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Stream: learning: questions

Topic: weak/lax fibers of a functor

Tobias Schmude (May 30 2022 at 17:55):

Has the concept of weak or (op)lax fibers of a functor received any attention? I couldn't find anything about it.
By a weak or (op)lax fiber of a functor $F: \mathcal{C} \to \mathcal{B}$ over an object $B$ I mean a category consisting not of data in $\mathcal{C}$ that lies (strictly) over the object in the base category, but which is connected to it via isomorphisms or arbitrary morphisms in the appropriate sense.

Recently I had a few thoughts about this, motivated by the bicategorical case. There the construction of strict fibers only works for local isofibrations, while weak fibers can be defined for any pseudofunctor, but are equivalent to the former under some conditions.
Before I continue with this I'd like to know if there's already some work about this, to me it just seems too canonical for there not to be any!

Mike Shulman (May 30 2022 at 22:29):

Well, for instance, the pseudo-fibers of a [[Street fibration]] form a pseudofunctor.

Mike Shulman (May 30 2022 at 22:29):

Pseudo-pullbacks and comma objects are ubiquitous in 2-category theory, and these fibers are a special case of those.

Zhen Lin Low (May 30 2022 at 22:32):

There's also Quillen's Theorems A and B.

Mike Shulman (May 30 2022 at 22:36):

If there's a specific fact about such fibers that you're wondering about, you might be able to get more specific answers about whether it's in the literature.

Tobias Schmude (May 31 2022 at 06:42):

Mike Shulman said:

If there's a specific fact about such fibers that you're wondering about, you might be able to get more specific answers about whether it's in the literature.

What I've shown is that strict and weak fibers are equivalent for isofibrations (where in the bicategorical case I purpose-built a notion of isofibration to facilitate surjectivity up to equivalence of the canonical functor from strict to weak). What I'm specifically interested in are the following:

• A similar result for lax fibers. Fibrations only seem to produce a functor from lax to strict fibers, and a natural transformation from the composite with the canonical inclusion in the opposite direction into the identity on the lax fiber.
• Pseudoinvertibility of the generalized inverse Grothendieck constructions $\mathbf{Cat}/\mathcal{B} \to \mathbf{Lax}_{\textrm{norm}}(\mathcal{B}, \mathbf{Prof})$ resulting from using weak/(op)lax fibers, as well as analogues for bicategories.

Zhen Lin Low (May 31 2022 at 06:55):

There is a definition of cartesian morphism in terms of these comma objects. The upshot is that you get an adjunction between the strict fibre and the comma object.

Mike Shulman (May 31 2022 at 07:01):

For lax fibers, you won't get more than an adjunction as Zhen says, and the Grothendieck construction won't be an equivalence. For pseudo-fibers, the equivalence between strict and weak fibers for isofibrations is a special case of the fact that strict pullbacks of isofibrations are also pseudo pullbacks, which I think appears in various places in the literature; perhaps Makkai-Pare? I don't recall seeing a Grothendieck construction for mere isofibrations stated in this way before, though.

Tobias Schmude (May 31 2022 at 07:02):

Zhen Lin Low said:

There is a definition of cartesian morphism in terms of these comma objects. The upshot is that you get an adjunction between the strict fibre and the comma object.

That makes sense. Thanks!

Tobias Schmude (May 31 2022 at 07:36):

Mike Shulman said:

For lax fibers, you won't get more than an adjunction as Zhen says, and the Grothendieck construction won't be an equivalence. For pseudo-fibers, the equivalence between strict and weak fibers for isofibrations is a special case of the fact that strict pullbacks of isofibrations are also pseudo pullbacks, which I think appears in various places in the literature; perhaps Makkai-Pare? I don't recall seeing a Grothendieck construction for mere isofibrations stated in this way before, though.

Pullbacks of isofibrations:
At least it's on the nlab! Without a reference though.

Inverse Grothendieck construction using weak fibers:
For isofibrations, it's equivalent to the "usual" generalized (profunctorial) inverse Grothendieck construction anyway. I don't really see if pseudoinvertibility might fail for arbitrary functors. If it still works, I'm not sure what the pseudoinverse would look like.

Mike Shulman (May 31 2022 at 07:39):

At least it's on the nlab! Without a reference though.

Hah. It's probably "folklore"...

Mike Shulman (May 31 2022 at 07:40):

I doubt you would get a Grothendieck equivalence using pseudofunctors for non-isofibrations.

Tobias Schmude (May 31 2022 at 07:48):

Mike Shulman said:

I doubt you would get a Grothendieck equivalence using pseudofunctors for non-isofibrations.

I'm thinking of normal lax functors into $\mathbf{Prof}$.
The functors corresponding to pseudofunctors under the generalized Grothendieck construction (using strict fibers) are exactly the Conduché functors, and as far as I can see isofibrations are not necessarily Conduché.
Or am I misunderstanding what you mean by "using pseudofunctors"?

Mike Shulman (May 31 2022 at 14:33):

Sorry, I misspoke; that's what I meant.

Tobias Schmude (May 31 2022 at 14:44):

If that transfers to bicategories I'd find that very interesting, since it would mean that not every pseudofunctor can be equivalently represented in an "indexed" way, contrary to the 1-dimensional case.
I'll check if that's actually the case.

Mike Shulman (May 31 2022 at 14:48):

Why would it mean that?

Tobias Schmude (May 31 2022 at 14:53):

The construction of strict fibers only applies to local isofibrations (I don't know if it yields a Grothendieck equivalence even if the pseudofunctor is not also a global isofibration). If the inverse Grothendieck construction using weak fibers only gives us an equivalence on isofibrations, neither of them works for pseudofunctors which aren't local isofibrations.

Mike Shulman (May 31 2022 at 15:52):

I see your point. One important thing to note is that the notion of (normal) lax functor is not equivalence-invariant in its domain. Accordingly, the 1-categorical Grothendieck equivalence for non-fibrations is an equivalence of 1-categories, not an equivalence of 2-categories. So in seeking such a thing for bicategories, you're going to have to be very careful about strictness.

Mike Shulman (May 31 2022 at 15:53):

This is one reason I think you won't get such an equivalence using pseudofibers, because pseudofibers are equivalence-invariant.

Mike Shulman (May 31 2022 at 15:53):

Perhaps there is a variant of the notion of lax functor that would work?

Mike Shulman (May 31 2022 at 15:55):

If you haven't already seen it, you may be interested in the notion of "displayed bicategory" introduced in this paper. Displayed 1-categories are another name for lax functors into Span (which are equivalent to normal lax functors into Prof), so probably displayed bicategories are an appropriate notion of lax functor in that case for a Grothendieck equivalence.

Tobias Schmude (May 31 2022 at 15:56):

Mike Shulman said:

I see your point. One important thing to note is that the notion of (normal) lax functor is not equivalence-invariant in its domain. Accordingly, the 1-categorical Grothendieck equivalence for non-fibrations is an equivalence of 1-categories, not an equivalence of 2-categories. So in seeking such a thing for bicategories, you're going to have to be very careful about strictness.

Oof, good point.

Tobias Schmude (May 31 2022 at 16:07):

Mike Shulman said:

Perhaps there is a variant of the notion of lax functor that would work?

That would be quite nice! I don't immediately have an idea though.

Tobias Schmude (May 31 2022 at 16:12):

Mike Shulman said:

If you haven't already seen it, you may be interested in the notion of "displayed bicategory" introduced in this paper. Displayed 1-categories are another name for lax functors into Span (which are equivalent to normal lax functors into Prof), so probably displayed bicategories are an appropriate notion of lax functor in that case for a Grothendieck equivalence.

I'm quite familiar with them by now (displayed 1-categories, not so much displayed bicategories) :grinning_face_with_smiling_eyes:
When we're already there: is there a fundamental reason why lax functors into $\mathbf{Span}$ correspond to normal lax functors into $\mathbf{Prof}$?
I'm seeing this from the perspective that spans are just profunctors between discrete categories. So I'd be somewhat surprised if this is an "accident".

Mike Shulman (May 31 2022 at 16:32):

Tobias Schmude said:

When we're already there: is there a fundamental reason why lax functors into $\mathbf{Span}$ correspond to normal lax functors into $\mathbf{Prof}$?
I'm seeing this from the perspective that spans are just profunctors between discrete categories. So I'd be somehow surprised if this is an "accident".

Right, it's not an accident: the operation $\rm Mod$ such that $\rm Mod(Span)=Prof$ is a right adjoint to the inclusion of a category of normal lax functors into a category of lax functors. It's a little tricky to state this precisely since a bicategory $B$ has to have local coequalizers in order for $\rm Mod(B)$ to be a bicategory. The nicest way I know of to state it is to generalize from bicategories to virtual double categories, where composites don't have to exist; this version of it is Proposition 5.14 of A unified framework for generalized multicategories.

Graham Manuell (May 31 2022 at 19:59):

Mike Shulman said:

I see your point. One important thing to note is that the notion of (normal) lax functor is not equivalence-invariant in its domain. Accordingly, the 1-categorical Grothendieck equivalence for non-fibrations is an equivalence of 1-categories, not an equivalence of 2-categories. So in seeking such a thing for bicategories, you're going to have to be very careful about strictness.

Oof. Are you sure? I thought this was true. I mean, I know that lax functors generally don't respect equivalence of their domain, but I was under the impression in this specific case we do end up getting an equivalence of 2-categories. At least that was my understanding and it seems to be claimed in Powerful functors by Ross Street. Possibly I'm misunderstanding what you meant?

James Deikun (May 31 2022 at 20:23):

I thought this was an important reason displayed categories are useful--they provide a way of talking about strict fibers in a context where their lack of equivalence-invariance makes sense.

EDIT: Was thinking of the wrong lack of equivalence-invariance ...

Mike Shulman (May 31 2022 at 21:21):

I don't remember what Street wrote and I don't have time to check it now. Right now it seems to me that even to get an equivalence of 1-categories you need to look at lax functors into the double category Prof, so that the components of the transformations can be functors (tight morphisms) rather than profunctors (loose morphisms). That's a good way of solving the invariance issue, since (loose-)lax double functors do preserve tight isomorphisms. Maybe at that point you can come up with a suitable notion of modification between tight transformations to get an equivalence of 2-categories. But it seems unlikely to me that Street did that...

Graham Manuell (May 31 2022 at 21:46):

Okay. Yes, you definitely need to restrict the morphisms are you suggest, though without doing this I don't think you even get an equivalence of 1-categories. I don't think there is any need to restrict the modifications -- i.e. it is a locally full sub-2-category. Of course, it was Benabou who did this before Street, but I wanted something published I could reference. In any case, Street does indeed mention the necessary restrictions to the 1-morphisms. There is no mention of double categories, but it amounts to the same thing.

Though I had been wondering before how this all related to the non-invariance under equivalence of lax functors, so your point about the lax double functors is interesting to me. Interesting that that's what (probably) ends up curing the non-equivalence in this case.

Tobias Schmude (Jun 02 2022 at 17:13):

I've let this sink in for a bit now, but there's still a lot that I don't get:

1. Why are lax functors not invariant under equivalence of the domain? The problem with normality is clear.
2. The 1-categorical Grothendieck equivalence relates $\mathrm{Cat}/\mathcal{B}$ with $\mathrm{Lax}_\mathrm{norm}(\mathcal{B}, \mathrm{Prof})$, the latter of which is a non-strict bicategory. So what does it mean for the Grothendieck equivalence to be an equivalence of 1-categories? Is there some source where I could read up on the details?
3. When reading about the generalized Grothendieck construction in Benabou's "Distributors at Work", I interpreted the formulation "mutually inverse to each other" to mean an equivalence of bicategories, and the actual proof to be folklore. Indeed, Street's paper claims an equivalence of bicategories, but gives only a proof sketch and cites lectures by Bénabou which I haven't found any notes of. Is it known that there's some subtle mistake in the proof sketch, or does it actually hold up?
4. Is there a more compact way to view displayed bicategories, analogously to displayed 1-categories as lax functors into $\mathrm{Span}$? That would already be half the way to a bicategorical generalized Grothendieck construction :grinning:

Mike Shulman (Jun 02 2022 at 17:19):

1. For example, if $A$ is any set and $IA$ denotes the indiscrete category on $A$ (with $A$ as its objects and all homsets singletons), then a lax functor $IA \to \rm Span$ is the same as a category with object set $A$. (It's normal iff that category contains no nonidentity endomorphisms.) But $IA\simeq IB$ for any nonempty sets $A,B$, while evidently categories with object set $A$ are not equivalent to categories with object set $B$ in any sense.

Mike Shulman (Jun 02 2022 at 17:31):

2. The 1-categorical version I alluded to is that the 1-category ${\rm Cat}/B$ is equivalent to the 1-category of normal lax double functors from $B$, regarded as a double category horizontally, to $\rm Prof$, which has functors as vertical arrows and profunctors as horizontal arrows. (I wasn't clear about this at first because I'd forgotten that this is the way you have to do it; sorry.)

Graham then pointed out that you can also get an equivalence of bicategories between ${\rm Cat}/B$ (now regarded as a 2-category) and the bicategory whose objects are normal lax functors $B\to \rm Prof$, whose morphisms are colax natural transformations whose components are representable profunctors, and whose 2-cells are arbitrary modifications. This is the equivalence that Street states.

I don't know of a place where the 1-categorical version appears explicitly in the literature. The version that refers to not-necessarily-normal lax functors into $\rm Span$ is a consequence of the Grothendieck equivalence for double categories proven by Lambert in Discrete Double Fibrations, where the double category is vertically discrete; and you can then deduce the version involving $\rm Prof$ from the universal characterization of $\rm Mod$ in virtual double categories that I mentioned above.

Tobias Schmude (Jun 02 2022 at 17:32):

Mike Shulman said:

1. For example, if $A$ is any set and $IA$ denotes the indiscrete category on $A$ (with $A$ as its objects and all homsets singletons), then a lax functor $IA \to \rm Span$ is the same as a category with object set $A$. (It's normal iff that category contains no nonidentity endomorphisms.) But $IA\simeq IB$ for any nonempty sets $A,B$, while evidently categories with object set $A$ are not equivalent to categories with object set $B$ in any sense.

Ah yes, in the light of that example it's obvious. My train of thought was that we could transport lax morphisms by precomposition with equivalences, but I see how that fails.

Mike Shulman (Jun 02 2022 at 17:33):

3. As far as I know, the proof works.

Mike Shulman (Jun 02 2022 at 17:34):

4. I don't know of anywhere that this has been defined, but you could ask the authors of the displayed bicategories paper!

Tobias Schmude (Jun 02 2022 at 17:44):

Mike Shulman said:

2. The 1-categorical version I alluded to is that the 1-category ${\rm Cat}/B$ is equivalent to the 1-category of normal lax double functors from $B$, regarded as a double category horizontally, to $\rm Prof$, which has functors as vertical arrows and profunctors as horizontal arrows. (I wasn't clear about this at first because I'd forgotten that this is the way you have to do it; sorry.)

Graham then pointed out that you can also get an equivalence of bicategories between ${\rm Cat}/B$ (now regarded as a 2-category) and the bicategory whose objects are normal lax functors $B\to \rm Prof$, whose morphisms are colax natural transformations whose components are representable profunctors, and whose 2-cells are arbitrary modifications. This is the equivalence that Street states.

I don't know of a place where the 1-categorical version appears explicitly in the literature. The version that refers to not-necessarily-normal lax functors into $\rm Span$ is a consequence of the Grothendieck equivalence for double categories proven by Lambert in Discrete Double Fibrations, where the double category is vertically discrete; and you can then deduce the version involving $\rm Prof$ from the universal characterization of $\rm Mod$ in virtual double categories that I mentioned above.

Thanks a lot for the thorough explanation!
Since the bicategorical equivalence seems to work, I'll stick to that for now, and keep the (virtual) double categorical version in the back of my mind.

Tobias Schmude (Jun 02 2022 at 17:46):

Mike Shulman said:

4. I don't know of anywhere that this has been defined, but you could ask the authors of the displayed bicategories paper!

Already did so (well, for one of the authors) :upside_down:

Tobias Schmude (Jun 24 2022 at 09:49):

Update: I think using weak fibers should lead to an equivalence, the key is using the weak slice 2-category $\mathrm{Cat}/_w\mathcal{B}$ (1-morphisms aren't strictly commuting triangles, but filled with an isomorphism, 2-morphism are compatible with the triangle fillers) instead of the strict one. I'm not quite done with all the details though, so take this with a grain of salt. I suspect the lax case might lead to a lax 2-adjunction.
@Mike Shulman: I'm not sure if the equivalence (if it is correct) contradicts your argument about non-invariance of lax functors. What do you think of this?

Mike Shulman (Jun 24 2022 at 16:01):

Can you state precisely the equivalence you think holds?

Tobias Schmude (Jun 24 2022 at 16:05):

It should be $\mathrm{Cat}/_w \mathcal{B} \cong \mathrm{Lax}_{norm, rep}(\mathcal{B}^{\text{op}}, \mathrm{Prof})$, where the right hand side is the bicategory of normal lax functors with 1-morphisms given by oplax transformations with representable components and 2-morphisms given by modifications.

Tobias Schmude (Jun 24 2022 at 16:10):

The pseudoinverse of the weak fibers functor is a variation of the usual Grothendieck construction, where we tack on a further isomorphic object in the category of elements construction: the objects of the category of weak elements of $P$ are given by tuples $(B, B', f, C)$ where $f: B \to B'$ is an isomorphism in $\mathcal{B}$, $C \in PB'$, the morphisms are the obvious thing, and the functor to $\mathcal{B}$ is projection onto the first component.

Tobias Schmude (Jun 24 2022 at 16:11):

The lax case works similarly, but doesn't yield an equivalence.

Mike Shulman (Jun 24 2022 at 16:25):

Hmm. Suppose $\mathcal{B}$ is the contractible groupoid with 2 objects $a\cong b$, and $\mathcal{A}$ is an arbitrary category. There's a lax normal functor $\mathcal{B}^{\rm op}\to \rm Prof$ that sends $a$ to $\mathcal{A}$ and sends $b$ to $\emptyset$. But for any functor into $\mathcal{B}$ the pseudofibers over $a$ and $b$ are equivalent, so it seems to me that this lax functor can't be in the image of an equivalence of that sort.

Tobias Schmude (Jun 24 2022 at 16:39):

Hm, that sounds very plausible as well. The machinery definitely spits out a functor from $PB$ to the fiber of the associated pseudofunctor into $\mathcal{B}$ over $B$, I'll have to have another look at the proof that this is an equivalence in the case of weak fibers.