## November 28, 2008

### Groupoidfest 08 - A Brief Report

#### Posted by John Baez

There were a lot of interesting talks at last weekend’s Groupoidfest here at Riverside. However, due to a lack of organization and shortage of time on my part, I can give only a brief report.

I especially enjoyed Weinstein’s talk on the volume of a differentiable stack. Someday these ideas will lead to great things. I’d heard him explain them before, but they sunk in deeper this time. I’ll try to explain someday in This Week’s Finds. In the meantime, try his paper.

Xiang Tang’s talk on group extensions and duality for gerbes was also very interesting, since it started with a review of the ‘Mackey machine’ for studying induced group representations, and then generalized this in exciting ways. Unfortunately, it zipped by too fast for me to fully understand. This wasn’t his fault: he only had 20 minutes! He hasn’t written up this work yet, so we’ll just have to wait.

I’m also happy to say that thanks to a talk by John Quigg, I’m no longer terrified of ‘Fell bundles’.

To work your way towards Fell bundles, first you should ask “If a monoid is a category with one object, what’s a C*-algebra?” The answer: “It’s a C*-category with one object!” But now suppose we want to think of a C*-category made into a kind of ‘vector bundle’ over a Lie groupoid — with the same objects, just different morphisms. This is a Fell bundle.

Some special cases may help. A Fell bundle over the terminal groupoid is just a C*-algebra. A Fell bundle over a topological group is sometimes called a “C*-algebraic bundle”. It’s a bundle of Banach spaces over the group, along with a multiplication that makes it act a lot like a C*-algebra.

But let me try to actually define a Fell bundle!

A Fell bundle starts out being a topological groupoid $X$ and a bundle of Banach spaces over the space of morphisms of $X$:

$p: A \to Mor(X)$

Let $A_f$ be the Banach space over the morphism $f$. Then, we equip $A$ with composition maps

$\circ : A_f \otimes A_g \to A_{f g}$

that are associative and unital. This makes $A$ into a category over $X$. In other words, we now have a category $C$ with $A = Mor(C)$, and $p$ gets promoted to a functor

$p: C \to X$

Then we demand that $p$ be a continuous functor between topological categories. Then we demand that $C$ be a C*-category, and demand that $p$ be a $*$-functor.

I hope the definition I’ve just given matches the standard one. I’ve tried to make it more ‘conceptual’ — that is, build it out of ideas rather than lists of equations. But I hope it could be polished further.

There were also some tantalizing talks on Lie groupoids in discrete-time mechanics and discrete field theories — especially gauge theories and nonlinear sigma-models — by Joris Vankerschaver and Melvin Leok. Someday this work should fit very nicely into Urs Schreiber’s grand plan to treat physics $n$-categorically in such an abstract way that we can make spacetime either discrete or continuous as a kind of afterthought.

There were also three talks on groupoidification. My student Christopher Walker introduced the basic machinery:

Alex Hoffnung explained the application to Hecke algebras:

I gave an overview focusing on the oldest application: the harmonic oscillator. I didn’t use slides, but this paper will eventually cover everything I said:

Jeffrey Morton spoke about his new paper, which opens up a big new branch of the groupoidification program. He showed how to turn groupoids not into vector spaces but 2-vector spaces. Spans of groupoids give linear maps between 2-vector spaces, and spans of spans of groupoids give linear maps between linear maps!

Part of what made this conference fun for me was seeing a bunch of old students of mine — Toby Bartels, Alissa Crans, Jeffrey Morton and Derek Wise — and watching them get to know the new ones: Alex Hoffnung, John Huerta, Christopher Walker and Chris Rogers.

I also met some new people, including Arlan Ramsay, father of Keith Ramsay of sci.math fame. See how many folks you can recognize here:

Extra credit if you don’t need to click on the photo to enlarge it!

For more on the Groupoidfest, see Jeffrey Morton’s report.

Posted at November 28, 2008 8:46 PM UTC

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### Re: Groupoidfest 08 - A Brief Report

John wrote:

See how many folks you can recognize here: Extra credit if you don’t need to click on the photo to enlarge it!

Enlarging the photo doesn’t allow me to read very many name tags. For now, one can refer to the list of participants. But I edited that photo to replace the faces with numbered white ovals and sent it to Aviv, so hopefully there it will soon appear online with a numbered list of names, so then you’ll be able to check for yourself.

Posted by: Toby Bartels on November 28, 2008 11:25 PM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

Anyway, Toby is the bald bearded smiling guy between the super-tall guys in the very back.

Posted by: John Baez on November 29, 2008 1:59 AM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

Now that events have caught up, I’ll repost the comment and question I had:

I suppose Joris Vakershaver spoke about material as in his 2005 article Discrete Lagrangian field theories on Lie groupoids?

Did he present stuff not to be found in that article? Are there any notes available anywhere?

He doesn’t by any chance look at Dijkgraaf-Witten theory as Chern-Simons field theory with parameter space and target space discrete groupoids #?

I wonder if he is aware of the close relation between his definition 3.5, p. 13 (a field as a graph morphism from a graph to the graph underlying a groupoid) with Anders Kock’s synthetic description of parallel transport?

Posted by: Urs Schreiber on November 30, 2008 7:52 PM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

Hi Urs,

The talk that I gave at the groupoidfest consisted essentially of the contents of the 2005 paper, together with some applications. The paper is a bit terse and convoluted in places and I’ve reworked a little bit since. However, the essential definitions remain the same. I’ve put the slides of my talk online.

Since noticing your question, I’ve spent the entire afternoon reading up on these old posts and I agree that the similarities are indeed remarkable but I can’t really comment on them yet.

By comparison, our aim in writing the paper was very modest. We wanted to come up with a discrete framework for field theories that would have all of the properties of the continuous one. Numerical integrators based on such an approach will typically outperform classical ones, since the former are symplectic/energy/momentum preserving (pick 2 out of 3). We succeeded in doing so (this is the content of the paper) and even came up with a nice example of soliton scattering in the nonlinear sigma model that shows some of these conservation properties at work. The last bit of work remains unpublished since it appears to be not that useful from a numerical point of view: symplectic integrators usually shine in computing averages of ensembles of trajectories, or over long integration time scales, and neither is really relevant for soliton scattering. Still, it is a nice proof of concept.

I find your remark about the Dijkgraaf-Witten model particularly interesting, and especially (in the post that you link to) the link between parallel transport and “nonabelian differential cohomology”. Again from a pedestrian point of view, in doing the nonlinear sigma model, if we took the target space to be a nonabelian group everything still worked out without any major changes compared to the Abelian case, except for the fact that the terminology was different. I don’t want to claim that these ideas are particularly deep or new, but they appeared quite naturally in the discrete context. I’m really interested in fleshing this out some more!

Posted by: Joris Vankerschaver on December 1, 2008 6:04 AM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

Thanks for the summary, John! That’s a good set of links to sniff through.

I recognise three people in that picture… it should be four! I guess Jeff did a runner.

Posted by: Jamie Vicary on December 4, 2008 9:31 PM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

‘Did a runner’ — ah, yet another quaint British expression that means absolutely nothing to me.

Luckily I can guess by context that it means something like ‘slipped away’. No, the people who slipped away are Alissa Crans and John Huerta. Jeff is there — but he probably has less hair than when you last saw him. Can you spot him?

Posted by: John Baez on December 4, 2008 9:45 PM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

Crikey! Hi, Jeff! Looks like the guy to his right still hasn’t got over the shock. The guy to his right can’t even look at Jeff any more.

Posted by: Jamie Vicary on December 4, 2008 10:09 PM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

Yes, great photo! Toby reminds me a bit of Aubrey de Grey, whose talk I watched recently on Ted.com, Why we age and how we can avoid it. It blew my mind. There’s a Methuselah Mouse Prize!

Posted by: Bruce Bartlett on December 5, 2008 12:41 AM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

British slang for hurriedly leaving as if from a crime scene. If my reading UK crime thrillers is correct…

As in The Sunday Mirror: “DRUGS baron John Barton did a runner while on pounds 10m heroin charges…”

“I wouldn’t quite say he did a runner. I just don’t think it occurred to him to pay,” a more diplomatic landlord Bell said later when the matter was settled.

Sorry, I’m more interested in the Groupoids as such, but being an American married to a Brit makes me notice these things.

Posted by: Jonathan Vos Post on December 5, 2008 4:20 AM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

“Doing a runner” a general term from leaving before something you would find unpleasant, which may or may not be involve anyone else being (ahem) “disadvantaged” (eg, not paying a restaurant bill). Eg, “I hate icing so I did a runner before they bought round the cake”.

Posted by: bane on December 5, 2008 6:39 AM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

… but with an implication that it’s something that, morally, one really ought to have stayed for, whether it’s paying for a meal, facing the music for ones crimes, or doing ones photogenic duty as an upstanding conference participant.

Posted by: Tim Silverman on December 5, 2008 12:46 PM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

For another view of the Fest, read Jeff Morton’s post.

Posted by: David Corfield on December 5, 2008 10:39 AM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

John, I can’t access the slides in your /groupoidification subdirectory — it looks like some sort of permission problem.

Posted by: Jamie Vicary on December 12, 2008 2:52 PM | Permalink | Reply to this

### Re: Groupoidfest 08 - A Brief Report

Sorry — I don’t know why my directories sometimes mysteriously switch from readable-by-all to not-readable-by-all. It should be fixed now.

Posted by: John Baez on December 12, 2008 4:46 PM | Permalink | Reply to this

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