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Stream: theory: category theory

Topic: fibrancy of double categories with companions

Matteo Capucci (he/him) (Jan 30 2024 at 14:07):

A framed bicategory/proarrow equipment/fibrant double category is one such that $C_1 \xrightarrow{(s,t)} C_0 \times C_0$ is a fibration, and this amounts to having all companions and conjoints in the double category.
Is there a similar characterization for double categories with all companions? I can't think of an obvious one on my two feet. It seems that asking for $s$ to be a fibration should be enough, but pulling back a loose map $A \to B$ along a tight map $A' \to A$ might yield a square whose bottom map isn't necessarily the loose unit, unless I'm missing something.

Bryce Clarke (Jan 30 2024 at 14:25):

A double category $\mathbb{C}$ may also be characterised as having all companions if there exists a pseudo double functor $\mathbb{S}q(C_{0}) \rightarrow \mathbb{C}$ that is the identity on objects and tight morphisms.

Matteo Capucci (he/him) (Jan 30 2024 at 14:26):

Yeah but I was hoping for something more 'formal', i.e. that I can express for internal categories more generally

Bryce Clarke (Jan 30 2024 at 14:27):

The condition I gave can be expressed for any pseudo-category object in a $2$-category with comma objects.

Bryce Clarke (Jan 30 2024 at 14:28):

There are probably other nice characterisations though

Matteo Capucci (he/him) (Jan 30 2024 at 14:29):

I think I found it by the way: a double category has companions iff the span $C_0 \xleftarrow{s} C_1 \xrightarrow{t} C_0$ is a left fibration in the sense of Street. This means $s$ is a fibration and $t$ inverts $s$-cartesian morphisms, which means that the target of a cartesian square has to be an isomorphism, which is as close to an identity as I can possibly hope to get.
I suspect conjoints amount to the span being a right fibration, and then having both means being a two-sided fibration (a thing which, IIRC, it's in Shulman's paper already if one remembers the characterization of two-sided fibrations as fibrations over the product)

Matteo Capucci (he/him) (Jan 30 2024 at 14:29):

Bryce Clarke said:

The condition I gave can be expressed for any pseudo-category object in a $2$-category with comma objects.

Uhm indeed, I dismissed it too quickly...

Matteo Capucci (he/him) (Jan 30 2024 at 14:35):

Having such a functor means having a 2-cell in the double category of spans as follows:
image.png

Matteo Capucci (he/him) (Jan 30 2024 at 14:36):

which makes $C_1$ into a promonad in Span

Matteo Capucci (he/him) (Jan 30 2024 at 14:37):

Incidentally, this is actually the thing I was trying to get to, so thanks @Bryce Clarke :grinning_face_with_smiling_eyes:

Bryce Clarke (Jan 30 2024 at 14:43):

I haven't thought about it in that way before, cool! Thanks for the link to Evan's blog post too.

Matteo Capucci (he/him) (Jan 30 2024 at 14:43):

Indeed, I think by Proposition 1 in @Evan Patterson's post I linked above, we see that the two descriptions we found are equivalent. This is because Evan there proves that (I'm going to translate his result into this specific instance), alternatively, a promonad on $C_0$ can be presented as a pair of loose monads ($C^\downarrow$ and $C_1$ in my case) such that the second is a left module over the first. But being a left module over $C^\downarrow$ is the definition of left fibration!

Matteo Capucci (he/him) (Jan 30 2024 at 14:45):

I'm giggling because I was kicking myself for spending an hour this morning reading Evan's post and playing with promonads instead of working, then this afternoon I work and this pops up... call it providence!

Matteo Capucci (he/him) (Jan 30 2024 at 16:14):

I just found out this remark in @Mike Shulman's paper on framed bicategories:
image.png
So my conjecture is wrong in some way. I suspect that left fibrancy only buys you (the universal property of) the counit of the companionship, and that one needs the span to be a left bifibration to also get a cocartesian cell witnessing the unit.
Then a right bifibration gives conjoints, and a two-sided fibration is a proarrow equipment.

Mike Shulman (Jan 30 2024 at 17:05):

The span being a two-sided fibration is precisely the condition you need to get companions only. To also have conjoints you need it to also be a two-sided fibration in the other direction.

Mike Shulman (Jan 30 2024 at 17:06):

Well, actually, I don't know whether something weaker than a full two-sided fibration suffices to get companions due to the double-categorical structure. But a double category with only companions is a two-sided fibration, and conversely, so if so then that weaker structure would (in the case of a double category) imply a two-sided fibration.

Nathanael Arkor (Jan 30 2024 at 18:56):

(Might be worth mentioning that @Christian Williams pointed out this characterisation of double categories with companions or conjoints respectively in https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category-theory/topic/exponentiable.20double.20categories/near/410453084.)

Matteo Capucci (he/him) (Jan 31 2024 at 08:10):

Matteo Capucci (he/him) said:

Indeed, I think by Proposition 1 in Evan Patterson's post I linked above, we see that the two descriptions we found are equivalent. This is because Evan there proves that (I'm going to translate his result into this specific instance), alternatively, a promonad on $C_0$ can be presented as a pair of loose monads ($C^\downarrow$ and $C_1$ in my case) such that the second is a left module over the first. But being a left module over $C^\downarrow$ is the definition of left fibration!

So here I was being too hasty... $C_1$ would actually be a bimodule (that's what profunctor means :face_palm: ) for $C^\downarrow$, hence a two-sided fbration

Evan Patterson (Feb 07 2024 at 05:54):

Bryce Clarke said:

A double category $\mathbb{C}$ may also be characterised as having all companions if there exists a pseudo double functor $\mathbb{S}q(C_{0}) \rightarrow \mathbb{C}$ that is the identity on objects and tight morphisms.

Hi @Bryce Clarke, do you have a reference for this statement? I know your thesis has the variant that a functorial choice of companions gives a strict double functor of the above form, but I couldn't immediately locate the quoted characterization in the literature.

Bryce Clarke (Feb 07 2024 at 06:43):

I do not have a reference for this statement. I was simply relaxing the strict version of the statement from my thesis. It doesn't seem to be written down in the early Grandis-Paré papers either, although it would not surprise me if it is written down somewhere in the double category theory literature. However, to me at least, this feels closer to an observation than a proposition.

Bryce Clarke (Feb 07 2024 at 06:47):

The nlab page for [[companion pairs]] has the following statement:

If every vertical arrow in some double category D has a companion, then the functor $f \mapsto f_{\ast*}$$ is a pseudofunctor $VD \rightarrow HD$ from the vertical 2-category to the horizontal one, which is the identity on objects, and locally fully faithful by the mate correspondence.

Bryce Clarke (Feb 07 2024 at 06:51):

As an aside, I think the nlab pages for [[companion pairs]] and [[conjunction]] (why not conjoint pairs?) could use a bit of updating. I might add this to my todo list, but anyone reading this should feel encouraged to make edits too.

Bryce Clarke (Feb 07 2024 at 06:57):

I also wonder if anyone is brave enough to change every instance of vertical and horizontal on the nlab to tight or loose (as appropriate). It is impossible to use the convention of horizontal vs. vertical consistently (after all, many concepts in double category theory are introduced in different papers which have different conventions).

Evan Patterson (Feb 07 2024 at 06:57):

Thanks for clarification, Bryce. I agree with you that this result should not be hard to prove. I think it's not entirely trivial, though: one needs to use the sliding correspondence for companions to furnish the action on cells, then do calculations to show that this is suitably functorial and pseudofunctorial. It might run to some length if spelled out in any detail. But ultimately routine, yes.

Evan Patterson (Feb 07 2024 at 07:04):

Thanks also for the pointer to the related characterization on nLab. It makes me suspect that these two results can be combined to characterize the presence of companions by the existence of an identity-on-objects-and-arrows (pseudo) double functor from the double category of quintets in the underlying 2-category of $\mathbb{C}$ to the double category $\mathbb{C}$ itself. That statement would pack quite a lot of information about what you can do with companions!

Bryce Clarke (Feb 07 2024 at 07:16):

Evan Patterson said:

Thanks also for the pointer to the related characterization on nLab. It makes me suspect that these two results can be combined to characterize the presence of companions by the existence of an identity-on-objects-and-arrows (pseudo) double functor from the double category of quintets in the underlying 2-category of $\mathbb{C}$ to the double category $\mathbb{C}$ itself. That statement would pack quite a lot of information about what you can do with companions!

I agree that this characterisation seems very likely. In cases like $Span$ it amounts to the same thing, but in cases like $Prof$ it would appear to carry a bit more information.

Bryce Clarke (Feb 07 2024 at 08:57):

@Evan Patterson Perhaps this result is really about a coreflective adjunction between the category of 2-categories and unitary pseudofunctors (those which preserve identities strictly) and the category of (pseudo) double categories with companions and unitary double functors. I guess this is almost what theorem 1.7 of "Limits in double categories" says.

Evan Patterson (Feb 07 2024 at 18:25):

Nice connection, I'll have to think about that. Thanks! (For anyone else following, the paper in question in "Adjoint for double categories.")