## June 12, 2007

### Two ArXiv Papers

#### Posted by David Corfield

Two papers of interest on the ArXiv today.

1) Framed bicategories and monoidal fibrations by Michael Shulman: what’s missing when you treat (rings, bimodules, bimodule homomorphisms) as an ordinary bicategory.

2) A universal property of the monoidal 2-category of cospans of finite linear orders and surjections by M. Menni, N. Sabadini, and R. F. C. Walters: extending the programme of capturing the universal properties of cospan-like categories to a 2-category.

Posted at June 12, 2007 12:31 PM UTC

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### Re: Two ArXiv Papers

David Corfield wrote elsewhere on this blog:

…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.

I know that Simon (my supervisor) has thought quite a bit about these things, since they also crop up in his work. He seemed quite receptive to the philosophy presented in the paper of Mike Shulman. Here at Sheffield there is also Johann Sigurdsson, coauthor of the book on Parametrized Homotopy Theory, in which these issues crop up, and which I think served partly as a motivation for Mike Shulman’s introduction of ‘framed bicategories’.

I don’t know enough about these things to comment. But I find the interplay between completely algebraic formulations of categorical concepts, non-algebraic formulations, and philosophies which combine the two, as the most fascinating but also the most difficult part of higher category theory. It seems to me that all three of the papers recently mentioned at the n-cafe (the two mentioned above and the new paper by Eugenia Cheng and Nick Gurski on degenerate tricategories) have this theme running through them, to some extent.

Posted by: Bruce Bartlett on June 20, 2007 2:35 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

Mike Shulman’s abstracts says:

Many bicategorical notions do not work well in these cases [for instance bimodules], because the ‘morphisms between 0-cells’, such as ring homomorphisms, are missing.

Every ring homomorphism gives rise to a bimodule, and composition of these bimodules corresponds to composition of the corresponding morphisms. What am I missing?

Posted by: urs on June 20, 2007 3:25 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

I think the problem is that you are heading for a world of pain! :-)

For example, we would like to say that $\mathcal{V}$-$\mathrm{Mod}$ is a symmetric monoidal bicategory. But if you try to actually exhibit the symmetric monoidal structure, and verify the axioms, using just the bicategory structure, then you will rapidly go mad.

What makes it particularly frustrating is that the associator maps, for example, are really isomorphisms; but all you are allowed say is that they are Morita equivalences!

Using Mike Shulman’s approach, one can easily show that $\mathcal{V}$-$\mathrm{Mod}$ is a symmetric monoidal framed bicategory, and then that (by the argument in one of his appendices) this induces a symmetric monoidal structure on the horizontal bicategory. Much easier!

For another example of a useful theory that would be difficult or impossible to build without a Shulman-style approach, see Richard Garner’s work on Double Clubs.

Posted by: Robin on June 20, 2007 5:35 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

What makes it particularly frustrating is that the associator maps, for example, are really isomorphisms; but all you are allowed say is that they are Morita equivalences!

Hm. Maybe that’s a point I am missing: I’d think we have the freedom to consider structural morphisms which are mere Morita equivalences. But if we are lucky and find that these morphisms are even isomorphisms (instead of a mere equivalences) then this is possible just as well.

I mean: in a 2-category, I can still say “isomorphism”. Why not?

Posted by: urs on June 20, 2007 6:26 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

Okay, I guess the point is probably that with algebra morphisms turned into bimodules, their compositions and inverses behave as usual only up to (canonical, I’d think) 2-isomorphism.

Posted by: urs on June 20, 2007 7:15 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

Sure, you can say isomorphism. But it doesn’t mean very much! I mean, in general an object of a bicategory is not even isomorphic to itself.

More specifically, as you pointed out, the inclusion $\mathrm{Cat}\to\mathrm{Mod}$ is only a pseudofunctor, and it doesn’t preserve isomorphisms. (Like all pseudofunctors, it does of course preserve equivalences.)

Posted by: Robin on June 20, 2007 10:50 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

Posted by: David Corfield on June 20, 2007 6:43 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

I’m not a `profunctory’ Australian, but one of the classic non-trivial examples of a double category is made from functors and profunctors as horizontal and vertical morphisms respectively - so such a thing has been thought of, if not axiomatized until now.

Posted by: David Roberts on June 22, 2007 2:19 AM | Permalink | Reply to this

### Re: Two ArXiv Papers

Mike Shulman’s been busy. Two more papers today:

Comparing composites of left and right derived functors

and

Posted by: David Corfield on June 21, 2007 10:05 AM | Permalink | Reply to this

### Re: Two ArXiv Papers

Just noticed that y’all had a conversation about my paper a while ago. I don’t know whether anyone will read this now, but I feel the need to respond anyway.

First of all, I want to stress that I still believe there are many uses for ordinary bicategories, that Morita equivalence is very important, that there may be monoidal bicategories whose structural equivalences don’t arise from a monoidal framed bicategory, and so on. You may indeed be headed for a world of pain when you delve into such matters, but sometimes the rule is no pain no gain; I have a great deal of respect for the work that tricategory theorists (and tetracategory theorists!) do. However, if you can avoid the pain in some cases, why go through it unnecessarily?

Secondly, even if we only cared about the bimodule which a ring map gives rise to, rather than the ring map itself, I maintain that there are still good reasons to have an abstract structure which includes both, and which gives a formal description of the process by which ring maps give rise to bimodules. If nothing else, this is dictated by the desire for category theory to describe mathematics, because such a process is undeniably mathematics, and if we only care about ring maps qua bimodules, this process only gains in importance. In practice, there are many ring homomorphisms, and I’d be pretty disappointed if category theorists could only say “sorry, you’ve got to make all your ring homomorphisms into bimodules before I can even talk about them.” (This is, of course, hyperbole, but I hope you know what I mean.)

Thirdly, there are situations in which it’s essential to maintain the distinction between ring homomorphisms and bimodules. This is especially true when you start to do category theory: every functor gives rise to a distributor, but it’s useful to know which distributors came from functors. (I prefer “distributor” to “profunctor” because a distributor is nothing like a pro-object in a functor category.)

For example, suppose you want to talk about limits. Well, a limit is a representing object, which in your 2-category/bicategory/framed-bicategory is going to have to be a morphism from something (like a terminal or unit category) to the category in which you’re talking about limits. However, if by “morphism” you mean “distributor”, then all “limits” “exist”, because the thing you’re trying to find a representing object for just is a distributor already. The statement that it’s representable (i.e. the limit really exists) says precisely that this distributor arises from an actual functor. If you don’t know the difference between functors and distributors, you can’t even say what it means for something to be representable, and representability is undeniably a notion at the core of category theory. Similar problems arise when you want to talk about diagram or presheaf categories: their objects really need to be honest functors, not distributors. Distributors are more important than they are often given credit for outside Australia, but functors are also important.

The Australians even already have a device to keep track of which distributors arise from functors; they call it an equipment. In the case of rings, it amounts to giving also the pseudofunctor from the category of rings and homomorphisms to the bicategory of rings and bimodules. I showed in my appendix that framed bicategories are, in some sense, equivalent to equipments. So in theory, you can do anything with equipments that you could do with framed bicategories. But my experience is that in many cases, the framed-bicategorical perspective leads one to the right notion, while the equipment perspective is unproductive.

More philosophically, I believe that framed bicategories explain why we associate a ring homomorphism to that particular bimodule, in a way which equipments do not. An equipment comes equipped with (hence the name, I suppose) the pseudofunctor assigning to each ring homomorphism its associated bimodule, whereas in a framed bicategory the existence and properties of such bimodules are necessary consequences of the fact that you can extend and restrict scalars along a ring homomorphism. And extension and restriction, in turn, are defined by universal properties of bimodule maps which are equivariant with respect to ring homomorphisms. I think it’s not unreasonable to say that in category theory, we always understand something better when we know what universal property it satisfies; this is another advantage of framed bicategories over equipments.

Now, sometimes you can sort-of recover the functors as the distributors which have adjoints. This is only really true when the categories involved are “Cauchy-complete”. Often you will hear people saying that Cauchy-completeness is a “mild” condition on a category (for example, in the unenriched case it just means that idempotents split). Thus (they say), even though you care about the functors, there’s really no loss in only working with the bicategory of distributors, because you can always find the functors as the adjoint distributors.

My first response to this is that it’s like point two, above. If you only want to work with Cauchy-complete categories, this can only make the process of Cauchy-completion more important. But by only working with distributors, you’ve prevented yourself from even being able to talk about the difference between Cauchy-complete and non. Since non-Cauchy-complete categories do arise in practice, I’d be disappointed if category theorists could only say “sorry, you’ve got to Cauchy-complete your categories before I can talk about them.” In a framed bicategory, on the other hand, we can define in abstract generality what it means for an object to be Cauchy-complete, and prove theorems about the properties of Cauchy-completion.

My second response is that it’s true that Cauchy-completeness is a mild condition in some cases. No ring, for instance, is ever Cauchy-complete; the Cauchy-completion of a ring (qua one-object Ab-category) is its category of finitely generated projective modules. Maybe that’s not so bad; it’s a nice small category, and a very important one in its own right! But there exist monoidal categories $V$ (such as the category Sup of sup-lattices) such that no small $V$-category is Cauchy-complete (and in fact, whether any Sup-category can be Cauchy-complete depends on how you phrase the definition).

In “relative” situations, Cauchy-completeness is known to also include stackification. Again, you may only want to work with stacks rather than prestacks, but this makes the process of stackification no less important.

Fourthly, allow me to return to the fundamental dictum of category theory (or whatever we call it): morphisms are more important than objects. That is, a type of object is defined more by the morphisms between them than by the objects themselves. In this case, we have to go one step further; the essential difference between bicategories and framed bicategories (and the importance of the latter) doesn’t really lie in the difference between framed functors and 2-functors (be they lax, oplax, or pseudo), since the former always give rise to the latter. Rather, it lies in the difference between framed transformations and bicategorical transformations, as reflected in the 2-category of framed bicategories, framed functors, and framed transformations.

This 2-category comes with a notion of “framed equivalence”, and any framed equivalence gives rise to a biequivalence of horizontal bicategories. However, it also comes with a notion of “framed adjunction”, which does not, in general, give rise to a classical biadjunction. In practice, framed adjunctions arise in many “base change” situations, such as transferring between different types of enrichment or comparing enriched to internal categories (which was Verity’s goal in using double categories). Geometric morphisms between topoi also give rise to framed adjunctions.

Using only the language of bicategories, framed adjunctions are practically impossible to describe. Using the language of equipments, they are possible to describe in theory, and there is some partial work in this direction, but you are far less likely to come up with the right definitions unless you think in terms of double categories. In his thesis, Dominic Verity defined a tricategory of equipments in which adjunctions capture the same idea as framed adjunctions—but he did it by first making his equipments into double categories!

(The inability of bicategories to capture framed adjunctions has nothing to do with the fact that framed bicategories form a 2-category rather than a tricategory; that’s more about the avoidance of unnecessary pain. The point has to do with the nature of framed transformations, not their strictness. And in fact, framed bicategories do sometimes need to live in a tricategory, like Verity’s.)

Whew! I think that’s the end of my propaganda spiel. If you made it all the way through, thanks for reading. Though I sound like I have strong opinions, I’m happy to entertain alternate views too and I’m willing to be convinced, so please let me know what you think.

Posted by: Mike on November 9, 2007 11:03 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

Mike,

I suppose I will provide an example for a very delayed reply. But better late than never:

Today I went back to your article on framed bicategories because, even though I didn’t take the time to read more than a bit of the introduction back then, it now dawned on me that probably what you describe in there must be closely related to what Konrad and myself call a locally strict 2-functor. And indeed, that seems to be the case.

We are concerned with the problem of having a 2-functor of the kind

$tra : P_2(X) \to Bitor$ or $tra : P_2(X) \to Bimod$ and the like, i.e. 2-functors (=bifunctors) with codomain a “bicategory of the second sort, being a many-object monoidal category” in the terminology of the introduction of your article, which have the property that locally, i.e. after pulling them back along a surjection

$\pi : P_2(Y) \to P_2(X)$

they factor through a 2-functor

$i : Gr \to Bitor$ or $i : Gr \to Bimod$

with $Gr$ a strict 2-category:

$\array{ P_2(Y) &\stackrel{\pi}{\to}& P_2(X) \\ \downarrow^{triv} &\Downarrow^t_\simeq& \downarrow^{tra} \\ Gr &\stackrel{i}{\to}& Bitor }$

You have seen me draw this kind of diagram before and heard me say before: 2-functors $tra$ for which such a diagram exists encode nonabelian gerbes with connection. That’s why we are interested in them. You can think of this as a special case of what Ross Street discusses in section 6 here.

Now, without loss of generality, we can assume in both cases that $i : Gr \to ...$ is surjective on objects, locally full and faithful, etc. hence a proarrow equipment the way you recall in def C.1.

But since moreover $Gr$ is taken to be strict I gather from your discussion that considering that 2-functor $i$ is pretty much tantamount to looking at the framed bicategories of bimodules/bitorsors which you discuss.

If you can agree with that so far, I have a question for you:

we know how to do the following construction, but it it looks as if there is some general abstract framed bicategorical nonsense underlying it, which deserves to be highlighed further:

namely, given a “locally strict 2-functor” as above, we can pull back further to the kernel pair of $\pi$ where we canonically obtain an pseudonatural transformation:

$g : i \circ (\pi_1^* triv) \to i \circ (\pi_2^* triv) \,.$

This is a transformation between 2-functors with values in a weak 2-category. However, both of them factor through the strict 2-category $Gr$. So this transformation deserves to come from a component map which is an honest 1-functor

$g : P_1(Y \times_X Y) \to i_* Hom(I,Gr) \,,$

(where $I = \{\bullet \to \circ\}$ is the standard interval category)

or something very similar.

As I said, it is not hard to achieve that by more or less brute force. But I am wondering if this resonates with something you have thought about, which might allow us to refer to your abstract theory instead of writing this out the pedestrian way.

Posted by: Urs Schreiber on March 25, 2008 6:52 PM | Permalink | Reply to this

### Re: Two ArXiv Papers

For my own benefit, I have written a lightning review of the basic idea of framed bicategories:

Framed bicategories and locally strict 2-functors (pdf).

At the end I make the remark that the “proarrow equipment” of the bicategory of bimodules which is one of the archetypical examples of framed bicategories is really precisely the mechanism that underlies the canonical 2-representation of strict 2-groups # on Vect-module categories which I emphasized a lot here on the blog. For a bit more see

The canonical 2-representation (pdf).

Posted by: Urs Schreiber on March 26, 2008 6:09 PM | Permalink | Reply to this
Read the post HIM Trimester on Geometry and Physics, Week 4
Weblog: The n-Category Café
Excerpt: Talk in Stanford on nonabelian differential cohomology.
Tracked: May 29, 2008 11:30 PM
Weblog: The n-Category Café
Excerpt: An enhanced structure on a 2-category, called a "proarrow equipment," lets us define weighted limits and develop a good deal of "formal category theory."
Tracked: November 23, 2009 5:25 AM

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