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November 13, 2006

Breen on Gerbes and 2-Gerbes

Posted by John Baez

Back in the summer of 2004, at the Institute for Mathematics and its Applications, there was a workshop on nn-categories. It was an intense, exhausting affair. Amid endless talks on various definitions of weak nn-category, Larry Breen gave two talks introducing us to gerbes and 2-gerbes. As the conference proceedings slouch slowly towards completion, you can now read his presentation, which has been polished into an excellent paper:

  • Lawrence Breen, Notes on 1- and 2-gerbes, to appear in nn-Categories: Foundations and Applications, eds. J. Baez and P. May.

    Abstract: These notes discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are mainly based on the author’s previous work in this area, which is reviewed here, and to some extent improved upon. The main emphasis is on the description of the explicit manner in which one associates an appropriately defined non-abelian cocycle to a given 1- or 2-gerbe with chosen local trivializations.
Posted at November 13, 2006 5:46 PM UTC

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4 Comments & 2 Trackbacks

Re: Breen on Gerbes and 2-Gerbes

Breen also mentions crossed modules as something
similar but I don’t see the direct comparison. What am I missing?
Since a gerbe is `on a space X’
I assumed the cross module should be over
C^\infty(X)??

Posted by: jim stasheff on November 19, 2006 8:01 PM | Permalink | Reply to this

crossed modules

Breen also mentions crossed modules as something similar but I don’t see the direct comparison.

The definition (1.9 in Breen’s text) of a crossed module of groups seems to be the standard one:

a crossed module of groups is

a group homomorphism

(1)t:H→G t : H \to G

together with a homomorphism

(2)α:G→Aut(H) \alpha : G \to \mathrm{Aut}(H)

such that

(3)H →t G Ad↘ ↓α Aut(H) \array{ H &\stackrel{t}{\to}& G \\ & {}_{\mathrm{Ad}}\searrow & \;\downarrow \alpha \\ && \mathrm{Aut}(H) }

and

(4)G×H →Id G×t G×G α(⋅)(⋅)↓ ↓Ad(⋅)(⋅) H →t G \array{ G \times H &\stackrel{\mathrm{Id}_G \times t}{\to}& G \times G \\ \alpha(\cdot)(\cdot) \downarrow\;\;\;\; && \;\;\;\; \downarrow \mathrm{Ad}(\cdot)(\cdot) \\ H &\stackrel{t}{\to}& G }

commute.

A Lie crossed module is a crossed module in the world of Lie groups.

I assume in Breen’s setup we don’t want to or need to specify the precise world we work in (compare the third paragraph in the introduction).

Posted by: urs on November 20, 2006 1:50 PM | Permalink | Reply to this

Re: crossed modules

Call me Jim the Obscure.
I was trying to ask about the relationship
(implicit in Breen but I don’t see it made explicit) between crossed modules and gerbes.

If I identify a vector bundle with its space of sections as a module over C^\infty X,
then can I identify a vector gerbe as a
crossed module similarly?

Posted by: jim stasheff on November 20, 2006 3:57 PM | Permalink | Reply to this

modules of sections

I was trying to ask […]

identify a vector bundle with its space of sections as a module over C ∞XC^\infty X,

Sorry! I misunderstood your question, then.

Taking the risk of misinterpreting your question once again (please bear with me), I would first like to remark that despite its name, the “crossed module” appearing in the theory of gerbes does not play a role like the “module of sections” of a vector bundle does.

Instead, it plays the role of the structure group of fiber bundle. A crossed module of groups is nothing but a strict 2-group. And like a bundle has a structure group, it’s higher generalization has a structure 2-group.

Moreover, I don’t think that Larry Breen ever talks about the vector bundle analog of gerbes, i.e. of something like “2-vector bundles” of “vector 2-bundles”, or the like.

But other people do (for instance these). Even though this is still pretty much in progress.

Recently I have talked with Bruce Bartlett about what should replace the monoid C ∞XC^\infty X acting on sections of a vector bundle (with connection) when we pass to vector 2-bundles.

So if you are willing to follow me and agree that a 2-vector bundle (with connection) is a (suitable) 2-functor from 2-paths to 2-vector spaces, and if triv\mathrm{triv} is the tensor unit in the monoidal 3-category of such functors (as described here), then the 2-monoid acting on 2-sections, replacing C ∞(X)C^\infty(X), is

(1)End(triv). \mathrm{End}(\mathrm{triv}) \,.

Notice that, using the canonical 2-rep of any strict 2-group (crossed mdoule) I can thus built an associated 2-vector bundle (with connection, and recall my terminmology here) for a given (nonabelian) gerbe. (I can spell out details on request.)

So my personal answer to your question would be: the categorification of C ∞(X)C^\infty(X) acting on the module of sections of a vector bundle is End(triv)\mathrm{End}(\mathrm{triv}) acting on the space of 2-sections, which is Hom(triv,tra)\mathrm{Hom}(\mathrm{triv},\mathrm{tra}).

But that’s just me.

Posted by: urs on November 20, 2006 4:22 PM | Permalink | Reply to this
Read the post 2-Monoid of Observables on String-G
Weblog: The n-Category Café
Excerpt: Rep(L G) from 2-sections.
Tracked: November 24, 2006 5:41 PM
Read the post Obstructions for n-Bundle Lifts
Weblog: The n-Category Café
Excerpt: On obstructions to lifting the structure n-group of n-bundles.
Tracked: September 12, 2007 11:26 PM

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