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November 29, 2011

Higher Structures Along The Lower Rhine

Posted by Urs Schreiber

This is to announce the workshop

Higher Geometric Structures along the Lower Rhine

Organisers: Christian Blohmann, Marius Crainic, Ieke Moerdijk

Date: Thu, 2012-01-12 – Fri, 2012-01-13

Details: Program, Abstracts

This is the first in a series of short workshops jointly organized by the Geometry/Topology groups in Bonn, Nijmegen, and Utrecht, all situated along the Lower Rhine.

Posted at November 29, 2011 1:09 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2464

12 Comments & 1 Trackback

Re: Higher Structures Along The Lower Rhine

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


Posted by: Thomas on January 1, 2012 8:39 AM | Permalink | Reply to this

Re: Higher Structures Along The Lower Rhine

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)!

I’ll see what I can do. If things go well, I’ll be posting notes, as usual. (But sometimes things don’t go well :-)

Posted by: Urs Schreiber on January 11, 2012 8:21 AM | Permalink | Reply to this
Read the post Mathematical Aspects of String and M-Theory in Oxford
Weblog: The n-Category Café
Excerpt: A workshop in Oxford on mathematical aspects of string theory.
Tracked: January 11, 2012 7:26 PM

Mikhail Kapranov: “Higher Segal spaces”

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 nnLab entry higher Segal space.

This looks like a model for some “cyclic” version of (,2)(\infty,2)-categories. However, in discussion afterwards we heard that the relation to (,2)(\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.

Posted by: Urs Schreiber on January 12, 2012 2:58 PM | Permalink | Reply to this

Dennis Borisov, “Simplicial approaches to derived C-infinity geometry”

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-(,1)(\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.

Posted by: Urs Schreiber on January 12, 2012 4:00 PM | Permalink | Reply to this

David Carchedi, “Sheaf Theory for Étale Stacks”

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 CC a site “similar” to the site of (smooth) manifolds, let XSh (C)X \in Sh_\infty(C) be an \infty-stack on that site, hence let XX be a smooth ∞-groupoid, or similar.

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

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

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

Sh (X)Et/XSh (C)/X Sh_\infty(X) \simeq Et/X \hookrightarrow Sh_\infty(C)/X

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

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.

Posted by: Urs Schreiber on January 13, 2012 3:01 PM | Permalink | Reply to this

Re: David Carchedi, “Sheaf Theory for Étale Stacks”

This looks interesting, but I don’t understand it. What is the functor Sh ():C(,1)ToposSh_\infty(-)\colon C\to (\infty,1)Topos? Aren’t we trying to define what we mean by sheaves “on” an object of the site CC?

Posted by: Mike Shulman on January 15, 2012 5:23 AM | Permalink | Reply to this

Re: David Carchedi, “Sheaf Theory for Étale Stacks”

Aren’t we trying to define what we mean by sheaves “on” an object of the site CC?

For the special sites under consideration, namely C=ManifoldsC = Manifolds and its variants (such as supermanifolds, etc.), there is a canonical such notion. So this is used and extended to \infty-stacks over CC.

I am thinking that, more generally, what is being made use of is that these sites are canonically “geometries”, namely equipped with a notion of “admissible”=étale covers. That’s where a notion of étaleness is specified and then Yoneda-extended.

Posted by: Urs Schreiber on January 15, 2012 8:43 AM | Permalink | Reply to this

Re: David Carchedi, “Sheaf Theory for Étale Stacks”

What is going on is, the sites I consider CC are assumed to be a class of structured spaces in that they can be viewed as (sober) topological spaces TT together with a morphism to some topos Sh(T).Sh(T) \to \mathcal{E}. (For instance for manifolds, \mathcal{E} is the classifying topos for local rings, and the map is the classifying map for the sheaf of smooth functions). I then put the Grothendieck topology on this category which is given by open covers of the underlying spaces. Then for an object XX of the site CC, I have a canonical “small site” of open covers, and infinity sheaves over this site is the topos of “small infinity sheaves” of XX. This gives me my functor Sh: :C(,1)Topos.Sh:_{\infty}:C \to \left(\infty,1\right)Topos. With this functor defined, I can take its homotopy Kan extension alonge Yoneda to get a functor Sh (C)(,1)Topos.Sh_\infty\left(C\right) \to \left(\infty,1\right)Topos.
Posted by: David Carchedi on January 15, 2012 5:07 PM | Permalink | Reply to this

Re: David Carchedi, “Sheaf Theory for Étale Stacks”

Oh, okay. (Like Urs, I would find it more satisfying to characterize these “little toposes” in canonical terms of the “big topos” rather than just specifying what they are supposed to be by hand – I’d hoped that maybe you had an answer to this.)

Posted by: Mike Shulman on January 15, 2012 11:44 PM | Permalink | Reply to this

Re: David Carchedi, “Sheaf Theory for Étale Stacks”

As I have mentioned to Dave before, I keep wanting to see if we can give an abstract characterization of étale \infty-stacks as objects in a given ambient cohesive \infty-topos.

In that context we already have a good notion of what should be a sub-class of all étale \infty-stacks, namely the locally constant \infty-stacks over some object XX. And we know which sub-category of the slice \infty-topos these correspond to, namely to that on the “Π\Pi-closed” morphisms EXE \to X (recalled in sec. 2.3.13).

It feels like there should be a way to characterize the étale \infty-stacks in terms of the locally constant ones. But I am not seeing clearly yet.

Posted by: Urs Schreiber on January 15, 2012 6:52 PM | Permalink | Reply to this

Re: David Carchedi, “Sheaf Theory for Étale Stacks”

For instance, what is this here:

objects AA with the property that there is an atlas UAU \to A (effective epi out of a 0-truncated object) such that there is a diagram

E U× AU U U \array{ E &\to& U \times_A U \\ \downarrow && \downarrow \\ U' &\to& U }

with the left morphism a locally constant \infty-stack over UU' and the two horizontal morphisms effective epis.

So U× AUUU \times_A U \to U is to be thought of as the source map out of the morphism \infty-groupoid, and we want to say that this is étale. We say this by saying that it is covered by covering spaces.

Hm…

Posted by: Urs Schreiber on January 15, 2012 8:00 PM | Permalink | Reply to this

Klaas Landsman, “Topos theory and C-star algebras”

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

Posted by: Urs Schreiber on January 13, 2012 3:19 PM | Permalink | Reply to this

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