## March 10, 2009

### Synergism

#### Posted by David Corfield

When I first read in John’s writings about 2-groups I naïvely imagined that people would rapidly find analogues for everything done with ordinary groups. Hence my call for a Klein 2-geometry in 2006.

Later in that year, I’m to be found on the recently formed Café musing:

I presume people are wondering about which equivalents of features of group representation theory might be found? Are there ‘locally compact 2-groups’, and if so, are there Haar measure and Peter-Weyl theorem equivalents? For finite 2-groups, is there an equivalent of orthogonality in the character table? In general, is there an equivalent of the adjunction between the restricting and inducing functors? What about the branching rules? There must be dozens more questions like these.

To this John replied:

…right now there are very few people working on 2-groups. There are no “2-group theorists”: all these people are doing lots of other interesting things too. So, instead of trying to set up a vast edifice of machinery, the right approach is to find some really exciting examples, with connections to other branches of math, which will get more people interested in the subject. Then the machinery will practically build itself.

That’s why I’m working on specific examples, only developing enough general machinery to exhibit these examples. The string 2-group is the best one so far, since it hooks on to loop groups, affine Lie algebras, the WZW model - in short, the whole apparatus of postmodern Lie theory. The Poincare 2-group may also be really interesting - we’ll see.

Here I am, several months later, wondering if things couldn’t move along a little faster. Over in the other ‘culture’ it looks like they’ve just jointly solved a problem. Perhaps the ability to focus attention on a single problem was conducive to progress. On the other hand, having a couple of Fields’ medallists on the job must help.

Still, I’m wondering whether we’re maximising the synergy possible with the talented people who drop by here. John reported, again in 2006, that he and Jeff Morton

…figured out how to categorify the algebraic integers in any algebraic extension of the rationals, getting an “algebraic extension” of the category of finite sets. We figured out the beginnings of a theory that associates a “Galois 2-group” to any such algebraic extension.

I take it that work’s left idly gathering dust somewhere. Shouldn’t we be able to move that theory on quite rapidly?

Or, if there’s more interest in the representation theory of Lie 2-groups, then the observation that special functions, Bessel, hypergeometric, etc., appear as matrix entries of representations of particular Lie groups, might be suggestive. Does, say, the string 2-group have something special appear as matrix entries of representations?

Posted at March 10, 2009 12:12 PM UTC

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### Re: Synergism

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

(1)$cos(x) cos(y) = \frac{1}{2} [cos(x-y) + cos(x+y) ]$

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.

Posted by: Bruce Bartlett on March 10, 2009 3:34 PM | Permalink | Reply to this

### Re: Synergism

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.

Posted by: John Baez on March 10, 2009 9:25 PM | Permalink | Reply to this

### Re: Synergism

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…

It wasn’t supposed to be a complaint, though I can see it sounds like one. And it wasn’t supposed to be about any one piece of theory in particular - I just mentioned ones I knew about.

What I was trying to draw attention to was the possibility that the full potential of collaborative research was not being realised on this blog. I could be wrong - it’s possible that the usual forms of interaction suffice. But we now have one instance of successful blog synergy.

Gowers reports yesterday on his polymath project:

First, let me say that for me personally this has been one of the most exciting six weeks of my mathematical life.

He is surprised at the speed of progress, attributed to

how often I found myself having thoughts that I would not have had without some chance remark of another contributor. I think it is mainly this that sped up the process so much.

There’s a fair amount of nudging people to have new thoughts here, but I like the idea of an intense burst of focused activity.

However, rather tellingly:

There seemed to be such a lot of interest in the whole idea that I thought that there would be dozens of contributors, but instead the number settled down to a handful, all of whom I knew personally.

So, you might ask, why not just fund them to spend 6 weeks together? Is anything gained having the details available online? At the very least, as Gowers says, we now have

the first fully documented account of how a serious research problem was solved, complete with false starts, dead ends etc.

But he also gives the impression that the blog/wiki format helped the process.

Perhaps it would be hard to find a sufficiently precise problem to people here to work on. If it’s a ‘develop the theory of $X$’ sort of search, there may be too many ways to generalise it to its ‘proper’ setting. But then we have had specific challenges, such as the first $n$-Café Millennium Prize.

Posted by: David Corfield on March 11, 2009 10:22 AM | Permalink | Reply to this

### Re: Synergism

There are perhaps some lines of study of 2-groups that have not been mentioned yet in this discussion and might provide a more optimistic view.

Ronnie Brown and Jean-Louis Loday defined the non-abelian tensor product of two groups as an offshoot of their work on van Kampen theorems for cat$^n$ groups. It is very closely linked to crossed modules and to crossed squares and hence to both 2- and 3-groups. Ronnie has collected a bibliography of papers on this subject which is impressive. It shows a lot of interaction with results in more classical areas of groups theory and, I suspect, also would reveal, if looked at more closely, some very honest and interesting applications of 2-groups. (By ‘honest’ I mean ‘non-contrived’, as sometimes one may do something just because it can be done, and that is often not going to give one somthing of importance outside the immediate area)

Perhaps this tensor product is something that could be looked at more closely. (I would volunteer to give an nLab entry but may be too near the wood to see more than the trees and would therefore ask for some active interaction!)

The bibliography is at

http://www.bangor.ac.uk/~mas010/nonabtens.html

Another area that may be of interest is that of computational crossed modules. Chris Wensley has put in a lot of work on this

see

http://www.gap-system.org/Packages/xmod.html

This may provide a useful tool for further investigation of finite 2-groups and their uses.

Posted by: Tim Porter on March 11, 2009 10:54 AM | Permalink | Reply to this

### Re: Synergism

Perhaps one approach to take is to look for 2-groupish structures in some parts of, say, combinatorics. A prime source of groups is as subgroups of symmetric groups, for obvious reasons. Symmetric groups are the full group of self bijections of some finite set. Replacing finite set by either finite category, or a free category on a finite graph, you can try to see how to define a corresponding 2-group. How sensitive is this 2-group to special features of the graph or category. For instance, start with a poset (e.g. the poset of subgroups of some group) is there a 2-group of automorphisms of the corresponding category and if so what structure on the poset is detectable in the structure of the 2-group? What I am suggesting is thus a `bottom up’ approach extending some of the features that John and the group at Riverside seemed to have started some time ago. This would be led by the examples and perhaps that way we will be surprised at what we find!

Posted by: Tim Porter on March 12, 2009 6:13 PM | Permalink | Reply to this

### Re: Synergism

I forgot to mention a paper:

BROWN, R., MORRIS, I., SHRIMPTON, J. and WENSLEY, C.D. Graphs of morphisms of graphs,

in Elec. J. Combinatorics, article A1 of volume 15(1), April 3, 2008, which might provide food for thought in this area.

Posted by: Tim Porter on March 12, 2009 6:49 PM | Permalink | Reply to this

### Re: Synergism

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.

Posted by: Zoran Skoda on March 10, 2009 11:56 PM | Permalink | Reply to this

### Re: Synergism

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!

Posted by: David Ben-Zvi on March 11, 2009 2:28 AM | Permalink | Reply to this

### Re: Synergism

I wrote a bit about 2-group representations on ‘measurable categories’ in week274 of This Week’s Finds — and much more in the paper discussed there. This framework uses a lot of analysis — it’s a bit like the study of infinite-dimensional continuous unitary representations of topological groups. So, it’s superficially very different than the approaches based on algebraic geometry. But they should be related somehow. I made a tiny guess about how…

Posted by: John Baez on March 12, 2009 8:51 PM | Permalink | Reply to this

### Re: Synergism

A new paper on 2-representations, but still merely of groups – Explicit Formulas for 2-Characters.

Posted by: David Corfield on April 29, 2009 9:53 AM | Permalink | Reply to this

### Re: Synergism

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.

Posted by: stefan on March 11, 2009 7:26 PM | Permalink | Reply to this

### Re: Synergism

One possible source of ideas on 2-group rings can be found in

2-groups, trialgebras and their Hopf categories of representations
Author: Pfeiffer, H.
Year:2007
Volume: 212
Number: 1
Pages: 62 - 108
MR2319763 (2008d:16060)

Posted by: Tim Porter on March 11, 2009 9:25 PM | Permalink | Reply to this

### Re: Synergism

Again on 2-Hopf algebras, you might look at
INTERNAL CATEGORIES AND QUANTUM GROUPS
the 1997 dissertation of Marcelo Aguiar.

Posted by: Tim Porter on March 14, 2009 1:00 PM | Permalink | Reply to this

### Re: Synergism

It’s nice to give links, so people can just click. Here are some of the first papers on categorified Hopf algebras:

Ever since Khovanov got into the act, categorifying various specific Hopf algebras — ‘quantum groups’ — became a big business!

Posted by: John Baez on March 16, 2009 1:18 AM | Permalink | Reply to this

### Re: Synergism

Just a quick belated thanks! The suggested links and references therein should be enough to keep me quite busy.

Posted by: stefan on March 19, 2009 5:52 PM | Permalink | Reply to this

### Re: Synergism

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.