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Stream: theory: category theory

Topic: Effective epi of topoi not stable by base change

Fernando Yamauti (May 31 2024 at 12:28):

Hi. Does anyone know any example of an effective epi (also called of effective descent) between topoi that is not universal?

Well. Actually, I don't even know whether that is true, but, since all proofs of stability by base change that I've seen in the literature usually restrict to an specific class of effective epic (e.g., open surjections), I'm inclined to believe that it's indeed the case. I'm having a hard time finding such an example, though.

Morgan Rogers (he/him) (May 31 2024 at 12:47):

What do you mean by "not universal" here?

Fernando Yamauti (May 31 2024 at 12:51):

Morgan Rogers (he/him) said:

What do you mean by "not universal" here?

Ah. Sorry. I mean not stable by base change.

Morgan Rogers (he/him) (May 31 2024 at 13:06):

Hmm the examples available in, say, Sketches of an Elephant are indeed members of stable classes. It's worth noting that hyperconnected morphisms are shown to be descent morphisms (they have the stronger property that "locales descend along them"), so any counterexample will be essentially localic in nature. It is also mentioned that descent morphisms have to be surjections. So if you want to find an example that doesn't fall into those classes, you could test some simple surjections of locales which are neither open nor proper.
Having said all that, is your question "are descent morphisms stable under pullback?"

Morgan Rogers (he/him) (May 31 2024 at 13:07):

Oh I realised that I assumed that being "of effective descent" corresponds to what Johnstone calls "descent morphisms" but I haven't checked; is that the case?

Fernando Yamauti (May 31 2024 at 13:45):

Morgan Rogers (he/him) said:

Having said all that, is your question "are descent morphisms stable under pullback?"

Yes. It's just that I'm skeptic they do so. This is why I asked for an example failing to satisfy descent after base change.

Fernando Yamauti (May 31 2024 at 13:48):

Morgan Rogers (he/him) said:

Oh I realised that I assumed that being "of effective descent" corresponds to what Johnstone calls "descent morphisms" but I haven't checked; is that the case?

I think so. At least, the version on part C of the book.

Well. It would be nice of there was a synctatic characterisation of effective epis in terms of internal sites. But that's probably too much to ask for.

Morgan Rogers (he/him) (May 31 2024 at 13:51):

At one point early on in Part C there is a class of locale maps called "triquotient maps" mentioned in passing which are a common generalization of open and proper maps. That's probably a good first class to check for descent properties.

Morgan Rogers (he/him) (May 31 2024 at 13:52):

I have a hunch that those are not stable under pullback, but I haven't thought about them in a long time.

Fernando Yamauti (May 31 2024 at 14:00):

Morgan Rogers (he/him) said:

I have a hunch that those are not stable under pullback, but I haven't thought about them in a long time.

Hmm...isn't Plewe's result in remark 3.2.7 claiming that they are stable under pullbacks in the localic case? If so, are you referring to a non-localic version thereof?

Morgan Rogers (he/him) (May 31 2024 at 14:19):

Oh you're right, it's a slightly stronger condition than I remember; in particular, it imposes surjectivity(!). Well that's convenient. It would certainly be interesting if triquotient implies descent. I don't dare hope for the converse, but maybe..?

Fernando Yamauti (May 31 2024 at 15:00):

Morgan Rogers (he/him) said:

Oh you're right, it's a slightly stronger condition than I remember; in particular, it imposes surjectivity(!). Well that's convenient. It would certainly be interesting if triquotient implies descent. I don't dare hope for the converse, but maybe..?

I think triquotient implies descent by noticing that it defines a linear retract of $O(Y)$-algebras and, then, the equalisers after tensoring by modules will be split. The converse I think fails. For instance, a surjection between compact Hausdorff spaces seems to satisfy descent (not entirely sure of that last statement, though).

Morgan Rogers (he/him) (May 31 2024 at 15:28):

That observation alone is definitely something worth putting into the world somewhere. What's your interest in descent morphisms? I have never understood them in any depth (I just know where I could read about them if I wanted to). Is there a reason for you expecting/not expecting them to be stable?

Fernando Yamauti (May 31 2024 at 18:29):

Well. That observation is not very original. The same argument ist already in Joyal-Tierney's monograph on Galois Theory. And I've just noticed that my "counter example" is not a counter example: a continuous map between compacta is always proper.

My interest is because I'm trying to prove some generalisations of Joyal-Tierney's theorem and I want to get a feel of what kind of effective epis (if any) I should be careful of.

I don't really have any good reason to not expect they are not stable besides the one I've already mentioned:in the literature, people usually restrict to a subcollection when they need stability by base change.

But regarding your previous conjecture: effective descent implies triquotient. Maybe a frame morphism defining a faithfully flat module that doesn't have a linear retract might do the job (as a counter example). The problem is finding one (in case it exists).

Morgan Rogers (he/him) (May 31 2024 at 18:56):

I guess your last message contains a mistake, since it seems like you have both implications of triquotient iff descent in the above?

Fernando Yamauti (May 31 2024 at 19:06):

Morgan Rogers (he/him) said:

I guess your last message contains a mistake, since it seems like you have both implications of triquotient iff descent in the above?

I think my initial claim is: triquotient implies split equaliser and, therefore, descent. The problem is: effective descent implies triquotient. The suggested possible counterexample is: fully faithful without a retract. Did I mess up something?

Fernando Yamauti (May 31 2024 at 21:19):

Fernando Yamauti said:

Morgan Rogers (he/him) said:

I guess your last message contains a mistake, since it seems like you have both implications of triquotient iff descent in the above?

I think my initial claim is: triquotient implies split equaliser and, therefore, descent. The problem is: effective descent implies triquotient. The suggested possible counterexample is: fully faithful without a retract. Did I mess up something?

Nah. That won't work. All suplattices are dualisable. With that, one can prove pure morphisms have retracts. Since any morphism of effective descent must be pure, my suggestion is wrong. So I think triquotient is really the same as effective descent. And therefore effective descent is really stable by base change.

I keep forgetting that suplattices are not like usual modules over rings since they are always dualisable.

Mike Shulman (May 31 2024 at 23:43):

What do you mean by "dualisable"? The category of suplattices is $\ast$-autonomous, so every object is "dualisable" in that sense, but I don't think it is compact closed.

Graham Manuell (Jun 01 2024 at 11:38):

Yes, suplattices are dualisable in the compact closed sense if and only if they are (constructively) completely distributive.

Graham Manuell (Jun 01 2024 at 11:50):

I will restrict my attention to effective descent morphisms of locales, since I understand those better and it should be no weaker a condition than the topos-theoretic one. Certainly in this setting it is true that effective descent morphisms are stable under pullback by a general theorem about morphisms that are effective descent with respect to the codomain fibration.

@Fernando Yamauti What makes you think that every effective descent morphism is a triquotient map? It is definitely false for topological spaces, for example. It is also false for finite locales.
On the other hand I think it might be true that no one has given an example of an effective descent morphism in the category of locales that is not a triquotient, though I would be surprised if one doesn't exit. I have been thinking about this myself recently.

Fernando Yamauti (Jun 01 2024 at 22:09):

Mike Shulman said:

What do you mean by "dualisable"? The category of suplattices is $\ast$-autonomous, so every object is "dualisable" in that sense, but I don't think it is compact closed.

I only need that $Hom (M, N) \cong (M \otimes N^o)^o$. Compact closedness is not required. By the way, I'm not very familiar with the definition of star autonomous categories. Are ordinary modules over any ring already star autonomous?

Fernando Yamauti (Jun 01 2024 at 22:13):

Graham Manuell said:

I will restrict my attention to effective descent morphisms of locales, since I understand those better and it should be no weaker a condition than the topos-theoretic one. Certainly in this setting it is true that effective descent morphisms are stable under pullback by a general theorem about morphisms that are effective descent with respect to the codomain fibration.

Nice. What's the statement of that general theorem? Where can I read about that?

Mike Shulman (Jun 01 2024 at 22:16):

Fernando Yamauti said:

I only need that $Hom (M, N) \cong (M \otimes N^o)^o$. Compact closedness is not required.

Ok, yes, you have that in suplattices and any $\ast$-autonomous category.

Are ordinary modules over any ring already star autonomous?

No, they're not. There are several equivalent definitions and a number of examples at [[star-autonomous category]].

Fernando Yamauti (Jun 01 2024 at 22:56):

Graham Manuell said:

Fernando Yamauti What makes you think that every effective descent morphism is a triquotient map? It is definitely false for topological spaces, for example. It is also false for finite locales.
On the other hand I think it might be true that no one has given an example of an effective descent morphism in the category of locales that is not a triquotient, though I would be surprised if one doesn't exit. I have been thinking about this myself recently.

Maybe I'm missing something on the definition of a triquotient. My reasoning is the following:

A morphism of locales $f \colon X \to Y$ will induce an $O(Y)$-algebra structure on $O (X)$ by the action of $f^*$. Now, if $f$ is of effective descent, it's known that it has an $O(Y)$-linear retract $r \colon O (X) \to O (Y)$. By taking $f_\# \colon = r$, I think we are done. $O (Y)$-linearity implies the equation containing the intersection. The other equation follows from the fact that $r$ preserves joins and $rf^* = 1$.

No? Am I messing something up? I usually oversee very trivial mistakes, so be careful...

Fernando Yamauti (Jun 01 2024 at 23:08):

Mike Shulman said:

Are ordinary modules over any ring already star autonomous?

No, they're not. There are several equivalent definitions and a number of examples at [[star-autonomous category]].

Thanks for the clarification. Is there any subcategory of the category of modules over a ring $A$ (aside from the obvious one of dualisable $A$-modules), that will be star autonomous with dualisation functor the usual one $Hom_A (-. A)$?

Or better, how good is the analogy between suplattices and (possibly a subcat of the cat) modules over an arbitrary commutative ring? Or, even better, can I extend in a meaningful way that analogy (if any) to the $(n + 1, 1)$-cat of locally presentable $(n, 1)$-categories?

Mike Shulman (Jun 01 2024 at 23:25):

Fernando Yamauti said:

Is there any subcategory of the category of modules over a ring $A$ (aside from the obvious one of dualisable $A$-modules), that will be star autonomous with dualisation functor the usual one $Hom_A (-. A)$?

I don't know of any, and I doubt it when $A$ is completely arbitrary.

Or better, how good is the analogy between suplattices and (possibly a subcat of the cat) modules over an arbitrary commutative ring?

Like most analogies, it's good for some things but not others. I think $\ast$-autonomy is a place where it is not good.

Or, even better, can I extend in a meaningful way that analogy (if any) to the $(n + 1, 1)$-cat of locally presentable $(n, 1)$-categories?

The analogy between suplattices and locally presentable categories is somewhat better, as some formulas like $\hom(M,N) = (M \otimes N^*)^*$ do have partial analogues in that case. But it has limitations, since the opposite of a locally presentable category is not (usually) locally presentable, so the category of locally presentable categories is also not $\ast$-autonomous.

Fernando Yamauti (Jun 01 2024 at 23:44):

Mike Shulman said:

The analogy between suplattices and locally presentable categories is somewhat better, as some formulas like $\hom(M,N) = (M \otimes N^*)^*$ do have partial analogues in that case. But it has limitations, since the opposite of a locally presentable category is not (usually) locally presentable, so the category of locally presentable categories is also not $\ast$-autonomous.

Thanks. The problem of taking opposites is not really a problem as the correct higher analogue should be the $(2, 1)$-cat of $Mod_A$-modules. But if the star autonomous property already breaks for the analogy "locally presentable $(0, 1)$-categories are like certain abelian groups", any such extension would indeed be really hopeless.

Graham Manuell (Jun 02 2024 at 09:03):

Fernando Yamauti said:

Graham Manuell said:

I will restrict my attention to effective descent morphisms of locales, since I understand those better and it should be no weaker a condition than the topos-theoretic one. Certainly in this setting it is true that effective descent morphisms are stable under pullback by a general theorem about morphisms that are effective descent with respect to the codomain fibration.

Nice. What's the statement of that general theorem? Where can I read about that?

It's on the nlab page: https://ncatlab.org/nlab/show/descent+morphism#properties

Graham Manuell (Jun 02 2024 at 09:07):

Fernando Yamauti said:

Graham Manuell said:

Fernando Yamauti What makes you think that every effective descent morphism is a triquotient map? It is definitely false for topological spaces, for example. It is also false for finite locales.
On the other hand I think it might be true that no one has given an example of an effective descent morphism in the category of locales that is not a triquotient, though I would be surprised if one doesn't exit. I have been thinking about this myself recently.

Maybe I'm missing something on the definition of a triquotient. My reasoning is the following:

A morphism of locales $f \colon X \to Y$ will induce an $O(Y)$-algebra structure on $O (X)$ by the action of $f^*$. Now, if $f$ is of effective descent, it's known that it has an $O(Y)$-linear retract $r \colon O (X) \to O (Y)$. By taking $f_\# \colon = r$, I think we are done. $O (Y)$-linearity implies the equation containing the intersection. The other equation follows from the fact that $r$ preserves joins and $rf^* = 1$.

No? Am I messing something up? I usually oversee very trivial mistakes, so be careful...

You are using a different definition of effective descent morphism compared to what people usually talk about. You are considering descent with respect to modules over the frames instead of with respect to the codomain fibration. I think your version is also stable under pullback though. You can probably prove it directly without much difficultly.

On the other hand, I don't think triquotients or proper surjections will be effective descent in this sense.

Graham Manuell (Jun 02 2024 at 09:22):

Fernando Yamauti said:

Mike Shulman said:

Are ordinary modules over any ring already star autonomous?

No, they're not. There are several equivalent definitions and a number of examples at [[star-autonomous category]].

Thanks for the clarification. Is there any subcategory of the category of modules over a ring $A$ (aside from the obvious one of dualisable $A$-modules), that will be star autonomous with dualisation functor the usual one $Hom_A (-. A)$?

Even for modules over a frame the category is not *-autonomous with respect to this functor, so this isn't really a disanalogy. Instead the dualising object is given by $A^\mathrm{op}$. Though there is, of course, no notion of opposite for a module over a ring.

Fernando Yamauti (Jun 02 2024 at 10:38):

Graham Manuell said:

You are using a different definition of effective descent morphism compared to what people usually talk about. You are considering descent with respect to modules over the frames instead of with respect to the codomain fibration. I think your version is also stable under pullback though. You can probably prove it directly without much difficultly.

On the other hand, I don't think triquotients or proper surjections will be effective descent in this sense.

Am I? The definition I want is the one so that $f \colon X \to Y$ is an map of effective descent between locales iff the induced map after taking sheaves, $\tilde{X} \to \tilde{Y}$, is an effective $2$-epimorphism in the $2$-category of topoi. Explicitly, $\tilde{Y}$ should be equivalent (by the canonical map) to the colimit of the induced \v{C}ech simplicial object.

On the other side, sheaves over $X$ are equivalent to $O (X)$-modules corresponding to frames defining an étale morphism of locales (i.e., open with an open diagonal). But all these properties (algebra structure, idempotency, affineness and existence of a left adjoint) are preserved by descent, i.e., an $O (X)$-module coming from a sheaf satisfying descent descends uniquely to an $O (Y)$-module that also comes from a sheaf.

You mentioned finite locales as a counter example of effective descent that is not triquotient, can you expand on that? Maybe using your counter example I can work my way back to see what breaks in my argument.

Fernando Yamauti (Jun 02 2024 at 10:45):

Graham Manuell said:

Even for modules over a frame the category is not *-autonomous with respect to this functor, so this isn't really a disanalogy. Instead the dualising object is given by $A^\mathrm{op}$. Though there is, of course, no notion of opposite for a module over a ring.

Thanks. That's a nice remark. I mean the analogue of $A^o$ should be $Hom_{\mathbf{Z}} (A, \mathbf{Z})$, but that still make the analogy fail. Do you know if the analogy gets better for Boolean rings $A$: something like modules over Boolean algebras (in suplattices) are the same as modules over Boolean rings (in abelian groups)? I'm somehow frustrated that the analogy breaks so soon

Graham Manuell (Jun 02 2024 at 10:50):

Fernando Yamauti said:

Graham Manuell said:

Even for modules over a frame the category is not *-autonomous with respect to this functor, so this isn't really a disanalogy. Instead the dualising object is given by $A^\mathrm{op}$. Though there is, of course, no notion of opposite for a module over a ring.

Thanks. That's a nice remark. I mean the analogue of $A^o$ should be $Hom_{\mathbf{Z}} (A, \mathbf{Z})$, but that still make the analogy fail. Do you know if the analogy gets better for Boolean rings $A$: something like modules over Boolean algebras (in suplattices) are the same as modules over Boolean rings (in abelian groups)? I'm somehow frustrated that the analogy breaks so soon

These are definitely different. The category of suplattices behaves differently to the category of abelian groups, so I don't know how reasonable it is to expect categories of the corresponding modules to behave similarly in every respect.

Graham Manuell (Jun 02 2024 at 11:01):

Fernando Yamauti said:

Graham Manuell said:

You are using a different definition of effective descent morphism compared to what people usually talk about. You are considering descent with respect to modules over the frames instead of with respect to the codomain fibration. I think your version is also stable under pullback though. You can probably prove it directly without much difficultly.

On the other hand, I don't think triquotients or proper surjections will be effective descent in this sense.

Am I? The definition I want is the one so that $f \colon X \to Y$ is an map of effective descent between locales iff the induced map after taking sheaves, $\tilde{X} \to \tilde{Y}$, is an effective $2$-epimorphism in the $2$-category of topoi. Explicitly, $\tilde{Y}$ should be equivalent (by the canonical map) to the colimit of the induced \v{C}ech simplicial object.

On the other side, sheaves over $X$ are equivalent to $O (X)$-modules corresponding to frames defining an étale morphism of locales (i.e., open with an open diagonal). But all these properties (algebra structure, idempotency, affineness and existence of a left adjoint) are preserved by descent, i.e., an $O (X)$-module coming from a sheaf satisfying descent descends uniquely to an $O (Y)$-module that also comes from a sheaf.

You mentioned finite locales as a counter example of effective descent that is not triquotient, can you expand on that? Maybe using your counter example I can work my way back to see what breaks in my argument.

Hmm. Okay. This is indeed the usual notion for toposes that I was trying to avoid above because I understand it less well than notion usually considered for locales. (By the way, where did you get the terminology effective 2-epimorphism? I think it is good to have a name for this and this name is reasonable, though I haven't heard of this being used before.)

However, what I am not convinced by what you said here:

Fernando Yamauti said:

A morphism of locales $f \colon X \to Y$ will induce an $O(Y)$-algebra structure on $O (X)$ by the action of $f^*$. Now, if $f$ is of effective descent, it's known that it has an $O(Y)$-linear retract $r \colon O (X) \to O (Y)$. By taking $f_\# \colon = r$, I think we are done. $O (Y)$-linearity implies the equation containing the intersection. The other equation follows from the fact that $r$ preserves joins and $rf^* = 1$.

I don't see why an effective descent morphism in your sense gives a retraction in this way and I feel like it should not be true. Where did you get this result from?

(But also, the triquotient assignment doesn't preserve joins, so unless I misunderstand what you mean it doesn't give a 'linear' retract.)

Fernando Yamauti (Jun 02 2024 at 11:23):

Graham Manuell said:

Hmm. Okay. This is indeed the usual notion for toposes that I was trying to avoid above because I understand it less well than notion usually considered for locales. (By the way, where did you get the terminology effective 2-epimorphism? I think it is good to have a name for this and this name is reasonable, though I haven't heard of this being used before.)

That's the usual terminology for $(\infty, 1)$-categories (for instance, in Lurie's HTT): a morphism is an effective epimorphism if it's the geometric realisation/colimit of the induced \v{C}ech simplicial object.

However, what I am not convinced by what you said here:

A morphism of locales $f \colon X \to Y$ will induce an $O(Y)$-algebra structure on $O (X)$ by the action of $f^*$. Now, if $f$ is of effective descent, it's known that it has an $O(Y)$-linear retract $r \colon O (X) \to O (Y)$. By taking $f_\# \colon = r$, I think we are done. $O (Y)$-linearity implies the equation containing the intersection. The other equation follows from the fact that $r$ preserves joins and $rf^* = 1$.

I don't see why an effective descent morphism in your sense gives a retraction in this way and I feel like it should not be true. Where did you get this result from?

(But also, the triquotient assignment doesn't preserve joins, so unless I misunderstand what you mean it doesn't give a 'linear' retract.)

That's Prop.3 at p.12 (pure morphisms have linear retracts) and Thm. 1 at p.18 (descent iff pure) in Joyal-Tierney's AMS memoirs on Galois theory. In that case, the triquotient assignment will preserve all small joins, so, in particular, also the directed ones.

Graham Manuell (Jun 02 2024 at 11:42):

Okay, this is why I said before you were using a different notion of effective descent. Pure morphisms are not the same as the effective descent morphisms you are talking about, they are effective descent for modules, not for sheaves. (Well, they are also effective descent for sheaves, but they aren't a characterisation in that case.)

Graham Manuell (Jun 02 2024 at 11:43):

I think you have your second sentence backwards, triquotient assignments preserve directed joins, not all small joins, but you need them to preserve all joins.

Fernando Yamauti (Jun 02 2024 at 12:42):

Graham Manuell said:

Okay, this is why I said before you were using a different notion of effective descent. Pure morphisms are not the same as the effective descent morphisms you are talking about, they are effective descent for modules, not for sheaves. (Well, they are also effective descent for sheaves, but they aren't a characterisation in that case.)

Hmm...I see. So a morphism of locales that is of effective descent for modules also is of effective descent. However, a morphism of locales that is of effective descent might not be always enlarged to a morphism of effective descent for modules (since it, a priori, just descends specific modules). Thanks. That seems helpful.

Now. Regarding my argument, I didn't get anything backwards. What I'm saying is that if $f$ is of effective descent for modules, then the respective linear retract defines a triquotient assignment. That assignment preserves all joins because it was already chosen to do so.

What doesn't work is the converse. A triquotient assignment might not preserve arbitrary joins, so I cannot deduce that it is of effective descent for modules. But, by Plewe's result, it is of effective descent (in the usual sense). The proof seems very non-trivial, though.

That's clarifies a lot of things. Still, the reduction from topoi to locales for claiming the stability under base change requires that the argument should be valid for any base topos (first, use the hyperconnected-localic factorisation as @Morgan Rogers (he/him) suggested and, then, claim the result for the respective internal locales). That I'm not very comfortable with. Maybe there's a more direct argument... But, even for $1$-cats, effective epimorphisms shouldn't always be stable by base change, right? I wonder if I there's a higher version of that theorem for codomain fibrations you've mentioned and, then, I can work directly with the $2$-cat of topoi.

Graham Manuell (Jun 02 2024 at 12:54):

Effective epimorphisms don't need to be stable under pullback, but effective descent morphisms (with respect the codomain fibration) do. If you use a different notion of effective descent morphism or an effective 2-epimorphism in a general 2-category, it might not be stable under base change. What is probably is true is that the 2-dimensional version of effective descent morphism is stable under base change, but this isn't exactly what you are talking about.

Effective descent morphisms of locales are not the same as effective 2-epimorphisms of the corresponding localic toposes. Effective decent morphisms of locales are a stronger condition since they allow descent of all localic bundles instead of just sheaves. Probably the notion you want is also pullback stable, but that would require it's own proof.

Fernando Yamauti (Jun 02 2024 at 13:04):

Graham Manuell said:

Effective epimorphisms don't need to be stable under pullback, but effective descent morphisms (with respect the codomain fibration) do. If you use a different notion of effective descent morphism or an effective 2-epimorphism in a general 2-category, it might not be stable under base change. What is probably is true is that the 2-dimensional version of effective descent morphism is stable under base change, but this isn't exactly what you are talking about.

Effective descent morphisms of locales are not the same as effective 2-epimorphisms of the corresponding localic toposes. Effective decent morphisms of locales are a stronger condition since they allow descent of all localic bundles instead of just sheaves. Probably the notion you want is also pullback stable, but that would require its own proof.

Oh! So there's yet a third notion of effective descent. In total, descent for: modules, locales and étale locales (or sheaves). That theorem you've mentioned work for more general bifibrations, no? If it does work, I can restrict it to the (sub)bifibration with fibers over the locale $X$ the étale morphisms (of locales) $Y \to X$.

Graham Manuell (Jun 02 2024 at 19:00):

Unfortunately, I am now a the limit of my knowledge. I don't know if the result still holds for sub-bifibrations or not.

Fernando Yamauti (Jun 02 2024 at 19:10):

Graham Manuell said:

Unfortunately, I am now a the limit of my knowledge. I don't know if the result still holds for sub-bifibrations or not.

No problems. It was a nice exchange. I've already learnt a lot. Thanks. I will try working that out. If I get to anything meaningful, I will post it here.

Morgan Rogers (he/him) (Jun 02 2024 at 21:05):

Nice, I'm glad someone who has more actively worked with descent morphisms stepped in, thanks @Graham Manuell . The fact that we can characterize descent morphisms via the Čech complex makes them a lot clearer to me; this discussion is about whether the 2-category of topoi is "regular" (for one of the 2-categorical generalizations of that concept). This reminds me of a discussion I listened in on between Mattieu Anel and @Simon Henry years ago where they were trying to figure out a characterization of the 2-category of (1-)topoi... fun!

Graham Manuell (Jun 03 2024 at 11:14):

Yes, I agree this is a nice way to think about descent. For whatever reason, the version of regular 2-categories people usually study involves what I think amounts to lax descent rather than descent, but yes, this would be the analogue that uses bipullbacks / iso-comma objects instead of comma objects for the kernel pairs.