The n-Category Café

Well, I imagine that the representation theory of Lie 2-groups indeed does have interesting ‘matrix elements’… although I haven’t really thought of things in this way. After all, we know that the representation theory of the String 2-group is intimately tied up with the representation theory of loop groups. Instead of being a collection of numbers, the ‘matrix elements’ are now a collection of vector spaces — aren’t these related to the loop group modules graded by their energy? The identities between the special functions, like

now get replaced by isomorphisms, according to the usual mantra. Aren’t these something like the fusion rules?

The current strategy seems okay though: work out lots of interesting examples, and see what crops up.

David wrote:

Here I am, several months later, wondering if things couldn’t move along a little faster.

Only if 1) some people already working on 2-groups decide to put more time into it or 2) we get more new people working on this subject.

I’m not pessimistic about the rate at which 2-group theory is being developed. Three coauthors and I recently finished a 90-page paper on infinite-dimensional representations of 2-groups. This opens up a big new field of study with fascinating ties to existing work on analysis, quantum theory and group representation theory. But what makes me happiest of all about this paper is that the physicist Laurent Freidel and his graduate student Aristide Baratin were coauthors! They got interested because they want to develop physics applications of 2-group representation theory, following the groundbreaking work of Crane, Sheppeard and Yetter. The general theory took longer to develop than we expected, but now it’s there, and it’s ready to be applied. I think that Freidel, Baratin, and my student Derek Wise (another coauthor on this paper) plan to go ahead and develop the applications. Once physicists get going on this stuff, progress will speed up tremendously.

I was also delighted to attend the 2-group workshop in Barcelona last summer — the first of its kind — and see people from many fields of mathematics coming together and talking about 2-groups from many different perspectives. The 2-group photo gives some sense of this…

For example, at this workshop I met Mathieu Dupont, who recently finished a thesis categorifying homological algebra so that it applies to what you might call ‘abelian 2-groups’. I’ve described a lot of related work by his advisor Enrico Vitale and also the Granadan school of 2-group theorists in week267.

So, I think things are bubbling along quite nicely. 2-group theory hasn’t become a ‘hot topic’ of the kind that makes dozens of mathematicians drop what they’re doing and switch to working on this. But that’s fine with me! I actually find it quite unpleasant when something I’m working on becomes a ‘hot topic’: a lot of confusion and clutter get generated.

Hmm. I guess your complaint is not about the progress of 2-group theory in general, but about how various intriguing ideas mentioned on this blog have not blossomed as rapidly as you’d like: most notably, Klein 2-geometry, but also the theory of Galois 2-groups.

I think the problem is that you need someone working full-time on a subject for it to move forward quickly. So the question is: who is going to do it?

I can imagine enjoying concentrated work on Klein 2-geometry and its connection with categorified logic. But I don’t have time to actually do that, now.

For one thing, all my grad students are working on other projects: categorifying the $\lambda$-calculus (Mike Stay), groupoidifying Hecke algebras (Alex Hoffnung), groupoidifying Hall algebras (Christopher Walker), categorifying symplectic geometry (Chris Rogers) and understanding what the octonions do for physics (John Huerta). Whenever I have a moment to be creative, I want to use my creativity to push these projects forwards.

For another thing, I’ve been helping James Dolan develop some new ways of thinking about algebraic geometry. We’ll release a preliminary version of a paper on that any day now. I’m very excited about that!

So, Klein 2-geometry may simmer slowly on the back burner until someone — and probably not me — decides to spend one or two hours a day thinking about it.

If such a person existed, and posted specific concrete problems on the $n$-café whenever they got stuck, things would move much faster.

And here by ‘specific concrete problem’ I don’t mean things like “For finite 2-groups, is there an equivalent of orthogonality in the character table?”

I mean things like “Is the following proposition true or false?” That’s when most mathematicians get excited.

1. On homological 2-algebra and applications: Apart from very interesting foundational work of Vitale and his students, Hans-Joachim Baues and collaborators have very practical results (including the best results so far in computing stable homotopy groups of sphere – classical and important problem; the calculations involve new information on 3rd term of Adams spectral sequence) when developing (for a number of years!) 2-categorical homological algebra of a little different sort, they call it “secondary” homological algebra; they have even secondary triangulated categories, whose definiion is shorter than of the usual one. There is a very nontrivial and important 2-Hopf algebra in this connection, which is a 2-categorical analogue of the Steenrod algebra of (primary) cohomological operations; Baues wrote somewhere that any fact on secondary cohomological operations (what is also a classical subject) is a fact about the secondary Steenrod algebra. He already published a book solely on that example of a categorified Hopf algebra.

Hans-Joachim Baues, The Algebra of Secondary Cohomology Operations
Series: Progress in Mathematics , Vol. 247,
2006, XXXII, 483 p.

It is a pity that people from other 2-schools are not looking at that work. I heard Larry Breen has some interesting unpublished observations about the work of Baues school from another perspective, I hope we will hear soon about it. Some of the newer Baues’s papers are at the arXiv now
( listings ), but there are many relevant earlier opera including those about universal Toda brackets.

2. Above there is a quotation about induction/restriction for 2-representations; in this connections I would like to mention that for coherent actions of monoidal categories or bicategories there is such a pseudoadjunctions; it can be also done equivariantly with respect to yet another monoidal action which commutes with the first up to distributive law: the induction is then a special case of a tensor product, for what I call “biactegories” (left and right actions of two monoidal categories on the same 1-category, which commute up to invertible binatural transformation which satisfies the diagrams for a distributive law for such actions) and this tensor product is obtained using pseudocoequalizers in Cat. I started writing about “biactegories” in 2006 and stopped after 30 pages of technicalities, I hope to resume when less busy, in the meantime I wrote few further hints in a paper on equivariance in noncommutative geometry arXiv:0811.0470 , v2, cf. page 3-6.

I think the data indicates that 2-representation theory looks very different than ordinary representation theory. For example if G is a finite group and H is any subgroup, there are indeed induction and restriction operations between 2-representations (geometrically these are pushforward and pullback along the map from BH to BG), but the usual yoga of decomposing representations doesn’t apply. Namely EVERY representation already appears in the decomposition of the trivial representation – or any other (induced) representation for that matter. This follows from the Morita equivalence of Muger and Ostrik. Put in other words, the functor of H-invariants on G-representations defines an equivalence between representations of G and those of the double cosets H\G/H.

(This is closely related I think to the discussion on the thread about stabilizers, regarding the breakdown of the idea of “subgroup” in homotopical contexts.)

This Morita picture gets more complicated for Lie groups (it depends sensitively on the “smoothness” properties of the representations - it holds for “algebraic” representations, but not for “locally constant” or “smooth” ones), but to it indicates that simple analogies between group actions on vector spaces and on 2-vector spaces don’t apply. It would be fascinating to see what kind of picture does apply, and any hints people can share would be much appreciated!

First I’d like to second David’s opinion that online collaboration is a great idea, at least for certain types of problems. I followed the work going on on Gowers’s blog, and there was a lot of discussion of just what types of problem might be good candidates.

Second, since we are talking about 2-groups, I wonder if anyone could point me to current research on the categorification of Hopf algebras. Here are some of the questions that leap out:

Is an internal category in the category of Hopf algebras a 2-Hopf algebra?

Is there a concept of “2-group ring?”

Is a 2-Hopf algebra a special 2-vector space?

I know these aren’t concrete–I’m hoping to get some reading material suggested to me and then maybe…we’ll see.

Could it be that we need to develop all of algebra in 2 analogue before we have an idea of what 2-geometry is? With that in mind I think the following would be worth a shot:
1: Look at the preadditive subcategory of 2-groups. Call it the category of abelian 2-groups.
2:The structure of the arrows pointing from a 2-group to itself in that subcategory is a 2-ring.
3: A 2-field is a 2-ring where all arrows as in 2 have inverses.
The big question is can we develop enough of (2) to make this work, and if the supposed subcategory exists and is unique.