September 13, 2006

Higher Categories and their Applications

Posted by John Baez

As part of the Fields Institute program on Geometric Applications of Homotopy Theory, there there will be a workshop on:

There will be a strong emphasis on applications to homotopy theory and physics. Speakers include John Baez, Julie Bergner, Eugenia Cheng, Alissa Crans, Nick Gurski, Andre Henriques, André Joyal, Steve Lack, Aaron Lauda, Tom Leinster, Peter May, Joshua Nichols-Barrer, Simona Paoli, Urs Schreiber, Mike Shulman and Danny Stevenson.

For talk titles, abstracts and other information, see this blog entry.

The program also features some other interesting workshops (listed here) and courses, listed below.

Here are the short courses being taught as part of the Geometric Applications of Homotopy Theory program at the Fields Institute from January to June 2007. Many of them are related to n-categories!

This schedule is a bit tentative.

Eugenia Cheng: Higher categories (6 lectures, January)

Andre Joyal: Basic aspects of quasi-categories (6 hours, late January)

The fundamental category of a simplicial set. Categorical equivalences. Adjoint maps. Quasi-categories. Weak categorical equivalences. Inner fibrations, isofibrations. Join and slice. Left and right fibrations. The model structure for quasi-categories. Relation with the classical model structure on simplicial sets. Equivalence with complete Segal spaces, Segal categories and simplicial categories. The fibered model structures. Homotopy factorisation systems. The covariant and contravariant model structures over a base. Initial and terminal objects. Initial and final maps. Fully faithful maps, dominant maps. Minimal quasi-categories. Minimal fibrations. Morita equivalences. Localisation.

J.E. Bergner, A survey of $(\infty,1)$-categories. In preparation.

A. Joyal, Quasi-categories and Kan complexes, JPAA vol 175 (2002), 207-222.

A. Joyal, The theory of quasi-categories I. In preparation.

A. Joyal and M. Tierney, Quasi-categories vs Segal spaces. To appear.

A. Joyal, Quasi-categories vs simplicial categories. In preparation.

André Joyal: Extension of category theory to quasi-categories (6 hours, early February)

Adjoint maps. Diagrams. Limits and colimits. Large quasi-categories. The quasi-category K of Kan complexes. Complete and cocomplete quasi-categories. Grothendieck fibrations. Proper and smooth maps. Kan extensions. Cylinders, distributors and spans. Duality. Yoneda Lemma. The universal left fibration over K. Trace and cotrace. Factorisation systems in quasi-categories. Quasi-algebra. Locally presentable quasi-categories. Quasi-varieties. Internal categories. Descent. Exact quasi-categories. Quasi-topos. Higher quasi-categories.

A. Joyal, Quasi-categories and Kan complexes, JPAA vol 175 (2002), 207-222.

A. Joyal, The theory of quasi-categories II. In preparation.

A. Joyal, The theory of quasi-categories in perspective. To appear.

J. Lurie, Higher topos theory.

J.F. Jardine: Simplicial presheaves (6 lectures, late January)

Simplicial sheaves and presheaves, homology and cohomology, descent, localization and motivic homotopy theory, sheaves and presheaves of groupoids, stacks, cocycle categories, torsors, non-abelian cohomology.

P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Mathematics Vol. 174, Birkhäuser Basel-Boston-Berlin (1999).

J.F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology, Homotopy and Applications 3(2) (2001), 361-384.

J.F. Jardine, Cocycle categories.

J.F. Jardine, Lectures on simplicial presheaves.

Fabien Morel and Vladimir Voevodsky, $A^{1}$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math., 90:45-143 (2001), 1999.

Max Karoubi: Hermitian K-theory (6 lectures, early March)

Hermitian forms and quadratic forms, positive and negative hermitian K-theory, various versions of the periodicity theorem, homotopy invariance, topological analogs, the case when 2 is not invertible, a new homology theory on rings : the stabilized Witt group.

M. Karoubi and O. Villamayor, K-théorie algébrique et K-théorie topologique II. Math. Scand. 32, pp. 57-86 (1973).

A. Bak, K-theory of forms, Annals of Math. Studies 98. Princeton University Press (1981).

M. Karoubi, Théorie de Quillen et homologie du groupe orthogonal. Le théoreme fondamental de la K-théorie hermitienne. Annals of Math. 112, pp. 207-282 (1980).

F.J-B.J Clauwens, The K-theory of almost hermitian forms, Topological structures II, Mathematical Centre Tracts 115, pp. 41-49 (1979).

M. Karoubi, Stabilization of the Witt group, C.R. Math. Acad. Sci. Paris 342, pp. 165-168 (2006)

Fabien Morel: Structure and computations of $A^1$-homotopy sheaves, with applications (6 lectures, early March)

Basic structure of $A^1$-homotopy sheaves as unramified sheaves on the category of smooth $k$-schemes. $A^1$-motives and $A^1$-homology of spaces. The Hurewicz theorem. Description of the $H_0$ of smash powers of $G_m$’s in terms of Milnor-Witt K-theory. Consequences and examples of computations. The first non-trivial $A^1$-homotopy sheaf of algebraic spheres. The Brouwer degree.

$A^1$-homotopy classification of rank n vector bundles. Application to the Euler class for rank n vector bundles over dim n affine smooth n-dimensional schemes and to stably free vector bundles.

The theory of $A^1$-coverings, $A^1$-universal coverings and $A^1$-fundamental group. Examples of computations (for surfaces for instance). Some perspectives towards a “surgery classification” of smooth projective $A^1$-connected varieties and more.

References:

F. Morel, An introduction to $A^1$-homotopy theory, In Contemporary Developments in Algebraic K-theory, ICTP Lecture notes, 15 (2003), pp. 357-441, M. Karoubi, A.O. Kuku , C. Pedrini (ed.).

F. Morel, On the structure of $A^1$-homotopy sheaves, part I and part II.

F. Morel and V. Voevodsky, $A^1$-homotopy theory of schemes, Publications Mathématiques de l’I.H.E.S, volume 90.

J.F. Jardine: Presheaves of spectra (6 lectures, early May)

Presheaves of spectra and symmetric spectra, stable categories, homology and cohomology, motivic stable categories, chain complexes and simplicial abelian presheaves, derived categories, Voevodsky’s category of motives (maybe).

J.F. Jardine, Generalized Etale Cohomology Theories, Progress in Mathematics Vol. 146, Birkhäuser, Basel-Boston-Berlin (1997).

J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445-552.

J.F. Jardine, Presheaves of chain complexes, K-Theory 30(4) (2003), 365-420.

J.F. Jardine, Generalised sheaf cohomology theories, in Axiomatic, Enriched and Motivic Homotopy Theory, NATO Science Series II 131 (2004), 29-68.

J.F. Jardine, Lectures on presheaves of spectra.

Jacob Lurie (6 hours, late May - early June)

Derived algebraic geometry, representability of derived moduli functors, virtual fundamental classes, derived group schemes and equivariant cohomology theories, elliptic cohomology, topological modular forms.

M. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 554–565, Birkäuser, Basel, 1995.

B. Toen and G. Vezzosi, Algebraic geometry over model categories (a general approach to derived algebraic geometry).

J. Lurie, Higher topos theory.

J. Lurie, Derived algebraic geometry. Being rewritten; old version available at http://www.math.harvard.edu/~lurie/.

J. Lurie. A survey of elliptic cohomology. Available at http://www.math.harvard.edu/~lurie/.

Paul Goerss: The moduli stack of formal groups and homotopy theory (late May - early June)

The interplay between the geometry of formal groups and homotopy theory has been a guiding influence since the work of Morava in ’70s, and it has been a thread in homotopy theory ever since. The current language of algebraic geometry allows us to make very concise and natural statements about this relationship.

In these lectures, I will discuss how the geometry of the moduli stack M of smooth one-dimensional formal groups dictates the chromatic structure of stable homotopy theory. At a prime, there is an essentially unique filtration of M by the open substacks U(n) of formal groups of height no more than n and the resulting decomposition of coherent sheaves on M gives exactly the chromatic filtration. Topics may include formal groups, the relationship between quasi-coherent sheaves and complex cobordism, the role of coordinates, the height filtration, closed points and Lubin-Tate deformation theory, and algebraic chromatic convergence. Since M is in some sense very large and cumbersome, I hope also to give some discussion of small (i.e. Deligne-Mumford) stacks over M; these include the moduli stack of elliptic curves and certain Shimura varities, thus bringing us to the current research of Hopkins, Miller, Lurie, Behrens, Lawson, Naumann, Ravenel, Hovey, Strickland, and many others.

Let me remark that I am merely the expositor here, building on the work of many people, especially Jack Morava and Mike Hopkins.

Behrens, Mark, A modular description of the $K(2)$-local sphere at the prime 3, Topology 45 No. 2 (2006), 343–402.

Goerss, Paul, (Pre-)sheaves of ring spectra over the moduli stack of formal group laws, Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., 131, 101-131, Kluwer Acad. Publ., Dordecht 2004.

Hopkins, M. J., Algebraic topology and modular forms, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 291–317, Higher Ed. Press, Beijing, 2002.

Hopkins, M. J. and Gross, B. H., The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc. (N.S.), 30 No. 1 (1994), 76-86.

Hovey, Mark and Strickland, Neil, Comodules and Landweber exact homology theories, Adv. Math., 192 No. 2 (2005), 427-456.

Ezra Getzler: Lie n-groupoids (June)

Kai Behrend (June)

Posted at September 13, 2006 9:09 AM UTC

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Tracked: December 5, 2006 6:07 PM

Re: Higher Categories and their Applications

I’ll arrive in Toronto Monday (Jan. 8) late afternoon. If possible, I’ll try to meet Igor then, but he may be occupied. Will anyone else be around that evening and in need of a beer or something? Bruce? Danny? John? Simon?

Posted by: urs on January 2, 2007 7:39 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

Tom Leinster and I are arriving on the evening of January 5th, and we’re staying at Annex Quest House. Where are you staying? We’ll be talking about n-categories, so I’ll probably need a beer by then.

Posted by: John Baez on January 2, 2007 9:09 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

Where are you staying?

I have a room in

“Days Hotel & Conference Centre” (30 Carlton Street)

If you just let me know where I can find you, I’ll come by.

We’ll be talking about $n$-categories, so I’ll probably need a beer by then.

Due to jet lag, I might even need a 2-beer. But a strict one.

Posted by: urs on January 2, 2007 9:20 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

John wrote:

Tom Leinster and I are arriving on the evening of January 5th

Actually, I’m arriving on the evening of 6th (Saturday). I fully expect to be jet-lagged and exhausted from having stayed up packing and preparing all of the night before, as seems to be inevitable. But it would be great to meet up with people, and the beer will go to my head even faster than usual.

Posted by: Tom Leinster on January 3, 2007 3:37 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

Should we arrange then that I’ll come by the Annex Quest House after I have dropped my luggage?

My plane is supposed to arrive 15:40 at Pearson International.

I don’t know how long it will take me to get from that point to my hotel and then to some place where we meet. Maybe 18:00 is reasonable.

Posted by: urs on January 3, 2007 3:56 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

This might be off to the side, but are there any other Café regulars going to the New Orleans meeting? A meet-up at some point? Or is everyone else going up to the Toronto event?

Posted by: John Armstrong on January 3, 2007 1:35 AM | Permalink | Reply to this

Re: Higher Categories and their Applications

John Armstrong wrote:

[…] are there any other Café regulars going to the New Orleans meeting? A meet-up at some point? Or is everyone else going up to the Toronto event?

The 2007 Joint Mathematics Meetings in New Orleans will end before Higher Categories and their Applications starts — and my student Jeffrey Morton is going to both! I’ll send him an email saying you’ll be in New Orleans. You can recognize him from the blurry picture on his webpage, or this. He’s a lot less scary than he looks.

I don’t know any other Café regulars going to New Orleans.

Posted by: John Baez on January 3, 2007 1:53 AM | Permalink | Reply to this

Re: Higher Categories and their Applications

I remember him from his talk at Union. Thanks for the heads-up, I’ll try to run into him.

Posted by: John Armstrong on January 3, 2007 3:29 AM | Permalink | Reply to this

Re: Higher Categories and their Applications

Of course! We all met at Union College!

Posted by: John Baez on January 3, 2007 5:50 AM | Permalink | Reply to this

Re: Higher Categories and their Applications

Will anyone else be around that [Monday] evening and in need of a beer or something?

I should be arriving (downtown, by Greyhound) Monday morning at 9:25 EST. I won’t be jet-lagged exactly, but I’ll still be tired. So I really should meet somebody that morning for coffee (or tea, whatever they drink in Canada). I don’t drink alcohol, but I’m sure that I’ll find something to substitute by the evening.

Posted by: Toby Bartels on January 3, 2007 9:35 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

I’m sure that I’ll find something to substitute by the evening.

Okay, good. Where should we all meet, then, Monday evening?

Since I don’t know Toronto, I cannot make any good suggestions.

P.S.

Konrad Waldorf, a colleague of mine, will also arrive that evening. But later. It would maybe be good if we could agree on a meeting place where it is agreeable to spend some time. That would enlarge the chances of our intercontinental worldlines to get close enough in spacetime.

Posted by: urs on January 3, 2007 10:02 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

I don’t know Toronto, but here’s one option for a place to meet:

It’s relatively cheap and it has food, coffee and booze — though I only see wine and other drinks on their menu, not beer. It stays open late, and I believe it has plenty of indoor seating, though they brag about their patio. With high temperatures in Toronto averaging 5 Celsius, I don’t think we want to sit outside!

(It’s really called the Café Diplomatico.)

It’s marked ‘B’ on this map. The grey stuff above the eastern end of College Street on this map is the University of Toronto. The Fields Institute is 222 College St., one block east of the intersection of College and Spadina.

Annex Quest House is roughly ten blocks north on Spadina — it’s the point marked ‘A’ here. 30 Carlton Street, where Toby is staying, is about ten blocks east on College — it’s the point marked ‘F’ here.

Since Urs may arrive around 18:00 on Monday January 8th, I suggest that we all meet at the Café Diplomatico around this time. Konrad Waldorf could show up later.

Does this seem like a good plan?

I also suggest that Tom try to reach me at the hotel when he arrives the evening of the 6th. It seems at this time nobody n-categorical will be there but me.

Perhaps Toby should call me when he arrives in Toronto on the morning of Monday the 8th. The phone number at Annex Quest House is 1-416-922-1934. We can try to have brunch somewhere on College Street.

Posted by: John Baez on January 3, 2007 11:31 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

Here is a single map with all of the things on John’s map (designed to use John’s key), and you don’t even need Javascript to see it as intended. (But John’s maps have external links.) Additionally, it has the Fields Institute marked with “C”, and it’s eminently hackable.

30 Carlton Street, where Toby is staying […]

You got this information from Jim?

I suggest that we all meet there around this time.

Is “there” the Café Diplomatico again (which makes more sense), or still 30 Carlton Street (which an English teacher would insist on)?

Perhaps Toby should call me when he arrives in Toronto on the morning of Monday the 8th.

All right, I will!

Posted by: Toby Bartels on January 4, 2007 1:17 AM | Permalink | Reply to this

Re: Higher Categories and their Applications

I wrote above:

Here is a single map with all of the things on John’s map (designed to use John’s key)

To clarify that key: * A is the Annex Quest House, 83 Spadina Rd * B is the Café Diplomatico, 594 College St * C is the Fields Institute, 222 College St * F is the Days Hotel & Conference Centre, 30 Carleton St

Posted by: Toby Bartels on January 7, 2007 1:28 AM | Permalink | Reply to this

Re: Higher Categories and their Applications

Toby wrote:

30 Carlton Street, where Toby is staying […]

You got this information from Jim?

No, I got mixed up — I read that Urs is staying there, but thought you were. So you and Jim are also staying there, eh?

I suggest that we all meet there around this time.

Is “there” the Café Diplomatico again […] ?

Yes. I’ve fixed my comment to make it 100% clear.

Posted by: John Baez on January 4, 2007 1:41 AM | Permalink | Reply to this

Re: Higher Categories and their Applications

So you and Jim are also staying there, eh?

I’m not sure, but I don’t think so. That’s why I asked if Jim told you – because he didn’t tell me.

Posted by: Toby Bartels on January 5, 2007 3:53 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

Does this seem like a good plan?

Very well. See you there then! I am looking forward to it.

Posted by: urs on January 4, 2007 10:42 AM | Permalink | Reply to this

Re: Higher Categories and their Applications

Incidentally, are there any n-categorical social meetings happening in Toronto today (Sun 7th Jan)? Sitting here at an internet cafe near Yonge street… wishing I had a laptop.

Posted by: Bruce Bartlett on January 7, 2007 6:28 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

In case anyone is curious, I met Toby around noon today at the Future Bakery and Café on Bloor St., which seems to be the place to hang out, drink coffee, have breakfast at any time, and talk about math.

Later, Tom showed up and met Toby for the first time. We had a great conversation about category algebras, Möbius functions of categories, and using these to solve problems in combinatorics and number theory — topics I need to study in preparation for writing an issue of This Week’s Finds about Tom’s work on the Euler characteristic of a category.

Later, Toby and I walked to the Fields Institute, leaving Tom to finish preparing his talk on bicategories — the first talk of the workshop tomorrow!

Here’s Tom last night after dinner at Urban Thai on College Street:

He isn’t smiling like that now!

At the Fields Institute I ran into Simona Paoli, Dorette Pronk and (for the first time) Bruce Bartlett, who is now merrily talking to Toby.

At 6 pm today, a bunch of people will meet at the Café n-Categorico… the show is starting!

Posted by: John Baez on January 8, 2007 8:48 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

In case anyone is curious…

Yes, keep those of us left behind informed about what’s going on.

…topics I need to study in preparation for writing an issue of This Week’s Finds about Tom’s work on the Euler characteristic of a category.

Great. If you could get to the bottom of what was said about graph Laplacians on that thread that would be wonderful.

Posted by: David Corfield on January 8, 2007 11:02 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

keep those of us left behind informed about what’s going on.

Hi David,

I did hope to find the time to write some reports. But it turned out to be impossible so far, for me. Too many interesting things going on.

I am hoping to say something when I am back home.

(When and if. Currently there seems to be reason for concern about bad weather at various airports. Hope I don’t have to spend the weekend at Pearson International.)

Posted by: urs on January 12, 2007 2:23 PM | Permalink | Reply to this

Re: Higher Categories and their Applications

You can see a bunch of photos of people at the workshop. I’ll sort them out and label them later.

I spent a day talking to Tom Leinster about his ‘Euler characteristic of a category’ (or $n$-category!) — now I understand it much better. I don’t think Laplacians are crucial here; Möbius inversion is crucial.

Posted by: John Baez on January 12, 2007 2:20 PM | Permalink | Reply to this

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