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June 23, 2008

Aldrovandi on Non-Abelian Gerbes and 2-Bundles

Posted by Urs Schreiber

guest post by Bruce Bartlett

Just a quick report back on one of the talks given at the workshop on categorical groups in Barcelona - the one given on Thursday afternoon by Ettore Aldrovandi. Tomorrow I hope to report back on part II of Tim Porter’s talk (slides) on Friday. (The other talks on Friday were very important and impressive too of course, focusing on stable homotopy theory… but I don’t know enough to say anything intelligent there. I hope that a certain Professor Porter will report back on the morning talk by Behrang Noohi. [he did - urs])

Aldrovandi on non-abelian gerbes and 2-bundles

Ettore Aldrovandi’s talk was titled Butterflies, morphisms between gr-stacks, and non-abelian cohomology There were lots of ideas in this talk, but what came through loudest and clearest for me is how the stacky picture of 2-bundles contributes two important additions which should really be added to the picture of 2-bundles John presented in his talk (page 17 of John’s slides). I’ll recall the theorem for you here:

If 𝒢\mathcal{G} is a strict 2-group and XX a manifold, then there is a 1-1 correspondence between

(a) equivalence classes of principal 𝒢\mathcal{G}-2-bundles over XX

(b) elements of the Cech cohomology H(X,𝒢)H(X, \mathcal{G})

(c) homotopy classes of maps f:XB|N𝒢|f : X \rightarrow B |N \mathcal{G}|

(d) elements of the Cech cohomology H(X,|N𝒢|)H(X, |N \mathcal{G}|)

(e) isomorphism classes of principal |N𝒢||N\mathcal{G}|-bundles over XX.

What’s missing here are the following:

(f) isomorphism classes of gerbes which are torsors over the gr-stack represented by 𝒢\mathcal{G}
(in the sense of Breen)

(g) isomorphism classes of gerbes bounded by the crossed module associated to 𝒢\mathcal{G} (in the sense of Debremaeker).

What is a gerbe bounded by a crossed module? It was defined by Debremaeker in the 70’s. A gerbe EE over XX banded by a crossed module HGH \rightarrow G consists of a gerbe EE over XX and a functor J:EGTorJ : E \rightarrow G-\mathrm{Tor} compatible with the crossed-module structure maps. I think that’s a pretty neat definition… how does it relate to previous definitions of nonabelian gerbes? Actually Google has just shown me a paper by Jurco on Crossed module bundle gerbes; classification, string group and differential geometry which seems to answer these questions… mmm.

Anyhow, Ettore’s talk was very interesting to me because he spoke a language “closer to Brylinski” than I’ve experienced in other formulations of 2-bundles and gerbes. I like old-school geometry in the form of bundles, sections, vector fields, connections, and so on. I am a big fan of the higher gauge theory program, but it seems to me that none of the ways to talk about 2-bundles listed in (a)-(e) give a hands-on geometric way to handle them - and that’s what we need, if we want to go through with John and Urs’s programme of computing characteristic classes for String bundles using 2-connections, and so on.

Here’s another question about higher gauge theory I have: we know that the strict version string 2-group corresponds to the crossed module P^ k(G)P k(G)\hat{P}_k(G) \rightarrow P_k(G) of paths in GG. How are we going to do differential geometry on this big thing? For 2-connection purposes, should we rather be working with the smaller, weak model?

Coming back to Ettore’s talk. He recalled how these two descriptions (noabelian gerbes versus gerbes banded by a crossed module) were related, and he showed that the whole thing is one huge equivalence of 2-categories… in fact it’s so canonical this equivalence works locally so it’s really an equivalence of 2-stacks. He ended the talk by showing how one can use the beautiful language of butterflies to represent morphisms of weak 2-groups and hence make this whole thing functorial in the 2-group 𝒢\mathcal{G}.

Anyhow sorry about the higgledy-piggledy nature of this post but I’m having some trouble with the server (probably the fault is on my side).

Posted at June 23, 2008 12:35 PM UTC

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6 Comments & 1 Trackback

Re: Aldrovandi on Non-Abelian Gerbes and 2-Bundles

Hi Bruce,

thanks for the report!

Here is a very quick reply:

1) A gerbe by itself always corresponds to an AUT(H)AUT(H)-2-bundle. To get structure crossed modules different from AUT(H):=(HAdAut(H))AUT(H) := (H \stackrel{Ad}{\to} Aut(H)) one needs to play the tricks that you mention.

Notice that every crossed module (HG)(H \to G) has a canonical morphism into AUT(H)AUT(H) which on HH is the identity and on GG the map GAut(H)G \to Aut(H) that is part of the data of a crossed module.

2) Even though the String 2-group is big, it is made of (Fréchet) manifolds. So if you are happy with differential geometry albeit infinite dimensional, there is nothing to stop you here.

The infinite-dimensionality here may appear to be a pain, but in many cases it is actually very useful: the centrally extended loop groups appearing in the strict String 2-group appear all over the place in related topics and in many cases the existence of the inf. dimensional strict version of the String 2-group can be understood as providing the reason why these affine groups appear.

3) But it is true that one may want to make use of the fact that the big strict String Lie 2-group is entirely governed by a finite dimensional albeit weak Lie 2-algebra. One wants to be able to describe String 2-bundles with connections using locally differential forms with values in the small weak Lie 2-algebra.

The necessary prerequisite for doing so we have developed in L L_\infty-connections, as you know.

There, everything is plain differential geometry. We show how to explicitly construct String 2-connections from Lifts of ordinary gg-connections when a certain Chern-Simons 3-connection trivializes.

But what you probably want to see eventually is the integration of these 2-connections to full nonabelian differential 2-cocycles. There is a systematic method for doing so which I describe in section 7. As pointed out there, it reproduces in special cases a procedure once given by Brylinski and MacLaughlin.

With Hisham Sati in Bonn our plan is to spell out section 7.5 and then spell out the explcit construction of String 2-bundles and Fivebrane 6-bundles with connection along these lines.

Posted by: Urs Schreiber on June 23, 2008 1:15 PM | Permalink | Reply to this
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 1:23 PM

Re: Aldrovandi on Non-Abelian Gerbes and 2-Bundles

Hi Bruce,

there was a slight typesetting issue where you recall the definition of a gerbe bounded by a crossed module.

Please check if the way I resolved it is correct: a functor from the gerbe to the category of GG-torsors.

That’s what I could make of what you typed, but I am not sure that’s what was meant.

I suppose we are to think of the “gerbe” as an extended groupoid then, here in this context?

Posted by: Urs Schreiber on June 23, 2008 9:24 PM | Permalink | Reply to this

Re: Aldrovandi on Non-Abelian Gerbes and 2-Bundles

Yes that’s right, a functor to the category of GG-torsors. I always think of gerbes (stacks) as “smooth groupoids”… the basic data is the groupoid “assigned to the point”, and the rest just tells us how to make sense of smooth families. Thanks for your comments above; it’s going to take me sometime I fear to understand what’s going on in the world of nonabelian gerbes. Everytime I come back to this subject I’ve forgotten fundamental stuff you’ve explained to me before.

Posted by: Bruce Bartlett on June 23, 2008 11:27 PM | Permalink | Reply to this

Re: Aldrovandi on Non-Abelian Gerbes and 2-Bundles

I always think of gerbes (stacks) as “smooth groupoids”… the basic data is the groupoid “assigned to the point”

I agree that a stack on an abstract site such as that of manifolds is a good way to talk about smooth groupoids. This is precisely the point of view of Chen and diffeological smooth spaces, once categoriefied.

But “gerbe” is not the same as “smooth groupoid”. And not every stack is one on manifolds.

Once upon a time, a gerbe on a space XX was defined to be a locally non-empty and transitive stack on the site of open subsets of XX. That kind of gerbe cannot even be evaluated on a point!

Many things which live in (2-)categories equivalent to (certain) gerbes are nowadays addressed as “gerbes”. I am not a fan of that habit. Nobody says “sheaf” when what they really have is a bundle, nobody says “sheaf” when what they really have is a 1-cocycle, even though all these things live in equivalent categories (when suitably defined).

Why be less precise in a more complicated situation?

The model of a gerbe which consists of a certain groupoid over base space is really a special case of a cocycle. (And it’s a coincidence of high abelianness that in the abelian case that groupoid happens to be the same as the total space of a 2-bundle.)

The entire topic would be less mysterious if these difference were being taken into account in the terminology. Or so I think.

Posted by: Urs Schreiber on June 24, 2008 3:16 PM | Permalink | Reply to this

Re: Aldrovandi on Non-Abelian Gerbes and 2-Bundles

I agree wholeheartedly. Another misuse is to refer to gerbes as being merely the categorification of linebundles. Those provide a useful class of examples but to restrict to those is like restricting to 1-dimensional vector spaces and trying to do linear algebra!

Posted by: Tim Porter on June 24, 2008 3:45 PM | Permalink | Reply to this

Re: Aldrovandi on Non-Abelian Gerbes and 2-Bundles

Here are the references to Debremaeker:

R. Debremaeker, 1976, Cohomologie met Waarden in een Gekruiste Groepenschoof op ein Situs, Ph.D. thesis, Katholieke Universiteit te Leuven.

R. Debremaeker, Cohomologie à valeurs dans un faisceau de groupes croisés sur un site. I , Acad. Roy. Belg. Bull. Cl. Sci. (5), 63, (1977), 758-764.

R. Debremaeker, Cohomologie à valeurs dans un faisceau de groupes croisés sur un site. II, Acad. Roy. Belg. Bull. Cl. Sci. (5), 63, (1977), 765-772.

R. Debremaeker, Non abelian cohomology, Bull. Soc. Math. Belg., 29,(1977), 57-72.

You may also be interested in the preprints by Ettore and James Milne:

E. Aldrovandi, 2005, 2-Gerbes bound by complexes of gr-stacks, and cohomology


J. S. Milne, 2003, Gerbes and abelian motives.

Posted by: Tim Porter on June 24, 2008 2:55 PM | Permalink | Reply to this

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