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January 8, 2009

Categories, Logic and Physics in London

Posted by David Corfield

I and many others braved the British winter and failing rail network to attend the 4th CLP Workshop, held yesterday at Imperial College London.

I was going to say something about Ieke Moerdijk’s talk on ‘Infinity categories and infinity operads’, but I see Urs has already published notes on a very similar talk given in Lausanne last November. (Those operads in the diagram at the top of page 3 are coloured, by the way.) This is part of an effort to provide a new definition of nn-category.

Perhaps it’ll be time soon for an update to Tom Leinster’s survey of definitions of nn-category. There he looked at ten definitions. In his talk yesterday he mentioned that there are at least twelve. But, as he pointed out to me, counting definitions as different is not so straightforward.

Tom promised to bore all the experts present with his introductory talk. So we all strove to be bored. Not, as he noted, that boredom would establish expertise.

Eugenia Cheng gave us a talk on the periodic table, and the work she has done with Nick Gurski on degeneracy. Seeing the old table again, I was wondering what happens in the limit down that diagonal where nn-categories stabilise. John and Jim call them stable ω\omega-categories (p. 25). The free such one on one generator they call Braid Braid_{\infty}. Presumably one would call the analogous one with duals Tang Tang_{\infty}, but it would turn out to be equivalent to Π(Ω S )\Pi(\Omega^{\infty} \S^{\infty}) in view of Eugenia’s result.

These are not to be confused with Lurie’s stable infinity categories, which are stable (,1)\infty, 1)-categories.

An idle thought occurred to me on (n,rn, r)-categories as to whether you might have an additional interval where jj-morphisms have duals up to a certain point, then are equivalences, before becoming trivial.

Posted at January 8, 2009 9:55 AM UTC

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Re: Categories, Logic and Physics in London

Come to think of it, maybe Lurie’s stable \infty-categories aren’t even stable in the Baez-Dolan sense, let alone \infty-categories in their sense.

If so, another opportunity for a Leinster growl?

Posted by: David Corfield on January 8, 2009 4:47 PM | Permalink | Reply to this

Re: Categories, Logic and Physics in London

Yeah, they are stable in a different sense. However, I don’t think this one can be blamed on Lurie; the word “stable” has been used in algebraic topology with this meaning for quite some time.

Actually, there is a connection between the two usages. Namely…

  • Baez-Dolan stable \infty- groupoids can be identified with
  • E E_\infty-spaces, of which the grouplike ones can in turn be identified with
  • connective spectra, and
  • an (,1)(\infty,1)-category is stable in Lurie’s sense if it behaves similarly to the (,1)(\infty,1)-category of all spectra (not just connective ones).

So in Lurie’s sense a “stable” (,1)(\infty,1)-category means “a category of stable objects” rather than a category which is itself stable.

Posted by: Mike Shulman on January 9, 2009 6:47 PM | Permalink | Reply to this

Re: Categories, Logic and Physics in London

so it’s worth the other key strokes to use
“a category of stable objects”

Posted by: jim stasheff on January 10, 2009 2:30 PM | Permalink | Reply to this

Re: Categories, Logic and Physics in London

An idle thought occurred to me on (n,r)(n,r)-categories as to whether you might have an additional interval where jj-morphisms have duals up to a certain point, then are equivalences, before becoming trivial.

Certainly, but I would wait until somebody actually used these (at least in low nn) before I'd make up a name and a table for them. When you think about it, there are a lot of possible variations!

Posted by: Toby Bartels on January 8, 2009 4:53 PM | Permalink | Reply to this

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