Adjunctions with respect to profunctors

Question

Let $P : W° \times Y \to \mathbf{Set}$ and $Q : X° \times V \to \mathbf{Set}$ be profunctors, and let $L : X \to W$ and $R : Y \to V$ be functors. Suppose that $$P(Lx, y) \cong Q(x, Ry)$$ natural in $x \in X$ and $y \in Y$ [1]. Is there a name for such a relationship, and is it subsumed by some other concept, e.g. adjunctions in $\mathbf{Prof}$? In particular, it would be useful to prove statements about these "adjunctions with respect to profunctors", for instance giving characterisations of (co)reflective adjunctions with respect to profunctors, using known statements about adjunctions more generally.

[1] Such situations were actually studied in the original paper on adjoint functors, Kan's Adjoint functors, under the term "rel. adjointness". However, Kan did not have the richness of abstraction that we have today, so his treatment of them is quite concrete. The only examples given are essentially representable, in that they're induced by closed structure.

Answer 1

I remember I tried to work with this definition for a while when I still believed in the notion of relative category. Under some coherence assumptions, your notion is related to the classical notion of adjunction. Follow me in the presentation below.

Def. A virtual relative category is a couple $(\mathsf{C}, \rho)$ where $\mathsf{C}$ is a category and $\rho: \mathsf{C}(-,-) \Rightarrow \mathsf{P}(-,-) $ is an epimorphism in $\mathsf{Psh}(\mathsf{C}^\circ \times \mathsf{C}).$

In my original definition $\mathsf{P}$ was assumed to be small, and the whole definition had the advantage that I could use Kan extensions freely, because size issues where taken care of by this smallness assumption.

Rem. Every virtual relative category has an homotopy category, $\mathsf{Ho}_\rho(\mathsf{C})$ having the same objects of $C$ and where $\mathsf{Ho}_\rho(\mathsf{C})(a,b) = \mathsf{P }(a,b)$. Moreover, $\rho$ indudes a quotient functor $C \to \mathsf{Ho}_\rho(\mathsf{C})$.

This Rem. shows the perks of this notion. For a honest relative category $(\mathsf{C},\mathcal{W})$ one cannot guarantee for the existence of an homotopy category. Of course, smallness is not used at this point, but one starts to see that the whole structure is tamer than the usual notion.

Def. An homotopical functor $f$ between virtual relative categories is a functor $f: \mathsf{C} \to \mathsf{D}$ such that there exists a (necessary unique) $\bar{f}: \mathsf{P}_\mathsf{C}(-,-) \Rightarrow \mathsf{P}_\mathsf{D}(-,-)$ making the obvious diagram involving $\rho$ commutative.

Def. A Quillen adjunction between virtual homotopy categories is a couple of homotopical functors (in opposite directions) such that $\mathsf{P}_\mathsf{C}(\mathsf{L}c,d) \cong \mathsf{P}_\mathsf{D}(c, \mathsf{R}d)$.

Prop. A Quillen adjunction between virtual relative categories induces an adjunction between their homotopy categories.

Rem. The proposition above shows that, to the price of passing to the homotopy category, this kind of $\mathsf{P}$-relative adjunctions are actually subsumed by honest adjunctions. Moreover, this idea (which has a very homotpical flavour) can be stretched to the case in which $\mathsf{P}$ is just an endoprofunctor (without $\rho$ you lose the quotient functor). Otherwise, I do not see how to define Homotopy categories.