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Stream: theory: category theory

Topic: adjointness wrt profunctor(s)

Matteo Capucci (he/him) (Apr 09 2020 at 09:47):

Let $\bf{C}$ and $\bf{D}$ be categories, an adjoint pair $F \dashv G$ is a pair of functors $\bf{C} \underset{G}{\overset{F}\rightleftarrows} \bf{D}$ such that
$\rm{Hom}_{\bf{D}}(Fx, y) \simeq \rm{Hom}_{\bf{C}}(x, Gy)$
Suppose we have instead profunctors $P_{\bf{C}} : \bf{C}^{\rm{op}} \times \bf{C} \to \bf{Set}$ , and $P_{\bf{D}}:\bf{D}^{\rm{op}} \times \bf{D} \to \bf{Set}$ . Is there a name for a pair of functor $F$ and $G$ as above satisfying:
$P_{\bf{D}}(Fx, y) \simeq P_{\bf C}(x, Gy)$
?

Matteo Capucci (he/him) (Apr 09 2020 at 09:52):

This notion should finally generalize both categorical adjointness and its grandparent 'algebraic' adjointness, i.e. the relationship $\langle T u, v \rangle_U = \langle u, T^\perp v \rangle_V$ where $T:U \to V$ is a linear operator, $U$ and $V$ (real) vector spaces, and $\langle -, - \rangle$ denotes inner product. I bet you would get something similar by considering an adjunction in the $\R$ -enriched setting.

Matteo Capucci (he/him) (Apr 09 2020 at 09:53):

I do not think is hard to come up with an elementary theory of these things, but I hardly believe nobody smarter than me already thought it out

Amar Hadzihasanovic (Apr 09 2020 at 10:01):

I think those are morphisms between particular objects of the 2-Chu construction $Chu(\mathrm{Cat}, \mathrm{Set})$ -- see this paper by @Mike Shulman .

Paolo Capriotti (Apr 09 2020 at 10:05):

Also note that the condition is equivalent to the commutativity of the square of profunctors:

$\begin{array}{c} \mathbf D & \xrightarrow{P_{\mathrm D}} & \mathbf D \\ \downarrow & & \downarrow \\ \mathbf C & \xrightarrow{P_{\mathrm C}} & \mathbf C \end{array}$

where the vertical profunctors are $G_*$ and $F^*$ .

Amar Hadzihasanovic (Apr 09 2020 at 10:08):

As for your other point, I doubt there is a good way of turning, say, Hilbert spaces into enriched categories so that adjoint operators become enriched adjunctions or something. I think it's one of those things that everyone tries once when learning CT, and then realises that one needs to generalise things to the point where they are kind of tautological.

Oscar Cunningham (Apr 09 2020 at 10:21):

This would also be a 2-cell in the double category of profunctors and functors.

Mike Shulman (Apr 09 2020 at 13:53):

Chu constructions in general, however, do generalize both categorical adjointness and algebraic adjointness. It's perhaps debatable how tautological that is.

Amar Hadzihasanovic (Apr 09 2020 at 14:18):

I agree.
I was thinking more specifically about Matteo's reference to “the $\mathbb{R}$ -enriched setting”. I don't think I've seen any convincing description of inner product spaces or Hilbert spaces as some kind of “enriched category”-like objects, with the inner product being a hom-object. But maybe I haven't looked hard enough.

Mike Shulman (Apr 09 2020 at 14:44):

Ah, yes. No, I don't think there is such a description either.

Mike Shulman (Apr 09 2020 at 14:47):

sarahzrf (Apr 11 2020 at 19:11):

Oscar Cunningham said:

This would also be a 2-cell in the double category of profunctors and functors.

isnt one of the functors going the wrong way?

fosco (May 30 2020 at 21:23):

I'm late to the profunctor party! I suspect that the profunctors P_C and P_D have to satisfy some sort of condition in order to give a meaningful theory. Like, if P_C = P_D = the terminal presheaf, then F,G are arbitrary functors