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Stream: deprecated: id my structure

Topic: higher profunctors

John Baez (Jun 23 2023 at 23:04):

I'm wondering: since we can define profunctors between categories, can we define "pronatural transformations" between "2-profunctors" between 2-categories?

John Baez (Jun 23 2023 at 23:06):

By a 2-profunctor from a 2-category $\mathbf{A}$ to a 2-category $\mathbf{B}$ I mean a 2-functor

$F: \mathbf{A} \times \mathbf{B}^{\text{op}} \to \mathbf{Cat}$

John Baez (Jun 23 2023 at 23:08):

And I don't exactly know what a "pronatural transformation" should be, but it should look more profunctor-ish than an ordinary natural transformation.

John Baez (Jun 23 2023 at 23:11):

What do I even mean by that? Well, let's see. Given two 2-profunctors

$F, G\colon \mathbf{A} \times \mathbf{B}^{\text{op}} \to \mathbf{Cat}$

I want a thing going from $F$ to $G$. A natural transformation $\alpha$ from $F$ to $G$ would give, for any objects $a \in \mathbf{A}$ and $b \in \mathbf{B}$, a functor

$\alpha_{a,b} \colon F(a,b) \to G(a,b)$

John Baez (Jun 23 2023 at 23:12):

So I guess a "pronatural transformation" should instead give a profunctor $\alpha_{a,b}$ from $F(a,b)$ to $G(a,b)$. Has anyone thought about something like this?

Dylan Braithwaite (Jun 23 2023 at 23:27):

I believe this should be a special case of the double profunctors that @Christian Williams is working on. Namely if you regard $\mathbf A, \mathbf B$ as vertically-discrete double categories then horizontal(?) profunctors between them are 2-functors $\mathbf A \times \mathbf B^\text{op} \to \mathbf{Cat}$, and your 'pronatural transformations' should be the square-cells which have $F$ and $G$ on the top and bottom edges and identity-vertical-profunctors on the other two edges.

Dylan Braithwaite (Jun 23 2023 at 23:34):

In the case that $\mathbf{A, B}$ have only 1-object, pronatural transformations correspond to [[Tambara modules]].

We speculated about a tricategory of such pronatural transformations in the appendix of this paper but outside of Christian's threads here I wasn't able to find anything written about them to cite.

John Baez (Jun 23 2023 at 23:50):

Dylan Braithwaite said:

I believe this should be a special case of the double profunctors that Christian Williams is working on.

Thanks. That's amusing - my own grad student! I got interested in this idea for reasons that seem quite different.

John Baez (Jun 23 2023 at 23:54):

Your paper looks very interesting. Here's a tiny comment. When you write

Even when collages of string diagrams are our novel contribution...

did you mean

Even though collages of string diagrams are our novel contribution...

?

John Baez (Jun 23 2023 at 23:57):

Another minor comment:

Coends and profunctors [32, 33], far from being a obscure concept from category theory,

should be

Coends and profunctors [32, 33], far from being an obscure concept from category theory,

or even better

Coends and profunctors [32, 33], far from being obscure concepts from category theory,

Mike Shulman (Jun 24 2023 at 00:01):

Staying in the world of 2-categories, couldn't you just consider a pseudonatural transformation between the composite pseudofunctors $A\times B^{\rm op} \to \rm Cat \to Prof$?

John Baez (Jun 24 2023 at 00:07):

I guess you're saying this is a way to make my idea precise. It looks like it: given two 2-profunctors

$F, G\colon \mathbf{A} \times \mathbf{B}^{\text{op}} \to \mathbf{Cat}$

a pseudonatural transformation between their composites with $\mathbf{Cat} \to \mathbf{Prof}$ indeed gives a profunctor $\alpha_{a,b}: F(a,b) \nrightarrow G(a,b)$ for each $a \in \mathbf{A}, b \in \mathbf{B}$, with some nice pseudonaturality holding.

Dylan Braithwaite (Jun 24 2023 at 00:08):

John Baez said:

Your paper looks very interesting. Here's a tiny comment. [...]

Thanks! I've got a list of small corrections to be done before the camera-ready version for ACT, but I'll make sure they're on the list!

John Baez (Jun 24 2023 at 00:12):

Great! Here's a comment that's a tiny bit bigger: when you define 'bimodular profunctors' in Definition 4.1, you use $t_M$ for the left strength and $t_N$ for the right strength, distinguishing them only by the fact that $M$ is in the monoidal category $\mathbb{M}$ and $N$ is in the monoidal category $\mathbb{N}$.

John Baez (Jun 24 2023 at 00:13):

But this makes your later notation $t_{M \otimes N}$ ambiguous, since here you must mean that either $M, N \in \mathbb{M}$ or $M, N \in \mathbb{N}$; you can't tensor an object of $\mathbb{M}$ with an object of $\mathbb{N}$.

John Baez (Jun 24 2023 at 00:19):

So I believe the law you're stating here, $t_M ; t_N = t_{M \otimes N}$, is really two separate laws, one for the left strength and one for the right strength. I was able to figure this out, and you explain everything more precisely in Appendix B, but I think it's a bit confusing to the reader to use $t$ both for the right and left strength and to sometimes but not always assume $M$ is an object of $\mathbb{M}$ and $N$ is an object of $\mathbb{N}$ (if that's really what's going on here). I think it's possible to be clearer without being too much longer.

John Baez (Jun 24 2023 at 00:22):

Anyway, this is very interesting stuff!

John Baez (Jun 24 2023 at 00:33):

Mike Shulman said:

Staying in the world of 2-categories, couldn't you just consider a pseudonatural transformation between the composite pseudofunctors $A\times B^{\rm op} \to \rm Cat \to Prof$?

So maybe I should define a 2-profunctor between 2-categories (or bicategories) to be a pseudofunctor $A\times B^{\rm op} \to \mathrm{Prof}$, to achieve the effect I want.

Mike Shulman (Jun 24 2023 at 01:44):

It depends on what the effect you want is! (-:

Mike Shulman (Jun 24 2023 at 01:45):

I wouldn't tend to call that a 2-profunctor, though. I think of Prof as a categorification of Rel or Span, and a 1-profunctor is a functor to Set, not a functor to Rel or Span.

John Baez (Jun 24 2023 at 04:19):

My goals are pretty long-range - I'm trying to create something analogous to the Brauer 3-group which is actually a 4-group. For this I need to categorify the concept of bimodule, and thus enriched profunctor, and I think I should do it in such a way that the morphisms between bimodules again resemble bimodules (not maps).

So, as a warmup, I'm trying to dream up some sort of tricategory where the objects are bicategories, the morphisms are categorified profunctors of some sort, but the 2-morphisms also resemble profunctors.

Mike Shulman (Jun 26 2023 at 16:53):

My guess would be that in that case you would still want the "categorified profunctors" to be functors to Cat rather than Prof. If it seems yucky to have the profunctors be maps into Cat but then have to embed them in Prof before considering maps between them, you could do it double-categorically: regard A and B as double categories with only identity loose arrows, and consider pseudo double functors $A\times B^{\rm op} \to \mathbb{P}\rm rof$ with loose transformations between them.

Mike Shulman (Jun 26 2023 at 16:54):

(By $\mathbb{P}\rm rof$ I mean the double category whose objects are categories, whose tight morphisms are functors, and whose loose morphisms are profunctors.)