The n-Category Café

Hi Urs,

it would be great if one could read more about the talks (and if one could read texts or slides of them after they were hold)!

Best wishes and a happy new year,
Thomas

Mikhail Kapranov is talking about joint work with Tobias Dyckerhoff on a definition of 2-Segal spaces.

I have added some brief notes to an $n$Lab entry higher Segal space.

This looks like a model for some “cyclic” version of $(\infty,2)$-categories. However, in discussion afterwards we heard that the relation to $(\infty,2)$-categories – or whatever it is – still needs to be understood.

The bulk of the talk is a list of natural examples. Maybe I’ll have more notes on these later.

Dennis Borisov is talking about derived differential geometry.

In his article with Justin Noel, Simplicial approach to derived differential manifolds he discusses that David Spivak’s derived smooth manifolds are, in direct analogy to ordinary smooth manifolds, entirely determined by their global sections of simplicial smooth functions, hence that they are equivalent to a full sub-$(\infty,1)$-category of the opposite of simplicial C-infinity rings.

That’s what one would expect, but it is good to have it worked out. I haven’t looked at the article yet.

David Carchedi gave a nice talk on theory that has grown out of his thesis,

Small Sheaves, Stacks, and Gerbes over Étale Topological and Differentiable Stacks

all revolving around the generalization of the notion of étale groupoids (hence orbifolds) to fully fledged (higher) topos theory.

One central theorem (out of a long list of theorems he gave) goes like this. For $C$ a site “similar” to the site of (smooth) manifolds, let $X \in Sh_\infty(C)$ be an $\infty$-stack on that site, hence let $X$ be a smooth ∞-groupoid, or similar.

Then there is a natural notion of the $\infty$-topos $Sh_\infty(X)$ of $\infty$-stacks on $X$ . This is obtained as the left $\infty$-Yoneda extension of the canonical $Sh_\infty(-) : C \to (\infty,1)Topos$.

But there is also the slice $\infty$-topos $Sh_\infty(C)/X$, which also feels like being about $\infty$-stacks over $X$ in a sense.

This one theorem of Dave’s states that there is a reflective embedding

$$Sh_\infty(X) \simeq Et/X \hookrightarrow Sh_\infty(C)/X$$

whose essential image $Et/X$ is identified as the full sub-$\infty$-category of the slice over $X$ on those morphisms which deserve to be called étale morphisms out of étale ∞-stacks into $X$.

He gives an explicit characterization of these $\infty$-étale structures in terms of presentations by étale-simplicial manifolds: an étale $\infty$-stack over (smooth) manifolds is one that has a presentation in the local model structure of simplicial presheaves presented by a simplicial manifold all whose structure maps are ordinary étale maps of manifolds.

There was much more in his talk. Maybe more notes later.

Klaas Landsman gave a nice introduction to ideas of and motivations for the Bohr topos corresponding to a non-commutative $C^\ast$-algebra.

He amplified the theorems by Harding-Döring, Hamhalter and by Alfsen-Shultz, see here, which state that the Bohr topos is in a way a spatial realization of the Jordan algebra corresponding to the $C^\ast$-algebra.