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August 15, 2008

Connections on Nonabelian Gerbes and their Holonomy

Posted by Urs Schreiber

Finally this sees the light of day:

U.S. and Konrad Waldorf
Connections on non-abelian Gerbes and their Holonomy
arXiv:0808.1923

Abstract: We introduce transport 2-functors as a new way to describe connections on gerbes with arbitrary strict structure 2-groups. On the one hand, transport 2-functors provide a manifest notion of parallel transport and holonomy along surfaces. On the other hand, they have a concrete local description in terms of differential forms and smooth functions.

We prove that Breen-Messing gerbes, abelian and non-abelian bundles gerbes with connection, as well as further concepts arise as particular cases of transport 2-functors for appropriate choices of structure 2-group. Via such identifications transport 2-functors induce well-defined notions of parallel transport and holonomy for all these gerbes. For abelian bundle gerbes with connection, this induced holonomy coincides with the existing definition. In all other cases, finding an appropriate definition of holonomy is an interesting open problem to which our induced notion offers a systematical solution.

This builds on

Smooth functors vs. differential forms - which establishes the relation between smooth 2-functors with values in Lie 2-groups and differential L L_\infty-algebraic connection data;

Parallel transport and functors - which establishes the relation between transport nn-functors and nn-bundles/(n1n-1)-gerbes with connection for n=1n=1

and realizes the construction I did with John in Higher gauge theory as a cocycle in second nonabelian differential cohomology which represents a globally defined transport 2-functor.

nn-Café regulars will remember some discussion of the development of the notion of locally trivializable transport nn-functors and their classification in nonabelian differential cohomology in early Café entries such as

On Transport, Part I
On Transport, Part II

and many other ones, to some extent summarized in this big set of slides. You might enjoy Konrad’s more readable slides .

For instance you can read about 2-vector transport for associated String 2-bundles (discussed for instance here), twisted vector bundles with connection as quasi-trivializations of 2-vector transport (which I talked about at the Fields institute), local formulas for abelian and nonabelian surface holonomy as vaguely conceived before here, see all this now related to Street’s theory of codescent in the context of differential cohomology which is the right formalization of those 2-paths in the Cech 2-groupoid which I kept talking about once upon a time.

Alas, a couple of things didn’t quite make it into this file or are only indicated. But if you could wait that long you probably can also wait a little longer still…

Posted at August 15, 2008 6:59 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1764

19 Comments & 6 Trackbacks

Read the post Waldorf on Transport Functors and Connections on Gerbes
Weblog: The n-Category Café
Excerpt: A talk on parallel 2-transport.
Tracked: September 6, 2008 4:39 PM
Read the post Talk in Göttingen: Second Nonabelian Differential Cohomology
Weblog: The n-Category Café
Excerpt: A talk on second nonabelian differential cohomology.
Tracked: October 19, 2008 9:42 PM
Read the post Codescent and the van Kampen Theorem
Weblog: The n-Category Café
Excerpt: On codescent, infinity-co-stacks, fundamental infinity-groupoids, natural differential geometry and the van Kampen theorem
Tracked: October 21, 2008 9:33 PM
Read the post Twisted Differential Nonabelian Cohomology
Weblog: The n-Category Café
Excerpt: Work on theory and applications of twisted nonabelian differential cohomology.
Tracked: October 30, 2008 7:49 PM
Read the post Higher Structures in Math and Physics in Lausanne
Weblog: The n-Category Café
Excerpt: A workshop in Lausanne on Higher Structures in Mathematics and Physics
Tracked: November 2, 2008 5:20 PM
Read the post Bär on Fiber Integration in Differential Cohomology
Weblog: The n-Category Café
Excerpt: On fiber integration in differential cohomology and the notion of generalized smooth spaces used for that.
Tracked: November 26, 2008 7:57 AM

Re: Connections on Nonabelian Gerbes and their Holonomy

Would it be asking too much if someone can open an entry on Gerbes on ncatlab? I would like to learn about them…

Posted by: Daniel de Franša MTd2 on June 24, 2009 6:36 AM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

There does seem to be a lack of GERBE in the nLab. Until something is added on this, you can find an introduction to gerbes (very much derived from Larry Breen’s notes) in the Menagerie notes that you can access from my nLab page.

Breen’s notes are on the archive as math.CT/0611317. They are good and clear.

Posted by: Tim Porter on June 24, 2009 2:10 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

I don’t think Breen’s notes are clear. They sure aren’t clear to me. They may be clear if you know what he means. But what if you don’t?

Posted by: Eugene Lerman on June 24, 2009 4:24 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

I suppose that, as always, it depends from where you are coming! If we were to launch into Gerbes for nLab what would be the starting point that would be useful. What slant should be prioritised, e.g. gerbes from non-abelian cohomology, which is what Breen is using (is that the problem?) Should things be on a space with an open cover or is a topos and an approach from that angle better. I know which I like since my background was in homotopy theory (spatial with open covers) but that is just me.

Of course one way to do this is to start an entry in the nLab yourself. You need not worry, it will quickly fill up and you will be able to complain when it does not explain! (and will be encouraged to do so).

Posted by: Tim Porter on June 24, 2009 4:48 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

I have created a stub on the nLab and will start writing something later on today if possible… but feel free to start without me!

Posted by: Tim Porter on June 24, 2009 4:57 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

I am afraid that I will be cheering from the sidelines for the next few weeks.

If I were to write such an entry I would start by writing out the example of principal G-bundles (G fixed) over the category of manifolds (big site). Follow up by principal G-bundles with connections.

Mention why this is a different piece of an elephant than the one Hitchin described in “What is a gerbe?”…

Posted by: Eugene Lerman on June 24, 2009 7:57 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

Ordinary bundles in their Cech incarnation I find the most accessible route into gerbish.

Posted by: jim stasheff on June 25, 2009 2:11 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

Do you know if anyone has done it this way?
I mean by this: took Cech cocycles on some space with coefficients in a fixed group and checked the axioms for a gerbe?

Posted by: Eugene Lerman on June 25, 2009 9:34 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

Do you know if anyone has done it this way? I mean by this: took Cech cocycles on some space with coefficients in a fixed group and checked the axioms for a gerbe?

Hm, what do you mean? The only way I currently see how to interpret this question it has the obvious answer:

the basic theorem of GG-gerbes is that they are classified by nonabelian cohomology with coefficients in the automorphism 2-group AUT(G)AUT(G).

This says that they are given by nonabelian Cech cocycles given by “Morita generalized” morphisms or anafunctor (or whatever you like to call them)

XC(Y)BAUT(G) X \stackrel{\simeq}{\leftarrow} C(Y) \to \mathbf{B}AUT(G)

with C(Y)C(Y) the Cech 2-groupoid of a cover YXY \to X.

So, what Jim Stasheff is asking for is indeed the starting pivotal point of [[gerbe (general idea)]]:

a GG-gerbe on XX is whatever is classified by a cocycle

XBAUT(G) X \to \mathbf{B}AUT(G)

just like a GG-principal bundle is whatever is classified by a cocycle

XBG. X \to \mathbf{B}G \,.

And what is it that is classified by cocycles? Their [[homtopy fiber]]s PXP \to X.

Regarded in the world of stacks, that gives gerbes.

(To Jim: I didn’t forget Stasheff-Wirth at [[gerbe (general idea)]]. I thought that would be the perspective you’d like most… :-)

Posted by: Urs Schreiber on June 25, 2009 9:53 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

I meant my question literally: is there a textbook/paper (preferably in English, but I suppose I can manage Russian too) where this is spelled out?

Posted by: Eugene Lerman on June 26, 2009 2:30 AM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

I meant my question literally: is there a textbook/paper (preferably in English, but I suppose I can manage Russian too) where this is spelled out?

But let me know what “this” is. My impression so far is that you are asking for the nonabelian Cech-cocycle description of gerbes. If so, this is in every standard text on gerbes, the first theorem to be proven about them.

For instance in Breen’s Notes on 1- and 2-gerbes it is section 5 cocycles and coboundaries for gerbes.

The cocycle description itself is equation (5.1.10), the classification theorem is mentioned and referenced on the bottom of page 14.

Or Ieke Moerdijk’s Introduction to the language of gerbes and stacks discusses the Cech-cocycle description of gerbes from page 16 on, and the classification theorem appears as theorem 3.1 on p. 21.

The statement is originally due to Giraud’s work Cohomologie non-abélienne.

In Brylinski’s book Loop spaces, characteristic classes and geometric quantization the cocycle description of gerbes is extracted in chapter 5.2 Sheaves of groupoids and gerbes and the classification theorem is theorem 5.2.8 on p. 200, 201.

The discussion of cocycles for gerbes is traditionally complicated by the fact that general sheaves of groups are used, instead of just a group, then there is the discussion of band, etc., all of which somewhat contributes to tending to hide a simple idea behind non-essential technical details.

Another thing that gerby tradition has is to express in linear formulas or rectangular diagrams what is intrinsically a nice geometric higher dimensional structure. The funny-looking nonabelian cocycle for a gerbe is really just a tetrahedron (the 3-simplex, since we are talking about a 2-cocycle) in BAUT(G)\mathbf{B} AUT(G).

I find this helpful, since it makes at once clear a lot of structure, such as for instance the nature of coboundaries. You can find these tetrahedra drawn in my work with John Baez, for instance the gerbe 2-cocycle tetrahedron is the title piece of John’s Namboodiri lecture slides Higher Categories, Higher Gauge Theory.

John recalls the theorem in question there on slide 10.

Finally, gerbes, in as far as they are nonabelian, are really objects associated to principal 2-bundles. The cocycle description of principal 2-bundles is more transparent, conceptually, as it is the 2-bundle that is associated by abstract nonsense to the 2-cocycle, whereas the gerbe comes from that only after some fiddling.

Accordingly, the nonabelian Cech cocycles in question here are discussed at length and in detail in the literature on 2-bundles by Toby Bartels, Igor Baković and Christoph Wockel. The relevant links are collected at [[principal 2-bundle]].

Finally, in case I am misunderstanding your question, maybe you are asking for literature that describes nonabelian Cech cocycles as nn-functors out of Cech nn-groupoids?

This is described in some detail for instance in the article that this thread here is about. Another discussion more in the style of the Lie-groupoid community is in section 2 of Ginot, Stiénon, GG-gerbes, principal 2-group bundles and characteristic classes.

Please let me know if that serves to answer the question.

Posted by: Urs Schreiber on June 26, 2009 7:05 AM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

This answers my question.
Thank you.

Posted by: Eugene Lerman on June 26, 2009 4:41 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

This answers my question. Thank you.

Okay, thanks for letting me know. So this shows that apparently one crucial point of our nnLab entries on gerbes had been missing.

To remedy this, I have now created [[gerbe (in nonabelian cohomology)]] and started filling in the above material.

Posted by: Urs Schreiber on June 26, 2009 5:02 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

Maybe to amplify:

[[Cohomology]] (Cech-, sheaf-, nonabelian-, whatever) is just homs H(X,A)=Hom(X,A)H(X,A) = Hom(X,A) in the right context.

A cocycle is just a morphism c:XAc : X \to A in the right context.

A coboundary just a 2-morphism ccc \Rightarrow c'.

A cohomology class just an equivalence class [c][c].

The thing classified by a cocycle c:XPc : X \to P just its [[homotopy fiber]].

Where throughout:

the “right context” where these statements are true/ make sense is not just the (,1)(\infty,1)-catgeory Top, but any (,1)(\infty,1)-category that “looks like Top”.

This means: any [[(infinity,1)-topos]].

Which in turn means: any [[(,1)(\infty,1)-category of (,1)(\infty,1)-sheaves]].

Which in plain English means: any context of [[\infty-stacks]].

Which in practice means: [[simplicial sheaves]] with weak equivalence remembered.

Which in the language favored around here means (for the case that the site is DiffDiff):

\infty-groupoids internal to [[diffeological spaces]] with \infty-[[anafunctors]] between them.

Posted by: Urs Schreiber on June 25, 2009 10:29 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

Well, the reason for my request is to study non cohomological homology, because I’d like to understand the stringy stuff Urs writes.

Posted by: Daniel de Franša MTd2 on June 24, 2009 5:19 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

the reason for my request is to study non cohomological homology

I guess you mean [[nonabelian cohomology]]. (?)

Would it be asking too much if someone can open an entry on Gerbes on ncatlab?

I have now added to the entry [[gerbe]] that Timothy Porter kindly created a chunk of material which I consider as the “general idea” of gerbes.

Please have a look and let me know about whatever questions arise, so that we can proceed with working the answers in.

Tim Porter and others planning to work on this entry I’d kindly ask to see my log about my changes at [[latest changes]].

Posted by: Urs Schreiber on June 25, 2009 10:45 AM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

“I guess you mean nonabelian cohomology”.

Sure, this is why I chose this blog post. But as you can see all the time in my posts, I make silly mistakes in writting, even in my mother language. Since I was a little child, It seems that sometimes words scramble in my mind. I’m thankful you took the effort to understand me.

Posted by: Daniel de Franša MTd2 on June 25, 2009 1:06 PM | Permalink | Reply to this

Re: Connections on Nonabelian Gerbes and their Holonomy

We have split [[gerbe]] apart into

[[gerbe (as a stack)]],

[[gerbe (general idea)]]

and

[[bundle gerbe]].

I also started

[[principal 2-bundle]],

[[principal infinity-bundle]]

and

[[fibration sequence]].

Everything here boils down to the discussion of fibration sequences in (,1)(\infty,1)-toposes of \infty-sheaves, really.

Unfortunately the server is really not responsive today. It must be a problem with the server, because also ssh-ing into the machine is a pain.

So I need to migrate the Lab to a different hosting company. Does anyone have any experience/suggestions/advice?

Posted by: Urs Schreiber on June 25, 2009 3:23 PM | Permalink | Reply to this

nLab responsiveness

I wrote:

Unfortunately the server is really not responsive today.

It’s working consistently fine now for a few hours already, unless I am dreaming.

And before that I did something that Jacques Distler urged me to do anyway, but which I hadn’t done in a while:

I removed the >\gt 70 MB backup copy of the total nnLab database file that was still sitting in “my web directory” waiting for being downloaded to my local machine.

I am not sure if this is just a coincidence, but maybe the wiki software is being slowed down by such a huge file just sitting around?

Posted by: Urs Schreiber on June 25, 2009 8:50 PM | Permalink | Reply to this

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