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Proarrow equipments (also known as framed bicategories) are identity-on-objects locally fully faithful pseudofunctors $({-})_* \colon \mathcal K \to \mathcal M$ for which every 1-cell $f_*$ in the image has a right adjoint.

Though there is no reason in general that, for a given 2-category $\mathcal K$, there should be a canonical proarrow equipment structure, in many cases of interest, this is the case. For instance, proarrow equipments $({-})_* \colon \mathcal K \to \mathcal M$ satisfying a certain exactness property (Axiom C of Rosebrugh–Wood's Proarrows and cofibrations) are determined by the codiscrete cofibrations in $\mathcal K$. In practice, there often seems to be an evident choice of $\mathcal M$ and $({-})_*$.

What are some examples of 2-categories $\mathcal K$ for which there are distinct interesting proarrow equipment structures $({-})_* \colon \mathcal K \to \mathcal M$?

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    $\begingroup$ I have thought about this for some time, especially in light of your last remark: most often, the equipment is a Kleisli bicategory of an obvious 2-monad, and it would be natural to reverse-engineer this: find properties of a 2-monad ensuring that the free functor to the Kleisli bicategory equips the domain with proarrows. (This comment isn't even helpful :D just a selfish way to be sure I receive updates if someone answers) $\endgroup$
    – fosco
    Commented Nov 28, 2021 at 8:40

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I'm not sure whether this counts as "interesting", but if $(-)_* : \mathcal{K} \to \mathcal{M}$ is any proarrow equipment and $i:\mathcal{K}' \to \mathcal{K}$ is any locally fully faithful pseudofunctor, then the composite $\mathcal{K}'\to \mathcal{M}$ is again a proarrow equipment.

For instance, $(-)_*$ could be the proarrow equipment of $V$-enriched categories and profunctors, and $i$ could be the 2-functor $W\text{-Cat} \to V\text{-Cat}$ induced by some monoidal functor $W\to V$. Such a functor isn't always locally fully faithful, but there are cases where it is, such as the "discrete objects" inclusions $\rm Set\to Cat$ or $\rm Set \to Top$.

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  • $\begingroup$ Upvoted as this does provide an example of distinct equipment structures that one might care about, but it's not quite what I'm looking for, as in these cases there still appears to be a "canonical" equipment, from which the others are induced. $\endgroup$
    – varkor
    Commented Nov 28, 2021 at 16:58
  • $\begingroup$ Right, I suspected you might be actually looking not just for a $K$ that can be equipped with proarrows in more than one way, but for such a $K$ where it's not obvious that one or the other such equipment is preferred. $\endgroup$
    – Mike Shulman
    Commented Nov 28, 2021 at 21:13
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Here is a different sort of non-answer: the category $\rm Dbl$ of (strict) double categories can be enhanced to a virtual equipment in two different ways, neither of which is canonical: using "horizontal" or "vertical" double profunctors respectively. Indeed the lack of canonicity can be made precise, as the transposition automorphism of $\rm Dbl$ interchanges the two equipments.

This fails to answer the question not only because the equipments are only virtual, but because we are here talking only about the 1-category $\rm Dbl$. This 1-category can be enhanced to a 2-category in two different ways, which correspond exactly to the two virtual equipments, in the usual codiscrete-cofibrations way.

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  • $\begingroup$ Thanks, this is an interesting example. Given that neither horizontal nor vertical profunctors are primary, it seems plausible that the right kind of structure to consider here is a "double equipment" (by which I don't mean the usual "double category with companions and conjoints" perspective of an equipment, but rather an identity-on-objects double functor between double categories with some appropriate structure). $\endgroup$
    – varkor
    Commented Nov 30, 2021 at 19:36
  • $\begingroup$ Haha, yes, you could try to define something like that. Often in applications of double categories there is a distinction between the two kinds of arrows and it's clear which kind of profunctor would be most appropriate. But I could imagine some other applications where one might want to consider both together. $\endgroup$
    – Mike Shulman
    Commented Nov 30, 2021 at 19:41

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