 So the next talk is by Axel Osmond and he will talk about the over-topos at a model and this is joint work with Olivia Caramello. Okay, Axel. Thank you very much. Thank you very much for being there for this talk where I have the pleasure to present this joint work with Olivia Caramello. So just as we go from screen. So let me first introduce my topic. So if you have a topological space, then you have a specialization order between points. And at a given point, you can look at the upset or the downside of this point or the specialization order. The upset contains all points that lie above a given point X and the downside contains the point that are below a given point. And of course, this construction generalizes to arbitrary subset of your topological space. And in this talk, we are interested in the toposteoretic analog of the downside. So recall that a groten diktopos has a category of points which are the geometric morphism from set into this topos. And also you can look at the under category and the over category at a given point. And those are the analog of the upset and the downside respectively. And of course, this generalized to arbitrary geometric morphisms where you can look at the under category of the home category of geometric morphism between two topos at a fixed geometric morphism. In this talk, we are interested at the over category at a given point. This led us to the notion of totally connected topos. So a geometric morphism is said to be totally connected if the inverse image part has a Cartesian left adjoint that is which preserve finite limit. And then the pair of adjoint given by this left adjoint and the inverse image is a terminal object in the category of section of your geometric morphism. And in particular, as it is quite common to classify the geometric properties of topoy by the property of their terminal geometric morphism to set the function of global section. We say that a groten diktopos is totally connected if its terminal geometric morphism is totally connected as a geometric morphism. And this means that your topos has a terminal point. In particular, if it is a classifying topos of a geometric theory, then this means that this geometric theory has a terminal model in set and in fact in any arbitrary topos. So just I would like first to recall something which is already known but which will be useful to understand what we are doing to do, which is we can construct canonically a totally connected geometric morphism at a given arbitrary geometric morphism. To do so recall that the category of groten diktopoi has power with two, which is a universal two cell classifying natural transformation between geometric morphism into a given topos. And the codomen part of this universal two cell, which is called the universal codomen is always totally connected. This is in some sense a generic totally connected morphism. And actually using the fact that totally connected geometric morphisms are stable under pullbacks, we can construct, we can use this generic morphism to construct totally free totally connected geometric morphism at another geometric morphism. To do so, we just have to take the two pullback of the universal codomen along a point if we want to compute the over topos at a point or at an arbitrary geometric morphism. In both cases, the universal property of this construction is that its point will be exactly the category of the over category at the corresponding point. And for an arbitrary geometric morphism, the over category at this geometric morphism. In particular, in the case where this geometric morphism is the name of a model of a geometric theory, that is if our topos is the classifying topos of a geometric theory, then the universal property of the over topos is that it classifies a homomorphism of T model into the inverse image of this model along arbitrary geometric morphisms. So the purpose of this talk is to provide a site description of the construction of the over topos because as we as it is done this way, it is an abstract universal construction which does not retain any information about the site, and in particular the syntactic property of encoded in the model structure and the structure of the classifying topos of a theory. So we want to provide a canonical site for this construction and we will have to do in two steps. First, we will have to process for the case of a model in set that is for a point of a topos and then we can do this for the general case for an arbitrary geometric morphism, but it will use a bit more involved the technologies about starting and next category and so on. So, for the set-valuated case, just recall before that for geometric theory, you have its syntactic site where objects are formulas in context, morphisms are probability equivalence classes of probably functional formula between those formulas in context, and the topology, the syntactic topology on this category is generated by the cover that are encoding the disjunctive sequence of your theory. Now, a model in set is a flat function for the syntactic topology into set sending a formula in context to its interpretation and we can look at the category of element of global element of this function whose objects are a pair made of a formula in context and the global element of this formula in context and morphisms between two global elements are just an underlying morphism between the formulas in context that may commute the corresponding global elements. So, in set we are going to use an interesting properties that make the construction easier, which is that one, the one object set is generating in set that is any set is a co-product of one indexed by its global element. And in particular, this will be true for the interpretation of formulas in context by a model M, and this will also be preserved under inverse image. That is, if you look at the inverse image of a model in set, then the decomposition is still true after applying inverse image. So, in particular, so moreover, if you consider a model that is a flat functor for the syntactic topology, then covering family for the syntactic topology are sent to jointly epimorphic family, which means that if you take a global element of this interpretation, then there is at least one member of the cover such that this global element has an antecedent along this member of the cover. This leads us to consider the following topology on the category of global element, which consists of families of all possible antecedent of a global element along a cover, an interpretation in M of a syntactic cover. And this generate a topology will recall the antecedent topology on the category of global element. And our main result is that the category of global element and the syntactic, the antecedent topology on this category of global element is a presentation for the overtopos. So, to prove this, we have to show that it possesses the universal property of the overtopos, which is that we must prove that any geometric morphism for an arbitrary topos into this shift opos is the name of a homomorphism of T model into the inverse image of M. The inverse image along the global section function of your arbitrary topos. So to do so, suppose we have a homomorphism to the inverse image of your model. This is the same thing as a natural transformation between a flat for the syntactic topology into where the code of man of this transformation is the inverse image of your model. This natural transformation has components and text by object of the syntactic site. And now, if you take an element of an interpretation in M of a formula in context, then you can look at its inverse image and you can take the fiber of its inverse image along the component of your natural transformation at the corresponding formula in context. This return you an object, a fiber object at each element of M. And you can do this in a functorial way thanks to the naturality of your natural transformation and using pullback property. And this return you a functor from the category of element of M into G. And then a proving flatness for the antecedent topology is just is something which is mostly an application of stability of co-products and epimorphisms in gothic toposes. On the other hand, if you have a geometric morphism into the shift opos over this site. This is the same thing as a flat functor from the category of element for the antecedent topology. So in particular, this return you a familiar object and text by global element of interpretation and for a given interpretation and choice of an element. You can first compose the object corresponding to a given element and take its terminal map into one and compose it canonically with the name of the inverse image of this element. And this return you a map into the interpretation in the inverse image of M. And then you can glue all those objects. You can do all the objects that correspond to global element of a fixed sort. And this return you a map corresponding to a given formula in context. And this is a way to construct. Actually, a model of T in G, whose interpretation of the formula in context, a given formula in context is given by the corresponding co-product reconstructed above. And the homomorphism of T model into the inverse image of M just is provided as you can see there by the universal property of the co-product at each formula in context. So the reversibility of this process is mostly actually an application of extensiveness and stability in gothendic topos. So I won't be too long on it. I prefer to turn to the logical aspect of this construction. So it isn't as we know that this new overtopos classifies homomorphisms into fixed model to inverse image. We would like to know what is the geometric theory, which is classified by the overtopos. So to do so, we have to define for this over theory, we have to define over language in which this new theory will live and an auxiliary language in which we will test with what kind of sequence we want to have in this over theory. So the over language will consist of a new sort for each global element of your model and a new function symbol for each morphism in your category of global element of this model. And on the other hand, the auxiliary language will be an extension on the ambient language in which you will add a new constant symbol for each global element of your model. And then your model is canonically equipped with a new structure, an extended structure for this extension of the language where you interpret the constant corresponding to an element into by the corresponding element. And you have also a canonical interpretation from the over language to the auxiliary language, which replace in a formula, all instance of free variable of the sort corresponding to an element with the corresponding constant in the auxiliary language for this element. And then the new theories over theory associated to a model will have as axioms all the geometric sequence in the over language was interpretation in the auxiliary language are valid in the extended structure. Your model have canonically for the auxiliary language. And, actually, we can also prove that the over topos as we constructed it is the classifier is the classifying topos for this over theory. So now I turn to the general construction in the general case. So in general case, the complication is mostly that one is not anymore a generator in an arbitrary growth in the topos. This means that we cannot just restrict to global elements of a model, we have to consider arbitrary generalized element. So and they have morphism between them they have more complicated structure. But at least we can in the following we will also restrict to the ones that are indexed by basic element. That is, if you have if you have a presentation site for your topos, you will be able to at least to restrict to generalized element that are indexed by an object coming for the form the site of presentation of your topos. But in any case, the complication will be that not only the formula of the interpretation may be able to vary, but we must we must also take in account variability of the indexing object in the generalized element. So to address this problem properly, I have first to give some word about the notion of stocks and the construction of zero topology. So as Ricardo told you in the last last talk, there is a notion of zero topology for an index category. And in particular, if you have a Cartesian stack on a growth index topos, then you can consider the growth index construction associated to this Cartesian stack and equip the corresponding fiber category with the zero topology, which is the smallest topology which is basically making the corresponding fibration a common fism of site. And in particular, if you have a small site of presentation for you based to post, you can compose the your Cartesian stack with the unit I'm bidding. This return you a fibration over the site of your topos, and you can restrict the geo topology over for the site. And actually you can give the concrete description of covering family for the geo topology as the Cartesian lift of cover for the g topology of your basis. And actually, you have those defined the same topos. This is the same thing if you take as a topos with the geo topology or its restriction for the unit for along the unit are bidding off site of presentation. And what is important is that you get a geometric morphism induced by the comorphism you have, thanks to geo topology. So now, something which was marked by Olivia recently is that if you have a geometric morphism is then you can construct canonically to indexed category associated to it. One indexed by the domain topos, which send an object to the category of indexed generalized element of the universe image part of your geometric morphism. And on the other hand, for the codoment opos, you have an index category on the codoment opos which send an object to the category of generalized element of its image. And so we and what happened is that both those indexed category and use the same construction and use the same category and there are the growth and deconstruction, which is the comma of the domain topos and the inverse image and which is equipped with two fibrations on the left over the domain category and on the right over the codoment category. And now you can equip this comma category with the smallest topology, which makes simultaneously those two fibrations comorphism of sites. And this topology, which we call the lifted topology is jointly generated by the two geo topologies associated to those two fibrations. Important to know something we would like to have is a way to restrict us to small data to site of presentation for the domain codoment opos because in practice we want small site because the category I defined there is a large site. So to construct a small site, if we have a site of presentation for the codoment opos, then we can look at the comma where we are on the on the right, we consider the inverse image of basic object that is the compose of the inverse image with the unit and bidding. And we can restrict the lifted topology to this category and we still have a fibrations over the site for the codoment opos and this restriction allow you to have a comorphism of site. On the left, you can also restrict, if you have a presentation site for your domain topos, you can also restrict along the unit and bidding. So on the on the left, and again, you have an innocent way to restrict your topology, the lifted topology on this coma category. And the interest is that the new coma category is small and you have a canonical topology on it derived from the lifted topology. So a problem with this construction is that the new construction is a bit less pure because it's not any breaks the properties that we have a fibrations because we will see that inverse image of basic generalized elements are not necessarily basic, but this is not very important for our proposal. So just a word on the concrete description of the lifted topology, the restriction of the lifted topology in this context, it will consist of restriction, it will consist of families where you take first a generalized element of the inverse image of the basic element. Then you look at the inverse image of a cover of the object you look at the inverse image of, and then you take a cover of each, you take a cover for each fibers of this generalized element. So now I would like to apply this to a Cartesian to the case where you consider a model of a geometric theory in an arbitrary topos. So, if you have a T model for a T model into a groten diktopos with a standard site of presentation. This is the same thing as a flat flat for the syntactic topology into your into this topos and this define Cartesian Cartesian stack, sending sending an object of your base site to an object seek into the category of C indexed generalized element of the the functor coding for your model. And now again, you can define the the the category of generalized element of your model, which is the coma category between the unedited embedding and the geodesic. And again, you can define the the the category of generalized element of your model, which is the coma category between the unedited embedding and the geodesic, the the flat functor coding for your model. And again, you can equip it with the restriction of the lifted topology we described above. And in this case, it has the following presentation. You take a generalized element of an interpretation of formula in context. You look at all at at the interpretation in M of a syntactic cover of the corresponding the underlying formula in context. And then you ask for covers of each fibers of this generalized element, along the numbers of your syntactic cover. In some sense, you ask for having a fiber wise covering of the fiber. And this, this is a generalization of the antecedent topology we constructed in the set evaluated case. And this return you again. A com of ism of site forms the coma category into the base is the site for the topos in which you interpret your model. And this com of ism of site returns you a shift oppose and a canonical geometric morphism. And the result is that again, this category, this shift oppose over the coma category together with the antecedent topology is has the universal property of the over topos at the at the model and in particular the geometric morphism induced by the com of ism is the totally connected universal morphism at this model. So, again, a proving that the, this topos has a universal property of the over topos is quite similar to the set evaluated case. Now the principal difference is just that when you will, you will construct a model from, from a natural transformation. As in the set evaluated case, you will have to glue object index by diagram of generalized element rather than just the coproduct of discrete set of global element, but this is not a very important difference actually. And so there is no need to be too long about that. So I would prefer to finish now. So just perhaps a last remark on this construction, which is that at some point for practical reasons, we want to restrict us because we want site description, we want to restrict us to the restriction of the lifted topology, which are concrete but are a bit cumbersome and may seem a bit ad hoc. But actually it's again a situation in which if you drop the site, the site information and go in a more abstract level, a more invariant level of the growth index topos themselves, you have a more pure description in which the universal properties of your object are more evident. So now just to finish, I would give a few perspective for this work. First, which is that actually totally connected geometric morphism and topos are a dual notion to the notion of a local geometric morphism topos. And it is well known that for a geometric morphism point of a topos, you can look at the local topos over this geometric morphism, which is the growth index value localization. And it is obtained as this is a dual of the totally connected to the over topos. Actually, this is a dual construction of the over topos and it is obtained through a formula describing it as a cofiltrated to limit of etal geometric morphism and something we would like to know is if the over topos also has a similar description as a cofiltrated to limit but of what kind of morphism it would be a generalization of the fact that the closure of a point is an intersection of the closed neighborhood it has. Finally, I would just perhaps also there is a question to know what is the theories geometric theory which is classified by the over topos in the arbitrary case. Perhaps this is for a notion of relativized geometric theory, you don't know. And finally, it would be interesting to have some description of the functor, which is the analog of the functor externalizing a topos into etal geometric morphism over it, but for totally connected topos. So, thank you for your attention. Okay, thanks a lot, Axel for this very nice talk.