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.

## Re: Two ArXiv Papers

David Corfield wrote elsewhere on this blog:

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.