The n-Category Café

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

It is divided into three parts, titled

• Thomas Strobl (Lyon)

  Higher gauge theory – $Q$-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

• Sept 8, Thomas: Connections on $Q$-Bundles I

• Sept 9, Urs: $\infty$-Lie groupoids

• Sept 14, Camille: Nonabelian gerbes in terms of Lie groupoids

• Sept 16, Urs: Differential cohomology

• Sept 20, Thomas: Connections on $Q$-Bundles II

• Sept 22, Urs: (an example from) $\infty$-Chern-Weil theory

• Sept 24, Camille: Connections on nonabelian gerbes in terms of Lie groupoids

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 $\Delta[1]$ to a 1-category of spans of simplicial sets

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

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

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

such that the first copy of $[k]$ lands over $0$ and the second over $1$.

Then they prove that

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

Moreover, they show that left fibrations $K \to \Delta[1]$ 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 $(\infty,0)$-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.