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Graham Manuell (May 21 2026 at 08:11):@Joshua Wrigley and I have finally put out our paper on localic categories that classify models of geometric / dual geometric logic: https://arxiv.org/abs/2605.20407.
This is the the sequel to our largely expository paper "The representing localic groupoid for a geometric theory", which gives a concrete description of the localic groupoids in the famous Joyal-Tierney paper. I also spoke about this at CT2023 in Louvain-la-Neuve. This paper (together with the previous one) has been more than 4 years in the making, but only for logistical reasons, not because it is particularly complicated -- I believe it is actually all very straightforward when seen from the correct perspective.
The position we advocate for in the paper is that localic categories provide a more flexible framework than Grothendieck toposes in that they can classify not only geometric theories, but also 'dual' geometric theories. The ones that classify dual geometric theories can be thought of as a mirror world to that of toposes that has not been studied before, but I believe it definitely deserves to be.
In this paper we construct classifying localic categories and groupoids for various bundles equipped with logical structure. When these bundles are local homeomorphisms, we recover the localic groupoids that classify geometric theories, demonstrating that these groupoids satisfy a stronger universal property than that of their corresponding classifying toposes. We also prove a dual result that there exist classifying localic categories and groupoids for proper separated bundles satisfying a dual geometric theory. Thus, localic groupoids classify strictly more kinds of logical theories than toposes. Our approach provides a concrete construction of the localic categories and the generic bundles involved in terms of generalised frame presentations. To accommodate our approach, we prove en passant a constructive, pointfree version of the Alexandroff--Hausdorff theorem and that internal functors that are fully faithful and effective descent morphisms on objects induce equivalences between the categories of discrete opfibrations over the source and target categories.
David Michael Roberts (May 22 2026 at 06:24):Nice! (I'm not biased at all, I promise)
I'm very much of the opinion that internal categories need to be taken more seriously by people in the stacks community, not just internal groupoids, and the "geometry" has a lot of things ripe for the picking.
Graham Manuell (May 22 2026 at 12:47):Yes, I agree. Also, your work on anafunctors was useful to us, so thank you for that.
The case of localic groupoids is actually an interesting one because there is also an order enrichment and this can often be used together with the groupoid structure to recover non-invertible morphisms. But this works differently in the 'topos' and 'dual topos' cases and so it's more uniform (as well as being simpler) to just work with categories directly. In any case, I want to look more about the role of order enrichment in future.
Also, I personally find the stack approach difficult to understand, but internal categories/groupoids much easier, and I wish more people phrased things in these terms. It feels much more concrete.
Fernando Yamauti (May 22 2026 at 16:31):Graham Manuell said:
Joshua Wrigley and I have finally put out our paper on localic categories that classify models of geometric / dual geometric logic: https://arxiv.org/abs/2605.20407.
Congratulations on the preprint! It's nice to see this dual thing written down.
I didn't have time to read everything carefully and I likely still won't have enough time anytime soon, but I can't resist my temptation to ask questions anyway.
1) What do you mean by the universal property of the JT groupoid among étale complete ones hinted in the introduction? Is that just what it classifies?
2) In that characterisation of the cat of topoi as a localisation of the cat of étale-complete groupoids restated in terms of anafunctors, how did you get rid of the openness of the left leg?
3) Is there a definition of a dual topos that fits your presentation by groupoids via proper stuff? Related to that, should objects of this (possibly conjectural) thing be thought as compacta as opposed to sets?
As a remark related to 3, there's a way to obtain internal compacta out of a (maybe coherent only) topos by applying the functor taking right ultrafunctors to the terminal topos. Left ultrafunctors gives instead finite objects.
Joshua Wrigley (May 22 2026 at 19:35):@Fernando Yamauti I can reply to 1) and 2).
Fernando Yamauti said:
1) What do you mean by the universal property of the JT groupoid among étale complete ones hinted in the introduction? Is that just what it classifies?
The JT groupoid of a geometric theory is étale complete, and so it remains a representing object for the restriction of the pseudofunctor to étale complete groupoids. Essentially, using classifying topos theory alone you would only see (at least instantaneously) the universal property of with respect to this restricted pseudofunctor.
Fernando Yamauti said:
2) In that characterisation of the cat of topoi as a localisation of the cat of étale-complete groupoids restated in terms of anafunctors, how did you get rid of the openness of the left leg?
You're right that here we are perhaps being less precise than maybe we ought to be. Let be the class of locale morphisms which are effective descent in and in (i.e. the corresponding geometric morphism is). Then all the machinery of Moerdijk's paper goes through with open surjection replaced by lies in . In particular, open surjections and proper surjections are both contained in .
It is perhaps a surprising quality of topoi that we can restrict our attention to open surjections, in that any topos admits an open surjection from a localic topos (and in some nebulous way, this is related to the privileged role of geometric logic within topos theory).
Graham Manuell (May 22 2026 at 19:47):@Fernando Yamauti Thanks for your interest!
1) Yes, it is what it classifies. It's the normal universal property of a classifying topos rephrased in terms of localic groupoids. It's the restriction universal property we prove in section 6 without for all localic categories/groupoids restricted to étale-complete case.
2) I assume you are referring to how Moerdijk uses anafunctors where the equivalences need to be open surjections on objects? It turns out this is not necessary, because being slightly more liberal with the notion of equivalence does not identify any two étale-complete groupoids that were previously not identified. This follows from our Theorem 5.4 on the one side together with Moerdijk's correspondence with toposes on the other side.
(I was writing this when Josh replied, so I'll keep it even though he also answered.)
3) In the paper we just defined the 2-category of dual toposes, not the dual toposes themselves, but you could define these to be the categories of proper separated bundles over the appropriate classes of localic categories. And yes, these should somehow model universes of compact Hausdorff locales in the same way toposes model universes of sets. Exactly what the axiomatisation is (a la Giraud) I do not know yet. I have some vague ideas, but nothing concrete as of yet.
As to your remark, that's interesting. I don't know much about ultracategories, but @Joshua Wrigley does. I feel like they should have a strong relation to some kind of localic categories.
Fernando Yamauti (May 27 2026 at 14:51):@Graham Manuell @Joshua Wrigley Thanks for the clarifications. Regarding 2, I guess I somehow forgot proper morphisms are of effective descent.
Regarding 3, not sure how you get the corresponding geometric object out of the dual of a geometric theory. For instance, for a given object of the base topos, the topos defines another theory (or a discrete category). What's the corresponding universal model of the dual geometric theory and should it define a proper separated locale in after taking the geometric realisation of such groupoid inside the cat of topoi?
Is that all just supposed to boil down to ?
Graham Manuell (May 27 2026 at 15:56):@Fernando Yamauti We construct the universal proper separated bundle over the representing localic category of the dual geometric theory in Section 4.4. This would be an object in the category of all proper separated bundles over the category, so if you take that category to the the 'dual' topoes, then that object is what acts as the universal model.
You could take the topos of sheaves over the localic category like you suggest, but this will lose information. I'm not sure if the proper separated bundle becomes an internal compact Hausdorff in this topos or not. But even if it does, it won't generally satisfy any obvious universal property. One of the main reasons for the discussion of dual geometric theories in the paper is to argue that toposes, which are inherently tied to sets / discrete locales, cannot handle these kinds of theories, but localic categories can.
I'm not completely sure what you mean by it boiling down to , but the answer is probably no.
Fernando Yamauti (May 27 2026 at 17:22):Graham Manuell said:
I'm not completely sure what you mean by it boiling down to , but the answer is probably no.
I had the impression , say some ordinary set seen as a theory, would have the universal model of the respective dual theory equivalent to or maybe (as cats with source and target the identity), up to inverting ff surjections of effective descent, under your construction.
My point was not to try putting everything in a topos (like in the case of étale models), but, instead, to understand whether models of both the proper and étale versions can be symmetrically represented by topoi.
No information should be lost, right? As proper separated (étale complete) groupoids should be equivalent, up to inverting ff surjections of effective descent, to a subcat of the cat of topoi.
Graham Manuell (May 27 2026 at 19:04):I see. So firstly, I do not believe that the bicategories of toposes and dual toposes will be equivalent, but both do sit inside the bicategory of localic categories and anafunctors.
When you say you view a set as a theory, do you mean the theory of a single sort, or some particular fixed ? If it's the latter the theory is classified by the terminal localic category and that would indeed also classify a dual geometric theory, but perhaps not in a very interesting way. If it's the former, I don't think it would be possible to specify the that the sort of the dual geometric theory is of the form using just dual geometric logic. Of course, it is possible to talk about the dual geometric theory with a single sort, but the models of that will be compact Hausdorff locales without restriction and the classifying localic category won't be equivalent to the one classifying a single object for a geometric theory.
Also, a rather surprising aspect of the theory is that while the groupoids that represent geometric theories are open, the groupoids that representing geometric theories are not necessarily proper. This is small breakdown of the duality. And in the dual case, we'd need a dual version of étale completeness, since étale completeness is tied to toposes specifically.
But you are right that if you consider proper étale-complete groupoids (in the usual sense), they ought to give some kind of subcategory of the bicategory of toposes, but that's not what we are doing with when we work with classifying groupoids for dual geometric logic.
Fernando Yamauti (Jun 10 2026 at 05:09):Sorry the for the extremely late reply. Past few weeks have been very hectic. I'm still, however, interested in understanding this thing. I will be more verbose to try avoiding trivial miscommunication issues.
I will say cotopos and cogeometric instead of adding the prefix "dual" everywhere.
Graham Manuell said:
When you say you view a set as a theory, do you mean the theory of a single sort, or some particular fixed ? If it's the latter the theory is classified by the terminal localic category and that would indeed also classify a dual geometric theory, but perhaps not in a very interesting way. If it's the former, I don't think it would be possible to specify the that the sort of the dual geometric theory is of the form using just dual geometric logic. Of course, it is possible to talk about the dual geometric theory with a single sort, but the models of that will be compact Hausdorff locales without restriction and the classifying localic category won't be equivalent to the one classifying a single object for a geometric theory.
The base topos will always be . What I'd meant was considering some set (in ) as a discrete locale with frame . This frame is the synctactic category of a geometric theory. Its opposite category is the synctactic category of a cogeometric theory. What's the geometric realisation in of the proper separated groupoid classifying such cogeometric theory?
Would it even be localic again? If yes, would it be compact Hausdorff?
Graham Manuell said:
Also, a rather surprising aspect of the theory is that while the groupoids that represent geometric theories are open, the groupoids that representing geometric theories are not necessarily proper. This is small breakdown of the duality. And in the dual case, we'd need a dual version of étale completeness, since étale completeness is tied to toposes specifically.
Let me ask two questions that I think will clarify if there's any hope of representing models as topoi.
1) Can the cogeometric theory for one sort cannot be represented by a topos?
2) Is your functor (and the variant valued in groupoids) invariant under Morita equivalences?
Both cannot be true, otherwise some topos would classify the theory of compacta and, from what I understood in the discussion here, the theory of compacta should be cogeometric (?). But if 2 is definitely no, then I guess what I wanted doesn't even make sense: to find, for each cogeometric theory , a geometric morphism such that any model of over a localic groupoid could be represented by pulling back along some map .
Graham Manuell (Jun 10 2026 at 09:54):I haven't checked the details, but I think locales should embed into cotoposes just like they embed into toposes and the classifying groupoid for the cogeometric theory should be equivalent to the locale viewed as a discrete category. It will not be a compact Hausdorff locale.
For your next questions:
David Michael Roberts (Jun 10 2026 at 10:09):I would assume by 2. Fernando means that if H and H' are weakly equivalent localic categories, then T-Mod^PS sends the (span of) weak equivalence/s between them to an equivalence of categories (or span thereof).
David Michael Roberts (Jun 10 2026 at 10:10):So that really T-Mod^PS descends to a 2-functor on the bicategory of localic stacks (of categories).
Graham Manuell (Jun 10 2026 at 10:35):Yes, in that case, we prove this in our paper.
Fernando Yamauti (Jun 11 2026 at 00:03):David Michael Roberts said:
I would assume by 2. Fernando means that if H and H' are weakly equivalent localic categories, then T-Mod^PS sends the (span of) weak equivalence/s between them to an equivalence of categories (or span thereof).
Well. By Morita equivalence, I meant really the one with bibundles . I was interested in knowing whether the functor would descend to something having domain the cat of topoi. But I had forgotten at the time a bibundle need to be open in order to send étale things to étale things. The dual argument seems to show tensoring by proper bibundles preserve proper separated stuff.
Fernando Yamauti (Jun 11 2026 at 00:04):Both topoi and cotopoi embed into geometric localic stacks. Étale completion gives reflection for the former and I guess there's a reflection back to cotopoi if locales really embed into cotopoi (just take the geometric realisation inside cotopoi).
Anyways, this whole business make me feel optimistic in hoping geometric stacks on locales could perhaps classify any sketch (arbitrary limits and colimits).
David Michael Roberts (Jun 11 2026 at 03:12):Fernando Yamauti said:
Well. By Morita equivalence, I meant really the one with bibundles .
They are the same, at least for internal groupoids. What's your definition of a bibundle for a pair of internal categories?
David Michael Roberts (Jun 11 2026 at 03:18):@Fernando Yamauti I mean the equivalence relation on internal groupoids called "Morita equivalence" is the same if defined using bibundles or using weak equivalence functors. I have views on what a bibundle is for internal categories, but it amounts to the thing that makes this statement also true. But I would like to know your version so I don't make a wrong claim
Fernando Yamauti (Jun 11 2026 at 10:11):David Michael Roberts said:
They are the same, at least for internal groupoids. What's your definition of a bibundle for a pair of internal categories?
Yes. They are the same. My confusion came from, as I've just said, the fact non-open bibundles might not preserve étale things, so the map of stacks won't be come necessarily from a geom morphism.
The def of a bibundle I have in mind can be defined for any two simplicial objects (no Kan or quasi-Kan condition required), say an -bibundle. I don't want to write a large diagram here, but it should be an augmented bisimplicial object such that each column is a colimiting cone and the object on is equivalent to the stack coming from . That all induces a map from the geom real of to the geom real of . So it's as you said: bibundles are the same thing as maps between the stacks and Morita equ are isos of stacks.