## April 20, 2012

### Higher Structures in Topology and Geometry VI

#### Posted by Urs Schreiber

This July takes place in Göttingen the next Higher Structures meeting

Courant Research Centre workshop
Higher Structures in Topology and Geometry VI
July 9-11
Göttingen, Germany.
www.crcg.de/wiki/HSTG6

The invited speakers are

The workshop will consist of mini-courses given by the invited speakers plus a small number of shorter talks given by participants. Limited funds are available, in particular for phd students and postdocs. The deadline for registration is June 15.

Posted at April 20, 2012 11:01 PM UTC

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### Re: Higher Structures in Topology and Geometry VI

The titles for the mini-courses are now available:

• Julie Bergner: “To $(\infty,1)$-categories and beyond”
• Dennis Borisov: “Simplicial techniques in derived $C^{\infty}$-geometry”
• Thomas Schick: “Differential $K$-theory”
Posted by: Chris Rogers on May 7, 2012 11:04 AM | Permalink | Reply to this
Read the post Geometric quantization on moduli ∞-stacks
Weblog: The n-Category Café
Excerpt: A talk on higher geometric quantization on higher moduli stacks.
Tracked: July 9, 2012 9:12 AM

### Henriques on von Neumann 2-algebras

Some lightning summaries of some of the talks, here in Göttingen:

André Henriques discussed a proposal for what the higher categorical analog of a von Neumann algebra should be.

He provided evidence that the pattern is like this:

in analogy to how every von Neumann algebra is a suitable subalgebra of suitable endomorphisms of the essentially unique infinite dimensional separable Hilbert space $H$

$A \subset B(H) \subset End(H) \,,$

so a von Neumann 2-algebra should be a suitable topological sub-monoidal category of the monoidal category

$Bimod(R) = End(R)$

of endomorphisms of the essentially unique type $III_1$ von Neumann algebra factor – and endomorphisms here are endo-bimodules.

It helps if you think of this in terms of n-vector spaces given by algebras and bimodules: think of $R$ as presenting the essentially unique “most infinite dimensional” separable 2-Hilbert space. Then a von Neumann 2-algebra is a suitable sub-2-algebra of its endomorphisms.

Posted by: Urs Schreiber on July 9, 2012 4:13 PM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

If abelian von Neumann algebras are like measure spaces (or algebras of bounded functions on them), what would symmetric von Neumann 2-algebras be like?

Posted by: David Corfield on July 9, 2012 5:58 PM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

If abelian von Neumann algebras are like measure spaces (or algebras of bounded functions on them), what would symmetric von Neumann 2-algebras be like?

You might want to turn this around and say: ah, so now I know what a measurable topological stack is! (Namely the formal dual of a von-Neumann 2-algebra).

But I’ll check with André if he sees more of a “higher measure theory” appearing here.

Posted by: Urs Schreiber on July 9, 2012 8:28 PM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

Urs, I presume you mean ‘topological category’ as in enriched in $Top$?

Posted by: David Roberts on July 10, 2012 1:06 AM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

‘topological category’ as in enriched in Top?

No, I mean the categorical analog of a topological space: a stack in categories on $Top$.

Posted by: Urs Schreiber on July 10, 2012 6:21 AM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

Then why not say topological stack? (Fibred in groupoids or categories?) There are already two conflicting definitions of the phrase ‘topological category’, neither of which is a generalisation of the other. This is what comes of calling a stack of groupoids on $Mfld$ a Lie groupoid :-)

Whinge over.

Posted by: David Roberts on July 11, 2012 12:50 AM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

…a stack in categories on Top

ah, a stack of categories, not groupoids. Missed that.

Posted by: David Roberts on July 11, 2012 12:51 AM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

This is what comes of calling a stack of groupoids on $Mfld$ a Lie groupoid :-)

Not every stack of groupoids on smooth manifolds is a Lie groupoid. But, yeah, I like to call a stack of groupoids on smooth manifolds a smooth groupoid. In the same vein, a stack of groupoids on topological spaces I like to call a topological groupoid. It is the only sane way to give good meaning to the more naive definition of topolical groupoids. Similarly then for topological categories and 2-sheaves on topological spaces.

What is not my failt is that it is common to say “simplicial category” for an $sSet$-enriched category, and “topological category” for a $Top$-enriched category. But since this habit is in so evident violation of systematic terminology (as one notices as soon as one is working with actual simplicial categories) I don’t feel guilty about using the term in its systematic sense.

But let’s not further debate over terminology. We all do and have to live with a community using terminology in a locally but usually not globally consistent way. For communication we find the local equivalence, and that’s fine then.

Posted by: Urs Schreiber on July 11, 2012 12:24 PM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

It is the only sane way to give good meaning to the more naive definition of topological groupoids.

While I like stacks as much as the next guy, I think this sentence is unjustifiable. Ordinary internal categories in Top are a fine notion in their own right.

Posted by: Mike Shulman on July 11, 2012 10:04 PM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

It is the only sane way to give good meaning to the more naive definition of topological groupoids.

I think this sentence is unjustifiable.

“The only sane way” in the sense of Hawking:

I adopt the Euclidean approach, the only sane way to do quantum gravity non-perturbatively. [He grinned at this point.]

In the sense of: I know you will disagree, but if time is short, it is fun to state it provocatively this way.

Seriously, do we want to flesh out the 1001 ways to speak about topological categories here? All three of us know all the aspects of it. Is the idea that we stage a discussion for bystanders to get interested in? If so, let’s do it in a dedicated thread!

Posted by: Urs Schreiber on July 12, 2012 1:47 PM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

I’m just saying that your sentence was missing the “[He grinned at this point]”, so it would be easy for a bystander to take it seriously. Try adding a smiley face (-: or something when you have your tongue in your cheek.

Posted by: Mike Shulman on July 12, 2012 5:34 PM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

To turn this back to mathematics (I was only originally asking for clarification) do you know if this topological stack has an atlas which is a topological space? Then it would be presented by a category internal to $Top$ [Roberts, forthcoming], and then even I would be happy to conflate the two concepts.

Posted by: David Roberts on July 13, 2012 7:17 AM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

an atlas which is a topological space? Then it would be presented by a category internal to Top [Roberts, forthcoming],

Isn’t that essentially by definition of atlas of a geometric stack, here: of a topological stack?

For stacks in groupoids a morphism $U \to X$ is an atlas if it is effective epi and representable, which directly implies that the beginning of its Cech nerve

$U \times_X U \stackrel{\to}{\to} U$

is the groupoid internal to $Top$ that is equivalent to $X$ as a stack on $Top$, where on the left we have the homotopy fiber product of $U$ with itself over $X$.

So when you now talk about atlases for category-valued stacks, one would expect that whichever definition you choose, it got to be such as to make the above true, with the homotopy fiber product above replaced with the “comma-fiber product” or whatever that should be called.

Posted by: Urs Schreiber on July 13, 2012 1:37 PM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

Well, I’ve been playing around with stacks of categories, and I think I have a reasonable way to do this, with a little hint from Mike. It’s not exactly as you guessed. Let’s go back a step though: does $Bimod(R)$ admit an epimorphism from a topological space?

Posted by: David Roberts on July 14, 2012 4:49 AM | Permalink | Reply to this

### Re: Henriques on von Neumann 2-algebras

does $Bimod(R)$ admit an epimorphism from a topological space?

So the presheaf of objects of $Bimod(R)$ is supposed to assign to a given topological test space $U$ the set of “Hilbert $R$-bimodule bundles over $U$”, for some definition of that.

I don’t know if that will be representable, and it will probably depend on fine-tuning of the definitions such as to make André’s story work. If you do want to know this, you should contact André about it!

Posted by: Urs Schreiber on July 14, 2012 11:20 AM | Permalink | Reply to this

### Lean on ∞-categorical enrichment of

Rohan Lean talks about KK-theory. This is a certain category whose objects are $C^\ast$-algebras, such that the hom-sets/groups behave like $C^\ast$-algebraic K-homology in the first argument, and like $C^\ast$-alegrbaic cohomology in the second.

He claims to show the following, which is apparently a generalization of a somewhat more restrictive similar theorem known before:

there is the structure of a stable infinity-category and hence in particular that of a triangulated category on $C^\ast$-algebras such that the KK-theory category is the corresponding homotopy category.

More details on the general framework are in the lecture notes

• Ralf Meyer, KK-theory as a triangulated category (2009) (pdf)
Posted by: Urs Schreiber on July 9, 2012 4:25 PM | Permalink | Reply to this

### Blohmann on Hamiltonian Lie algebroid actions and equivariant cohomology

Christian Blohman talks about generalizing the Atiyah-Bott result on the relation betweeen Hamiltonian actions of Lie algebras and equivariant cohomology to the case where we have Hamiltonian actions of Lie algebroids.

Posted by: Urs Schreiber on July 10, 2012 2:59 PM | Permalink | Reply to this

### Re: Blohmann on Hamiltonian Lie algebroid actions and equivariant cohomology

Oh, my last link above is misleading. In the question session Christian says that his definition of Hamiltonian Lie algebroid action is very different from Bos’s.

Posted by: Urs Schreiber on July 10, 2012 3:30 PM | Permalink | Reply to this

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