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June 19, 2007

Degeneracy

Posted by David Corfield

Eugenia Cheng and Nick Gurski have just put on the ArXiv a sequel to their

The periodic table in low dimensions I: degenerate categories and degenerate bicategories,

entitled

The periodic table of nn-categories for low dimensions II: degenerate tricategories.

Posted at June 19, 2007 1:24 PM UTC

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Re: Degeneracy

Only had time to very quickly look at this paper. But here is a quick question.

One of the puzzles mentioned in the introduction is that thinking of a kk-fold monoidal rr-catgory as a (k+r)(k+r)-category with all (ik)(i \leq k)-morphisms trivial runs us into the problem that the former gadget seems to want to live in a (r+1)(r+1)-category, while the latter lives in an (k+r+1)(k+r+1)-category.

But maybe this is rather telling us that our expectation about the home of kk-fold monoidal rr-categories is wrong?

The reason I am saying this is that only recently I had a long discussion with somebody which crucially involved the 2-category whose

- objects are groups

- morphisms are group homomorphisms

- 2-morphisms are “intertwiners” of these, namely precisely those 2-morphisms which we obtain by thinking of the groups as 1-object categories and of their morphisms as fucntors.

The application we were talking about crucially demanded to take this 2-category serious. I noticed that it took me a while to make the structure of this 2-category transparent to my discussion partner. And I thought by myself that we should all better get used to thinking of groups as 1-object groupoids generally.

Precisely the same issue, in its analogous incranation, plays a crucial role in the entire field of von Neumann algebras. There it is very important to consider 2-categories (even though these are not always identified as such) whose objects are algebras, whose morphisms are algebra homomorphisms and whose 2-morphisms are intertwiners. In other words, to regard algebras as 1-object Vect-enriched categories.

In a couple of introductory talks to von Neumann algebra theory that I heard recently on this subject lots of time was spent with explaining what these intertwiners are and how their horizontal and vertical compositon works.

To the uninitiated eyes, the entire construction here is bound to look intricate and ad hoc. But it all comes down to a triviality once we seriously think of algebras as 1-object categories: these intertwiners are precisely natural transformations, henece 2-morphisms in Cat. That explains everything that is ever done with them.

So, my question is this: maybe it is “evil” (in the sense John Baez uses this word) to regard kk-fold monoidal rr-categories as anything else than kk-tuply stabilized (k+r)(k+r)-categories. Maybe we should not try to do that in the first place. A couple of applications do suggest so.

(By the way: Bruce once had a remark/question on precisely this issue here: Algebras as 2-Categories and its Effect on Algebraic Geometry).

Posted by: urs on June 19, 2007 1:59 PM | Permalink | Reply to this

Re: Degeneracy

…the entire field of von Neumann algebras. There it is very important to consider 2-categories (even though these are not always identified as such) whose objects are algebras, whose morphisms are algebra homomorphisms and whose 2-morphisms are intertwiners. In other words, to regard algebras as 1-object Vect-enriched categories.

I’m confused about this. I thought you preferred to treat von Neumann algebras in a bicategory with bimodules and bimodule homomorphisms.

That points then to the question addressed by Mike Shulman in the paper here.

Posted by: David Corfield on June 20, 2007 8:54 AM | Permalink | Reply to this

Re: Degeneracy

I thought you preferred to treat von Neumann algebras in a bicategory with bimodules and bimodule homomorphisms.

That’s the next step then, I’d think.

First we realize that we should regard algebras as 1-object categories and their morphisms as functors. This gives us 2-morphisms: “intertwiners”.

Next, we may want to be even more sophisticated and replace functors by profunctors or the like. That would correspond to replacing functors by bimodules.

By the way, at least for certain von Neumann factors (I forget the details of the assumption going into this) it turns out that all bimodules come from algebra homomorphisms. Hence all bimodule homomorphisms from “intertwiners”. So in these cases the two options above even coincide.

Another remark: if we do the same for groups, we also have the option of replacing functors by bitorsors. This should correspond precisely to passing from functors to saturated anafunctors/Hilsum-Skandalis morphisms.

Posted by: urs on June 20, 2007 12:09 PM | Permalink | Reply to this

Re: Degeneracy

I wonder if this is part of a larger trend to ‘relations’ rather than ‘sets’, that better than (categories, functors, natural transformations) is the bicategory (categories, profunctors, natural transformations). (Is there a better term for the 2-morphisms?)

Sorry to harp on about this, but nobody has expressed an opinion on the paper of Mike Shulman I keep mentioning, which suggests that some relational kinds of bicategory seem to have left something out. In the case of von Neumann algebras is something missing if you consider bimodules and bimodule homomorphisms?

Posted by: David Corfield on June 20, 2007 1:27 PM | Permalink | Reply to this

Re: Degeneracy

In the case of von Neumann algebras is something missing if you consider bimodules and bimodule homomorphisms?

This is probably not the example you want to be looking at, regarding your question. Most people working in von Neumann algebras (actually, I know only one single exception) never mention bimodules at all. Only “intertwiners”.

One could imagine that somebody in von Neumann algebra theory once said: “We should be using bimodules, but, it turns out all bimodules here do come from algebra homomorphisms, so we may just as well stick with homomorphisms and intertwiners of these.”

I didn’t get the impression that anyone did say so. But it is apparently true nevertheless.

nobody has expressed an opinion on the paper of Mike Shulman I keep mentioning

I haven’t found the time to look at it at all! Sorry.

Posted by: urs on June 20, 2007 1:48 PM | Permalink | Reply to this

Re: Degeneracy

I am really confused here. Just had a quick look at Mike Shulman’s paper, which you mentioned. Just the abstract and part of the introduction.

It seems that what he says is “missing” when we pass to bimodules are the ordinary homomorphisms.

I am not sure I understand the point here. I’d think these homomorphisms are represented by certain bimodules, pretty much like functors are a special case of profunctors.

Could somebody help me see the problem that Mike Shulman tries to solve?

Posted by: urs on June 20, 2007 3:30 PM | Permalink | Reply to this

Re: Degeneracy

Urs wrote:

I am not sure I understand the point here. I’d think these homomorphisms are represented by certain bimodules, pretty much like functors are a special case of profunctors.

I think the problem is that this is not always a satisfactory solution. Mike explains some reasons on the bottom of page 2. Two things which stuck out for me are:

(a) We would like ModMod to be a monoidal bicategory. The nicest interpretation of this is that the associator should be a ring map, but the way things are currently set up, it’s just an invertible bimodule. Even though that can be made into a ring map, it seems slightly unsatisfactory.

(b) There are occasions when you come across two “ModMod-like” 2-categories AA and BB in the wild, and they’re kind of clearly equivalent in a very strong way (stronger than just saying they’re an equivalence of 2-categories), in the sense that their “ring homomorphisms sectors” also co-incide. The current language can’t express this phenomenon.

That was my impression when I read the introduction to this paper.

Posted by: Bruce Bartlett on June 20, 2007 4:32 PM | Permalink | Reply to this

Re: Degeneracy

Even though that can be made into a ring map, it seems slightly unsatisfactory.

This is what I don’t understand. What is unsatisfactory about it? The idea of profunctors seems to show that it is instead perfectly natural?!

Posted by: urs on June 20, 2007 5:04 PM | Permalink | Reply to this

Re: Degeneracy

Urs wrote:

It seems that what he says is “missing” when we pass to bimodules are the ordinary homomorphisms.

I am not sure I understand the point here. I’d think these homomorphisms are represented by certain bimodules, pretty much like functors are a special case of profunctors.

Could somebody help me see the problem that Mike Shulman tries to solve?

Hi! I’m at a public library, and I have a little time to read this blog…

Sure, you can think of ring homomorphisms as special bimodules. But, how do you single out these ‘special’ bimodules for special treatment? You can do it in an ad hoc way, but Mike Shulman, Peter May et al want to do it more systematically. To do this, they equip the usual bicategory with

  • rings as objects
  • bimodules as 1-morphisms
  • bimodule homomorphisms as 2-morphisms

with some extra structure. Very roughly (I’ve seen talks on this, but forgotten a lot), the idea is to keep track of two kinds of 1-morphisms between rings: the homomorphisms, and the bimodules. Every homomorphism gives rise to a bimodule, and a bunch of other stuff is true too. Shulman axiomatizes this stuff using the concept of ‘framed bicategory’.

You might hope that being ‘framed’ was a property of a bicategory, rather than extra structure. For example, you might try to start with the bicategory with

  • rings as objects
  • bimodules as 1-morphisms
  • bimodule homomorphisms as 2-morphisms

and find which 1-morphisms correspond to ring homomorphisms, using solely the bicategory structure.

However, this seems impossible to do — at least in a non-evil way. If any sort of 1-morphism corresponds to a ring homomorphism, certainly an equivalence should. An equivalence, after all, is ‘just as good as an identity 1-morphism’, and the identity should come from a ring homomorphism if anything does. But, equivalences in this bicategory are ‘Morita equivalences’, and rings can be Morita equivalent even when there are no homomorphisms between them!

I think this shows that ‘detecting which bimodules come from ring homomorphisms’ is not something you can do given just the above bicategory. That’s why Shulman introduces an extra structure: the ‘framing’.

Posted by: John Baez on June 21, 2007 10:32 PM | Permalink | Reply to this

Re: Degeneracy

Thanks! Very helpful.

Clearly, the problem is that I am being way too naive.

For any braided monoidal category CC, I used to be very fond of the fact that Bim(C) \mathrm{Bim}(C) is monoidal (because transformations between 3-functors with values in ΣBim(C)\Sigma \mathrm{Bim}(C) should describe “2-tiered” 2-dimensional CFT).

I believe internally I was always thinking of a kind of strictified version of this, where, for any three algebra objects AA, BB, CC in my braided monoidal category I could neglect the nontriviality of the isomorphism A(BC)(AB)C. A \otimes (B \otimes C) \simeq (A \otimes B) \otimes C \,. Similarly for a triple tensor product of bimodules between such algebras.

I guess I can consider myself lucky that just as I discover that this is too naive, somebody has worked out the solution to it. ;-)

Posted by: urs on June 22, 2007 7:25 AM | Permalink | Reply to this

Re: Degeneracy

You might hope that being ‘framed’ was a property of a bicategory

Can this be described as a double category with a property? And, if so, is this an argument for viewing double categories as more natural than bicategories?

Posted by: David Corfield on June 23, 2007 1:00 PM | Permalink | Reply to this

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