## June 8, 2008

### 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.

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.

Posted at June 8, 2008 7:50 PM UTC

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### 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.

Posted by: David Roberts on June 9, 2008 1:49 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

I say really attentive, because my name was on the list - last week, but not any more.

Posted by: David Roberts on June 9, 2008 2:02 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

This is especially for David but it is relevant for all I hope.

First it is a shame that David won’t be there as I was especially pleased to see his name on the list.

Secondly, thanks to John for the `excellent’, I hope I can live up to it! In any case I am working hard on extending the Menagerie (see this blog on March 21, 2008).

My talk at Barcelona will revisit Larry Breen’s Bitorsors paper (L. Breen, 1990, Bitorseurs et cohomologie non abélienne, in The Grothendieck Festschrift, Vol. I , volume 86 of Progr. Math., 401–476, Birkhauser Boston, Boston, MA.) There is so much in the way of useful techniques in that paper, even though it is 18 years old) that it needs a lot of time to digest them. (John made a related comment recently.) Some of the techniques Larry introduced there (and which are described also in his beautiful article based on his Minneapolis talks) have become common currency amongst the denizens of this café, but there are, I think, still more things to reveal and perhaps to push further and some questions that are still to be asked, especially with regard to the long exact sequences in non-Abelian cohomology.

I have reworked some of the material in the version of the menagerie that I put on the blog in March and have another section or two that are not yet typed up to my satisfaction, however once the meeting is over I will put some of the more finished material in a second edition of the menagerie here. Some of that new material will be discussed in my talk.

I will finish up my talk (if I have time!) in discussing the work of Turaev on HQFTs and its possible connections here. The joint paper is now finally published in JHRS May 20, so you can all see what I will say, at least in part, and by that time of the week you may be excused for needing to have a ‘power nap’ as it looks as if we will be busy.

Hopefully someone who is a faster typer than I am, and more versed in the mysteries of itex will put summaries of the talks on this blog for the benefit of those who can not be there.

Posted by: Tim Porter on June 9, 2008 7:15 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

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.

Sorry you can’t make it… I look forward to your world premiere on the cafe, and I will indeed try hard to absorb it and also point others to your work.

Posted by: Bruce Bartlett on June 9, 2008 11:26 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

Tim wrote:

My talk at Barcelona will revisit Larry Breen’s bitorsors paper:

• L. Breen, 1990, Bitorseurs et cohomologie non abélienne, in The Grothendieck Festschrift, Vol. I , volume 86 of Progr. Math., 401–476, Birkhauser Boston, Boston, MA.

There is so much in the way of useful techniques in that paper, even though it is 18 years old) that it needs a lot of time to digest them. (John made a related comment recently.) Some of the techniques Larry introduced there (and which are described also in his beautiful article based on his Minneapolis talks) have become common currency amongst the denizens of this café…

Breen’s Grothendieck Festschrift paper is in French, and I must sadly admit that this has kept me from studying it carefully. I can fight my through mathematical French, but it’s like swimming through molasses. Luckily I was able to attend his course on this subject in Minneapolis — and his course notes are available online in a polished form suitable for Americans:

(Someday this will appear in the proceedings of the Minneapolis conference, along with many other things including a long introduction to quasicategories by André Joyal.)

I’m really eager to hear about long exact sequences in nonabelian cohomology!

Hopefully someone who is a faster typer than I am, and more versed in the mysteries of itex will put summaries of the talks on this blog for the benefit of those who can not be there.

Alas, I can’t commit myself to doing this: I prefer to see Barcelona and talk math with old friends than sit in my hotel room typing up notes. I may discuss a few talks in This Week’s Finds — but with no pretensions to completeness.

With luck, many speakers will make slides of their talks available electronically. I did, I’m pretty sure Bruce Bartlett will, and I suspect Derek Wise will too.

(If your remark was a hint to me, mine is a hint back to you. )

Posted by: John Baez on June 9, 2008 6:20 PM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

My idea is to make the revised and extended version of the Menagerie available. I have not had the courage to prepare detailed slides yet having been moving around somewhat (slides are heavy!) and Beamer presentations are long to prepare, so … but the notes have all the stuff I will talk on in them. I may make a cut down version available slightly later.

The Puppe sequences that Larry Breen uses should be very useful, but there are difficulties and that is where it gets interesting!!!!! The simplicity of their use, however, is what makes me feel that they make a very useful tool.

Posted by: Tim Porter on June 10, 2008 6:44 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

The hint about someone giving a brief summary of the ideas on the blog was not directed at John nor anyone else. It was just me noting how useful these summaries are when someone does have the time to do them. We cannot all get to these meetings. I for one wish I could get to more. However, the important new ideas that emerge can be extremely useful, so thanks to everyone who has done this on this blog and on the various others akin to it.

Posted by: Tim Porter on June 10, 2008 6:49 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

With luck, many speakers will make slides of their talks available electronically.

The slides for my talk are now available here (abstract). It’s about how unitary 2-representations of finite groups and their 2-characters provide a discrete toy-model for a ‘categorified geometric quantization’ (a topic which has cropped up recently).

Where are the 2-groups? Well they are a bit in there but I’ve mostly been wimping out, because I’ve only been dealing with the so-called ‘untwisted’ finite group TQFT, where the underlying 2-group is trivial… though understanding the geometry even in this simplest of cases was tough enough for me! I don’t think I properly understand yet the geometry in the twisted (genuine 2-group) setting.

Posted by: Bruce Bartlett on June 14, 2008 12:14 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

I love the ‘decategorified Yoneda lemma’ in your talk.

It’s especially nice when you think about it quantum-mechanically. In plain English: to be something is to have an amplitude to be anything.

Posted by: John Baez on June 14, 2008 3:09 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

You have an extra ‘i’ in ‘ambiijunction’ on pp. 33-35. Otherwise the slides are perfect.

Posted by: David Corfield on June 14, 2008 11:22 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

Ok fixed, thanks. John - I like your motto. How about: I ample, therefore I am. $\quad$ :-)

Posted by: Bruce Bartlett on June 14, 2008 3:01 PM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

> long exact sequences in nonabelian cohomology!

Apart from nonabelian cohomology in the sense of Giraud, Dedecker etc. there is also nonabelian homological algebra, that is doing derived functors in the setup of categories which are not abelian. Thus one has there long exact sequences by definition (infinitely long in one direction, not just “9-term exact sequence”). Some people in western Europe (e.g. Bourn) and Georgian school of category theory (Janelidze etc. book by Innasaridze) axiomatised the conditions on a category to be able to do something along this way. For example there are notions like semiabelian category.

Recently A. L. Rosenberg went into another direction considering what is the needed structure to do derived functors and he adds this structure (as there is homology and cohomology, he defines left ‘exact’ and right ‘exact’ structures on categories as certain Grothendieck pre(co)topologies). So even for abelian categories one can choose an exotic structure. One of the applications is the definition (of a variant of) algebraic K-theory (K_i for nonnegative i) which does not use classifying spaces, but is fully categorical. The basic properties are the same as for Quillen K-theory (maybe it is even the same but neither a proof nor a counterexample is known).

The bulk of this work is yet not available. In May 2007, AR gave talks at the K-theory school in Trieste and posted there some lectures; now the extended lecture notes are at Max Planck Bonn preprint server:

Rosenberg, A.
Topics in Noncommutative algebraic geometry, homological algebra and K-theory
MPIM2008-57 (pdf)

(carefully: more than 100 pages).

The nonabelian homological algebra part and applications to K-theory are lectures 3-6, thus starting page 45.

Posted by: Zoran Skoda on June 14, 2008 5:31 PM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

And now for something completely different…

Here are some comments on the slides

1) On page 15, you mention well-pointed topological 2-groups, but then go on to explain what a well pointed topological 1-group is again. Would it be easiest to explain what a well-pointed 2-group is referring to the associated crossed module?

2) On page 21 - it isn’t necessary for the open cover to be a good one to get Segal’s result, only that the space is paracompact (actually only that the cover admits partitions of unity, but who’s quibbling).

Posted by: David Roberts on June 9, 2008 2:15 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

It’s too bad you won’t be attending, David! If you ever want to talk more about your work, we can set up a guest post. And if you have something you’d like me to point the Barcelonans to, just let me know.

On page 15, you mention well-pointed topological 2-groups, but then go on to explain what a well pointed topological 1-group is again.

Whoops! That’s an artifact of careless cutting and pasting. As you suggest, I meant to say a topological 2-group is well-pointed if both the topological groups in its corresponding crossed module are well-pointed.

(For people reading this and scratching their heads: a topological group is well-pointed if the identity — hence any other point — has some neighborhood of which its a deformation retract.)

(And for those people: I could have made my previous parenthetical remark more scary using jargon like ‘NDR pair’ or ‘closed cofibration’ — so, you should be glad I’m too nice for that.)

On page 21 - it isn’t necessary for the open cover to be a good one to get Segal’s result, only that the space is paracompact…

Whoops again! And this was not just a mere typo — more like a thinko. Thanks.

Posted by: John Baez on June 9, 2008 3:31 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

I wish I would come to Barcelona, too, for at least some of these events. But one can’t have everything (especially not if 250,- conference fee is being charged on top of everything:-/). Please keep us informed!

Posted by: Urs Schreiber on June 9, 2008 4:39 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

slide 5: is it “it is deformation retract” or “it is a deformation retract”?

Posted by: Urs Schreiber on June 9, 2008 5:07 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

The latter, since I’m not Russian. Thanks!

Posted by: John Baez on June 9, 2008 7:42 AM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

Here is just a hint about Deligne’s notion of virtual objects. Recall that, given a triangulated category $T$, one can define its Grothendieck group $K_0(T)$ as the free group generated by set of objects of $T$ divided by the following “additivity relations” (it will be an abelian group, so that the multiplication is written $+$).

1) Any null object in $T$ is a zero in $K_0(T)$.

2) For any distinguished triangle $X'' \to X \to X'\to X''[1]$ we set $X=X'+X''$.

The idea of Deligne is that we can consider the $2$-group freely generated by the objects of $T$ divided by the analog of 1) and 2): just replace equalities by isomorphisms (and some coherence axioms). In other words, we somehow categorify the definition of $K_0$. We then obtain a $2$-group (the category of virtual objects of $T$) which encodes enough information to catch $K_0(T)$ and $K_1(T)$. The automorphisms of objects of $T$ have a non trivial contribution now: they will define canonically some classes in $K_1(T)$. But, as we get in higher category theory, we also need some higher structure on $T$: instead of triangulated categories, we have to consider stable $(\infty,1)$-categories (or equivalently, stable model categories) $T$. This has been worked out by Muro and Tonks. A student of Neeman (Vaknin) studied this kind of things for genuine triangulated categories under the price of modifying the notion of distinguished triangle.

$2$-groups are simple enough to be understood rather explicitly, but the idea is that, if you replace the notion of group by a reasonnable notion of $\infty$-group, then you will obtain via a similar process the space $K(T)$ of $K$-theory of $T$ (whose $i$th homotopy group is $K_i(T)$). Note that, if you admit that connected Kan simplicial sets define a such a reasonable notion of (classifying spaces of) $\infty$-group, Waldhausen’s definition of $K$-theory is constructed exactly following this simple point of view: Waldhausen’s construction consists very precisely to describe the classifying space of the corresponding $\infty$-group. If you wish some more precise and conceptual point of view, you will see that the fact $K$-theory is the universal $\infty$-group generated by the additivity relations follows formally from the work of Tabuada: http://arxiv.org/abs/0706.2420

Posted by: Denis-Charles Cisinksi on June 9, 2008 1:19 PM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

Posted by: John Baez on June 9, 2008 6:30 PM | Permalink | Reply to this

### Re: 2-Groups in Barcelona

Everyone interested in — or at! — the categorical groups workshop should check out David Roberts’ talk on fundamental 2-groups and 2-covering spaces.

Posted by: John Baez on June 19, 2008 9:41 AM | Permalink | Reply to this
Read the post Aldrovandi on Non-Abelian Gerbes and 2-Bundles
Weblog: The n-Category Café
Excerpt: A talk by Aldrovandi on different perspectives on gerbes and 2-bundles.
Tracked: June 23, 2008 12:41 PM
Read the post Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups
Weblog: The n-Category Café
Excerpt: A talk by Behrang Noohi.
Tracked: June 23, 2008 1:24 PM
Read the post Tim Porter on Formal Homotopy Quantum Field Theories and 2-Groups
Weblog: The n-Category Café
Excerpt: Tim Porter on HQFT
Tracked: June 24, 2008 5:01 PM

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