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Stream: learning: questions

Topic: Equivalences of (Fibrant) Double Categories

Joshua Meyers (Feb 25 2024 at 18:33):

What does it mean for two pseudo double categories $C,D$ to be equivalent? Is it that there are pseudo functors $F:C\to D$ and $G:D\to D$ such that $GF\cong 1_C$ and $FG\cong 1_D$? Well what is $\cong$? Is it the existence of an invertible natural transformation? The existence of an invertible pseudo natural transformation? Or a pair of (pseudo?) natural transformations which compose in both directions to be equivalent to identity by means of an invertible pseudo-modification? (I am using the definitions in *Pseudo-Categories* by Martins-Ferreira.)

Moreover, what is the double categorical analogue of the theorem that a functor is an equivalence iff it is fully faithful and essentially surjective? Maybe the many different choices one could make in the previous paragraph give different answers to this question.

Finally, suppose two double categories are fibrant i.e. they are equipments. Does an equivalence in any of the above senses automatically preserve the equipment structure, or must that be stipulated additionally?

Where can I look to learn about this stuff?

Matteo Capucci (he/him) (Feb 25 2024 at 19:38):

Re the first, $\cong$ ought to be a tight natural isomorphism, i.e. a natural transformation of double functors with invertible components. I suspect tightly essentially surjective on objects and locally essentialy surjective (meaning essentially surjective on loose arrows, where the isomorphisms of loose arrows are squares with identity tight boundaries) should be the same, assuming choice. This is similar to 2-equivalence of categories. The only reason I'm not 100% sure is that I didn't verify it, and it might be the case that one has to weaken 'squares with identity tight boundaries' to 'squares with invertible tight boundaries'. This can be check with some patience.

Re the second question, no, you need extra an requirement on the functors, namely they need to preserve cartesian squares. This is because the 'fibrant' in 'fibrant double category' means that $(s,t):C_1 \to C_0 \times C_0$ is a fibration, and an internal functor need not to necessarily be a morphism of fibrations, i.e. preserve cartesian maps.

Matteo Capucci (he/him) (Feb 25 2024 at 19:40):

For a reference, Grandis works with weak double categories in his book 'Higher Dimensional Categories', though the things you ask for do not appear directly there---you have to work them out from the defs he gives. I would also try with Donaldson and Yau's book on 2-dimensional category theory.

Joshua Meyers (Feb 25 2024 at 19:49):

Why must $\cong$ be a tight natural isomorphism rather than the other alternative choices I listed?

Interesting that you need an additional condition for fibrant double categories -- I see your reasoning but if there are two fibrant double categories which are equivalent as double categories but not as vibrant double categories, wouldn't it be possible to migrate the Cartesian structure from one to the other, resulting in a double category which is fibrant in two different ways? This conclusion can't be right, as "fibrant" is a property of a double category rather than a structure.

Matteo Capucci (he/him) (Feb 25 2024 at 19:54):

Maybe the double category structure does some magic I'm not factoring in, but it is certainly not the case that you can 'pushforward a fibration structure' along a commutative square of fibrations. This is because the universal property of cartesian maps is universal in the ambient category, so a map which is cartesian in one might not be anymore once mapped with a functor. Surely there must be some sufficient conditions for a functor to preserve cartesian moprhisms.

Matteo Capucci (he/him) (Feb 25 2024 at 19:59):

Joshua Meyers said:

Why must $\cong$ be a tight natural isomorphism rather than the other alternative choices I listed?

Sorry I didn't see you listed choices. A tight natural isomorphism should be tight natural transformation with invertible components. You have to interpret this words 'right' however, The components of a tight natural transformation $f \to g$ are (1) tight maps $fa \to ga$ for every object $a$ (2) 2-cells $fp \to gp$ for every loose arrow $p$. So when you ask it to be invertible you get strictly invertible tight maps, strictly because the tight world works strictly, and likewise for squares. So no modifications involved.

Matteo Capucci (he/him) (Feb 25 2024 at 20:01):

To navigate this potential mess it's good to remember that the tight direction in a double category is the one where structural morphisms go. The loose arrows are extra algebraic structure, which isn't used for structural purposes. That's sort of the whole point of doing category theory with double category, you can keep separate tight (strict) and loose directions and push all the coherence for the latter in the former.

Joshua Meyers (Feb 25 2024 at 20:17):

Matteo Capucci (he/him) said:

Maybe the double category structure does some magic I'm not factoring in, but it is certainly not the case that you can 'pushforward a fibration structure' along a commutative square of fibrations. This is because the universal property of cartesian maps is universal in the ambient category, so a map which is cartesian in one might not be anymore once mapped with a functor. Surely there must be some sufficient conditions for a functor to preserve cartesian moprhisms.

I have yet to work out the details, but loosely my reasoning is as follows:

1. Being fibrant is a property (not a structure) of a double category.

2. If equivalence of double categories is properly defined, it should preserve all (non-evil) properties.

3. Therefore equivalence should preserve being fibrant.

The only ways out are 1) if equivalence is not properly defined or 2) if being fibrant is evil.

John Baez (Feb 25 2024 at 20:31):

Joshua Meyers said:

What does it mean for two pseudo double categories $C,D$ to be equivalent? Is it that there are pseudo functors $F:C\to D$ and $G:D\to D$ such that $GF\cong 1_C$ and $FG\cong 1_D$? Well what is $\cong$? Is it the existence of an invertible natural transformation? The existence of an invertible pseudo natural transformation? Or a pair of (pseudo?) natural transformations which compose in both directions to be equivalent to identity by means of an invertible pseudo-modification?

The last one seems like the safest option, though of course the least convenient. I guess you are familiar with the [[principle of equivalence]], the concept of "evil", and other aspects of the philosophy of higher structures, but just in case anyone lurking here is not, I'll talk about it a bit anyway.

To get results that are invariant under the most general notion of equivalence we shouldn't make definitions that impose equations between $k$-dimensional cells in $n$-dimensional structures unless $k = n$; instead we should assert equivalences between them.

Double categories themselves being 2-dimensional, the collection of them forms a 3-dimensional structure - something like a tricategory, or maybe something else. To follow the principle of equivalence we should only assert that a $k$-cell in this structure has an inverse if $k = 3$.

That's what you're doing in your last option. In your earlier options you are choosing to 'live dangerously' by cutting things off prematurely. You can do this as long as you either 1) know you're running the risk of problems later, or 2) prove (or learn) theorems that justify what you're doing.

John Baez (Feb 25 2024 at 20:33):

Joshua Meyers said:

vibrant double categories

I like that!

Joshua Meyers (Feb 25 2024 at 21:01):

@John Baez right! I would also tend to think the weakest option is what we should go with. That's why I'm surprised to see people like @Matteo Capucci (he/him) in this thread, and Grandis in his book 'Higher Dimensional Categories' advocate stronger notions of equivalence.

Joshua Meyers (Feb 25 2024 at 21:04):

In any case, if we make our equivalences too strong, they should a fortiori preserve all non-evil properties, since the "correct" weak equivalences should

James Deikun (Feb 25 2024 at 21:17):

If pseudo double categories form a pseudo triple category, then double functors should be the tight, i.e. strict direction in the pseudo triple category. In this sense even asking for isomorphism is not all that "evil". Nonetheless there should be a weaker version of equivalence that can be useful. These are double functors that right lift against the inclusion in each shape of "walking cell" of its "walking boundary" and against the obvious map from the "walking parallel pair of squares" to the "walking square".

Matteo Capucci (he/him) (Feb 25 2024 at 21:17):

Ugh, I realize I read your first message wrong. Of course fibrancy is preserved under equivalence of double categories, since such a functor exhibits an equivalence of fibrations first. It's not preserved under arbitrary functors though.

Matteo Capucci (he/him) (Feb 25 2024 at 21:23):

Re strictness, it's not at all stricter than it can be. For instance for discrete double categories, ie bicategories, it recovers biequivalence, which is the weakest notion of equivalence of bicategories. The point is that the underlying tight category of a double category is strict and stays strict.

Matteo Capucci (he/him) (Feb 25 2024 at 21:24):

Again, one must embrace the fact tight and loose directions of a double category are very different, though they both look like a category

Matteo Capucci (he/him) (Feb 25 2024 at 21:25):

But maybe this is easier to get now that the confusion about fibrancy being preserved is out of the way

James Deikun (Feb 25 2024 at 21:27):

Hm, maybe the version of equivalence I give above is too loose. I'm not used to thinking of pseudo categories as much. Probably the functors should have to lift against the obvious map from the "walking parallel pair of tight arrows" to the "walking tight arrow" as well.

James Deikun (Feb 25 2024 at 21:41):

If you want to translate this to something a bit more conventional, it is a functor that is essentially surjective on objects and loose arrows, and locally bijective on tight arrows and squares.

Joshua Meyers (Feb 25 2024 at 21:57):

So I am looking more at the Martins-Ferreira paper and I realized that natural transformations are vertical: for each object $A$ they give a vertical (strict) morphism $F(A)\to G(A)$, for each horizontal morphism, they give a 2-cell, and for each vertical morphism, they give an equation (there are more equations too) -- whereas pseudo natural transformations are horizontal: for each object they give a horizontal (weak) morphism, for each vertical morphism they give a 2-cell, and for each horizontal morphism they give another 2-cell, this one with identity frames (satisfying certain equations). Pseudo natural transformations compose vertically strictly and horizontally up to isomorphism. Now a pseudo-modification as it turns out is a 2-cell going between two natural transformations and two pseudo-natural transformations!

Grandis seems to focus his discussion of equivalence on what he calls "horizontal equivalence" --- this is his strict direction, opposite to the current standard. I suppose I can see why we would want this. Take the double category Prof, for example. To say that two pseudo double functors $F$ and $G$ valued in Prof are isomorphic, we would want some sort of functor, not profunctor $Fc\to Gc$. Unfortunately it seems like this functor will have to be an isomorphism rather than an equivalence, unless we venture into triple categorical structures...

Mike Shulman (Feb 25 2024 at 22:03):

I don't have time to read this entire thread right now, but I want to point out that there really is more than one notion of equivalence for double categories, and there isn't just one "right" one: different notions of equivalence have different applications.

Joshua Meyers (Feb 25 2024 at 22:04):

On a side note, both Grandis and Martins-Ferreira redundantly stipulate that $\lambda_1=\rho_1:1 \otimes 1\to 1$, whereas this is derivable from the pentagon and triangle identities

Joshua Meyers (Feb 25 2024 at 22:05):

@Mike Shulman that's fine, reading the whole thread is not necessary, where can I learn about these different notions of equivalence?

Mike Shulman (Feb 25 2024 at 22:05):

And regarding "fully faithful and essentially surjective" there is some discussion in section 7 of Framed bicategories and monoidal fibrations.

Kevin Arlin (Feb 25 2024 at 22:05):

I don’t understand why you say the functor $Fc\to Gc$ should be an isomorphism. If the domain is a point then these are just categories, and an equivalence in Prof itself is the usual notion of equivalence of categories.

Mike Shulman (Feb 25 2024 at 22:08):

Well, one place where people have to study notions of equivalence is in putting a Quillen model structure on double categories, so some of the notions of equivalence are defined and compared in the various papers about that. That's rather more technical than you probably want here, though.

Joshua Meyers (Feb 25 2024 at 22:09):

@Kevin Arlin natural transformations between categories are unfortunately not part of the structure of the double category Prof. You just have categories, functors, profunctors, and morphisms of profunctors.

Mike Shulman (Feb 25 2024 at 22:09):

I think most of the simple options have already been mentioned here: the natural transformations can be either tight or loose, and either strict or pseudo.

Mike Shulman (Feb 25 2024 at 22:10):

Natural transformations are visible in Prof: they are the morphisms of profunctors between identity profunctors with nonidentity functors on the sides.

Kevin Arlin (Feb 25 2024 at 22:10):

Hm? No, the natural transformations are totally there. The tight 2-category is Cat.

Joshua Meyers (Feb 25 2024 at 22:10):

Oh that's true

Kevin Arlin (Feb 25 2024 at 22:11):

This is one of the big reasons people like Prof, everything is there. Even better, Prof itself is decomposable as the double category of lax functors from a point to Span, Evan’s been advertising how nice it is to recover all of basic category theory in this structure recently.

Joshua Meyers (Feb 25 2024 at 22:17):

Okay so you can define modifications between (vertical) natural transformations, Martins-Ferreira and Grandis just don't. Given natural transformations $a,b: F\to G: C\to D$, we send each $c\in C$ to a 2-cell between identity profunctors with $a_c, b_c$ on the sides, and we have an equation of 2-cells for every horizontal 1-cell in $C$ (maybe more too?)

Joshua Meyers (Feb 25 2024 at 22:18):

Is there a reference for this construction?

Joshua Meyers (Feb 25 2024 at 22:19):

Actually I think this is just a pseudo-modification except with identity pseudo-transformations

Kevin Arlin (Feb 25 2024 at 22:31):

That seems likely. Are you sure modifications aren’t defined in Grandis’ book? I don’t have it at hand but it seems like the kind of thing he’d have in there.

Kevin Arlin (Feb 25 2024 at 22:33):

It’s easier to find references for modulations (Evan’s paper with Lambert last month, Paré’s papers on Yoneda theory and on composition of modules) as those are the general cells you want in lax functor double categories.

Joshua Meyers (Feb 25 2024 at 22:39):

They are -- as 2-cells between two horizontal and two vertical transformations

Kevin Arlin (Feb 25 2024 at 22:41):

Right right, so shouldn’t you just make the loose transformations identities and there’s your reference? I guess that’s what you’ve already done.