## August 17, 2007

### Future Gazing

#### Posted by David Corfield

A year and a day into this blog perhaps we should open up a thread to allow some speculation as to what the next year might bring. Restricting myself here to philosophy, but feel free to choose any discipline, what prospects are there for greater attention being paid to n-categories from that quarter? I reckon the most likely notice will come from philosophers of physics, at least at the $n = 1$ level. Hans Halvorson (Princeton) I know is interested, and is arranging a conference at which John is speaking (Oct 3-4).

But what about the central heartlands of philosophy? Can we hope for a first step towards something akin to the revolution wrought by Russell and then the Vienna Circle? A sympathetic philosopher, Mike Beaney, asks for the “potential philosophical pay-off”? As I reply there, what’s not clear to me is what and how to pay. Perhaps now I’ll be returning to a philosophy department I can get a better idea.

Something I’d like to know more about is the reception of Russell’s ideas. How quickly were his achievements recognised? How long was it before the average Anglo-Saxon philosopher felt it incumbent upon him or herself to learn some predicate logic? I ought to spend some time looking through back copies of Mind.

Posted at August 17, 2007 2:50 PM UTC

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### Re: Future Gazing

some speculation as to what the next year might bring

And end to the notion of requiring permission to educate, at least in Turkey.

Posted by: John Armstrong on August 17, 2007 3:27 PM | Permalink | Reply to this

### Re: Future Gazing

I am in the process of researching for an entry in which I would like to indicate how some amazingly cool stuff which is being done using BV formalism (a great generalization of BRST formalism) is – secretly – exactly the idea of quantizing on an $n$-category.

“Just” apply the map $\mathrm{qDGCA}\stackrel{\sim}{\to} L_{\infty} \stackrel{\sim}{\to} \omega\mathrm{Lie}$ whenever you see those physicists use the letter $Q$.

That might potentially change the philosophy of physics eventually, I guess.

More later.

Posted by: Urs Schreiber on August 17, 2007 3:30 PM | Permalink | Reply to this

### Re: Future Gazing

Hmm, what will happen in the world of n-categories next year? Prediction is tough…

I can predict what I’ll talk about at Hans Halvorson’s conference: the importance of spans in quantum theory, with an emphasis on groupoidification. I can’t predict the title — I’m tempted to use “A Spanish Interpretation of Quantum Mechanics”, or maybe even “Quantum Mechanics: A Spanish Inquisition”, but those may be too cute for such a distinguished conference. (Especially since I don’t use racks in this attempt to force quantum mechanics to reveal its secrets.)

I also predict that I’ll give a year-long course on groupoidification, pushing the Tale of Groupoidification towards much greater precision and detail.

I predict that Urs will continue revolutionizing the foundations of physics with n-categorical ideas.

I predict that Jeff Morton will make available his thesis describing certain extended TQFTs as 2-functors.

Less certainly, I predict that Jacob Lurie and Mike Hopkins may come out with a paper proving a version of the cobordism hypothesis in dimensions 1 and 2 — giving algebraic characterizations of the (∞,1)-category of 1d cobordisms and the (∞,2)-category of 2d cobordisms. Meanwhile, Soren Galatius, Ib Madsen, Ulrike Tillmann, Michael Weiss may come out with a paper generalizing their work so far to describe the nerves of cobordism n-categories.

I also predict that more mathematicians will use (∞,1)-topoi in work on algebraic geometry.

So, n-categories will continue to gradually spread their influence in mathematics and physics…

As for philosophy, I can’t tell.

Posted by: John Baez on August 19, 2007 10:27 AM | Permalink | Reply to this

### Re: Future Gazing

Remind me of the relationship between the cobordism hypothesis and the tangle hypothesis.

Posted by: David Corfield on August 19, 2007 12:50 PM | Permalink | Reply to this

### Re: Future Gazing

Ok, so I should have looked at the end of this.

The Cobordism Hypothesis: The free stable $n$-category with duals on one generator is $n$COB: $n$-morphisms here are $n$-dimensional framed cobordisms between framed manifolds with corners.

The Tangle Hypothesis: The free $k$-tuply monoidal $n$-category with duals on one generator is $n Tang_k$: $n$-dimensional morphisms are $n$-dimensional framed tangles in $n + k$ dimensions.

Given there’s a Generalized Tangle Hypotheses

The $k$-tuply monoidal $n$-category of G-structured $n$-tangles in the $(n + k)$-cube is the fundamental $(n + k)$-category of (MG, $Z$)

Then there should be a generalized cobordism hypothesis.

Hmm, would Urs’ construction of the universal $G$-bundle in terms of the nerve of INN($G$) shed any light on the Thom Space MG?

Posted by: David Corfield on August 21, 2007 10:03 AM | Permalink | Reply to this

### Re: Future Gazing

Hmm, would Urs’ construction of the universal $G$-bundle in terms of the nerve of $\mathrm{INN}(G)$

I should point out that the insight that the realization of the nerve of the groupoid which I know by the various names $\mathrm{INN}(G) = G // G = \mathrm{Codisc}(G) = (G \stackrel{\mathrm{Id}}{\to} G)$ is a model for $E G$ in that $|\cdot| : (G \to G // G \to \Sigma G) \mapsto (G \to E G \to BG)$ is due to Segal. I originally learned this from Danny Stevenson, who also pointed me to the relevant article:

G. B. Segal, Classifying spaces and spectral sequences, Publ. Math. IHES No. 34 (1968) 105-112 .

But I like to make the point that one can regard the groupoid $G // G$ as “playing the role of” the universal $G$-bundle even without passing to geometric realizations, but stayin entirely within the world of groupoids. And that this point of view simplifies and illuminates some points.

So, for instance, the fact that $G // G$ “is” the universal $G$-bundle is, in the world of groupoids, encoded by the facts that

a) $G // G$ it is (not equal but) equivalent to the trivial groupoid. This yields the fact that $E G$ is (not equal but) homotopy equivalent to the point, i.e. contractible.

ii) $G // G$ has in fact the structure of a 2-group (this is an important piece of information which becomes pretty much invisible when you pass to $E G$ in the usual way, but by Segal’s construction, $E G$ has in fact the structure of a topological group!, in contrast to $B G$, which is a group only if $G$ is abelian). Using this and the canonical injection $G \to G // G$ we get the $G$-action on $E G$. And the fact that $G \to G // G \to \Sigma G$ is exact then gives that this action is free.

In the very last section of my article with David Roberts we indicate more ways to that that and how $G // G$ plays the role of the universal $G$-bundle, without the need of ever passing to geometric realizations.

And using this point of view, we then find the analogous statements for 2-groups. David and I never ever try to geometrically realize this stuff here. But John Baez and Danny Stevenson have a paper almost finished (compare the remarks John made here) where they do consider the universal $G_{(2)}$-bundle realized as a topological space.

shed any light on the Thom Space $M G$?

I wish I knew the answer to that. I am lacking intuition for Thom spaces. Maybe if there is a “nice” way to think of them, one could try to make an educated guess for whether and how there is a way to think of them just on the level of groupoids. That would be cool.

Posted by: Urs Schreiber on August 21, 2007 12:07 PM | Permalink | Reply to this

### Re: Future Gazing

Where would the difficulty come in this recipe for the Thom space $M G$ of a subgroup $\mathrm{G} \subseteq O(k)$?

We build this by forming the universal $G$-bundle $EG \to BG,$ then forming the associated vector bundle $EG \times_\mathrm{G} \mathbb{R}^k \to BG,$ then taking the one-point compactification of each fiber to get a sphere bundle $EG \times_\mathrm{G} \mathbb{S}^k \to BG$ and finally collapsing all the points at infinity to a single point $*$, getting $MG$ with its basepoint $*$.

The first step’s OK, isn’t it, forming $G//G \times_G \mathbb{R}^k$ as a groupoid.

Posted by: David Corfield on August 21, 2007 1:12 PM | Permalink | Reply to this

### Re: Future Gazing

The first step’s OK, isn’t it, forming $(G // G) \times_G \mathbb{R}^k$ as a groupoid.

Ah, okay. That sounds good. Right. So I guess you have in mind actually the action groupoid $\mathbb{R}^k // G$ ?

Hence something like $(G // G) \times_G (\mathbb{R}^k // G)$ ?

Posted by: Urs Schreiber on August 21, 2007 1:29 PM | Permalink | Reply to this

### Re: Future Gazing

That’s it.

Is there any problem with extending the action groupoid to $S^k // G,$ with the point at infinity a fixed point for $G$?

Posted by: David Corfield on August 22, 2007 8:38 AM | Permalink | Reply to this

### Re: Future Gazing

Is there any problem with extending the action groupoid to $S^k // G$ with the point at infinity a fixed point for G?

Naively certainly not, though we need to take care to correctly respect possible extra structure, like topologies or smooth structures, if that’s the context we are working on.

But I am willing to ignore this for the moment. Then the question is:

if we first form the groupoid $(G // G) \times_G (\mathbb{R}^k // G)_{\mathrm{cp}}$ with $(\mathbb{R}^k // G)_{\mathrm{cp}}$ the one-point compactification of the action groupoid, as you suggested, and if we then further identify all these points at infinity in each fiber (crossing our fingers that this can be made sense of when the required extra structure is taken care of), how would be go about checking if the resulting groupoid qualifies as a groupoid “playing the role of” the Thom space?

Posted by: Urs Schreiber on August 22, 2007 1:19 PM | Permalink | Reply to this

### Re: Future Gazing

Posted by: David Corfield on August 23, 2007 10:44 AM | Permalink | Reply to this

### Re: Future Gazing

The main theme of next year’s IAS program, run by Roman Bezrukavnikov, is the interaction of derived algebraic geometry, representation theory and topological field theory, so hopefully one will see a lot more along the lines John mentions coming out of there! (There are also two week-long conferences being held during the program which should be fun.)
Posted by: David Ben-Zvi on August 19, 2007 5:58 PM | Permalink | Reply to this

### Re: Future Gazing

Do you know anything about how one wangles ones way into these IAS programs? I have a certain desire to visit the IAS in the spring, since my wife may be visiting there (for reasons related to Chinese and Greek classics, rather than math). And, it would be great to learn a bit about the topics you mention.

Feel free to email if you prefer to discuss ‘wangling’ in private… but I have no shame about these things. Also, I think it’s useful to the young kids watching this show to see that it’s not by sheer effortless magic that Prof. X winds up visiting the Institute for Astounding Scholars: often Prof. X needs to work a bit, though he does his best to hide it!

(In England such people are called ‘swans’: they look so elegant as they float upstream, but down beneath their little feet are busily paddling away.)

Posted by: John Baez on August 21, 2007 1:52 PM | Permalink | Reply to this

### Re: Future Gazing

John said:

(In England such people are called swans: they look so elegant as they float upstream, but down beneath their little feet are busily paddling away.)

Well, whilst the idea of a swan paddling away vigorously underwater is sometimes used as a simile, I’ve never actually heard anyone called a swan.
Posted by: Simon Willerton on August 21, 2007 2:49 PM | Permalink | Reply to this

### Re: Future Gazing

Simon writes:

Well, whilst the idea of a swan paddling away vigorously underwater is sometimes used as a simile, I’ve never actually heard anyone called a swan.

Oh, okay. I bow to your clearly superior knowledge of British English. You don’t know how much I’d pay to be able to say ‘whilst’ like that.

Posted by: John Baez on August 23, 2007 12:19 PM | Permalink | Reply to this

### Re: Future Gazing

I’d really like to use ‘whilst’ too, but you just can’t get away with it if you speak with an American accent :-(

I don’t believe Quinion’s claim that there is no difference in meaning between ‘while’ and ‘whilst’. I thought I had observed that Rightpondians do use ‘while’ to refer to the noun (“a long while”), but reserve ‘whilst’ for the conjunction. Right? I personally find that a useful distinction.

Posted by: Todd Trimble on August 23, 2007 3:35 PM | Permalink | Reply to this

### Re: Future Gazing

Well, you certainly can’t say “a long whilst”. On the other hand, you can use “while” as a conjunction. In fact, I don’t think I use “whilst” at all, although it doesn’t stand out for me when other people use it. But I use “while” as a conjunction a lot.

Posted by: Tim Silverman on August 24, 2007 8:00 PM | Permalink | Reply to this

### Re: Future Gazing

Well wangling theory is one of my favorites, but I don’t know if I have deep insight into it (there’s also the danger of cramps when moving feet too rapidly underwater).

In this case there was an announcement some time of ago of a yearlong program at IAS, which I heard of through the grapevine (sound effect of busy paddling), and it was eventually on the IAS website, at which point one could apply for a semester or year of membership (with a deadline sadly about a semester ago). Once you hear about it it’s a regular application procedure like a postdoc or something, no mystery there (maybe there’s a mystery in the selection process, but my sense was there weren’t so so many applicants specifically for the program – a “benefit” of not advertising too broadly!).

I imagine that there are still openings for shorter term visits - I would recommend writing directly to Roman Bezrukavnikov, who’s running the program, or to Mary Jane Hayes, who (as School Administrative Officer) runs everything. Of course the weeklong conferences are wide open and advertising now for visitors.. but for longer stays I think contacting an organizer is the way to go.

Maybe I should clarify that when I talk about openings, this has to do with institute housing (and salary). They are quite happy (within bounds of course) to have people come and attend things and interact, the most limited resource is the Institute housing. In ‘97 I simply tagged along with my advisor to the QFT and Math program, even got an office and building key, just had to find my own apartment in town (and I think some students will be doing so next year). So if the housing is not an issue I think this should be quite feasible.

I do find that one always hears about such things far too late (eg I’ve never seen a conference poster before registration actually closes) — that’s one reason to have adverts in blogs like this, and to frequent some places that list conferences – I like Ravi Vakil’s list in algebraic geometry for example (and theoretically the Notices listing, though that’s so broad I never have the patience to scan through for things I might like). It might be nice for people following this blog if there was a feature with a list of upcoming conferences in related areas that people could contribute to?

Posted by: David Ben-Zvi on August 21, 2007 3:53 PM | Permalink | Reply to this

### Re: Future Gazing

Presumably with all the Wiki-expertise being displayed here recently, it wouldn’t take much to set up a MathConferenceWatch Wiki (though with rather different objectives to JournalWatch !) that anybody can update. Kiran Kedlaya has already started an Arithmetic Geometry Conference Page

Posted by: Dan on August 22, 2007 4:11 PM | Permalink | Reply to this

### Re: Future Gazing

You’re right! In fact, upcoming conferences are precisely the kind of data which inspired the creators of Semantic MediaWiki. Go to the ontoworld.org main page, and look at the “Events” section. Click on a few. Click on “full list” to see a nifty timeline. I wouldn’t call it fully developed, but it’s an interesting idea.

With this kind of setup, one could easily have live wiki pages like “Upcoming topology conferences” and “Upcoming algebraic geometry conferences”; all derived data from the wiki pages for conferences people have entered. With a bit of extra magic, you can even tag interesting looking conferences as “Remind me later” and so on :-)

Sadly, I have too much on my plate to get involved in such a project… but I think it’s a great idea!

Posted by: Bruce Bartlett on August 22, 2007 6:26 PM | Permalink | Reply to this

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