It’s been a while since I posted last, but in there I described some issues related to talks I gave in Portugal recently. I’m beginning a postdoc at the Instituto Superior Tecnico, in Lisbon, in less than a month’s time. In the meantime, I’ve been two weeks in Portugal, including a conference and apartment hunting.  Then, last week, I got married. So not surprisingly, I’ve been a bit slow in updating.

The talks I gave are this one, which I gave at IST and this one at the XIX Oporto Meeting on Geometry, Topology and Physics which was held this year in Faro, which this year was a conference on the theme of Categorification!  These talks also appear on my new website, which I got because my hosting at UWO will expire sooner or later, and I wanted something portable (and a portable email address came with it).

## Lisbon

Lisbon is an interesting city.  I’ve visited Europe before for conferences and travel and so on, but never for long, and have only lived in North America, where most urban areas are much newer and ancient history more poorly documented.  This is even more so in the southern parts of Europe that were part of the Roman Empire (and even more so in areas of India I’ve travelled in).  I’m looking forward to getting more familiar with the place, which has an exciting and under-appreciated history.  At least I assume it’s underappreciated, since a majority of people in Canada who ask me where I’m moving have never even heard of Lisbon, which I find surprising.

Human settlement in Portugal actually pre-dates homo sapiens, going back to Neanderthals (we often forget there’ve been a few dozen human species before ours. and our era is unusual in human history for having just the one).  Among Sapiens, there have been various periods, most recently the ancient megalith-building cultures, Phoenecians, Greeks, Carthaginians, Romans, Visigoths, Arabs, and then the kingdom (now a republic) of Portugal, established during the Christian reconquest of Iberia.  Lisbon itself dates back at least to Roman times. The oldest surviving areas of Lisbon date back (in streetplan, if not actual buildings) to the Moorish kingdom, when Iberia was known as al-Andalus, some 800 years ago.  Lisbon’s downtown, immediately below this area, couldn’t be more different, being one of the first urban areas planned on a grid – this followed the original area being destroyed in an earthquake and resulting tsunami in 1755.  As the capital of Europe’s first overseas empire, which had reached Japan and Brazil by well over 400 years ago, Lisbon has been a “global city” for at least that long, with spells of boom and bust, and more recently, dictatorship and revolution.  Its location means it was historically a hub that linked the older Mediterranean trading world and the larger Atlantic and Indian Ocean world.

Here is a picture of the main pavillion on the IST campus:

And here is a picture of the neighborhood where I’ll be living, about 10-15 minutes’ walk or two metro stops away:

As you can also see from these pictures, Lisbon contains a number of hills.  It is occasionally reminiscent of San Francisco in that way, and the style of buildings, which also resembles New Orleans occasionally.  And of course, since this is Europe, public spaces that look like this:

And so on.

## Visit at IST

Anyway, in the visit at IST, we also had a little mini-conference on categorification, featuring some people who also spoke at Faro (including me) giving longer and more elaborate versions of our talks.  I already commented on mine, so I’ll mention the others:

Rafael Diaz gave a talk about how to categorify noncommutative or “quantum” algebras, in the sense of algebras of power series in noncommuting variables, using ideas similar to the way commutative polynomial algebras can be “categorified” by Joyal’s species.  This “quantum species” idea is laid out partially in this paper. This leads on into ideas about categorifying deformation quantization.

The basic point is to think of “a categorification of a ring R” as a distributive category $(C,\oplus,\otimes)$ whose Burnside ring (the ring of isomorphism classes of objects, with algebraic operations from $\oplus$ and $\otimes$) is $R$, or more generally has a “valuation” valued in $R$ that is surjective and gets along with the algebra operations.

The category chosen to describe a deformation of $R$ is then the category of functors from $FinSet_0^k$ into $C$.  The main point is then to find a noncommutative product operation $\star$, in place of the obvious one derived from $\otimes$, which gives a categorification of a polynomial ring.  This has to do with sticking structured sets together, where some elements of the set can form “links” between the elements of sets – this uses three-“coloured” sets, where one “colour” denotes elements associated to links.

Yazuyoshi Yonezawa gave a talk about some stuff related to link homology invariants such as Khovanov homology.  Such invariants are a major theme for people interested in categorification these days, for various specific reasons, but in general because tangle categories have some nice universal properties, so doing certain kinds of universal higher-dimensional algebra naturally has applications to studying tangles, hence links, hence knots.  In particular, invariants like the Jones, HOMFLY, and HOMFLYPT polynomials, and Reshitikhin-Turaev invariants.  Yazuyoshi’s talk was about an approach to these things based on – as I understood it – some representation theory of quantum $\mathfrak{sl}_n$, and a diagrammatic calculus that goes with it, for assigning data to strands and crossings of a knot.  (This sort of thing gives a knot invariant as long as it’s invariant under Reidemeister moves – that is, is unaffected by changing the presentation of the knot.  Many of the knot invariants that come up here arise from treating the knot using some sort of diagrammatic calculus – which is where the category theory comes in.)

Aleksandar Mikovic gave a talk about higher gauge theory in the form of 2-BF theory – also known as BFCG theory, this is sort of the “categorified” equivalent of the theory of a flat connection, now taking values in a Lie 2-group.  Actually, he speaks about these in terms of Lie crossed modules, which is a rather nice language for talking about higher-algebraic group-like gadgets in terms of chains of groups with some extra structure (actions of lower groups on higher, and some other things) – see Tim Porter’s “Crossed Menagerie” for a comprehensive look.  The talk was related to finding gauge invariant actions for theories of this sort – the paper it’s based on is one with Joao Faria Martins.

## XIX Oporto Meeting

The Oporto meeting on geometry and physics, specifically devoted this year to categorification, was very interesting, with a range of good speakers. Unfortunately, Faro is not optimal as a conference site: the accomodations are a half-hour bus ride from the campus where the conference is held, and the buses come only about once per hour and as a result (of that, and jet-lag, which could happen anyway), I missed some of the talks. Otherwise, it’s a pleasant town with a nice atmosphere, and it was interesting to see some of the variety of people working on categorification.  In particular, a lot of people are working on categorifying aspects of representation theory, which in turn is interesting to topologists, and knot theorists in particular.

One bunch of ideas about categorical representations which was referred to a lot is due to Chuang and Rouquier, substantially described in a paper from a few years ago – here is a post from the Secret Blogging Seminar a few years back describing some of the ideas a bit more succinctly.  The basis for the most popular program being discussed, and the big idea in recent years, is due to Khovanov and Lauda – see the bottom section of this post.

Now, the main invited speakers each gave a series of three hour-long classes on their topic in the mornings, while in the afternoons the other speakers gave 20-minute talks.  The main speakers were these:

Mikhail Khovanov wasn’t able to attend for personal reasons, but there was a great deal of discussion about work that builds on his categorification of quantum groups with Aaron Lauda, who however was there and gave a nice series of talks introducing the ideas (though I missed some because of the unfortunate bus infrastructure). Aaron collects a bunch of resources on this subject here, and I’ll explain a bit of this below.

Sabin Cautis talked about the categorification of $sl_2$ in terms of geometric representation theory; the idea here is that there are certain spaces that carry natural representations.  These are flag varieties – the simplest example being Grassmanians – spaces whose points are the $k$-dimensional subspaces of some fixed $V$. In general, flag varieties are spaces whose points consist of a nested sequence of subspaces $V_0 \subset V_1 \subset \dots \subset V_k = V$ (the terminology “flag” suggests a flagpole with a 2D rectangle, suspended from a 1D pole, rooted at a 0D point).  The talk was an overview of how to use this to categorify some representation theory.  Here is a recent related paper by Cautis, including Joel Kamnitzer, (I blogged his talk here at UWO a while ago on a similar subject in some more detail), and Anthony Licata.  The basic point is that categories of sheaves on such spaces carry a categorical representation of $\mathfrak{sl}_2$.

One thing I found interesting – this time, as with Joel’s talk, is that span constructions turn up in this stuff quite naturally, but there is both a similarity and a difference in how they’re used.  In particular, given a flag $V_0 \subset \dots \subset V_i \subset \dots \subset V_k$, we can project to a flag with one fewer entries just by omitting $V_i$.  So the various flag varieties associated to $V$ are connected by a bunch of projections.  Taking two different projections (dropping, say $V_i$ and $V_j$), we get a span of varieties – that is, one object with two maps out of it.  We’re talking about spaces of functions on these varieties, so pushing these through spans is of interest.  Lifting a function (by pre-composition – assign a flag the value of the function at its image) is easy – pushing forward is harder.  This involves taking a sum over the function values over the preimage – all the long flags that project to a given short one (to make sure this is tractable, we consider only constructible functions, with finitely many values).  But this sum is weighted.  In the groupoidification program, something similar happens, but the weight there is the groupoid cardinality of the preimage.  Here, it is the Euler characteristic of the preimage (or rather, for each function value, the part of the preimage taking a given value contributes its Eular char. as the weight for that value).  Since groupoid cardinality is like a multiplicative sort of Euler characteristic, there seems to be a close analog here I’d like to understand better.

Catharina Stroppel talked about how the subject relates to Soergel bimodules, and led up to categorifying 3j-symbols.  Soergel bimodules showed up in several different talks about this stuff.  These are the irreducible summands in the bimodule that comes from applying induction functions between module categories $Ind: R^{\lambda'}-mod \rightarrow R^{\lambda}-mod$ finitely many times.

Here, the $R^{\lambda}$ are  rings of functions invariant under $S_{\lambda}$, which is the subgroup of the permutation group $S_n$ which respects a particular composition $\lambda$ of $n$ (like a partition, but with order – compositions also specify flag varieties, by specifying the codimensions at each inclusion).  The point is that, if $S_{\lambda'} < S_{\lambda}$, we get inclusions of the rings of invariant functions, and then we can induce representations along those inclusions.  (Notice, by the way, that the correspondence between compositions and the signature of a flag means that this is actually much the same as the inclusions I just described under Sabin Cautis’ talk).  Then doing a finite chain of such inductions gives a functor between module categories.  This can be described by tensoring with some $(R^{\lambda'},R^{\lambda})$ bimodule – the direct summands in this are the Soergel bimodules.  So these are central in talking about these categorical actions and categorified representation theory.  This in turn ended up, in this series of talks, at a categorification of 3j-symbols (which can be built using representations and intertwiners).

Ben Webster talked about how diagrammatic methods used in the Khovanov-Lauda program can be used to categorify algebra representations, and through that, the Reshitikhin-Turaev invariant; the key diagrammatic element turns out to be marking special “red” lines with special rules allowing strands to “act” on them by concatenation.  I must admit Ben Webster’s talks, which ended up rather technical, went far enough over my head that I’m reluctant to summarize, since I was still catching up on the KL program, and this was carrying it quite a bit further.  I do recall that there was much discussion of cyclotomic quotients (partly because Alex Hoffnung later came back to the matter and I had a chance to talk to him about it briefly) – that is, the quotients imposing the relations forcing something to be a root of unity, which isn’t surprising since quantum groups at $q$ a root of unity are important and special.  Luckily for the reader who is more up on this stuff than I, the slides can be found here and here.

Dylan Thurston spoke on Heegard-Floer homology (slides here, here, and here – full of great pictures, by the way), which is a homology theory for 3-manifolds (then an invariant for a closed 4-manifold), due to Oszvath and Szabo.  It’s a bi-graded homology theory (i.e. homology theories give complexes for spaces – this gives a bicomplex, with grading in two directions).  This theory gives back the (Conway-)Alexander polynomial for a knot when you take the Euler characteristic of the bicomplex.  That is: there are two directions this complex is graded in: one (columns, say) will correspond to the degree of the variable $t$ in the Alexander polynomial; for each $k$, the coefficient of $t^k$ is the Euler characteristic (alternating sum of dimensions) of the entries in that column.  So this is a categorification of this polynomial, in somewhat the way that Khovanov homology categorifies the Jones polynomial.

HF homology can be defined for a knot can be defined in a combinatorial way: a 3-manifold can be represented by a “Heegard diagram” – a 2D surface marked with (coloured) curves, which is a way of keeping track of how a 3-manifold is built by splitting it into parts.  From this diagram, one gets “grid diagrams”, and by a combinatorial process (see the slides for more details) generates the complex.

Others.  I didn’t manage to attend all the other talks (partly because of aforementioned bus issues, and partly because I was still working on mine, having taken a lot of time in Lisbon doing useless things like finding a place to live), but among those I did, there were several that were based on the Khovanov-Lauda program for categorified quantum groups: Anna Beliakova in particular worked with them on categorifying the Casimir (generator of the centre) of the categorified quantum group; people working with Soergel bimodules and categorified Hecke algebras such as Ben Elias and Nicholas Liebedinski.  Then there were the connections to link homology: Christian Blandet and Geordie Williamson talked about things related to the HOMFLYPT polynomial; Krystof Putyra and Emmanuel Wagner gave talks related to Khovanov homology and link homology.  Alex Hoffnung talked about a combinatorial approach for dealing with categorification of cyclotomic quotients as discussed by Ben Webster.

## Categorification of Quantum Groups

The reason for categorifying quantum groups, at least in this context, has to do with the manifold invariants associated to them.  Often these come from categories of representations of groups or quantum groups – more generally ones with similar formal properties, meaning roughly monoidal categories with duals (and possibly more structure).  These give state sum invariants, by assigning data from the category to a triangulation of a manifold – objects on edges and morphisms on triangles, say.   The categorification of quantum groups means we pass from having a monoidal category with duals (of representations), to a monoidal 2-category with duals (of representations).  This would mean the state-sum invariants it’s natural to construct are now for 4-manifolds, rather than 3-manifolds.  This is the premise behind spin foam models in gravity, but also has its own life within quantum topology as tools for classifying manifolds, whether or not it accurately describes anything physical.  Marco Mackaay, one of the conference organizers (among several others), has written a bunch on this – for example, this constructs a state-sum invariant given any “spherical” 2-category (a property of certain monoidal 2-categories – see inside for details), and this gives a specific consstruction using the Khovanov-Lauda categorification of $\mathfrak{sl}_3$.

The Khovanov-Lauda approach to categorifying quantum groups (in particular, deformations of envelopoing algebras of classical Lie algebras, within the category of Hopf algebras)  is most basically about “categorifying” the presentation of an algebra in terms of generators and relations.  That is, we describe a set with some operations in terms of some elements of the set (generators), and some equations (relations) which they satisfy involving the operations.  The presentation used for $U_q(\mathfrak{sl}_n)$ is the standard one based on an n-vertex (type-A) Dynkin diagram: basically, $n$ dots in a row.  There’s a generator $e_i$ for the $i^{th}$ vertex; the generators for non-adjacent vertices all commute, and for adjacent generators, we have $(q + q^{-1}) e_i e_j e_i = e_j e_i e_j$.  (The factor involving $q$ is the quantum integer $[2]_q$, and becomes 2 in the limit).

To categorify this, we still give generators, but the equations are replaced with isomorphisms – this means we need to be working in some category $R$, hence one essential task is to describe the morphisms.  So: the objects are just rows of dots, labelled by vertices of the Dynkin diagram.  The morphisms are (linearly generated by) isotopy classes of braids from one row to another.  The essential thing is that we have to carefully define “isotopy” here to ensure we get the categorified version of the relations above.  So for non-adjacent-vertex labels, we have the usual Reidemeister moves (the key ones being: we can slide a strand past a crossing, straighten out two complementary crossings); for adjacent-vertex labels, though, we have to tweak this, imposing some relations on strands involving the factors of $q$.  The relations take up a few slides in the talk, but essentially are chosen so that:

Theorem (Khovanov-Lauda): There is an isomorphism of twisted bialgebras between the positive part of $U_q(\mathfrak{sl}_n)$ and the Grothendieck ring $K_0(R)$, where multiplication and comultiplication are given by the image of induction and restriction.

Obviously, much more could be said from a five-day conference, but this seems like a nice punchline.