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John Baez (Sep 20 2025 at 22:03):I'm needing to say a bit about double profunctors in my paper for the Paré volume. I'm using 'loose' double profunctors from a double category to a double category , which (for starters) give a set of loose heteromorphisms from any object of to any object of .
My question: did Bob Paré explore loose double profunctors in this paper?
He goes about studying double profunctors in some way that I'm having trouble understanding. I just want to know if I should thank him for initiating the subject of loose double profunctors.
Kevin Carlson (Sep 20 2025 at 22:22):A lax double functor from to Span, the kind of thing Paré highlights in that paper, is instead a tight double profunctor , and I don’t think Paré considers the two-sided tight bimodules, since he’s really after a Yoneda embedding. I’m about 90% sure no loose double profunctors show up in that paper. I’m not aware of anyone saying anything about them, other than the nLab author that observes at [[double profunctor]] that they definitely don’t compose, until David Jaz Myers’ program to use them for systems theory, which is finally becoming paper-ified starting here.
Kevin Carlson (Sep 20 2025 at 22:26):A key problem with the loose double profunctors, as you may already be aware, is that you really can’t represent them as double functors into Span, because you’d need to transpose the tight and weak arrows of the domain, which has to be made special sense of in a weak double category, plus the tight arrows in Span have no 2-cells so you also have to upgrade to codomain Prof. Evan gave a definition of a “twisted lax double functor” to follow this line through on the Topos blog and there’s a paper cooking regarding it, but David and Sophie found “a double category over the walking loose arrow” was a way easier definition to get off the ground.
John Baez (Sep 20 2025 at 22:48):Thanks! I'm trying to write a little section in my paper to create a bridge between my work on structured/decorated cospans and David and Sophie's paper. So I'm familiar with the "double category over the walking loose arrow" perspective, which seems extremely efficient. I haven't thought at all about the "double functor into Span" perspective (on tight bimodules), or the "twisted lax double functor" perspective (on loose bimodules). I'm actually giving a rough explanation of the "pseudo bimodule internal to the 2-category Cat" perspective.
John Baez (Sep 20 2025 at 22:49):Anyway, it's good to know that Paré was talking about a certain special class of tight double profunctors.
Nathanael Arkor (Sep 21 2025 at 21:38):I wrote something about the relationship between Paré's work and tight/loose distributors in the introduction to Exponentiable virtual double categories and presheaves for double categories, which you might find helpful, John.
Nathanael Arkor (Sep 21 2025 at 21:38):I think it's definitely fair to say Paré was the first to study tight presheaves.
Nathanael Arkor (Sep 21 2025 at 21:39):(In fact, this was Bob's original motivation to study pseudo double categories, rather than strict double categories.)