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Background

There is a bicategory where the objects are categories, the 1-morphisms are profunctors, and the 2-morphisms are morphisms of profunctors. The non-obvious part of this assertion is that profunctors admit a "good" (i.e., coherently associative) composition. The way one sees this is to use the fact that the Yoneda embedding is the free cocompletion, from which we may identify profunctors from $\mathcal{C}$ to $\mathcal{D}$ with cocontinuous functors from $\operatorname{Set}^{\mathcal{C}^{\operatorname{op}}}$ to $\operatorname{Set}^{\mathcal{D}^{\operatorname{op}}}$. Over here, there is a strictly associative composition, so life is good.

All of this extends easily to the enriched setting. In particular, if we enrich over $\mathcal{V} = \operatorname{Cat}$, we can talk about strict 2-profunctors between strict 2-categories, and we get a nice bicategory of 2-categories, strict 2-profunctors, and morphisms between these.

Question

I would like to know if there is an analogous way to obtain a tricategory of bicategories and biprofunctors. You can "follow your nose" and write down what the composition should be, but checking all the axioms of a tricategory does not seem like the simplest approach. Of course, we can strictify to get equivalent strict 2-categories and 2-profunctors, but I doubt the composition will be the same; I don't think the "strict" colimits will in general agree with the "weak" colimits. Also, using the standard enriched theory only gives a bicategory; the natural transformations are lost. It seems that an appropriate statement of the form "the bicategorical Yoneda embedding is a free cocompletion" will handle things just as it does in the setting of ordinary profunctors, but I am not sufficiently comfortable with limits and colimits in bicategories to try to figure out if such a statement makes sense. Has anybody worked all of this out?

(The closest thing I have seen to such a result is this paper by Justin Greenough, which more or less establishes this result in the case of sufficiently nice $k$-linear monoidal categories. But his proof seems to be very specific to that setting, whereas one would hope that a proof could be established along the same lines as the result for usual profunctors.)

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My guess is that the answer to your question will be "no: the tricategory of bicategories and biprofunctors has not yet been constructed". –  André Henriques Aug 27 '10 at 4:08
    
It might be helpful. S. Bozapalides. A note on 2-distributors. J. Ausral. Math. Soc.,23:Series A (1977), 242–246 –  Ma Ming Apr 11 at 18:09
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3 Answers

This recent paper comes within about $3\epsilon$ of constructing the tricategory of enriched bicategories and enriched profunctors. I'm kind of sad that we didn't get around to crossing the remaining distance.

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Depending on what you actually need for your application, there might be something useful already known. Do you really need a tricategory, as opposed to some other model of weak 3-categories? Even if you had that, would you be able to do much with it?

There are several equivalent ways to think of profunctors, and some of them lend themselves to categorification quite easily. One is this:

A profunctor from $C$ to $D$ is the same a a colimit-preserving ordinary functor between the presheaf categories $PSh(C)$ and $PSh(D)$.

(See here for details: http://ncatlab.org/nlab/show/profunctor#FuncsOnPresheaves)

That's good, because a lot is known about categorifications of categories of presheaves and of morphisms between them.

For instance it is straightforward to set this up over bicategories: take objects bicategories, and hom-bicategories to be the full sub-bicategory on bicolimit-preserving bifunctors between their bipresheaf categories.

In case that your application is such that you only need higher categories whose higher morphisms are all invertible, one can go much further and consider the (oo,1)-category of (oo,1)-profunctors. By the above, this is simply the gadget whose objects are small (oo,1)-categories and whose hom-oo-groupoids are the full sub-oo-groupoids on the (oo,1)-colimit preserving (oo,1)-functors between the corresponding (oo,1)-presheaf categories.

More generally, one can generalize here (oo,1)-presheaf categories and all in addition all their reflective sub-(oo,1)-categories. That structure of (oo,1)-profunctors has proven to play a major role as kind of categorification of the category of vector spaces: one thinks of a presentable (oo,1)-category as vector space, of colimits as being sums of vectors, as colimit preserving (oo,1)-functors as linear maps.

More on this is here: http://ncatlab.org/nlab/show/Pr(infinity,1)Cat .

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Unfortunately, I really do need profunctors in the sense of maps $\mathcal{C} \times \mathcal{D}^{\operatorname{op}} \to \operatorname{Cat}$. As you say, if we could show that "my" profunctors are the same as "your" profunctors (bicolimit-preserving bifunctors between bipresheaf categories), we'd be done. The $(\infty, 1)$-categorical version would be sufficient for my purposes, but I do not believe anybody has yet worked out how to compose "my" profunctors in this setting (although I know that there are people actively thinking about these things). –  Evan Jenkins Aug 27 '10 at 22:06
    
Do you need "your" profunctors to compose in a "coend" type way? Would it be enough to show that functors $C\times D^{op}\to Cat$ are equivalent to cocontinuous functors $Cat^{C^{op}} \to Cat^{D^{op}}$, and then use composition of the latter functors to induce a composition operation on the former? –  Mike Shulman Sep 3 '10 at 7:11
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If you're still interested, I've worked this out on my personal web at nLab here, with supporting material linked to from this page.

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