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January 23, 2007

Higher Categories and Their Applications

Posted by John Baez

The Fields Institute workshop on n-categories was a lot of fun. If you couldn’t make it, you can still see what it was like. Here’s a webpage with abstracts, transparencies and photos of lots of the talks:

Someday soon I want to describe some of these talks — but not tonight.

Here are a couple of the photos that Dan Christensen took at the workshop.

André Joyal:

Urs Schreiber, John Baez and Toby Bartels:

There are a lot more photos in disorganized form here. If you went there and took some pictures, please pass them on to me, or at least give me links!

Posted at January 23, 2007 5:23 AM UTC

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9 Comments & 3 Trackbacks

Re: Higher Categories and Their Applications

So how many of our questions got answered?

Posted by: David Corfield on January 23, 2007 12:16 PM | Permalink | Reply to this

Re: Higher Categories and Their Applications

So how many of our questions got answered?

I had long and very interesting discussions with Nils Baas, that I am very grateful for. I tried to attract Nils Baas to our first nn-Café Millenium Prize Question. But I am not sure if I did succeed.

Posted by: urs on January 23, 2007 12:31 PM | Permalink | Reply to this

Re: Higher Categories and Their Applications

So how many of our questions got answered?

As it goes with good conferences, there is always more to talk about than time permits.

And that even holds if one is crazy enough to visit Toronto and be content with College Street being the only part of the city one visits.

Among the things that I was looking forward to talk about was Toby’s reconstruction of 2-bundles from local data.

We did find time for that, if maybe not as much as the topic deserves.

Interestingly, it turns out that, as far as I could figure out, Toby’s reconstruction does produce a G 2G_2-2-bundle that is trivial as a fiber bundle of categories BXB \to X, i.e. where the category BB is of the form B=X×FB = X \times F. All the nontriviality is pushed into the G 2G_2-action B×G 2BB \times G_2 \to B.

In fact, something roughly similar happens for the reconstruction of global transport 2-functors that I am working on with Konrad Waldorf.

Posted by: urs on January 23, 2007 9:10 PM | Permalink | Reply to this

Answers without Questions

So how many of our questions got answered?

One nice thing was that I received a couple of answers to questions that I did not even know I should ask! :-)

One nice insight was this:

André Joyal, in his talk, mentioned various important concepts in category theory. One of the them was related to factorization systems.

In that business, one considers collections EE and MM of morphisms that (are supposed to) behave like epi- and monomorphisms.

In particular, one considers squares A epi B C mono D \array{ A &\stackrel{\mathrm{epi}}{\to}& B \\ \downarrow && \downarrow \\ C &\stackrel{\mathrm{mono}}{\to}& D } where the top morphism is an epimorphism and the bottom morphisms is a monomorphism.

Of special interest, then, is the situation where a square of the above kind admits a diagonal A e B C m D, \array{ A &\stackrel{e}{\to}& B \\ \downarrow &\swarrow& \downarrow \\ C &\stackrel{m}{\to}& D } \,, for every choice of vertical arrows, such that everything still commutes

If that is the case, one says that the epimorphism ee is orthogonal to the monomorphism mm.

There are more or less obvious generalizations of this concept for the case where everything lives in nn-categories.

(I am grateful to Igor Bakovic for helpful discussion of this and of the related literature. In as far as I have forgotten the details again, that’s completely my own fault.)

What made all this interesting for me was that I recognized my own pet structures in this general structure that people are considering:

In my talk in Toronto I described how we can put local structure on an nn-functor P n(X) tra T \array{ P_n(X) \\ \;\;\downarrow \mathrm{tra} \\ T } by putting it into a square P n(U) p P n(X) tra U tra T i T, \array{ P_n(U) &\stackrel{p}{\to}& P_n(X) \\ \mathrm{tra}_U \downarrow\;\; &\Downarrow \sim& \;\;\downarrow \mathrm{tra} \\ T' &\stackrel{i}{\to}& T } \,, where pp is epi and ii is mono.

If pp happens to be “orthogonal” to ii in the sense of factorization system theory, i.e. if we can fill this diagram P n(U) p P n(X) tra U tra T i T, \array{ P_n(U) &\stackrel{p}{\to}& P_n(X) \\ \mathrm{tra}_U \downarrow\;\; &\swarrow& \;\;\downarrow \mathrm{tra} \\ T' &\stackrel{i}{\to}& T } \,, then this means, in the context I consider, that tra\mathrm{tra} has the given structure not just locally, but already globally.

By itself that’s not deep or anything. But I enjoyed seeing this contact to a larger context.

Posted by: urs on January 30, 2007 6:11 PM | Permalink | Reply to this
Read the post Cocycle Category
Weblog: The n-Category Café
Excerpt: Bruce Bartlett reports on Rick Jardine's concept of 'cocycle categories' and their relation to anafunctors.
Tracked: January 24, 2007 12:52 PM

Re: Higher Categories and Their Applications

Thanks for the report.
Will try to do something similar for
the coherent ;-) one in Paris
but I don’t know if anyone took photos.
Thanks for all those - who was the photographer?

Posted by: jim stasheff on January 30, 2007 5:02 PM | Permalink | Reply to this

Re: Higher Categories and Their Applications

I took most of the photos; Dan Christensen took a few.

Posted by: John Baez on January 30, 2007 6:35 PM | Permalink | Reply to this

Re: Higher Categories and Their Applications

For the Higher Structures … IHP fest
talks, some of which are now available,
go to

http://www.math.psu.edu/ping/IHP07/

and click on Speakers (lower right side of page)

Posted by: jim stasheff on February 2, 2007 2:45 PM | Permalink | Reply to this
Read the post Higher Structure in Geometry and Physics -- and in Paris
Weblog: The n-Category Café
Excerpt: Slides of talks for a conference on higher structures in geometry and physics
Tracked: February 2, 2007 3:24 PM

Dorette Pronk’s talk

I was asked for notes taken in Dorette Pronk’s talk #.

Unfortunately, I don’t have any reasonable notes to offer. Partly because I was quickly lost in the technical details.

Does anyone have notes he or she would share (say, scan in)?

Posted by: urs on February 6, 2007 1:36 PM | Permalink | Reply to this

Re: Dorette Pronk’s talk

I don’t, alas. If someone does, they should give me a copy, so I can add them to my webpage for this workshop. Right now for her talk I only have a link to the most relevant available paper of hers, Entendues and stacks as bicategories of fractions. This does not include her new work on tricategories of fractions.

Posted by: John Baez on February 7, 2007 9:35 PM | Permalink | Reply to this
Read the post QFT of Charged n-particle: Chan-Paton Bundles
Weblog: The n-Category Café
Excerpt: Chan-Paton bundles from the pull-push quantization of the open 2-particle.
Tracked: February 7, 2007 10:00 PM

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