The String Coffee Table

In the morning Dale Husemöller reviewed the definition of fibred categories, stacks and gerbes. Those interested can find essentially all he said written up in slightly different language in

I. Moerdijk
Introduction to the language of stacks and gerbes
math.AT/0212266 .

After the talk there was the question how connections and curving on a gerbe would be formulated in this ‘relatively abstract’ (namely stack-theoretic instead of bundle-gerbe-theoretic) context. I mentioned

L. Breen & W. Messing
Differential Geometry of Gerbes
math.AG/0106083,

but apparently people felt that understanding that paper is comparably demanding to working out the problem themselves. I must say the same still holds for me, unfortunately, though I should try again someday.

After that Jarah Evslin introduced some M-theory stuff necessary to discuss ‘Dualities some of which preserve twisted K-theory’ and Hisham Sati followed that up with a talk on ‘The M-theory/type II Partition Function and Generalized Cohomology’. This was a review of hep-th/0312033, 0404013, 0410293, 0501060, 0501245, 0507106.

The afternoon was then devoted to K-theory. Mathai Varghese spoke about ‘Global Aspects of T-Duality in type II string Theory’, based on hep-th/9902102, 0306062, 0312052, 0312284, 0401168, 0508084, 0412092, 0412268.

Just picking one special item from that talk, let me mention that at some point the K-theory of a ‘non-associative torus’ was being hinted at. Apparently there is an interesting $C^*$-sort of algebra arising from the deformation of the torus algebra which is slightly non-associative. Christoph Schweigert remarked that the coresponding formula, which looked like

with $\phi$ a group 3-cocycle, seemed to indicate that there is an associator around here, such that we are actually dealing with a weakly associative algebra. Would that mean we actually have a weak 2-algebra appearing here?

Finally Ulrich Bunke talked about his joint work with Th. Schick on ‘T-duality of orbifolds’. I don’t have the precise reference handy and don’t have the energy to go into any details right now.

Instead, I would like to mention the major personal insight that this day brought for me.Brano Jurčo told me about how he uses nerves of 2-groups as classifying spaces of nonabelian gerbes, and that he expects those simplicial-map diagrams which I had drawn on the blackboard the day before to play a similar role. These look as follows:

The tetrahedron on the bottom denotes one 3-simplex of the ‘Čech 2-groupoid’ of a covering $\mathcal{U} = \bigsqcup_i U_i \to M$ of some base space $M$, and the tetrahedron on the top left (ignore the third tertrahedron on the right for the moment) denotes a similar simplex in the 2-category of local 2-holonomy 2-functors $\hol_i : \mathcal{P}_2(U_i) \to G_2$.

My claim is that principal 2-bundles with 2-connection and 2-holonomy over $\mathcal{U}\to M$ are natural isomorphism classes $\Omega$ of functors from that Čech-2-groupoid $Č(M)$ to the 2-functor 2-category

Now, Brano’s remark made me see what should have been very obvious all along: Taking nerves and geometric realizations on both sides, this implies that 2-bundles with 2-holonomy are classified by homotopy equivalence classes of maps from the (geometric realization of the) nerve $|Č(M)|$ of $Č(M)$ to the (geometric realization of the) nerve $|G_2^{\mathcal{P}_2(\mathcal{U})}|$.

(We can take the nerve of a 2-category in complete analogy to how we do it for a 1-category: Objects become vertices, morphisms become edges, 2-morphisms between triangles of 1-morphisms become faces of a simplicial space, and so on. (With all higher identity $n$-morphisms taken into account!) The details have been worked out by Duskin:

John W. Duskin
Simplicial matrices and the nerves of weak n-categories I: nerves of bicategories
Theory and Applications of Categories, Vol. 9, 2001, No. 10, pp 198-308.

I am indebted to Igor Baković for making me aware of this just a few days ago.)

Even better, we can regard $M$ itself as (the geometric realization of) a simplicial space and realize that a particular choice of cover is nothing but a map from $M$ to $|Č(M)|$. Still better, $M$ and $|Č(M)|$ are, I am being told, always homotopy equivalent. Since I furthermore claim that the 2-bundle with 2-connection defined by $\Omega$ is independent of the choice of covering $\mathcal{U}$, it should follow that 2-bundles with 2-holonomy over $M$ are classified by hom. equiv. classes of maps

Due to the isomorphism in the first step this should finally mean that 2-bundles with 2-holonomy are classified by

This should be the rather obvious consequence of the above picture, which, as usual, says more than all the words here.

To see that this makes sense let’s forget about the fact that objects in $|G_2^{\mathcal{P}_2(\mathcal{U})}|$ are 2-functors. In other words, consider the 2-functor

which sends all 2- holonomy 2-functors to the single object of our structure 2-group (regarded as a 2-category with a single object), sends all pseudo-natural transformations to their morphism part and all modifications of these to the 2-morphism part. This corresponds to forgetting that we have a 2-holonomy on our 2-bundle.

Then we find that the resulting structure (namely bare $G_2$-2-bundles) should be classified

by maps

from $M$ to the nerve of $G_2$.

Branislav Jurčo is apparently going to talk about this result tomorrow, in the (equivalent) context of nonabelian bundle gerbes.

It should be clear, but let me mention it anyway that the case of classification of ordinary $G_1$-bundles is the easy special case where we regard the (1-)group $G_1$ as a 2-group with all 2-morphisms being identities. In this case we find that ordinary $G_1$-bundles are classified by maps to the ‘classifying space’ $BG$ which is nothing but the nerve of $G_1$:

as is very well known.

I should have thought of that before. Many thanks to Brano for pushing me in that direction!