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December 22, 2008

Higher Structures in Göttingen - Part II

Posted by John Baez

In November last year there was a little workshop in Göttingen on Higher Structures in Differential Geometry. Urs blogged about it here. In February there will be a kind of continuation:

Ieke Moerdijk and I will give three lectures each. He’ll probably speak about orbifolds or stacks — I’m not sure. I’ll give talks combining the Streetfest lectures on higher gauge theory, the 2006 Barrett lectures, and a talk about classifying spaces for 2-groups.

That’s a lot of stuff to cover. Luckily I’ll be aided by Alex Hoffnung (who will introduce the sheaf-theoretic approach to smooth spaces) and Chris Rogers (who will talk about Lie 2-algebras from 2-plectic geometry).

Clemens Berger will speak on ‘Higher categories and Eilenberg-Mac Lane spaces’ — that’s sure to be interesting. And Camilo Arias Abad and David Martinez Torres will speak on subjects yet to be disclosed.

I’ve never been to Göttingen. For any mathematician who knows a little history, visiting Göttingen is a kind of pilgrimage to sacred ground! So that should be fascinating in its own right, even apart from the workshop itself.

Posted at December 22, 2008 8:52 PM UTC

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Re: Higher Structures in Göttingen - Part II

Here are slides of the talks I’ll be giving in Göttingen. Later I’ll add links to Alex Hoffnung’s talk on smooth spaces and Chris Rogers’ talk on 2-plectic geometry and Lie 2-algebras. All these talks are supposed to fit together and form a coherent — or at least quasicoherent — introduction to higher gauge theory.

Lectures on Higher Gauge Theory

Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of “higher gauge theory” that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must “categorify” concepts from topology and geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on. This overview of higher gauge theory will emphasize its relation to homotopy theory and the cohomology of groups and Lie algebras.

Click on these to see transparencies of the talks:

  • Lecture 1 - Lie 2-groups and Lie 2-algebras,
    in PDF.
  • Lecture 2 - 2-bundles and classifying spaces for 2-groups,
    in PDF.
  • Lecture 3 - 2-connections on 2-bundles,
    in PDF.

You can also see lots of references for further reading, here.

Posted by: John Baez on January 25, 2009 8:08 PM | Permalink | Reply to this
Read the post Categorified Symplectic Geometry and the String Lie 2-Algebra
Weblog: The n-Category Café
Excerpt: A new paper shows how to build the string Lie 2-algebra by taking a compact Lie group with its canonical closed 3-form and then using ideas from multisymplectic geometry.
Tracked: January 27, 2009 12:04 AM

Re: Higher Structures in Göttingen - Part II

Now you can see the slides of Alex Hoffnung’s talk at this workshop:

Smooth Spaces: Convenient Categories for Differential Geometry

In 1977 K.T. Chen introduced a notion of smooth spaces as a generalization of the category of smooth manifolds. In 1979 Souriau introduced another notion, ‘diffeological spaces’, serving the same purposes. Both of these categories have all limits and colimits, and are cartesian closed. In fact, following ideas of Dubuc, we give a unified proof that the categories of Chen spaces, diffeological spaces, and simplicial complexes are ‘quasitopoi’: locally cartesian closed categories with finite (and in these cases all) colimits and a weak subobject classifier.

Posted by: John Baez on January 27, 2009 2:32 AM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

The abstract implies ambiguously that the Chen and Souriau categories are/are not/not known to be equivalent. Sorry I don’t have time to read the slides to see which is true.

Posted by: jim stasheff on January 27, 2009 1:50 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Hi Jim,

Sorry for the ambiguity. Chen spaces play a very small role in these slides. In particular, you will not find this result. However, Andrew Stacey discusses the question of equivalence in his first comment here. He tells us that the categories of Diffeological (Souriau) spaces and Chen spaces are not equivalent.

Posted by: Alex Hoffnung on January 27, 2009 5:16 PM | Permalink | Reply to this

Dinner?

I’ll arrive in Göttingen afternoon of Feb 4 (or earlier, I am very flexible in this respect). I am staying at Hotel Kasseler Hof.

(Most other hotels where participants will probably stay, most restaurants in question, as well as the math institute itself are all within walking distance.)

If anyone is interested in meeting for a joint dinner, let me know!


map: surroundings of math institute in Göttingen

Posted by: Urs Schreiber on January 28, 2009 8:33 AM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Hi, Urs!

Chris Rogers and I are taking a flight from LAX that changes in Heathrow and is supposed to arrive in Frankfurt at 19:20 on Wednesday February 4th. The last train from Frankfurt to Göttingen leaves before 21:00. So, it will be somewhat exciting, trying to find the train station in time to catch the train. If we succeed, I think the train to Göttingen takes about 2 hours or more, so we’ll get in rather late. (If we fail, we’ll come to Göttingen the next morning.)

So, a nice dinner on Wednesday night is probably not in my fate. But I don’t know Alex Hoffnung’s schedule.

Posted by: John Baez on January 28, 2009 5:34 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

I imagine you guys will go straight to LAX from Riverside, but if you’ve got spare time either on the way or on the way back, I could possibly meet for coffee, tea, bear, dinner (whatever) depending on time (and interest).

Posted by: Eric on January 28, 2009 7:04 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Usually when I get back from a long flight my only desire is to get home and collapse. But the prospect of eating bear meat is intriguing.

Posted by: John Baez on January 28, 2009 10:01 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Hello everyone,

I will be arriving earlier than others it seems. Probably sometime Monday afternoon (Feb 2nd). If anyone is around and would like to meet for food, drinks or math let me know.

Posted by: Alex Hoffnung on January 28, 2009 7:22 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

All right, I’ll see if I can come Monday and Tuesday already. I may have to go back to Hamburg on Wednesday. But trains are my second home (third, really) anyway…

Posted by: Urs Schreiber on January 28, 2009 10:15 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

The real problem with the dates is that

David Ben-Zvi is giving the Center-for-Math-Phys colloquium talk in Hamburg on Thursday Feb 5

on his stuff on geometric function theory, I suppose.

It’s a pity. I wanted to hear that talk and meet David B.-Z. What should I do?

Posted by: Urs Schreiber on January 29, 2009 12:42 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

I don’t know what you should do, but I’ll say this: you’re not going to learn anything new about higher gauge theory from my talks. You know all that stuff.

You may however learn something from the other talks. In particular, Ieke Moerdijk is giving talks on operads and algebras up to homotopy, dendroidal sets (which let you take the nerve of an operad), and the homotopy theory of \infty-categories and \infty-operads. I’m also really looking forward to Clemens Berger’s talk on higher categories and Eilenberg–Mac Lane spaces.

And, of course, it would be fun to talk. Alex Hoffnung will be showing up on Monday, jet-lagged but eager for conversation. Chris Rogers and I show up on Wednesday evening. We’re all staying until Sunday morning.

But Ben–Zvi is an interesting guy, and we’ll have plenty to do, so don’t feel bad if you want to talk to him.

Posted by: John Baez on January 29, 2009 7:23 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Yeah, I know. I don’t come to Göttingen for the talks (I have also heard Moerdijk talk about his \infty-categories before, though this time he will give more details I suppose), but to meet the other participants.

Well, I hope to see Alex Hoffnung on Monday and Tuesday and will have to be in Hamburg on Wednesday anyway, then I see how I feel and where I go on Thursday.

Posted by: Urs Schreiber on January 29, 2009 7:36 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Is there a geometric realization functor for dendroidal sets?

Posted by: jim stasheff on January 29, 2009 9:58 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Moerdijk and I discussed this intensively about a year ago. We thought we had a good one, but eventually he realized that it was too simple to be right. As far as I know, the situation hasn’t progressed since then.

Posted by: Tom Leinster on January 30, 2009 3:30 AM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Can anyone tell us about Clemens Berger’s talk on higher categories and Eilenberg-Mac Lane spaces?

Posted by: David Corfield on February 16, 2009 10:33 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

I’ll just sketch Clemens Berger’s talk very briefly.

He began by discussing a nice generalization of simplicial sets called θ n\theta_n-sets. A θ 1\theta_1-set is just a simplicial set. The cells in a θ n\theta_n-set are what Jim Dolan called ‘globoplexes’ — they’re products of balls and simplexes.

Just as a 1-category gives a simplicial set called its nerve, an nn-category gives a θ n\theta_n-set called its nerve.

Just as we can take the geometric realization of the nerve of a group (thought of as 1-category) and get its classifying space, we can take the geometric realization of the nerve of an nn-group (thought of as an nn-category) and get a kind of classifying space for it.

Pick an abelian group AA and form the nn-group that’s trivial at levels except the top level, where the nn-morphisms form the group AA. Berger calls this nn-group b nAb^n A. (Urs would call it B nAB^n A.)

Now take the classifying space of b nAb^n A. We get a version of the Eilenberg–Mac Lane space K(A,n)K(A,n). Of course this space is already well-known. But now we get a version that’s built out of globoplexes in a very nice efficient way.

In particular, we can try to compute the Euler characteristic of K(A,n)K(A,n) by taking the alternating sum of the number of kk-dimensional globoplexes:

k(1) kN k \sum_k (-1)^k N_k

This sum diverges, but if we regularize it

kt kN k \sum_k t^k N_k

we get a rational function — at least for A=/2A = \mathbb{Z}/2, which was the only example Clemens talked about. And, if we evaluate this rational function at t=1t = -1, we get the ‘right answer’: that is, the homotopy cardinality of K(A,n)K(A,n).

And, the details depend heavily on Fibonacci numbers, as discovered earlier by Jim (part 1, part 2)!

Posted by: John Baez on February 19, 2009 1:51 AM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Here are the slides of Chris Rogers’ talk at this workshop:

Lie 2-algebras from 2-plectic geometry

Just as symplectic geometry is a natural setting for the classical mechanics of point particles, 2-plectic geometry can be used to describe classical strings. Just as a symplectic manifold is equipped with a closed non-degenerate 2-form, a “2-plectic manifold” is equipped with a closed non-degenerate 3-form.

The Poisson bracket makes the smooth functions on a symplectic manifold form a Lie algebra. Similarly, any 2-plectic manifold gives a “Lie 2-algebra”: the categorified analogue of a Lie algebra, where the usual laws hold only up to isomorphism. We explain these ideas and use them to give a new construction of the “string Lie 2-algebra” associated to a simple Lie group.

For more details, see:

Posted by: John Baez on January 29, 2009 5:37 AM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

The British Airways flight Chris and I were going to take to this workshop has been cancelled, so we’ll show up 24 hours late if at all. Not clear it’s worth it now; I’d like to ask Chenchang Zhu about this.

Posted by: John Baez on February 3, 2009 11:16 PM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

The British Airways flight Chris and I were going to take to this workshop has been cancelled

That’s too bad. I was wondering with Alex Hoffnung the other day if you’d fly over Heathrow.

Not clear it’s worth it now; I’d like to ask Chenchang Zhu about this.

Maybe there is a chance to move some of the activity to Saturday?

Posted by: Urs Schreiber on February 4, 2009 7:10 AM | Permalink | Reply to this

Re: Higher Structures in Göttingen - Part II

Chenchang wants to go on with the show, so Chris and I will show up Friday morning — unless we run into more problems. She changed the schedule so that Chris will still give his talk, and I’ll give 2 out of my 3, skipping the last one: the one about connections on 2-bundles.

I can talk endlessly, so I’d be perfectly happy to give that last talk on Friday night or Saturday. But listening is harder work than talking, so I should probably refrain from doing this.

Posted by: John Baez on February 4, 2009 5:07 PM | Permalink | Reply to this
Read the post Dendroidal Sets and Infinity-Operads
Weblog: The n-Category Café
Excerpt: Notes from a talk by Ieke Moerdijk on dendroidal sets, with a few remarks on presheaves on the category of posets.
Tracked: February 6, 2009 1:05 AM
Read the post Moerdijk on Infinity-Operads
Weblog: The n-Category Café
Excerpt: Ieke Moerdijk on (infinity,1)-operads.
Tracked: February 6, 2009 5:01 PM
Read the post Higher Structures in Göttingen IV
Weblog: The n-Category Café
Excerpt: Announcement of the fourth workshop "Higher Structures in Göttingen".
Tracked: February 15, 2010 10:31 PM

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