# Comma objects in the bicategory of profunctors

I learned that the bicategory $\bf Prof$, while having lots of interesting properties, does not admit inverters, so it does not admit arbitrary pseudolimits.

Does this imply that it is impossible to define some useful constructions, like comma objects?

More in general, is there a precise account of which co/limits, bi-co/limits, and weighted co/limits can be constructed in $\bf Prof$?

In fact, in the thread above I say that $\bf Prof$ admits $\bf Cat$-cotensors, and I'm quite sure that ${\cal A}\pitchfork \bf B$ is defined on objects by $({\cal A},{\bf B})\mapsto {\bf A}^°\otimes {\bf B}$ (boldface = 1-cells of $\bf Prof$; mathcal = 1-cells of $\bf Cat$); but I'm not able to see what how to build the canonical span $d_0,d_1 \colon {\cal I}\pitchfork {\bf B} \rightsquigarrow \bf B$ in the case ${\cal I}=\{0\to 1\}$: how is it possible to define two functors ${\cal B}\otimes{\cal I}^°\otimes {\cal B}^° \to {\cal V}$ that satisfy a similar property of evaluation on co/domain?

• I don't know about comma objects, and I don't think there is a characterization of which limits and colimits exist in Prof. But cotensors do exist and are given as you say; more generally, lax limits exist and coincide with lax colimits. See for instance Street's paper Cauchy characterization of enriched categories. Feb 24, 2017 at 10:27
• " I don't think there is a characterization of which limits and colimits exist in Prof" this absence seems pretty strange; is it difficult? Not interesting? I see from your answer then that whereas pseudo limits do not exist, lax (and colax?) limits exist and coincide with colimits. This seems a pretty rigid structure, although I see why it is true. Street's paper, that I completely removed from my memory, seems to address part of my question in full generality. Have his work had some sequel? If yes, add it and I'll consider it an answer :-) Feb 24, 2017 at 11:21
• Also, please, tell me how do you find projections from B^I to B... Feb 24, 2017 at 13:27

I don't know about comma objects, and I don't think there is a known characterization of which limits and colimits exist in Prof.

But cotensors do exist and are given as you say. More generally, lax limits of lax functors exist and coincide with lax colimits, the projections being the right adjoints of the coprojections. In particular, $d_0,d_1 : \mathcal{I} \pitchfork \mathbf{B} \nrightarrow \mathbf{B}$ are the corepresentable profunctors adjoint to the representable profunctors induced by the coprojections $i_0,i_1 : \mathbf{B} \to \mathbf{I}^{\mathrm{op}} \times \mathbf{B}$.

See for instance Street's paper Cauchy characterization of enriched categories and later work such as Carboni-Kasangian-Walters An axiomatics for bicategories of modules. Relatedly, this paper of my own with Garner exhibits Prof as a certain kind of free cocompletion.

• I am trying to organize the material you left me, but I'm still unable to see through it. I would really like to know whether comma objects or lax pullbacks (now that I learned that the two concepts are different) exist in $\bf Prof$ and what is their shape, but I feel stuck in the black hole of multiple references... Feb 25, 2017 at 14:50
• I've skimmed thorugh the entire Street paper and i wasn't able to find a reference for a result similar to the one I asked for. Maybe it's my fault; and yet I feel that there are few links with the basic case of profunctors. Feb 25, 2017 at 15:19
• One way to answer your specific question from Street's paper is this: Proposition 4.2 proves that Prof, being a "cosmos", has tensors. Proposition 4.1 says that Prof^op is also a cosmos, hence also has tensors (i.e. Prof has cotensors), given by tensoring in Prof and then taking opposites. Your description of the cotensors follows. Feb 26, 2017 at 5:08
• As I said, I don't know about comma objects. But lax pullbacks are a lax limit of a lax functor, so they can be obtained as cocollages. Feb 26, 2017 at 5:12