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August 19, 2010

Minicourse on Nonabelian Differential Cohomology

Posted by Urs Schreiber

During September I’ll be at the ESI-institute in Vienna to give one third of a Minicourse on higher gauge theory .

My plan is at

Meanwhile with Domenico Fiorenza and Zoran Škoda we are discussing various aspects of the \infty-Chern-Weil theory that is part of this here on the nForum.

Posted at August 19, 2010 12:15 AM UTC

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Re: Minicourse on Nonabelian Differential Cohomology

Here is the planned schedule for the minicourse Higher Gauge Theory.

It is divided into three parts, titled

  • Thomas Strobl (Lyon)

    Higher gauge theory – QQ-Bundle perspective

  • Camille Laurent-Gengoux (Coimbra)

    Nonabelian gerbes with connection in terms of Lie groupoids

  • Urs Schreiber (Utrecht)

    Higher gauge theory – Category-theoretic perspective

And the sessions themselves are planned as follows

Posted by: Urs Schreiber on September 3, 2010 2:23 PM | Permalink | Reply to this

infinity-Bibundles and infinity-Anafunctors

Yesterday at ESI Christian Blohmann gave a talk on a result he got with Chenchang Zhu, to appear soon.

Their statement should – and that’s their motivation – serve to unify a bunch of constructions that are currently present in the literature, and a plethora of more such constructions that would certainly keep being invented until somebody gives a general statement such as they do now.

Which is this:

they construct a natural functor from the category of simplicial sets over Δ[1]\Delta[1] to a 1-category of spans of simplicial sets

{K Δ[1]}{K^ K 1 K 0} \left\{ \array{ K \\ \downarrow \\ \Delta[1] } \right\} \;\;\; \to \;\;\; \left\{ \array{ \hat K &\to& K_1 \\ \downarrow \\ K_0 } \right\}

where K 0K_0 and K 1K_1 are the fibers over the endpoint, and K^\hat K is the simplicial set whose kk-cells are those maps out of the join of simplicial sets

[k][k]K [k] \star [k] \to K

such that the first copy of [k][k] lands over 00 and the second over 11.

Then they prove that

Proposition If KΔ[1]K \to \Delta[1] is a left fibration then K^K 0\hat K \to K_0 is an acyclic Kan fibration, hence K oK^K 1K_o \leftarrow \hat K \to K_1 an \infty-anafunctor between \infty-groupoids..

Moreover, they show that left fibrations KΔ[1]K \to \Delta[1] that are 2-coskeletal sort of in both degrees are precisely the nerves of groupoid-bibundles, or rather are the action groupoids of these.

So this is some perspective on the combinatorics used to present the (,0)(\infty,0)-Grothendieck construction. The point is that both anafunctors as well as groupoid bibundles are models for morphisms in the \infty-topos of \infty-Lie groupoids or similar \infty-sheaf \infty-toposes.

Accordingly, Christian ended by saying that they are still fiddling with how precisely to restate this with simplicial sets replaced by simplicial manifolds. But I’d think it is better to do the general abstract construction in presheaves, and only later check – if really necessary – whether certain objects are representable in some way.

Since because the construction is functorial, it extends straighforwardly to the projective model structure on simplicial presheaves (over any site) and gives us the relation between \infty-bibundles and \infty-anafunctors there. We will want this for the local model structure, but if we assume for instance that the topos has enough points and we look at the hyperlocalization, then it is straightforward again, with taking left fibrations and acyclic fibrations in the above statement to be stalkwise such.

Posted by: Urs Schreiber on September 11, 2010 8:43 AM | Permalink | Reply to this

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