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October 31, 2006

Bakovic on 2-Groupoid 2-Torsors

Posted by Urs Schreiber

Today Igor Bakovic gave a talk on his work on the categorification of the notion of principal groupoid fiber bundles.

Igor has a couple of nice results, of which his talk mentioned only few aspects and of which my report here will mention even fewer. When his thesis is finished there should be more to say.

In his talk Igor sketched the definition of torsors for bigroupoids G\mathbf{G}. He indicated how a nonabelian cocycle with values in G\mathbf{G} is obtained from any such torsor and how he hopes that these cocycles are in fact equivalent to G\mathbf{G}-torsors.

Notice that reconstructing a 2-bundle from its local data is usually more subtle than going the other way around. Compare for instance Wirth’s work recently recalled by Stasheff.

There are several ways to say what a principal fiber bundle PP over a space XX for a structure group GG is.

An elegant way it to let G=X×G\mathbf{G} = X \times G be the trivial bundle of groups, and require that the the morphism

(1)P×GP× XP P \times \mathbf{G} \to P \times_X P

which in terms of ordinary elements reads

(2)(p,g)(p,pg) (p,g) \mapsto (p, p\cdot g)

be an isomorphism.

When stated this way, this is straightforwardly generalized to the case where G\mathbf{G} is more generally a groupoid, with object space G 0G_0. We then equip PP with a morphism

(3)PG 0 P \to G_0

saying, heuristically, which fibers of PP are acted on by morphisms starting at which object in G\mathbf{G}. The action of the category G\mathbf{G} on PP is then defined in just the same way as the action of G\mathbf{G} on itself (by composition of morphisms) - only that PP behaves like consisting of morphisms that just have a target, but no source.

So then a principal groupoid bundle is a bundle PP equipped with an action by G\mathbf{G} and such that the natural map

(4)P× G 0GP× XP P \times_{G_0} \mathbf{G} \to P \times_X P

is an isomorphism.

What I call a “groupoid bundle” here is what Igor (and many others) call a G\mathbf{G}-torsor.

It is well known that these G\mathbf{G}-torsors are classified by nonabelian cocycles

(5)g ijg jk g ik. \array{ g_{ij}\nearrow \searrow g_{jk} \\ \stackrel{g_{ik}}{\rightarrow} } \,.

The goal is to categorify everything in sight.

So instead of an ordinary groupoid, we now let G\mathbf{G} be a weak 2-category (a bicategory) all of whose morphisms are equivalences: this is a weak 2-groupoid, or bigroupoid.

Everything generalizes accordingly. See this excerpt of Igor’s thesis for some details.

Igor shows how to extract from a local trivialization of a torsor for a bigroupoid a cocycle with values in that bigroupoid. These cocycles are not triangles as above, but 2-commuting tetrahedra labelled in G\mathbf{G}.

For bigroupoids with just a single object, this problem was also addressed by Toby Bartels.

Igor’s construction works with everything taking place internal to a fixed given topos E\mathbf{E}. Like for instance that of sheaves over smooth manifolds, for the case of smooth 2-bundles. The admissable notion of “total space” of a 2-bundle (reconstructed from a 2-cocycle, say) depends on the choice of this topos.

There’d be more to say. But so much for now.

Posted at October 31, 2006 9:52 PM UTC

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3 Comments & 4 Trackbacks

Re: Bakovic on 2-Groupoid 2-Torsors

Any clues as to when the thesis might be forthcoming? (:))

Posted by: David Roberts on October 31, 2006 11:41 PM | Permalink | Reply to this

Re: Bakovic on 2-Groupoid 2-Torsors

Any clues as to when the thesis might be forthcoming?

I know that I have announced this thesis before. The suspense is getting unbearable, isn’t it? :-)

I do have clues that it will be defended soon, though.

Posted by: urs on November 1, 2006 9:24 AM | Permalink | Reply to this

Re: Bakovic on 2-Groupoid 2-Torsors

An exceprt of the thesis with a couple of details can be found here.

Posted by: urs on November 1, 2006 1:10 PM | Permalink | Reply to this
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