 So the next talk will be by Ricardo Zanfa and he will explain extending the topological pre-ship bandala junction to sites and toposes and this is joint work with Olivia Carmelo. Thank you. I hope everybody can see the slides. Yeah. So here I will be presenting to you a part of the joint project with and working on Olivia and I. And so basically what I will present to you today is this. So we have this known result in topology. When we have a topological space, X, we know that there is an adjunction between the pre-ships over the space X and the bundles over X. That is topological spaces over it. We wonder if we can do the same for an arbitrary small site and in fact we do. If there is a site, CJ, a small site, we can build an adjunction in the same way where instead of topological spaces, we'll have toposes of a certain kind over our base topos. And the properties of these adjunctions do reflect what happens in the topological case. So as I was telling you, this is an extract of my joint work with Professor Carmelo, which is Relative Topos Theorical Biostacks, which will be available in a few days. And in particular, I will recall briefly the topological pre-ship bundle adjunction. Those of you who are interested and want to see it again, you can find it, for instance, in Sheets in Geometry and Logic, though the result is quite ancient in topology, let's say. And I will also cite the local version of the same adjunction. This is somehow harder to find in literature. You can find it though, still in Sheets in Geometry and Logic, it's kind of implicit in some of the exercises. So I've written it here. So for those of you who haven't seen Olivia's lectures last week, I will recall how the topological pre-ship bundle adjunction works. So we know that we have this adjunction where X is a topological space, any topological space. The functor lambda is called the bundle of germs. And basically, it takes a pre-ship p. It computes this set. These are the stocks of germs of the pre-ship. We won't need to see the definition again. But basically, you take the germs of the elements of p in a point text, you glue all these sets together into this co-product. You have a canonical projection. And you can topologize this set so that this map becomes a continuous map. And this is called the bundle of germs of the pre-ship p. On the opposite direction, the right adjoint gamma is called the local section sphincter. It takes a bundle over X. So we have our continuous map p from it to X. And we call local sections the continuous maps from an open U inside X, such that they work as a section of our map p. So they are the local sections of p at the open U. This defines a pre-ship. And the properties of this adjunction are the following. First of all, we can calculate the fixed points of the adjunction. The fixed points in the pre-ships for our topological space are the ships for the topology. While the fixed points in the bundles are what are called the etal bundles over X, which are basically the local homeomorphisms with co-domain X. And they are the fixed points of the adjunction. So in particular, we get an equivalence at this level. The second important property of the pre-ship bundle adjunction is that when we start from a pre-ship, we compute its bundle of germs and then go back to the local sections, we end up getting the shiftification or the associated shift of our original pre-ship. So the composite gamma lambda is the shiftification factor. So you see the basic points in this result are that pre-ships and most importantly, ships over X can be thought as a category of spaces, because we are really seeing them as etal spaces over our base space X. And also that the shiftification factor gets a geometric interpretation in terms of local sections. And this is interesting because when you compute shiftification for a general grotendic topos, in the literature you almost invariably find just the one description which goes back to grotendic with a double plus construction, which is I mean it's a very nice construction, but it's very algebraic and very hard to use because when you compute that, you take the matching families for a pre-ship, you quotient them with respect to an equivalence relation and then you do the same process again. So it's kind of hard to manipulate it in a concrete case to do computations with it. While for instance, when you have topological spaces, you have this very nice description. So you know that shiftifying means taking the local sections of a particular bundle over your base topos. And at the beginning I told you that there is also a localic version. This basically happens because even though when you build the bundle of germs, you use the points in the space, you don't really need points. So you can do the same things point free and you get the pre-shift bundle of junction in the formalism of locales. So if L is a base local, you can consider the pre-shifts over L, its shifts, the locales over L and the et al, locales over L, et al map of locales is the natural generalization of the et al map of topological spaces, and you get all the same properties. So a junction, a joint equivalence at the level of shifts, and you can recover the shiftification. So it is natural to want. Ricardo, we don't see your video. Maybe you can switch it on. You don't see my video? Is it only my problem or someone else? I can see it. I see it myself. Okay, so maybe it's just me. Okay, sorry for the interruption. Okay. So we wonder if we can do the same for a generic site. So we have our topological space, and we want to substitute it with a site. So as the title of the slide suggests, we think of topos is a generalization of spaces. And so the generalization of continuous maps over a base topological space will generalize to geometric morphisms from any topos to our topos of shifts. This is the most unnatural generalization one can think of. There is, however, a size issue here. Because when we define the local sections of a topological space, we are considering home sets. If we want to do the same here, of course, we cannot speak of home sets, because the class of geometric morphisms between two toposes is in general not just a set. So we have to restrict that in a certain way. And we propose this restriction. We call a topos e over our base topos, small or relatively to the base. If for each x in the site, and in a moment I will tell you why we choose this, this category of geometric morphisms. So you see these are the geometric morphisms from this slice topos over the representable, given by x, to our topos e over the base topos, shifts Cj. We ask for this class of geometric morphisms to be a set after equivalence of geometric morphisms, of course. So why do we choose the elements in the site? Of course, we won't appreciate over the site. That's obvious. But the reasoning is in the topological case, the elements in the site are the opens. So in this case, we are really saying we are taking something from an open in the topological space to the space we are considering to calculate its local sections. So given the notion of topos small or relatively to the base, which mind is referred, I say topos, but the object of interest is actually the geometric morphism, because of course we may have the same topos with different geometric morphisms over the same days. And we call topos with this superscript s, topos over shift Cj, the full subcategory of the topos is over our base that are small, that are relatively small. And this will allow us to define local sections and have them being a set. So appreciate. In particular, we also want to generalize the notion of et al, topological space. And of course, we already have a notion here of et al topos. A topos over a base is said to be et al or equivalent to a local homomorphism, the terminologies are equivalent. If it is the topos is equivalent to one of these form. So it is equivalent of a slice of our base topos. And also this geometric morphism is the canonical geometric morphism to the base. So it's the one whose adjoins from the right adjoint to the essential image are the dependent product going this way, the pullback factor going this way and the dependent sum in this way. We denote the category of et al topos is over a base with et al over shifts. And in particular, all et al toposes are small relatively to the base. So we have this inclusion. And in a moment, I will tell you why. So this completes the picture because we have our topological space becomes a site bundles over the space becomes relatively becomes relatively small toposes et al bundles become et al topos. Now to define the adjunction, the local sections functor at this point is straightforward with the generalization of what we have in the topological case. Given an essentially small topos e over shifts Cj, we consider its image by a gamma to be disparate shift. So you see it's the local sections of e at the elements of C. And we know that these are sets because he is small relatively to the base topos. So which we chose e to be sure that this lens in set and not somewhere else. The definition of the bundle of germs, the left adjoint is requires a bit more of work, not much. But basically, we start with a pre shift P. We consider it's grotendic construction, which is a vibration over C, discrete vibration in this case. And since C has a topology J, those of you who have followed the latest lectures last week already know this, we can endow this category with the smallest topology, which we call the Jp, making this functor of comorphism of sites. We call this topology Girod's topology for the pre shift P. We call the site Giro side and obviously the topos Giro topos for P. Since this is a comorphism of sites with this topology, it induces co-variantly a geometric morphism. So we have always this geometric morphism, which is furthermore in this case it's also an etal geometric morphism, but in general we have it and it's always essential. So this is the bundle of germs functor. And to prove the adjunction at this point, it's you will see it's quite easy to prove it because it's a couple of lemmas and the rest is pretty straightforward. The first lemma shows that the Giro topos for a pre shift P is a conical co-limit in the category of toposes over the base. So basically we have our category of elements here. You see every object in here is mapped to an etal topos, which is going to be shifts over the representable index by that element. And so what happens? Why do we want it to be a co-limit? Because we want lambda to be the left adjoint of a home functor, of a contravariant home functor. So it's going to behave like a co-limit. And this is in fact true. You can prove this, it's a computation. So you see if you start from an arrow, geometric morphism from lambda P2E over our base topos, since this is a co-limit, having an arrow here is the same as having a cocoon. The cocoon has legs starting from toposes of this kind, this etal toposes 2E. Each of these legs is indexed by elements in P. So for instance, you will have a geometric morphism, which is indexed by an element A in P of X. So a cocoon of this form is the same thing as a natural transformation from P to this set of geometric morphisms, which is gamma. And so lambda is the left adjoint of gamma. So we have the adjunction at the level of pre-shifts. We want to see what happens when we restrict to shifts. And for that, we can use these two lemmas on etal toposes. The first one tells you that if you work over a base topos, arrows between etal toposes can be presented directly with arrows inside the base topos. So any geometric morphism here above E is presented by an arrow in the base topos. The second lemma tells you that the gerotopos of a pre-shift P, which we described in this way, is actually equivalent to an etal topos, which is the etal topos over the shiftification of P. And with these two lemmas, you can prove all the nice consequences about the adjunction, the pre-shift bundle adjunction. Because, see, if you start with a etal topos, shifts Cj over F, and you compute its gamma, its image, the local sections are defined in this way. But using the first line, you can reduce these two arrows from this representable to F. You go from the etal toposes over the base to arrows inside the topos. And this is just F by unit a lemma. This equivalence allows you to conclude that fixed points in small toposes over the base are in fact the etal toposes, the fixed points for our adjunction. In the opposite direction, if you start with a pre-shift P and you consider its shiftification, again, we do the same thing in reverse. So we start from the shiftification, we use unit a lemma, we move to etal toposes. Arrows from here to here are the same as arrows from this etal topos to this etal topos. But this is just gamma of lambda of P, because we said here that lambda of P is precisely this last topos. So this again, this allows you to conclude that fixed points of the adjunction are the J-shifts and also that the composite of gamma and lambda is the shiftification factor. So we end up with the let's say the pre-shift bundle adjunction for general sites, which you see is really the same thing as what happens in the topological case. Pre-shifts with relatively small toposes restricts one equivalence between shifts and etal toposes, no longer spaces, and you can recover the shiftification by composing the two adjoints. Can you say at this point how you overcome this iteration twice for the shiftification in the topos case? Yes, I will tell you later how, in a certain sense, in a couple of slides. So let's say what is interesting about this adjunction is that you can move in different directions. You can go up, let's say go up and see it as a truncation or reduction of a wider phenomenon, two categorical phenomenon. You can also go down and describe the shiftification using sites and their morphisms or comorphisms, or you can cross the bridge, let's say. So in some cases, for instance, I will show you in pre-order sites, you can actually forget the topos theoretic information. So you end up with a description of the shiftification that is really topos-less, let's say. So for the two categorical adjunctions, I won't tell you much, Oliva's already told you about this on Saturday, but basically there is a two categorical adjunction which leaves at the level of pseudo-functors from our category two ket and the toposes over the base topos. We have these two adjunctions. The right adjoint is still a local sections functor. The bundle of germs left adjoint behaves in the same way as in the discrete case, so it takes a syntax category and builds its zero topos, which is the topos over its associated vibration. And when you take this adjunction and restrict it to the one-dimensional case, you get, again, our discrete adjunction. So for those of you who are interested, of course, you will find plenty more content on this on the forthcoming paper. Going down, what is interesting is that this provides a geometric interpretation of the shiftification as local sections, because you see the value of the shiftification of p-atex is set up to equivalence of geometric morphisms from this topos, which is the one indexed by our object to this topos. And of course, speaking about classes of geometric morphisms is not that simplifying, let's say. But at this point, since you have geometric morphisms, you can go to sites. You can reduce to sites in various ways. So one way is the most standard one is using flat J-continuous functors. So a geometric morphism from here to here will correspond to a flat J-continuous functor from this site to this topos, or equivalently to a morphism of sites from this site to the canonical site of the topos. Another way of seeing this, which is harder to see at first, but basically you can present this topos with this site. It's the slice category of the x. This is the gerotopology for this discrete vibration. And all the functors in here can be presented by comorphisms of sites from this site, which presents the first topos, to this site, which presents the second topos. Mind, it's not that all comorphisms of sites will give you geometric morphisms in here. Of course, they have to satisfy some conditions to give you something which leaves over the base. The third point, which somehow is connected to the question, is the following. It's hard to give you really a technical idea of why this happens. But if you think of elements in the shiftification, an element in the shiftification of P attacks is something which may not exist in P, but is given by gluing local information in P. In the same way, it happens that a geometric morphism in here may not be presented at the level of sites by a comorphism, but it is locally presented by a comorphism of sites, and even better by morphisms of vibrations. So what happens is that you can find for each geometric morphism in here, you can find a j covering the family of x, and a family of morphisms of vibrations from c over yi to the vibration of P. And this is really telling you, I'm taking elements of P, I'm seeing them locally. These are all comorphisms of sites, and these, let's say, jointly induce a geometric morphism over here. So we are saying you may not have a presentation at the level of sites in terms of comorphisms from the site presenting this topos to the site presenting this topos, but you can restrict to a j covering family so that you have this presentation. And this somehow connects with the double plus construction because when you do the double plus, what you have is at first you have matching families up to, let's say, the local equality, and then you do a matching family of locally matching families. So what you get, sorry, a matching family of matching families, modular local equality. So what you get is the information of a, it's hard to explain, but you see it's, you get basically a local information for which when you restrict nicely enough, you get the same geometric morphism. So the point in the plus construction is that you have your local information, but the local information doesn't induce something already at the level of the site. So not something from CJ to INTP, but at the level of toposis it does. That's where the double plus kind of strikes in. I know it's a bit foggy. I will try to be maybe clear later. I'll finish now. And for the last point, as I was telling you, there are some cases in which you can move away from toposis altogether. You can forget the toposteoretic information as it happens for pre-order sites. So I recall you this result of locales. So basically locales over a base include into toposis over the shifts for that local. This inclusion goes into essentially small toposis, sorry, relatively small toposis. And also, I recall you reminded that for a pre-order site, this happens, the shifts over that pre-order site. By pre-order I mean category in which between two elements there is at most one error. It's a locale, a locale topos, and it is represented by the locales of ideals of j ideals of the category. And you can do the same for every pre-shift over your category. The topos you get is locale, and you can see it as presented by the ideals of the vibration. Using this, if you consider in particular pre-shift over a locale, shiftification is described in this way. And this is what we said before. This is gamma of lambda. But then, since we are in toposis over a locale topos, this is locale, and this is locale, we reduce to locale. So this is the same as maps of locales from this locale, which presents this sliced topos to this locale. And you can plug this in in the adjunction. And basically, you get this for pre-orders. You get that your adjunction at the topos theoretic level already exists at the level of pre-shifts over your pre-order and the locales over the ideals. And again, this induces the equivalences we expect. In particular, we get the notion of, there is a notion of et al net of pre-orders. So you get the pre-shifts over a pre-order are the same thing as et al pre-orders over a pre-order, and shifts are j et al shifts, generalizing a result in this article by Hemme and Iro. And with this, I think, yes, you used all my time. So I thank you for your attention.