## December 16, 2010

### QVEST, Winter 2010

#### Posted by Urs Schreiber

On somewhat short notice, for those in the vicinity of central Europe:

This Monday we have the first of a series of seminars:

• Quarterly Seminar on Topology and Geometry

December 20, 2010,

Univ. Utrecht, Netherlands

With the following talks:

• Hessel Posthuma (Univ. Amsterdam)

Integrable hierarchies and Frobenius manifolds

Abstract: I will explain the construction of an integrable hierarchy out of a Frobenius manifold, given in terms of a partition function of a cohomological field theory. In the homogeneous case, these hierarchies coincide with the ones constructed by Dubrovin and Zhang. Finally I will explain how one can use the action of the Givental group to deduce properties of these hierachies from those of the KdV equation when the underlying Frobenius manifold is semisimple. This is joint work with A. Buryak and S. Shadrin.

• Julie Bergner (UC Riverside)

Generalized classifying space constructions

Abstract: The process of constructing a classifying space for a group can be generalized so that we can find classifying spaces for categories. This construction is useful for many purposes, but also can be considered to lose much information if we use classical approaches in algebraic topology. In this talk, we give two approaches to strengthening the classifying space construction: one by changing how we think about spaces, and the other by changing the construction.

• Christian Blohmann (MPI Bonn)

Homotopy equivalence of correspondences and anafunctors of higher groupoids

Abstract: An anafunctor (also called span or zigzag) in the category of higher groupoids can be viewed as local trivialization of a principal bundle on a hypercover of a higher stack. Correspondences, on the other hand, are the higher generalization of groupoid bibundles. In the setting of ordinary categories the notions of anafunctors and bibundles are equivalent, both providing a localization of the category of groupoids at Morita equivalences. I will report on the generalization of this result to the setting of quasi-categories.

Posted at December 16, 2010 8:51 PM UTC

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### Re: QVEST, Winter 2010

Argh! Julie’s and Christian’s talks look so good. Please someone take notes! I’ll be in Wagga Wagga from Saturday, so apart from being on the other side of the world, I’ll be out in the sticks.

Posted by: David Roberts on December 16, 2010 11:53 PM | Permalink | Reply to this

### Re: QVEST, Winter 2010

Nothing of relevance will go unnoticed by the $n$Lab. For the moment I can maybe say the following:

Posted by: Urs Schreiber on December 17, 2010 12:23 AM | Permalink | Reply to this

### Re: QVEST, Winter 2010

The Blohmann-Zhu result looks interesting; for some reason I didn’t notice that the first time you posted about it. An inner fibration over $\Delta^1$ is a natural quasicategorical version of a collage, representing a profunctor, so in that case the corresponding span ought to be a quasicategorical version of the two-sided fibration representing the same profunctor. Is it known how to make that precise and whether it’s true?

Posted by: Mike Shulman on December 18, 2010 4:39 AM | Permalink | Reply to this

### Re: QVEST, Winter 2010

An inner fibration over $\Delta[1]$ is a natural quasicategorical version of a collage, representing a profunctor, so in that case the corresponding span ought to be a quasicategorical version of the two-sided fibration representing the same profunctor.

That sounds good.

Is it known how to make that precise and whether it’s true?

I haven’t yet seen a writeup of the claim, but when Christian told me about it in Vienna, I got away with the impression (which might be wrong) that the span that they consider given a morphism of simplicial sets $p : K \to \Delta[1]$ is that whose tip is

$\hat K : [n] \mapsto p^{-1}(0)_n \times_{K_n}\times Hom([n] \star [n], K) \times_{K_n} p^{-1}(1)_n \,,$

where the pullback is along the two injections $[n] \to [n] \star [n] \leftarrow [n]$ of $[n]$ into the join with itself.

So one should check if this construction reduces to the 2-sided fibration associated to an ordinary profunctor when $C \simeq p^{-1}(0)$ and $D \simeq p^{-1}(1)$ are ordinary categories.

Posted by: Urs Schreiber on December 19, 2010 7:29 PM | Permalink | Reply to this

### Re: QVEST, Winter 2010

It now looks to me as if it should not be the join $[k] \star [k]$, but the cylinder $[k] \times\Delta[1]$. It seems that the discrete 2-sided fibration associated to a profunctor $F$ is the category of sections of its collage $E_F \to \Delta[1]$.

Posted by: Urs Schreiber on December 20, 2010 8:26 AM | Permalink | Reply to this

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