 Hello everybody. So today, this is my pleasure to introduce again, Olivia Caramello, where we will give a second part of her lectures on morphism of opposites, which means relative opposites on the stacks. So Olivia, you can begin. Yes, so last time we introduced these two fundamental classes of morphisms between sites for the purpose of inducing morphisms between the associated shift opposites. We talked about morphisms and comorphisms of sites. So I just recall this result here that every morphism of sites induces a geometric morphism between the shift opposites going in the converse direction, in a contra variant way, while every comorphism of sites induces a geometric morphism going in the same direction. So we also pointed out a connection with the theory of conextensions, which can be used for describing the geometric morphisms induced by such functions between sites. So we recalled this proposition, which in fact shows already some kind of duality existence between these two kinds of functions between sites. So we shall start by precisely discussing the relationship between morphisms and comorphisms of sites. We shall see, in fact, that in fact they provide just two different ways of understanding morphisms of opposites, one of algebraic nature, which is what morphisms of sites provide, the other one of more geometric vibrational nature provided by comorphisms of sites. So as in mathematics you have a lot of algebra geometry dualities, also in this setting we shall see that in fact morphisms and comorphisms of sites are in a sort of duality relation between them. So the way we shall unify these two notions will be an implementation of the bridge technique which we reviewed last time. More precisely, we shall interpret these notions of morphisms and comorphisms as two different ways of describing the same invariant, mainly the notion of morphism of opposites, which correspond one to the other under a bridge. So in fact the idea is if we want to turn any morphism of sites into a comorphism and conversely in such a way that their invariant remains fixed, namely that they present up to equivalent to the same morphism of opposites, we necessarily have to change sites, we have to switch to different presentations. And in fact it turns out to work very well, this idea, because in fact it is possible to introduce constructions which precisely do this job. More precisely we shall construct new sites such that new more equivalent sites such that thanks to these new sites we can attach to a given morphism of sites, a comorphism which in a sense is adjoined to it and conversely from comorphisms to morphisms. But this adjunction relation as we shall see will not take place concretely. So they will not be adjoined in a concrete sense, in the usual sense of the world, but they will be toposporetically adjoined in the sense that the comma categories associated with them will not even be equivalent as categories, but there will be opposites naturally associated with them which will be equivalent. And this bridge will be precisely what in some sense embodies the relationship between a morphism and the associated comorphism and conversely. And then starting from this bridge one can also derive a dual adjunction between a category of morphisms from a given site and the category of more of comorphisms towards this site. Okay, so how does this technically work? So suppose we start with a morphism, we want to get a comorphism out of that. So we consider the comma category whose objects are the triplets consisting of an object of the target category, an object of the source category, and a morphism in the target category from one to the image of the object under, of the other object under the front. Of course the arrows in this comma category are defined in the obvious way. Then we notice that we have a canonical functor going from the source category string to this comma category, which just does the obvious thing. And then of course we have two canonical projection functors, one to C and one to D. By using one of these projection functors, the one to D, we can actually induce a growth endic topology on the comma category from the growth endic topology K that we have on D. So we denote this by K tilde. So now here is the result. So we have an adjunction between the canonical projection functor to C and IF. One can show that IF is a morphism of sites, which implies that its left adjoint is necessarily a comorphism of sites. And so we indeed have found a comorphism of sites that we can attach to our morphism. And this does require the job because in fact, the following triangle of geometric morphisms come up to isomorphism. You see here what we have is that the canonical projection functor to D is both a morphism and the comorphism of sites. And such functors actually induce equivalences at the top of the level. So you see we indeed have a bridge and we have two more equivalent sites. And under this bridge, you see that the morphism induced by F as a morphism of sites corresponds precisely to the morphism induced by the associated comorphism of sites, which is given by the canonical projection to C. So this is the way to go from morphisms to comorphisms of sites. It is particularly interesting to apply this to an arbitrary geometric morphism, regarded as a morphism of sites by taking the inverse image. Because of course, the inverse image functor of any geometric morphism can be seen as a morphism between the canonical sites. Okay, so if we do that, what we get is that the canonical projection functor from this category to in is a comorphism of sites which induces the given geometric morphism. So this will be relevant in connection with relative toposphere as we shall see later. Okay, so now just a brief explanation of what happens in the other direction. If we start instead from a comorphism of sites, we can induce a morphism of sites in this way. So basically in this case, we replace the site decay with another Morita equivalent site, which is the site of pre-sheets on D, with the extension of the groten-dictopology k on D to this pre-sheet category. It's immediately seen that the yonida embedding actually is both a morphism and the comorphism of sites inducing an equivalence between these two toposes. So we indeed have a Morita equivalent between these two sites. And under this Morita equivalence, the morphism induced by f as a comorphism of sites correspond to the morphism induced by m of f, the corresponding morphism. So we have a similar picture to what we had in the previous slide. Okay, so as I mentioned, what is going on here is actually a very general form of adjunction between morphisms and comorphisms as expressed by these equivalences of properties. You see, one way of expressing the existence of an adjunction between two factors, just in the ordinary sense, is to consider the comma categories and to say that they should be equivalent over the product of the two categories. So here we have something less rigid than that. So the two comma categories are not even equivalent abstractly, but natural toposes or sheaves on them become equivalent. So in fact, this is a quite compelling illustration of the fact that, I mean, the kind of unification that bridges can achieve does not boil down to a sort of concrete unification, which you can easily contemplate by remaining at the level of size, you see. So you really have to go to this, to the level of toposes to better see the connections existing even between very concrete. Okay, so as I mentioned, in fact, from these bridges, one can actually derive an adjunction, a dual adjunction, between a category of morphisms of size from a given size and a category of comorphisms of size to work with. So the details of this you can find in the monograph draft, which I put on archive last year. Okay, so now I would like to turn to another important class of functions between sites, which play a very important role in the context of our vibrational study of relativity. This is the class of continuous functions. So in fact, this notion of the continuous function between sites was already introduced by Grotendijk, but has not really been used very much so far. So I shall just briefly recap that. So there are different perspectives one can have on this notion. So one way of defining a continuous function from a category to a Grotendijk topos is to consider the home function associated to that. You remember last time we discussed this general home tensor adjunction induced by an arbitrary function. So you may require the home function to take values in the subcategory rather than the whole specific category. So you may define a function to be J-continuous if this happens. And then you can generalize this to an arbitrary site of definition for the topos E. And so you get the notion of J-K-continuous function between sites. So this is one way of apprehending this notion, but there is another alternative characterization, which is the way the notion was originally introduced by Grotendijk. And it is clarified by this proposition. So the proposition says that a function is J-K-continuous if and only if composing with the opposite of it at the level of the pre-shift topos restricts to the level of shift topos. Okay, just a remark because we have already talked about J-continuous flat function. So you might wonder if this is the same notion or not. Yes, it is for flat funtors. So remember that we said that a flat funtor was J-continuous if it sent J-covering families to epimorphic families. Okay, so if a funtor is flat, then being J-continuous in the sense of Grotendijk is perfectly equivalent to being J-continuous in the sense of, say, McLean and more like of sending J-covering families to epimorphic families. In general, the condition of being continuous is stronger if you don't have flatness. Okay, so under the flatness assumption, they are equivalent. But in general, the notion is more subtle. One can find counter examples. So in fact, let's say a few more words about the notion of continuity. So in fact, we can interpret a continuity as a sort of co-finality condition because in fact, by using the first characterization of continuity, you can see that a funtor is continuous if and only if you have such a co-limit representation of the value of the funtor at a given object, C, as a co-limit of the values of the funtor and the domains of arrows in the covering sieve. So this is a diagram indexed by the category of elements of the sieve. And in fact, this reformulation in fact invites us to introduce a relative notion of co-final funtory in order to express continuity as a form of co-finality. So how does that work exactly? So we introduce a notion of relative co-finality for a growth in victimology on the target category like this. So it is just a very natural generalization of the usual notion of co-final funtory. In fact, it specializes to that if you take the trivial topology. And indeed, it corresponds precisely to the condition that when we consider the co-limit of the composite of the funtor with the canonical funtor from the site to the topos, the canonical arrow towards the terminal object is an isomorphism. Okay, so by using this notion, one can characterize continuous funtors like this. So given a funtor A from C to D and a sieve on the source category on an object C, we can define a diagram indexed by the category of elements of S with values in D, which does the obvious thing. So it sends a pair D F where F is an arrow from D to C in the sieve S to the value of D under the funtor A. Now notice that everything here takes place over A of C because the sieve is over C and therefore actually we have a lifting of this funtor to the slice category on A of C. And in terms of this lift, we can express the continuity condition as a relative co-finality condition as stated in this proposition. And this is important because it leads us to a fully explicit characterization of continuous funtors in terms of these connected components, etc. So as you can see here, in fact, there are two main conditions. So the first is cover preservation, so which is what corresponds to J continuity in the flat setting. So the fact that covering sieves, covering families should be sent to covering families. But you see here that that condition is not enough. You need another one, which really involves connected components of comma categories associated with these funtors, with this diagram associated with the funtor and covering sieves on the domain category. Okay, so I just wanted to give all of this because in fact, you can find very, very little in the literature about continuous funtors. And I really believe they should be better known. And thanks to this explicit characterization, which is contained in the monograph I mentioned, you can really play around very effectively with them. So you can prove several results. In fact, I have used this criterion to show, for instance, the continuity of any morphism of the vibrations when they are regarded as comorphisms of sites with the gerotopology. Okay, so all of this is relevant for what we shall say later on, relative to opposite. So indeed, in our approach to relative opposites, vibrations, of course, play a key role. And the idea is to, which in fact dates back already to Giron, and to his paper, classifying topos, in which he introduced the fundamental construction of the classifying topos of a stack. So the idea is really to regard the vibrations as suitable kinds of comorphisms of sites, and then also consider the geometric morphisms induced by such comorphisms. So the fundamental construction here is that of the so-called gerotopology. So more generally, whenever you have a functor A from C to D, and a growth endic topology K on D, you can prove that there is a smallest growth endic topology on the source category C, which makes the functor A a comorphism of sites towards the targets. So now this growth endic topology admits an explicit description, not like a full description of the covering sieves, but at least a family of generating sieves in general. In fact, in the case of a vibration, this description greatly simplifies as stated by this proposition. So in fact, you can characterize the covering sieves as those such that the collection of all the Cartesian arrows in them is sent by the functor to a covering family in the base. So you see, in fact, this is the sort of horizontal topology that you obtain by lifting from the base. Okay, so we shall call this the gerotopology induced by K in honor of Geron, who indeed used the bet for constructing the classifying topos of a stack A, stack regarded as a vibration. So here we are using the vibration point of view. So he actually built the classifying topos of a stack in this way by taking the topos of shifts on C with respect to this growth endic topology. Okay, so just a couple of remarks about the connection between vibrations and continuity. In fact, one can show that every morphism of vibrations yields a continuous comorbidance of sites when we endow the categories with gerotopology. In particular, a vibration itself yields a continuous comorbidance of sites to the base. Okay, so in fact, we can functorialize this construction of the gerotopology in a straightforward way, in such a way that we have a gerotopology functor, which in fact does the obvious thing. So it keeps a vibration with the corresponding gerotopology and in such a way it regards that as a continuous comorbidance of sites. Just a remark on the gerotopology, well it is a very natural topology that one can put on a vibration and indeed one can understand several features of the vibration in terms of this topology. For instance, one can check the pre-stack condition for a vibration by looking at the topology and whether it is subcanonical or not. So this is an example, but there are also other results in the same spirit. So in fact, it is a very nice geometrically motivated topology. Okay, so now before going to relative toposies, I would like just to mention a classification result for essential morphisms. In fact, essential morphisms also play a key role in relative topospiria because whenever you have a continuous comorbidance of sites, it induces an essential geometric morphism between the corresponding shift topologies. So what is an essential morphism? Well, it is simply a morphism such that the inverse image factor has a left adjoint. This left adjoint is called the essential image of the morphism. So here is a theorem again from that monograph which establishes a classification for all essential geometric morphisms from a shift topos to an arbitrary topos in terms of continuous comorbidance of sites. So you see this works in the following ways. So when you have an essential morphism, the way you attach to that the corresponding comorbidance is by composing the essential image of the morphism with the canonical factor from the site to the top. And well, in the conversed direction, it's just the construction of the morphism induced by a comorbidance site, visualized for continuous comorbidance. Okay, so I would like just to mention that in the particular case of a vibration, in fact, the corresponding geometric morphism is not just essential, but it's even locally connected, something which is not true for arbitrarily morphisms of vibration. But for vibrations, it is true. So in particular, all the zero topos will be locally connected. Okay, again, all of this can be found in the same text. Okay, so now another relevant result which we shall use is the terminally connected factorization of an essential geometric morphism. So for this week, we have to recall the two classes we are going to use in such factorization results, terminally connected morphisms and localomial morphisms. So an essential geometric morphism is said to be terminally connected if the image of the terminal object under the essential image of the morphism is the terminal object of the target stop. So this generalizes the notion of locally connected morphism. Then we have the notion of a natal morphism of topos is also called localomial morphisms. You can find both terminologies in the literature and also in these slides I use both of them. So what we have is an orthogonality property of terminally connected morphisms with respect to localomial morphisms in the two categories of dropping purposes. So this shows already that we have the best premises for a good factorization system. And indeed, we have such a factorization system. So if we have a so every essential geometric morphism can be factor uniquely up to equivalence to the terminally connected morphism followed by a localomial morphism. And in fact, the factorization can describe this explicitly because in fact, the localomial morphism you take is just the one associated with the object given by the image of the terminal object under the essential image of the morphism. So you do the obvious thing. And in fact, this is just a generalization of the well known factorization of a locally connected morphism as a connected and locally connected morphism followed by a localomial morphism. So what is interesting about this factorization is that it can be nicely understood at the site of theoretical level by using the language of common sense of of continuous morphism. More precisely, we have this. So we can introduce a generalization of the well known comprehensive factorization of the factor. So, so given a factor f from c to d and a groten-dictopology k and d, we can consider the k-shift given by the co-limit of the composite of f with the canonical factorial prime from d to the shift-topos on dk. Then of course, since this is a k-shift, we we we can attach a discrete vibration to that. So we shall call the discrete vibrations which correspond to shifts gluing vibrations. So in this case, k-gluing vibrations. And actually, we realize that by the universal property of the co-limit, we indeed have a factorization of our original factor f through the category of elements of this factor fk given by this co-limit. So we call this the k comprehensive factorization of f. So this is important for us because it allows us to understand, in particular, the terminally connected factorization for essential morphisms at the site level, in case we can describe such morphisms as induced by continuous comorphisms. So first of all, in fact, this comprehensive factorization satisfies a universal property because in fact, it can be characterized as a unique up to equivalence factorization of the given factor as a as a relative co-final factor. Here you see we are using the chiratopology followed by a gluing vibration. So it's also universal say at the site level. And at the topos level, what we have is this that if we have a continuous comorphism of sites and we consider the corresponding morphism of toposis, actually the factorization of this morphism, the relative comprehensive factorization of this comorphism induces the terminally connected localomomorphism factorization of the corresponding geometric morphism. So this is, we are going to apply these results in the last part of the talk in the context of relative talk. Okay, so now we are ready to describe our approach for developing topos theory over an arbitrary based topos. So all of this, as I mentioned last time, is joint work with my PhD student Ricardo Zampa. Okay, so just let's fix the basic definitions. So we are going to use the language of indexed categories and the vibrations. So we have a category of index, a two category of indexed categories for any fixed category. So this is a category whose zero cells are the pseudo functors. So a pseudo functor, for those of you who are not familiar, is just a weakening of the notion of functor where you don't require strict equality or a composition, but just a quality up to some isomorphisms, which should satisfy certain compatibility conditions. So it is natural to consider pseudo functors when you deal with two categories, when you have a non-trivial two-categorical structure. So here you see we shall take as our target categories the category of categories or small categories. Well, of course, here there are several size issues that one has to deal with because of course some things might happen in a certain universe, other things in another universe, etc. So you can address these size issues in different ways. So you can use growth in the universes as already done by Giraud or you can just restrict to some suitable subcategories if you want everything to take place at just one level. So I mean, I will not give details of all of these size issues in this presentation, but you will find them in the monograph draft we shall make available next week. So in any case, we have these two categories of C index categories whose zero cells are the pseudo functors. So these are supposed to generalize pre-ships, you see. So a pre-ship is a functor from C op to sets. So here we are going to consider pseudo functors from C op to cap. Then what are the one cells? Well, the pseudo natural transformations between such two functors. These are also sometimes called C index the functors. And the two cells are the modifications between them. On the other hand, we have a category of vibrations over the given category C. Again here, we can make this into a two category by defining the zero cells to be the growth in the core street vibrations towards the given category. We can take as one cells the morphisms of vibrations. So these are like functors which send Cartesian arrows to Cartesian arrows in such a way that the corresponding triangle commutes up to a specified isomorphisms. So in fact, we name such isomorphisms in defining one morphisms and one cells. And then the two cells are the natural transformations between one cells. Okay, so then we shall sometimes need to restrict them to the full two subcategory of the cloben vibrations, namely the vibrations they keep with the cleavage. So a cleavage is a way of canonically, well canonically, it's a way of selecting Cartesian lifts for arrows you see in a fiber. Okay, so it is well known that these two formalisms index categories on one hand and the vibrations on the other are essentially equivalent to each other. You see, in fact, this is a wide generalization of what has been also mentioned by Sharpe yesterday in his lectures, when you have a set X, you can consider bundles over X, so the functions towards X, or you can consider the functors from X to sets where X is regarded as a discrete category, you see, and the two categories you get in this way are equivalent. So all of this is just a much more general and sophisticated version of that, but the idea is the same. So we have a two categorical equivalence between these two categories of index categories and vibrations. And so how does it work? So in one direction, when you have an index category, you can apply the famous Groton-Dick construction to get a vibration from that, fiber category from that. So this generalizes the construction of the category of elements of a pre-shift, with which you are probably very familiar. And in the conversed direction, well, it does the obvious thing. So if you have a vibration, you take for each object of the base category the fiber over that, which will be a category. And because of the axioms of a vibration, you will be able to get actually a pseudo-functor out of it. So this is how it works. So now let's recall the fundamental notion of stack or pre-stack and also on a site. So we can of course define this notion either by using the language of vibrations or by using the language of index categories. So first of all, let's give the description in the vibrational language. So the condition is actually a higher categorical generalization of the condition, which defines the notion of shift on a site. So we have to consider sieves for the Groton-Dick topology we have. And recall that every sieve can be seen as a sub-object of the corresponding representable functor in the category of pre-shifts. And so, of course, you can also regard that as a vibration, which is related to the discrete vibration associated with the representable biomorphism of vibration, which we call MS. Well, sorry, the action of the Groton-Dick construction on MS, which is the canonical sub-object. So composing with that, we get a functor going from the category of morphisms of vibrations from C over X to the category of morphisms of vibrations from the category of the vibration associated with S to D. And so the pre-stack condition is that such a functor should be full and faithful, while the stack condition is that this should be an equivalence. Okay, so we can build the category of stacks, two categories of stacks actually, by just taking the two full and faithful sub-category of the category of index categories on the stacks. So we shall denote this by analogy with sheaths stacks on CG. Okay, so stacks indeed generalize sheaths as shown by this proposition. And we can also naturally rewrite the stack or pre-stack condition in the language of indexed categories. So here, instead of considering these categories of morphisms of vibrations, we consider these categories of pseudo natural transformations between indexed categories, which are pseudo functors. And again, the condition is the same, so the functor should be full and faithful in the case of the pre-stack condition or an equivalence in the case of the stack condition. Okay, so now let me just pause for a moment to illustrate like the general philosophy which inspires our approach to relative toposphere. So which kind of role is going to be played by stacks in our approach? Well, actually the role of stacks is very important for two different but related reasons. So first of all, as we have just remarked, stacks generalize sheaths. And therefore, we can expect categories of stacks on a site to enjoy properties which should be formally similar to the properties enjoyed by categories of sheaths on a site, namely the rotating topology. And indeed, we shall use this analogy for listing a certain number of constructions which we are familiar with in the context of rotating toposes to these categories of stacks. So in particular, we shall see that we can define direct and inverse images of stacks, etc. And also that the functions between sites that are relevant in the context of rotating toposes such asomorphisms and comorotisms of size also naturally behave with respect to stacks. Also continuous functions will behave very well with respect to stacks. So we don't have to introduce other classes of relevant functions. Those we have are enough. Okay, so this is one perspective we can have on the notion of stack, but there is another also very important and different perspective which plays a central role in our development of relative toposphere is the fact that we can regard a stack on site as a kind of category internal to the corresponding topos of sheaths. So it is not a category internal to the topos of sheaths in the strict sense of the word. You see, Lauren LaForghini's course has presented you categorical semantics. So he has presented you the notion of model of a geometric theory in some category possessing enough structures. So in particular, you can formalize the theory of categories in a first order way, and you can consider models of the theory of categories in whatever category with finite limits, in particular in a growth in the topos. So in this way, you get the notion of an internal category to a topos. So how does stack, how do stacks relate to that? Well, in fact, they are a much more flexible notion than the notion of an internal category. But still, they can be regarded as kind of, they should be regarded as the right notion adequately taking into account the two categorical structures present in this setting, the right notion of category internal to a topos. And so as we are used to keep categories with growth in the topologies, we should be able to keep stacks as well with growth in the topologies if we want to develop relative topos. So this is precisely what we are going to do. So we shall introduce a notion of relative site in which we shall have as in the classical setting two kinds of elements. So the first is a stack and the second will be some kind of an analog of a growth in the topology on that. Okay, so you see that stacks play this double role in the theory and you will understand more and more as I continue with the explanation. Of course, these two perspectives integrate very well with each other, even though they are quite distinct. So just a remark, a formal remark. So every stack actually can be rigidified in the sense that it can be shown to be equivalent to a split stack and therefore to an internal category. But in fact, most of the stacks which naturally arise in the mathematical practice, especially in geometric situations, are actually not split in the sense that these isomorphism conditions cannot be replaced by equalities. It suffices for you to think about the canonical stack over a topos, which we shall review later. So you see there, you have the pullback functions that between the slice topos and the commutation results, the functoriality results don't hold strictly. They hold only up to certain isomorphisms, which satisfy good compatibility conditions, but still they do not hold strictly. So it's very, very important not to restrict two internal categories and work more generally in this more flexible language of stack. So I mentioned this because in fact throughout the past decades, the number of categories have developed some parts of relative topos theory by using the language of internal categories and internal sites. But in fact, all of this has not really brought several applications in different areas of mathematics, because actually this formalism is, at least in my opinion, too rigid to be easily applicable in practice. So we hope that by using the language of stacks, we can get a more flexible and applicable formalism. Okay, so now let's present the big picture on which our approach is based. So actually we have a network of two adjunctions here, which relates index categories on one end and relative topos on the other. So in this diagram, you see a functor SJ going from index categories to stacks. So this is a stratification. Then topos indicates the category of growth index toposis and geometric morphisms. So again, I am ignoring size issues. So of course, one might need to restrict to some suitable categories, etc. But just for the sake of this presentation, not entering into these details. So okay, so we have this category, and then we can restrict to the full subcategory on the essential geometric morphisms, both at the level of the morphisms and also at the level of the triangles. We want all of them to be essential. And so what I am going to describe you in the last part of the course is this adjunction, which you see on the first line between these two functors lambda and gamma. You see, I have used the same notation that I had used for the pre-shift bundle adjunction for topological spacing. Why? Because this is actually a point-free generalization of that. It's a very wide generalization of that. Then we shall also briefly talk about the specialization of these two pre-shifts. But in fact, one has a version of that for index categories more generally. So we have this fundamental adjunction taken place there, which we shall describe in detail later. Now this adjunction actually restricts to an adjunction at the level of stacks, because in fact the gamma factor always takes values in stacks. And conversely, the lambda factor takes values in essential geometric morphisms. The reason why this is the case is because of the results I have recalled, that when you deal with vibrations and morphisms of vibrations, and you keep them with the gerotopology, you always end up with essential geometric morphisms at the top. Okay, so we have a restriction of this adjunction there. And it is interesting to restrict to essential morphisms, because at that level we can define some functors that are relevant for several purposes. So in particular, we have a functor E, which is what is involved in the terminally connected factorization. So it sends a given geometric morphism to the value of the terminal object under the essential image of the morphisms. So you see that here we really need to restrict it to essential morphisms to define that. And in fact, this functor has an interesting adjoint, which is the functor L sending an object of the topos of shifts on the side to the corresponding etal morphism of toposes. And in fact, one can show that these two functors are adjoint to each other. In fact, they are quite interesting because for instance, you can, by composing the restriction of lambda, so lambda prime, with E, you get a functor from stacks to shifts. And in fact, this functor is adjoint to the inclusion of shifts into stacks. And so it represents a sort of truncation functor, which can also be explicitly described at the site level. And so you see that all a number of fundamental ingredients fit into this framework. So now we are going to give more details about all of this. So before stopping for the pause, I would like just to focus on the functor L, because this is actually an interesting functor, especially when you define a growth and ectopology on the category of toposes, representing an analog of the open covert topology, which you can define on some, on the category of topological space, or some small category of topological spaces. So we recalled the definition of that in the first part of the lectures. So yeah, we can get an analog of that. We get one level higher in terms of complexity, but formally it is analogous. So we define this et alcovert topology by postulating that the sieves which cover are those which contain families of local homeomorphisms, such that the corresponding familiovaros is epimorphic in the topos. So you really see that this is the natural higher categorical analog of the open covert topology for topos. So basically what the volume of these developments is to be able to think of the category of toposes as really a generalization of the category of topological space. You see, and in fact it works. So in particular we can consider the composite of the functor L, which just gives the et al morphism, with the canonical functor L going from the site to the topos. So this actually gives a relatively full and relatively faithful bimorphism of sites. So bimorphism means both morphism and comorphism. So such functors actually enjoy some very pleasant properties. In particular, I approve the result in the monograph draft from last year about such functors, such bimorphisms that are relatively full and relatively faithful, showing that they induce co-adjoint retract. I mean they induce pairs, essentially pairs of petty and gross toposes attached to the given situation. And you see what you expect when for a pair of petty and gross toposes is that there should indeed be a retraction of one into the other and that the maps which the morphisms which realize this retraction should be on the one hand in essential inclusion of the petty topos into the gross topos and on the other hand a retraction given by a local morphism. So by using these results we can apply them in this situation. Of course, they can be applied also in the usual situation of the open covert topology for topological spaces and indeed you recover the usual pairs of gross toposes associated with the categories of topological spaces. And so here it's interesting because if we apply this to the functor L composed with little L, so this becomes actually relatively full, relatively faithful and the bimorphism of sites from the original site to this site endowed with the ethyl covert topology. Therefore, in fact, this allows us to see any grotendic topos, our topos of shifts on CJ as a petty topos associated with this gross topos, which of course you can present in these two different ways. Of course, these grotopos will belong to another grotendic universe, so you will not stay in the same, but you see in fact that we have shown that properties such as being essential or local for geometric morphisms behave well with respect to a change of the umidus. So this is not a problem, but just to make clear that they don't belong to the same. It's a very big topos, what you get, but still I mean formally this is interesting because it allows one to in particular view any object of the original topos as an ethyl morphism to the terminal object in the associated grotopos. And this was a question originally posed by Grotendic in the 70s in his Buffalo course, he specifically asked for that and this problem was recently brought again to the public attention by Colin McLarty in some of these recent talks and in fact when I heard that I thought that in fact we already had an answer to that without even having looked for that. So indeed I think Grotendic was really trying to develop geometric tools for studying topos. So he wanted actually to be able to regard any topos as sort of petty topos because you see in the petty grot formalism petty toposes are considered as generalized spaces, while grotoposes are considered as themselves categories of spaces. So if you really want to have a geometric perspective on a topos it is more natural to think of it as a petty topos, you see, rather than a grotopos. So you see this formalism justifies this point of view and indeed you see it's part of this wall framework summarized by this big picture which indeed provides geometric tools for investigating toposes by using this very geometric language of stacks and vibrations. Okay, so I'll stop here for the pose and then which I'll resume for the last part of the course.