. One was by Arash Pourkia, about Braided Categories and Hopf Algebras. The second was by Michael Misamore, on Galois Theory – from the point of view of Grothendieck, and could equally well be called “Covering Spaces”… from the point of view of Grothendieck.