I am pleased to announce the following paper, written in collaboration with Axel Osmond (@axel-osmond):

This work introduces a new topos-theoretic construction, that of the over-topos at a model of a geometric theory in a Grothendieck topos, and investigates both its logical and geometric aspects. Here is the abstract:

*With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the associated sheaf topos, which we call the over-topos at the given model, admits a canonical totally connected morphism to the given base topos and satisfies a universal property generalizing that of the colocalization of a topos at a point. We first treat the case of the base topos of sets, where global elements are sufficient to describe our site of definition; in this context, we also introduce a geometric theory classified by the over-topos, whose models can be identified with the model homomorphisms towards the (internalizations of the) model. Then we formulate and prove the general statement over an arbitrary topos, which involves the stack of generalized elements of the model. Lastly, we investigate the geometric and 2-categorical aspects of the over-topos construction, exhibiting it as a bilimit in the bicategory of Grothendieck toposes.*

The construction of the over-topos can also be dualized, providing a wide generalization of Grothendieck-Verdier's notion of localization of a topos at a point.

This paper combines a variety fo techniques and touches several distinct themes, introducing new ideas or constructions in connection with each of them :

- Syntactic categories and classifying toposes
- Totally connected toposes and colocalizations
- Grothendieck topologies on fibrations
- Computation of Grothendieck topologies generated by different families of sieves
- Geometric morphisms and stacks associated with them
- Giraud's construction of the classifying topos of a stack
- 2-categorical constructions in the bicategory of Grothendieck toposes

In my forthcoming joint work with Riccardo Zanfa (@rzanfa) we shall introduce a whole new framework for developing relative topos theroy via stacks, thereby providing a broad context where the results of this paper can be understood. Stay tuned! ;-)

]]>I'm M djo k'naan, and I recently started to study category theory and topos as a autodidact.

I'm interested in their applications in the logic field.

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As I started recently category theory and topos theory, I have some questions about unification theory and bridges, I precise that i'm an autodidact so i'm very far from being an expert.

My first question is, let's say that I have theory T1 , and somehow I manage to extend it to a more general one T2, by more general i mean that each sentences of T1 can be completely translated into the language of the T2 (ie. definitions, theorems, etc.), we can also imagine that the T2 contains more general quantifiers such as the sentences in the subtheory are just a particular case of the more general one.

Obviously in the super-theory you will have more theorems, more definitions, etc. What you don't want to do is to start over in your super-theory, you want some kind of "map" whereby you can 'canonically' convert definitions into super-definitions such as if you have a theorem in your sub-theory this theorem still holds in the extended one in a "general" way.

You also want to be able to see if you can translate those definitions to exotic definitions so as to have new theorems, once you have eliminated all those cases you can really focus on what is really specific to your super-theory.

So my question is, is there an easy way to do that, an object or a concept of the bridge or unification theory?

Here is a pseudo-example, for T1 take the set theory, for T2 take the fuzzy set theory, if you see a set E as a constant function whose value is always 1 (fuzzy set) ,each sentence regarding that set in the set theory can match a sentence in the fuzzy set theory, and what you want is to have a mapping between definitions of set theory to definitions in the fuzzy set theory such as the theorems in the set theory still hold through the mapping in the fuzzy set theory.

You don't want to try to find by yourself all the theorems manually. You want a natural way to go from T1 to T2.

My second question is, is there a more general structure than the topos one, in the sense that one can have some category structure, without a topos structure but still having some logic associated to it; i'm not really at ease with the terminal objects used to describe the truth map (correct me if i'm wrong), in general i'm not a ease with the way the terminal object is used, i'm more at ease with the notion of initial object.

Thanks for bearing with me.

Let J be a family of sieves on a category \mathcal C . We use a slightly different formulation of Grothendieck topology here. We say a sieve S on an object x **covers** f : y \to x iff f^* S \in J . Then for J to be a Grothendieck topology it must satisfy the following three conditions:

- for any f \in S , S covers f ;
- if S covers f then for any g with codomain identical to domain of f , S covers f \circ g ;
- if S covers f , and R covers any morphisms in S , then R covers f .

It is easy to see such a formulation is identical to the usual conditions for J to be a Grothendieck topology (e.g. see *Sheaves in Geometry and Logic*, p. 110).

Interesting things emerge if we now change our notation slightly: we use S \vdash_J f to mean S covers f (in J ). Then the above conditions look as follows:

- (Reflexivity) \forall f\in S.S\vdash_J f ;
- (Weakening) S \vdash_J f \Rightarrow S \vdash_J f \circ g ;
- (Transitivity) S \vdash_J f\ \&\ \forall s\in S. R \vdash_J s \Rightarrow R \vdash_J f .

1., 2., 3. are roughly reflexivity rule, weakening rule and transitivity rule of an abstract inference relation in logic, respectively.

If we view the cover relation (-)\vdash_J(-) as a genuine inference relation, it leads to the following several questions:

- In this interpretation, not objects but morphisms are treated as "syntactical object". In particular, it is not the same as how we understand the syntactic site of a theory; there the objects are of type \{\vec x.\phi\} , which are formulas equipped with a context, and morphisms are provably functional formulas. Can we understand such a setting here in a logically meaningful way?
- Related to the above question, we seem to have a composition operation on these syntactical object, and in the above formulation I suggest this composition should be understood as some kind of "weakening". Is this explanation morally correct, or perhaps a better way to ask is that is there a reasonable reading to make sense of this composition-as-weakening idea?
- If we've developed an intuitive understanding of this, how it would influence on our understanding of an arbitrary site?

Sorry if this question seems a bit long and perhaps does not make sense at all, but it's no harm to ask it here to see if anyone have any sparks on this.

]]>I have a very broad interest in mathematics, logic, physics, philosophy, etc. But in a word, I'm always trying to find more conceptual ways we can think about these subjects, resulting in deepening our understanding.

I first come across topos in Prof. Caramello and Prof. Joyal's talks on categorical logic and topos in the conference Topos à I'IHÉS, which I immediately fell in love with this subject! After that, I've been following *Sheaves in Geometry and Logic* and some other references by myself. I'm still relatively new to this subject, and I am looking forward to studying from all of you here in this forum.

The video of my recent talk "Toposes as unifying spaces: historical aspects and prospects" at the *Workshop in honor of Alexander Grothendieck’s legacy* at the Universidad Nacional de Colombia is now available on YouTube:

This talk (whose slides can be downloaded here) discusses how the unifying concept of topos was introduced and conceived by Grothendieck, as well as the future prospects provided by the theory of toposes as ‘bridges’.

It is a shorter (and partially different) version of the lecture on the same subject I gave in 2018 at the ENS for the series *Lectures Grothendieckiennes* organized by Frédéric Jaëck (whose slides are available here):

In fact, I have recently finished writing (in French) my contribution for the Proceedings volume of that lecture series, which will be published by Spartacus: this paper can be downloaded here. An English translation is also in preparation.

]]>- This question quotes Laurent Lafforgue (without naming him for some reason) comparing Caramello's program to the Langland's program. Their specific question is: "I wonder if there were any papers confirming the overall sentiment of , i.e. papers where Caramello's theory is applied to Langlands. Were there at least some long-standing problems that historically were not believed to have any serious connection with logic or topoi, and were solved using Caramello's techniques?"

Ignoring their dubious claim that, "The entire mathematical world has produced few, if any, correspondences which unify lots of non-trivial mathematics as elegantly as the Langlands program does," it would be nice to know what the current state of progress of linking these two programs is. - This question, by far the highest profile, is asking for insight into the quote "one can generate a huge number of new results in any mathematical field without any creative effort," and whether it is an exaggeration.

The answers are divided based on their interpretation of the quote, but Caramello's book (published since the question was asked back in 2010) certainly contains numerous illustrations of the kinds of result that can be extracted from the topos-theoretic bridge framework. As someone who works with Grothendieck toposes myself under Caramello's supervision, I know that "without any creative effort" doesn't mean "without any effort": extracting the results in a form that is comprehensible takes significant work, but the reason that effort is not 'creative' is because the toposes act as a guide, avoiding the need for one to rely on great leaps of inspiration to progress. I hope Prof Caramello will have more to say on the subject. - This question asks how Olivia's program relates to Jacob Lurie's work on higher topos theory and higher categories, specifically asking, "Has there been any work connecting these schools of thought?"

Most of the results in Lurie's*Higher Topos Theory*are extensions to oo-toposes of results for Grothendieck (1-)toposes, so there is some immediate relation between the background on which the programs rest, but it would be interesting to get some insight into how Prof Caramello views the relation between their research directions.

I may eventually report any interesting answers and discussion provided here as answers to the original questions.

]]>Thanks again to John Alexander Cruz Morales for organizing this course, and for making the recordings available with the kind help of Javier Gutiérrez.

]]>Thanks again to my excellent interviewer - mathematician Francesco Genovese - and to all the staff of Meet Science for the organization!

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