### 2-Groups in Barcelona

#### Posted by John Baez

Next Saturday, I’m flying to Spain. Then on Monday I’ll go to this:

- Workshop on Categorical Groups, Institut de Matemàtica de la Universitat de Barcelona, June 16-20, 2008, organized by Pilar Carrasco, Josep Elgueta, Joachim Kock and Antonio Rodríguez Garzón.

A categorical group, or ‘2-group’ for short, is a category equipped with structures mimicking those of a group. So, for example, you can ‘multiply’ objects, and every object has an inverse. I like 2-groups because they’re much easier to handle than more general $n$-groups — but they still offer many fresh opportunities for categorifying familiar math and discovering new connections between fields.

Let me tell you a bit about this workshop…

Pilar Carrasco has told me how the workshop will run: each day there will be one long talk in the morning, and a couple of shorter ones in the afternoon. The long talk will start with an hour of ‘introduction’ — warmup material, I guess. Then there will be a formal lecture for 45 minutes, and then 45 minutes of discussion. That’s great, because it’s enough time to really learn something!

On Monday I’ll start the show by explaining classifying spaces for topological 2-groups, based on a paper with Danny Stevenson. You can already see the slides on my website.

On Monday afternoon, fellow Café regular Bruce Bartlett will speak about representations of finite groups on finite-dimensional 2-Hilbert spaces. Here it’s not the groups but their representations that are getting categorified! His talk is available here. It’s also worth trying the slides for his talk Aspects of duality in 2-categories, which he gave at the recent workshop on Categories, Logic and Foundations of Physics. You can see a video of this talk, too. Even better, try his talk The geometry of 2-representations of finite groups at a conference in honor of Max Kelly, and the paper he wrote for that conference: The geometry of unitary 2-representations of finite groups and their 2-characters. This stuff is really cool; I should explain it here sometime, or coax him to do so.

Then Laiachi El Kaoutit Zerri will speak about joint work with C. Menini and A. Ardizzoni on coendomorphism bialgebroids.

On Tuesday morning, Enrico Vitale will teach us about homological algebra in 2-categories, leading up to some ideas about the correct definition of an abelian 2-category! He’ll start describing some constructions in the 2-category of (symmetric) categorical groups: kernel and cokernel, relative kernel and relative cokernel — and some mysterious things called the ‘pip’ and ‘root’. These constructions are all instances of bilimits (or co-bilimits).

On Tuesday afternoon, Aurora del Río — who visited Riverside for a year — will speak about 2-groups categorifying the usual $K$-theory groups $K_i R$ of a ring $R$. It will be good to see her again and find out what she’s up to.

Then Luis Javier Hernández will speak on joint work with Aurora del Río and M. T. Rivas on categorical groups and ‘$[n,n+1]$-types of exterior spaces’. I don’t know what those are, but he promises to give a new description of the classifying space of a categorical group, so I’m looking forward to that.

On Wednesday morning, my former student Derek Wise will talk about representations of 2-groups on infinite-dimensional 2-Hilbert spaces, based on a paper he’s writing with Aristide Baratin, Laurent Freidel and me. Actually, there’s no definition of an infinite-dimensional 2-Hilbert space yet! But the ‘measurable categories’ of Crane and Yetter come close, so we’re using those for now, and calling them ‘higher Hilbert spaces’. When full-fledged 2-Hilbert spaces come along, some of these will give examples.

It looks like Wednesday afternoon we get to take a break and look around the city.

On Thursday morning, Behrang Noohi will speak about a method for conveniently specifying maps between weak 2-groups. He’ll give applications to:

- the classification of group actions on stacks (and calculation of their quotients).
- The functorial study of principal 2-bundles (e.g., explicit description of ‘extension of the structure 2-group via a lax monoidal functor’).
- A nonabelian generalization of Deligne’s result relating additive functors between Picard stacks to the derived category of abelian sheaves.

On Thursday afternoon, his collaborator Ettore Aldrovandi will continue discussing these ideas.

Later on Thursday, Pietro Polesello will speak about a concept of ‘character’ for locally constant stacks. Locally constant stacks are a categorified version of locally constant sheaves, and they can be described by actions of the fundamental 2-group of the space these stacks are living on, just as locally constant sheaves are described by actions of the fundamental group. In this talk he’ll how how to associate to a locally constant stack a ‘character’, that is, a locally constant sheaf on the loop space which has character-like properties. This is done by using the notion of character of a representation of a 2-group, which was introduced by Ganter and Kapranov in the case of discrete 2-groups.

Friday morning we’ll have an excellent talk by another Café regular: Tim Porter. He’ll speak about classifying spaces for categorical groups, and their relations to non-abelian cohomology, topological quantum field theories, and homotopy quantum field theories.

Friday afternoon, Fernando Muro will speak on joint work he’s done with Hans-Joachim Baues on categorical groups in brave new algebra. This sounds like lots of fun! He says: “Elmendorf and Mandell (2006) modified Segal’s construction to obtain brave new rings (i.e. ring spectra) out of categories with ring structure. Similarly for modules, etc. The aim of this talk is to present a functor going in the opposite direction. We will introduce categorical (commutative) rings, categorical algebras, categorical modules… together with their graded versions, and we will show how to associate such 2-dimensional algebraic structures to a symmetric spectrum. This will be done by means of a theory of homotopy 2-groups for spectra.”

Later, Andy Tonks will give the final talk, on categorical groups in $K$-theory and number theory. This looks like lots of fun too — at least if I can understand it. Maybe someone can help me out, by explaining this key sentence in his abstract: “Deligne remarked that the first Postnikov piece of the $K$-theory spectrum of $R$ is in fact classified by a categorical group $V(R)$ of so-called virtual objects.” This sounds almost comprehensible. What’s the 2-group of virtual objects of a ring $R$?

So, if my brains don’t blow a fuse, I’m bound to learn a lot at this workshop. I hope to see some of you there!

After this workshop, I’ll spend some time visiting Pilar Carrasco in Granada. Then I’ll go back to Barcelona for the big workshop on homotopy theory and higher categories from June 30th to July 5th. Then to Paris.

## Re: 2-Groups in Barcelona

The really attentive people may have detected the appearance of another certain Cafe regular’s name on the list of attendees. This person sends their apologies, and grins and grinds their teeth only slightly at the funding methods for student travel in their university.

So instead of giving my talk in Barcelona, I will attempt to give Cafe patrons a world premiere. Hopefully those Cafe patrons present in Barcelona can point other workshoppers in the direction of my efforts.