We seek to understand topoi as a certain class of stacks on the big site Sp of locales with étale covers (if you restrict to topoi with enough points, you can take sober topological spaces), that are of the form X\mapsto \text{Core(Geom(Sh(}X),\mathcal{E})) for a topos \mathcal{E} . Here Core denotes the maximal subgroupoid of a category.

The first insight (which was essentially mentioned by Prof. Caramello in her talk) is that by weak left Kan extending the functor \text{Sh}:Sp\to \text{Core}_{(2,1)} Topos along the Yoneda embedding y:Sp\to Stacks(Sp), we get a (2,1)-adjunction ( \text{Core}_{(2,1)} is the maximal sub-(2,1)-category):

Lan_y\text{Sh}:\text{Core}_{(2,1)} Topos\to Stacks(Sp)

and

y:Stacks(Sp)\to \text{Core}_{(2,1)} Topos

defined by

X\mapsto \text{Core(Geom(Sh(}X),\mathcal{E})) .

The idea now is to characterize topoi as the* quotient stacks* of localic groupoids: We claim a stack \mathcal{X} on Sp is in the essential image of y iff:

- The diagonal \Delta:\mathcal{X}\to \mathcal{X}\times \mathcal{X} is representable i.e. for any representable functor yX for a locale X and map f:yX\to \mathcal{X}\times \mathcal{X}, the pullback along \Delta is again representable.

This corresponds to the fact, that the diagonal morphism of a topos \mathcal{E} is a localic geometric morphism. Thus, the base change of the diagonal along any map \text{Sh}(X)\to \mathcal{E}\times \mathcal{E} will be a localic geometric morphism with target \text{Sh}(X) and hence a localic topos itself. - There exists a locale X and a map p:yX\to\mathcal{X} with the following property: For any locale Y and map f:yY\to\mathcal{X}, the base-change (which is representable by a locale Y' which follows from (i)) \bar{p}: yY'\to yY corresponds to an open surjection of locales Y'\to Y. This corresponds to the fact, that any topos admits an open surjective geometric morphism from a localic one.

From this data, we can reconstruct a localic groupoid and hence a topos.

This presentation would be more convenient than the localic groupoids themselves, because they tend to be rather big and complicated. For the same reason, algebraic stacks instead of "smooth groupoids in algebraic spaces" are used in algebraic geometry.

My question is now, whether this is a well-known characterization. And if not, whether this is actually correct and if somebody wants to help me with working out the details.

]]>The definition of solid Abelian group in the notes uses signed measures. If you use non-negative measures, do you usefully get a similar notion of solid Abelian monoid?

(I’m interested in understanding how far the notions for condensed sets transfer to point-free topological spaces. For them we have some decent notions of measure and integration, but you have to be much more careful about whether reals are Dedekind or 1-sided, and that makes subtraction delicate.

By the way, Ming Ng’s talk this afternoon includes suggestions that the scaling factor p from Ostrowski’s theorem in the Archimedean case is best considered as an upper real, ie with the topology of upper semicontinuity.)

Steve.

]]>Can you say more about the problem of parameters?

On the face of it, geometric logic deals with parameters well, if the parameters are points of a space (ie models of a geometric theory T).

If we then define a space parametrized by p:, ie a geometric theory T’(p), then that is an internal geometric theory/site in S. We can form a semidirect product T’’ = T x| T’, and then get a bundle -> such that the fibre over any point p is . (Generalized point; generalized fibre = pullback.) can be though of as Σ_{p:} , as long as you understand that Σ is not a coproduct of spaces.

Ming Ng’s talk tomorrow is about trying to do this when T is a theory of places (on Q), and T’(p) is the theory of points of the completion for place p. It’s pretty clear how to do that for absolute values p, so the interesting mathematics lies in factoring out equivalence to get places.

You mentioned an issue of declaring “constant” sorts, but that’s fairly painless if you exploit the ability to declare derived sorts in a geometric theory. It’s roughly what you see in the Elephant B4.2.7 account of geometric theories, and in “AUs and classifying toposes” I showed how to exploit it in a base-independent way by using the AU constructions to approximate colimits.

All that assumes, of course, that T’(p) is constructed geometrically from (the generic model of) T. The big challenge to geometric reasoning is whether it can deal with the topos applications where at present the sites we have are not defined geometrically.

Steve.

]]>Link: https://gather.town/i/b18pcnTY

If you don't know, gathertown is an online platform that caters to more informal conversations. You will be able to create an avatar that you can move, and as you do so interact with other participants via a webcam.

]]>I was late in registration. My name appears on the "list of participants", but I do not have a link to the sessions. Is it possible to receive a link at this point?

]]>I would like to have some follow up on the question I asked professor Lafforgue about deriving results which should be valid for models in any category with enough structure. I asked about whether it would be worth it (for a non-logician) to go into the syntactic level to prove for example the isomorphism theorems of group theory, since this is supposed to be doable and these results are obviously of interest in diverse categories (not necessarily toposes, say in topological spaces, manifolds, schemes), or whether this level is so messy that it would not be worth it so that it would still be easier to derive the results individually. Lafforgue seems to think that it is not worth it to go to this level, so that perhaps non-logicians should keep working only on the semantic level (this might be an interesting discussion on its own), but instead referred to the world of classifying toposes, which would suit better since this is an object in the world of semantics. However, i do not yet understand this so well so I ask some naive follow up questions for those of you who actually know the subject we are dealing with:

- Is it somehow possible to derive the results for models of the theory in categories with weak structure, having only finite products, by proving them in the world of the classifying topos? If not, is it possible at least if the category is a topos, so that we can for example get the same standard algebraic results when working with sets, sheaves or sets acted on by a group?

- Are there other ways of doing it with only one proof when we just work semantically? I am not interested in working axiomatically with the formalism of ''group objects in a category'' since this is again very messy.

I am sorry about bothering the conference with things which should be obvious, but the first part is supposed to be a school to learn the subject after all, hehe.

]]>

Now suppose we have a functor $\mathcal{C} \xrightarrow{F} \mathcal{D}$. The identity profunctor on $\mathcal{D}$ is given by the Hom-functor $\mathcal{D}(-, =)$. Now $F$ embeds contravariantly as a profunctor from $\mathcal{D} \to \mathcal{C}$, by a left action on this Hom, i.e. it maps $(c, d) \mapsto \mathcal{D}(F c, d)$, and covariantly as a profunctor $\mathcal{C} \to \mathcal{D}$ by a right action similarly. These are denoted $F_*$ and $F^*$ respectively, and also have an adjointness relationship $F^* \dashv F_*$ . These profunctors are particularly important, and are called the (co)representable profunctors - morally, every profunctor is the same as some functor precisely when it is representable.

Is there some relationship between these two mappings, or is the connection merely superficial?

F. Borceux, *Handbook of categorical algebra*. Cambridge ; New York: Cambridge University Press, 1994.

If you haven't met geometric theories before, it can be helpful to see a lot of examples. Prof Lafforgue explained how to express the theories of groups and equivalence relations as geometric theories. What's your favourite kind of mathematical object? Can you work out how to express the axioms of those objects as a geometric theory?

]]>Feel free to pose any questions or discuss about the courses and talks given at this event, and also to introduce yourself in the relevant section of the Forum.

Please note that the Forum has built-in latex editors for writing mathematical formulas; just write

for each latex expression you need to insert, and it will be automatically displayed (you can check the result by clicking on the Preview button).

You can also, for a better graphical result, just type as if you were writing a normal latex source (but beware that in that case the Preview button will not show the compiled result - still, you will be able to edit your message for 60 minutes after your original submission, in particular to correct any typos or rendering mistakes). Here is an example:

$\textup{Hom}_{ }(y_{\cal C}(c), F)\cong F(c).$

We look forward to hearing from you!

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