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February 12, 2007

This Week’s Finds in Mathematical Physics (Week 245)

Posted by John Baez

In week245 of This Week’s Finds, read about the Fields Institute workshop on Higher Categories and Their Applications — and the piano Coxeter played at the age of three!


Here’s the model of the 120-cell hanging from the Fields Institute ceiling:

Posted at February 12, 2007 2:29 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 245)

Strict2Cat = [strict 2-categories, strict 2-functors, strict natural transformations, modifications]

Weak2Cat = [weak 2-categories, weak 2-functors, weak natural transformations, modifications]

Gray = [strict 2-categories, strict 2-functors, weak natural transformations, modifications]

We have inclusions of weak 3-categories: Strict2Cat → Gray → Weak2Cat: and Lack shows, not only that the second inclusion fails to be an equivalence, but that there’s no equivalence between Gray and Weak2Cat.

But to be fair, there’s another possibility intermediate between Gray and Weak2Cat:

Toby = [strict 2-categories, weak 2-functors, weak natural transformations, modifications]

Except nobody is ever going to talk about the 3category Toby, because it is equivalent to Weak2Cat!

And that’s why everybody kept citing some comment of Bénabou’s, to the effect that the really important thing was not weak 2categories (which he invented) but weak 2functors (which I guess he also invented at the same time, at least in full generality).

Posted by: Toby Bartels on February 12, 2007 7:20 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 245)

Model category theory formalizes this by speaking of a category C equipped with a classes of morphisms called “weak equivalences”. We can formally invert these and get a new category Ho(C) where the weak equivalences are isomorphisms: this is called the “homotopy category” or “derived category” of our model category. But this loses information, so it’s often good not to do this.

But since we have a notion of homotopy of homotopies of … in a model category, what we really ought to do is insert these inverses, but declare them inverses only up to a homotopy (an invertible 2morphism, which we insert), and this is invertible only up to a homotopy of homotopies (an invertible 3morphism, which we insert), and so on. I’m unsure to what extent the resulting (∞,1)category has lost information.

Posted by: Toby Bartels on February 12, 2007 7:24 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 245)

Toby wrote:

… what we really ought to do is …

I see where you’re coming from, but freely adjoining all these homotopies and homotopies between homotopies seems unlikely to get us the right answer.

The right answer is to use the simplicial localization of Dwyer and Kan:

  • William G. Dwyer and Daniel M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980), 267-284.
  • William G. Dwyer and Daniel M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980), 17-35.
  • William G. Dwyer and Daniel M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440.

(You can see these papers online if you use the UCR proxy server or some other method of proving to the journal that you have an academic affiliation with some institution that pays for this journal.)

Dwyer and Kan showed that for any category equipped with a notion of ‘weak equivalences’, we can construct a simplicially enriched category. This is the simplicial way of getting our hands on the ‘right’ (,1)(\infty,1)-category.

In the special case of a model category, they provide a bunch of equivalent ways to get their hands on this simplicially enriched category.

But, these guys (like you) are very fond of seeing how much can be done with just a concept of ‘weak equivalences’.

I explained this stuff in my talk at the Fields Institute, though quite sketchily. See pages 11–16.

Posted by: John Baez on February 13, 2007 2:36 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 245)

I see where you’re coming from, but [it’s] unlikely to get us the right answer.

The right answer is to use the simplicial localization of Dwyer and Kan:

I explained this stuff in my talk at the Fields Institute, though quite sketchily.

Right, I remember the slogan

Model categories are a trick for getting (∞,1)-categories.

but I forgot the detail

Dwyer and Kan’s ‘simplicial localization’.

So I should look at that to learn to do it right!

Posted by: Toby Bartels on February 13, 2007 9:45 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 245)

Toby wrote:

So I should look at that to learn to do it right!

Yes. It’s pretty simple if you know the bar construction, which is the universal method of ‘weakening’ any sort of structure that can be described as the algebras of a monad. If you don’t know the bar construction, you’d better start by studying that — it’s really cool!

Todd Trimble gave a nice description of the bar construction at the level of generality we need here — try it!

The bar construction is simplicial in nature; your idea of ‘freely throwing in homotopies between homotopies…’ sounds globular. Now that I think about it, your idea may be just the globular way of talking about Dwyer and Kan’s simplicial localization! I’m not sure — maybe you can figure it out.

Posted by: John Baez on February 16, 2007 3:00 AM | Permalink | Reply to this

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