## February 15, 2010

### Higher Structures in Göttingen IV

#### Posted by Urs Schreiber

guest post by Christoph Wockel

Dear all,

we cordially invite you to participate the CRCG Workshop “Higher Structures in Topology and Geometry IV”, which will take place June 2-4 at the Göttingen Mathematics Institute (Germany). Main speakers will be

• Ezra Getzler (Northwestern)
• Birgit Richter (Hamburg)
• Urs Schreiber (Utrecht)
• Christoph Schweigert (Hamburg)

In addition, there will be a minor amount of talks given by PhD students and postdocs. For additional information you may consult the

Best wishes,

Christoph Wockel (on behalf of the organisers Giorgio Trentinaglia and Chenchang Zhu)

On this blog we had also reported on

Posted at February 15, 2010 10:13 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2171

### Re: Higher Structures in Göttingen IV

first talk: Birgit Richter on Richter-Baas-Dundas-Rognes 2-vector bundles:

BDR 2-vector bundles

(rough first notes right after the talk)

Posted by: Urs Schreiber on June 2, 2010 2:02 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

second part of Birgit Richter’s talk: algebraic K-theory of bimonoidal categories

Posted by: Urs Schreiber on June 3, 2010 4:29 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

the first part of Ezra Getzler’s series of talks on descent for $L_\infty$-algebras

Posted by: Urs Schreiber on June 2, 2010 3:06 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

second part of Ezra Getzler’s talk: the homological category of filtered $L_\infty$-algebras

(this is a bit rough, I’ll try to fill in the missing details later, as time permits)

Posted by: Urs Schreiber on June 3, 2010 9:55 AM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Assume Getler = Getzler ;-)
but any chance of a defintion of a Getzler cat?

Posted by: jim stasheff on June 4, 2010 12:39 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

I think Urs might mean a Getzler-(category of fibrant objects), which was defined just above this subsection.

Posted by: Todd Trimble on June 4, 2010 1:33 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Thanks for catching the typo. It wasn’t the only one, but typos in names are (even more) unpleasant.

Yes, as Todd says, by a “Getzler-category of fibrant objects” I here mean his slight variation of the set of axioms of a Kenneth-Brown-category-of-fibrant-objects. As described roughly starting in the section A notion of category of fibrant objects.

Posted by: Urs Schreiber on June 4, 2010 2:09 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Last part of Ezra Getzler’s talk is still running, but now we have seen the main statement:

Posted by: Urs Schreiber on June 4, 2010 2:29 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

now Thomas Nikolaus is talking about his algebraic models for $\infty$-groupoids and $(\infty,1)$-categories

Posted by: Urs Schreiber on June 2, 2010 3:48 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

The pdf-slides for Thomas Nikolaus’ talk are here: Algebraic models for higher categories

Posted by: Urs Schreiber on June 4, 2010 8:01 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Is the terminology (Serre) cofibration standard in some subculture?

Posted by: jim stasheff on June 6, 2010 3:01 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Dear Jim,

maybe my use of terminology was wrong. Since the fibrations are called (Serre) fibrations, I thought that the corresponding cofibrations are called (Serre) cofibrations. Is that wrong? If you just call them cofibrations, how can they be distinguished from the Hurewicz-cofbrations (I mean the one in the Strom model structure).

Posted by: Thomas Nikolaus on June 6, 2010 8:57 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Unless I’ve taken leave of my senses, isn’t a Hurewicz cofibration just a closed cofibration ($\sim$ NDR-pair), where this is in the Quillen model structure? If so then cofibration = Serre cofibrations, and Hurewicz cofibration = closed cofibration can do it.

Posted by: David Roberts on June 7, 2010 6:27 AM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

I’m not sure how the language has evolved outside of Top. The original usage was Hurewicz fibration = arbitrary homotopy lifting property, while Serre =
homotopy lifting property for maps in of polyhedra’, as I recall. I had not seen these names applied to cofibrations. I guess you mean a similar distinction between the homotopy extension property. That would be fine, but where did this usage first appear?

Posted by: jim stasheff on June 7, 2010 1:14 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

jim wrote

I guess you mean a similar distinction between the homotopy extension property. That would be fine, but where did this usage first appear?

Yes, that is what I meant. I thought that is the most intuitiv name for the correspondin cofibrations. But maybe it is not the usual name.

David wrote:

If so then cofibration = Serre cofibrations, and Hurewicz cofibration = closed cofibration can do it.

I would also be fine with that.

Posted by: Thomas Nikolaus on June 7, 2010 1:48 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

The cofibrations in the Quillen model structure are not the maps that have the HEP for polyhedra; they are the retracts of relative cell complexes. But I could see calling them “Serre cofibrations” or “Quillen cofibrations”, or also “$q$-cofibrations” in line with the style that May-Sigurdsson introduced for distinguishing between different model structures.

And while $q$-cofibrations are in particular Hurewicz cofibrations, the notion of “Hurewicz cofibration” or “$h$-fibration,” like that of Hurewicz fibration, is a little bit sensitive to the ambient category of spaces that you choose. In particular, the $h$-cofibrations in the category of compactly generated weak Hausdorff spaces are all closed, but this is not true in the category of all topological spaces.

Posted by: Mike Shulman on June 7, 2010 8:17 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

In particular, the h-cofibrations in the category of compactly generated weak Hausdorff spaces are all closed, but this is not true in the category of all topological spaces.

Ah, thanks for that. I know that NDR-pairs provide examples of h-cofibrations, but I wasn’t 100% sure of the other way around.

Posted by: David Roberts on June 8, 2010 3:16 AM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Originally I recall Hurewicz fibration *meant* in the cat of all topological spaces
so changing the name when changing the cat seems appropriate

ditto for Serre

Posted by: jim stasheff on June 8, 2010 1:50 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

currently John Huerta is giving a nice talk on Division algebras and Lie $n$-superalgebras.

I’ll provide the link to his pdf slides later…

Posted by: Urs Schreiber on June 3, 2010 9:59 AM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

first notes from Chris Rogers’ talk on $n$-plectic geometry.

Posted by: Urs Schreiber on June 3, 2010 11:53 AM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Konrad Waldorf’s talk on: transgression of bundle gerbes to loop space

Posted by: Urs Schreiber on June 4, 2010 2:05 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

The slides for Christoph Schweigert’s talk are here: Gerbes on Lie groupoids on work with Thomas Nikolaus.

Posted by: Urs Schreiber on June 5, 2010 10:18 AM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

the slides for John Huerta’s talk are here: Lie n-algebras, supersymmetry and division algebras

Posted by: Urs Schreiber on June 6, 2010 9:42 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

I’ll put up the notes (and audio!) for Urs’s talk as soon as I can get to a scanner…

Posted by: Bruce Bartlett on June 7, 2010 11:22 AM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Here are handwritten notes (talk 1, talk 2) from Urs’s talk ‘On differential cohomology in an $(\infty,1)$-topos’. There are also audio recordings of the two talks (talk 1, talk 2, quality not great) available. Here is a pic of Urs at the blackboard!

The primary reference is Urs’s notes on his personal nLab page.

Posted by: Bruce Bartlett on June 12, 2010 2:24 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Nice notes. I hope John Huerta learns this stuff and applies it to his Lie super-2-algebras and Lie super-3-algebras.

Posted by: John Baez on June 14, 2010 6:16 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

I hope John Huerta learns this stuff and applies it to his Lie super-2-algebras and Lie super-3-algebras.

I talked with John H. about this a bit in Oberwolfach.

On he one hand it is very straightforward to generalize the setup to any notion of geometry, in particular to supergeometry. For instance in the main running example, replace Cartesian probe spaces $\mathbb{R}^n$ by super-Cartesian probe spaces $\mathbb{R}^{p|q}$. This is in fact essentially the same step that gives synthetic differential $\infty$-geometry only that in one case the nilpotent elements of the function rings on test spaces skew-commute, in the other they commute. One can also do both and consider synthetic super $\infty$-geometry and $\infty$-Lie super-groupoids and $\infty$-Lie super-algebroids.

Setting all this up is quite straightforward. Two summers ago while at the Hausdorff center in Bonn, I tried with my office mate back then to warm up for the evident notion of integration of $\infty$-Lie superalgebras to $\infty$-Lie supergroups by getting our hands dirty with working out the basic examples, such as the integration of the super-translation super Lie algebras using the abstract integration procedure. This was supposed to warm us up for the integration of the supergravity super Lie 3-algebra and its cousins. But it turned out that we found the integration of just the simple super-translation super Lie algebras in terms of the abstract procedure quite tedious. (Of course writing out the integration formally is a tautology, what is tedious is cutting it down to something equivalent but recognizable.) Possibly we were being dense, but it caused me to not further follow up on that.

And it also caused me to step back a bit and wonder if this is going in the right direction. With everybody doing derived algebraic geometry, concentrating on supergeometry feels a bit like pre-school: everybody knows how to count over the natural numbers, but you only know 1 and 2. ;-)

It is striking how many aspects of supergeometry that people are fond of are really secretly aspects of derived geometry. In lots of cases the $\mathbb {Z}_2$-grading of supergeometry is naturally refined to an $\mathbb{N}$-grading.

Not in all cases of course. But maybe those where it is not naturally refined this way, maybe it is secretly refined this way?

Be that as it may: doing differential cohomology in a super/derived $(\infty,1)$-topos should be a straightforward application of the general theory, though with some interesting but possibly tedious aspects.

Posted by: Urs Schreiber on June 14, 2010 7:24 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

The day after god created straightforward projects with interesting but possibly tedious aspects, he created grad students.

I wonder where the tedium was located, exactly?

Posted by: John Baez on June 14, 2010 9:18 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Or, as Henry (aka JHC) Whitehead said, grad students are like bacteria to digest cellulose for the supervisor (cellulose = the existing printed literature).

Posted by: jim stasheff on June 15, 2010 12:53 PM | Permalink | Reply to this

### Re: Higher Structures in Göttingen IV

Having grown up with integer grading, the popularity $Z_2$ always seemed an accident of either sexy’ naming or appearance in physics.

Posted by: jim stasheff on June 15, 2010 12:51 PM | Permalink | Reply to this

Having grown up with integer grading, the popularity $Z_2$ always seemed an accident of either ‘sexy’ naming or appearance in physics.

I think if you want to pinpoint the relevance, it is in the quantization step from Grassmann algebra to Clifford algebra. And that’s of course of some relevance.

To be explicit:

for $V$ a vector space, the Grassmann algebra $\wedge^\bullet V$ is nicely $\mathbb{N}$-graded.

Its quantization is obtained by putting an inner product $\langle -,-\rangle$ on $V$ and deforming the relation

$v \cdot w + w \cdot v = 0$

to the relation

$v \cdot w + w \cdot v = \langle v,w\rangle 1$

for $v, w \in V$.

This Clifford relation breaks the original $\mathbb{N}$-grading down to a $\mathbb{Z}_2$-grading. So all that springs from this, all spin geometry etc. , is not $\mathbb{N}$-graded but $\mathbb{Z}_2$-graded.

Posted by: Urs Schreiber on June 15, 2010 1:20 PM | Permalink | Reply to this

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