 heatingisten, we stopped on basic examples of Grohten dyktoposis. Now they are very present from the point of view of the category properties they satisfy. In fact, every grohten dyktopos is a kind of completion of a site presenting it. in kompleti v kategorijstvu. In vsega kratenica je obroženja in kukroženja. In tudi, ki so početne in početne, se početno je komputat v nekaj efektivno različenju. Kaj smo vse vse pravimo? submission. Sopoč učinem je, da sem bilo, da sem bilo učin, da imam zelo, da lahko sem bilo, da sem bilo učin, da sem bilo, da sem bilo, da sem bilo učin. Tako na vseto vse, da imamo prešlične zrata, zato vsej smo vse naredili, da so je dobro vse nekaj zrata, kaj je zelo sovjev'ščilja. Tko se dogodilo, je to po tebnih tem, da je to počunja, zelo štih prokratič, da se je štih, da je to štih, da se je štih, da je to štih, da je to štih, da je to štih. Vsi je počunje, da je to štih, da je to štih, for an arbitrary site. And so this is the universal characterization of the edification operation. Then they point, is how to explicitly construct that and as we should see, there are different approaches to this problem, some more algebraic other more geometric, but still it is an operation that can be explicitly completed. So a crucial feature of this factor je zelo, da je zelo, finačne liniče. Način, ne obječenje liniče, obječenje liniče, ali obječenje liniče. To je za njihoj obječenje. Zato, kako je izgleda liniče, v kategoriju šivs, včetno, nekaj je šiv, nekaj je zelo, da je šiv kondition, as we have seen, can be expressed in terms of an equalizer condition, so in terms of a limit condition. One can show that the category of sheaves is closed in the category of pre-sheaves with respect to all small limits. And therefore, since small limits exist for pre-sheaves and they are computed pointwise there, we can conclude that such small limits exist also for sheaves. In particular, we can consider the terminal object, which is a trivial kind of limit. The terminal object is simply the pre-sheaf sending every object to the singleton. And so this, of course, is a sheaf for any grot and ectopology. And it provides the terminal object in the category of sheaves on that side. Now concerning co-limits, a key observation we can make is that since the associated shift factor has a right adjoint, namely the inclusion factor, it necessarily preserves arbitrary co-limits. And so, since when we start from a sheaf, we can regard it as a pre-sheaf and then shift it high and we come back to the original sheaf, we can compute arbitrary co-limits in a shift-topos by considering the diagram as it had values in pre-sheaves. We compute the co-limit there and then we apply sheafification and this will give us the co-limit in our category of sheaves. And so, in this way, we can see that our topos of sheaves has also small co-limits. There are also other pleasant properties of grot and ectoposis, which I have not listed here. The fact that they satisfy very pleasant representability properties due to the fact that they have both separating sets and co-separating sets, et cetera, and that everything behaves well, well-poweredness, co-well-poweredness, et cetera. So, these are results which allow one to apply a special adjoint in terms of theorems in the context of toposis. And so, you see, you are really in a world where everything you would dream of actually takes place. So, in particular, you have constructions that are useful for building higher-order things inside a topos, for instance, exponentials. So, exponentials exist in any grotendic topos. In fact, they are computed as in the corresponding pre-shift topos. And also, another important object, which can be associated with any topos, is the so-called sub-object classifier, which, as the name tells it, classifies the sub-objects of objects in a topos. This is an object, which is usually denoted by omega, and which captures a great extent of the internal logic of a topos regarded as a mathematical universe. And this object can be built explicitly in terms of the site of presentation for a topos. So, you see, actually, all these constructions can be computed. I want really to stress this because it's very relevant for the applications. You see, the toposes are very big things, and so they might be scary because one might think, why do I have to deal with all these things if I am just interested in something very concrete? In fact, the advantage of working in such very big mathematical environments is that they have no holes, in a sense. So, the existence of all these operations makes it possible to make computations much better. So, you see, the idea is that when you cannot compute certain things in a restricted environment, such as a site or a theory, the idea is to complete this site or this theory into a topos, to compute the things there, and then to try to reinterpret the result of these calculations in terms of the presentations you have started with. And, in fact, this back and forth method, which, in fact, is essentially the bridge method, works very well. So, the relationship between a topos and its site presentations or, more generally, its presentations in terms of theories or other kind of objects is very natural. And so, you can really have this very effective back and forth between the level of, the concrete levels of sites or presentations and the abstract level of toposes where invariant lies. So, I will give some elementary illustrations of this interplay between the two levels in a moment. Of course, if you want to read more, you can take, for instance, my habilitation thesis, which contains a lot of examples of bridges. And so, you can see this back and forth technique in action in several different mathematical contexts. OK, so, this slide was just to summarize some of the main categorical features of toposes. Now, as I anticipated, one key feature of grotendic toposes is the fact that they can essentially be built from below. They can be built from the mathematical practice through sites or other kinds of presentations. So, this possibility of building them from below actually technically translates into the fact that such categories have separating sets of objects or sometimes in the literature, these are also called sets of generators. In fact, you can think of this as a basis for a topological space. So, as you can present a topological space by giving a base for this space, in the same way, you can present a topos by giving a site for this topos. Of course, the level of generality of sites with respect to bases is much broader. In fact, you can have, for given topos, infinitely many sites associated with that, belonging even to different areas of mathematics. So, this is going well beyond the context of topology or even the context of group theory. You see, grotendic compared to the existence of different sites for one topos to the existence of different presentations for a group of generators and relations. But this is formally correct, of course, but when you pass from groups or topological spaces to sites, you really go from some particular field of mathematics to essentially all mathematics. And so, this is why toposis can be so powerful in unifying ideas and techniques from different branches of mathematics. So, formally, what is a separating set of objects for a topos? Well, it is a set of objects, which, as the name suggests, separates what happens in the context of any other object, in the sense that for any object of the topos, the collection of arrows from objects in the separating set to this object is epimorphic. So, concretely, this means that suppose you have two arrows going from this object to another object and you want to understand when they are equal, what you can check is whether they're composite with all the arrows from objects in the separating set to the object A, if all these composites are equal or not, because it is precisely equivalent, given the fact that such a family is epimorphic. So, basically, these generators are able to detect any difference between two pairs of parallel arrows in the topos. OK, so, this proposition shows that, indeed, we can get canonically from any small site or presentation for a topos, a separating set of objects for the topos, just by taking the images of the objects in the site within the topos. So, you see, if you take the unit embedding, which brings you from a category to the corresponding pre-shift topos, and then you apply shiftification, you get a canonical functor going from the category to the topos. And so, basically, if you take the objects in the image of such functor, basically, you see immediately that these objects form a separating set for the topos. In fact, how do you prove this? Well, basically, this is an easy consequence of the fact that already at the pre-shift topo level you have that, because, actually, you can prove more precisely that every pre-shift is a co-limit of representable pre-shifts indexed over the category of elements of the pre-shift. Now, since shiftification preserves co-limits, this co-limit representation is preserved when you apply shiftification, and the representables are sent to the objects in the image of this functor. And so, of course, when you have a co-limit, it gives, in particular, epimorphic funges. So, this is how you see that such images of objects in the site actually generate the topos. So, indeed, you can really see this as a generalization of a topological space and some bases for this space. In fact, it is a strict generalization, because when you have a topological space x and the base is b for this space, in fact, such bases will give you a site of definition for the corresponding topos if you endow that with the grotendictopology induced by the canonical topology. OK, but we shall see this more precisely in a moment. So, now, I would like to talk about a special kind of sites that are important in toposphere, the so-called subcanonical sites. So, when you shiftify representable in general, you will get a functor, which is no longer representable. But there are some topologies such that all the representables are already shifts, which means that when you shiftify, you still remain with the same shift. So, these topologies are called subcanonica. So, a topology is said to be subcanonical if every representable functor is a shift. Now, is there a largest subcanonical topology on the given category? Yes, this is called the canonical topology. And such a canonical topology can be characterized explicitly in terms of what is called effective epimorphic or universal effective epimorphic states. So, one defines a sieve to be effective epimorphic if it forms a collimit cone under the canonical diagram, which is in the slice category over the object on which the sieve is taken. And then we say that such a sieve is universally effective epimorphic if not only itself, but also it's pulled back along an arbitrary arrow maintains these properties effective epimorphic. So, one can characterize the sieves for the canonical topology in particular we are going to consider the canonical topology on a grontendic topos, which is a locally small category, but in general not small. And it will be good to have such an explicit characterization of its covering sieves as the universally effective epimorphic ones. Now, it is possible, of course, to generate sites from separating sets of objects for a topos. So, we have seen that whenever you have a site of definition, this gives you a separating set of objects. But there is also a converse to that. So, if you have a separating set of objects, you can build a site out of that. How? Well, in a very natural way, you induce the canonical topology on the topos on these separating sets. And now, a natural question that one can pose is, OK, this topology, I define it. Ah, sorry, I request something. OK. So, this topology, I define it as an induced topology. So, its definition refers to the topology on the topos. And so, it is a natural question whether you can get an intrinsic characterization of this topology directly involving the category C. Now, depending on the situation, this is possible or not, there are a number of circumstances in which it is possible, in particular when the objects of the separating set can be characterized among the objects of the topos as objects which satisfy some generalized compactness conditions. So, they could be compactness, irreducibility, super compactness, et cetera. So, one can define several notions of compactness for objects in a topos. And in fact, there is a result in my monograph from last year, which shows that for a large class of such generalized compactness conditions, when you induce the canonical topology on such categories of such generalized compact objects, you get a topology which can be described intrinsically in terms of the category C. In fact, their sieves can be characterized as their covering sieves and be characterized as those which contain certain kinds of effective epimorphic families. So, this can be useful if you want to build dualities, for instance, between the site level and the topos level, and then possibly once you get to the topos level to switch to another representation. So, for instance, in my work on spontaneous dualities, I have used these generalized compactness conditions very much to establish dualities between, in that case, preorder categories and local sort of logical spaces, on the other hand. And so, by using this intrinsically defined grotendictopologies, it was possible to build bridges, giving rise to such dualities. So, in general, it is a nice question to you should pose yourself. Whenever you induce a topology, can you capture this topology intrinsically by just referring to the categorical structure present on the company? OK, so now I would like to say a few words about this general methodology that, in fact, inspires most of the results I'm going to present in this course and also several other developments you will see in this conference. So, I will be quite short and just summarize the basic principles around the behind the use of toposis. So, in this course, we shall be concerned with presentations of toposes in terms of sites, because we are going to focus on, say, the traditional geometric way of approaching the toposes. Stila, it's important to bear in mind that the toposes can be presented not just by using sites, but by using several different kinds of mathematical objects. For instance, groups we have seen, or more generally, topological or local group points, or as Loran has explained, they can be associated with suitable kinds of first order theories in a very meaningful way, or they can also be presented as associated with non-commutative structures such as quantals or quantaleoids, et cetera. And in fact, I think there is much more research that can be done in introducing new ways of building toposes from mathematical objects which naturally arise in practice. So, this is by no means the end of the story. So, in some sense, the toposes are mathematical crossroads, which can be encountered from different paths. So, they are places where different paths converge and reflect one into the other. So, I expect that in the future years, more and more ways will be introduced to construct meaningful toposes from concrete mathematical situations. But already, you can see how general this notion is, because you can really build the toposes from such a great variety of different concepts. And in any case, in each of these cases, the way of associating toposes with such presentations is very meaningful in the sense that there is a very nice back and forth between properties of these presentations and invariants defined at the toposes. So, this is a very important research line that should be developed, like trying to invent new presentations for toposes and also study in a systematic way the way topos theoretic invariants express in terms of different presentations. So, as Loran already stressed in this course, every topos is associated with infinitely many presentations, in particular with infinitely many sites of the future. So, this means that you can have completely different sites such that the categories of shifts on these sites are equivalent. And these presentations, it's very important to remark that they may belong and in fact, they do often belong to different areas of mathematics. So, this is an important realization because it is something you can really exploit to build the connections which do not boil down to simple dictionaries between different context. They can really allow you to transport results from one area to another in a very deep and surprising way because the way invariants at the topos level manifest in the context of different presentations can be very different. So, the kind of translations that bridges can realize are often very surprising. So, in fact, the key idea is really that of exploiting the duality and this back and forth, if you prefer, between toposes and their presentations in order to build bridges. So, bridges are induced by invariants. So, first one is to understand what an invariant is. So, by a topos priority invariant, we mean any notion which is invariant and we respect to categorical equivalence of properties. So, invariant means that if I have two toposes that are equivalent to each other, I can basically transfer across the equivalence of this notion. So, it could be a property or a construction, whatever thing the notion. So, I think it's pretty clear what I mean. Of course, this could be formalized but I think it's not necessary because in practice it is clear what is meant. In fact, generally speaking, whenever you have a genuine categorical property, well, since the notion of equivalence for toposes is just categorical equivalence, well, by a sort of general meta theorem, if your categorical property is natural enough, it will be automatically invariant with respect to categorical equivalence. So, this means we have a lot of invariants at disposal, which we can try to use to build bridges. So, how do we build bridges? Well, by investigating how a given invariant expresses in terms of different presentations for a given form. So, actually we look for sorts of unravelings of invariants in terms of different topos presentations. So, in fact, it's important to realize that toposes are instrumental for building these bridges, but at the end of the process, they completely disappear. So, one ends up with just concrete connections and correspondences between properties of different types of presentations. So, toposes in the end disappear and they should disappear. So, we really want to get rid of toposes in the end because our aim is to really get concrete insights. But, toposes are crucial as a sort of computing machines that allow us to have a privilege point of view on our subject. And so, to calculate in a most effective way and then later, as one has entered a bridge, one should exit the bridge. And so, in the end, one should end up with a completely concrete statement. But in most cases, one realizes that the use of toposes was really indispensable because, otherwise, one doesn't really know where to look for the right invariants. It's difficult to predict which kind of information can be transferred from one context to another because, of course, there are certain information that can be transferred and some other information which cannot be transferred. So, you see, the theory of topospheoretic invariants really allow you to have insights on what you can expect to be transferable and what, instead, is too concrete, too marginal to be transferred. In any case, that's the basic idea. So, one, indeed, can build bridges by calculating invariants in terms of different presentations, and this often leads. Even when you apply this technique to quite elemental invariants and relatively simple equivalences between different presentations of the same topos, you often get quite surprising and deeper results. Of course, the more complicated is the invariants that you consider and the more sophisticated are the equivalences between different presentations of the given topos you have, the deeper will be the bridges that you generate. So, of course, there are a greater variability in the complexity of calculations of topospheoretic invariants in terms of size. So, there are invariants that can be computed in a sort of even automatic way. Really, you have automatic methods for computing a large class of invariants. For other invariants, such as the homological invariants or the homotopy theoretic invariants, things tend to be more difficult, so there are some partial results, but we cannot say that we have a perfect algorithm that can be effectively used to compute them. But in any case, this is a methodology that it makes sense to try to use in any situation where you dispose of different presentations for a given topo. In fact, it is a method that you can also use for investigating a single theory by means of different points of view, because in fact, from a technical perspective, when you have different languages or different ways of describing a certain theory or of thinking about a certain concept, it's very likely that you can turn this into different presentations for one topo. And then, once you have that, you can apply the bridge technique. Okay, so, technically speaking, this is a typical bridge using size. So, you have two different sides presenting equivalent of this. And so, as I said, what you do is, you consider invariants at the topo theoretic level and you try to compute them on one side and on the other side. What is important to stress here is that we have a unification at the level of the toposis in the sense that we have one invariant at the topos level and different expressions of this invariant in the setting of the two different sides. So, you see here, I have written, I have called the invariant I and the two properties P and Q of the two sides which correspond to this invariant in a different way because, indeed, concretely, they can be completely different, you see? So, that's the interest of the technique. Even though abstractly, they are just one thing, concretely, they can be completely different. And so, for instance, suppose you are interested in property P and you are able to be the bridge which turns you into, that turns your property into another equivalent property Q, suppose you are able to solve the Q, then you get P, you see? So, in practice, you can really change the shape of and the level of complexity of your results significantly when you pass from one presentation to another. Even very small changes at the level of presentations can have a huge impact in terms of computability of invariants. I say this on the basis of the experience I have accumulated throughout the past years. In fact, the duality between toposes and their presentations is very, very subtle. So, even very small changes, small, I mean, from a concrete point of view, which could seem innocent, they could have a huge difference when you have to compute invariants. Okay, so, then, now I'm going just to give some elementary illustrations of the possibility of computing in a topos by expressing everything just in terms of a site, just to convince you in a purely elementary way of the fact that everything which happens in a topos can really be described at the site theoretically. Of course, as you shall see, such descriptions can be more or less complicated. In general, under the most natural assumptions, they could indeed be quite complicated. Very often they simplify in some particular cases, but if you want to do things at the highest possible level of generality, you will find some quite complex characterizations. But still what I insist on is the fact that such computations are generally feasible, at least for larger classes of invariants. Now, just to give such illustrations, I would like to start with a key observation about how to get rid in certain situations of the shitification process. Because, you see, there is one key ingredient in the passage from a topos of pre-chips to a topos of shifts, which is the shitification process. So this shitification process is a non-trivial process. If you take the most classical top of theory book, you will find the description of shitification in terms of the plus-plus construction, which really, I mean, I have to say I'm not very fond of that description. It's very formal. It doesn't have very geometric, intuitive understanding. In fact, I much prefer another presentation of the shitification that I'm going to present later. But in any case, still dealing with shitification is not completely striped for work. So an important remark that one can make is that the shitification operation actually induces a closure operation on sub-objects in the corresponding pre-chips. V's description is much simpler than shitification. In fact, you can find the description of this closure operation here in the slides. So the closure operation is defined by means of this pullback diagram. So this is the abstract definition, but what is important in relation to site characterization is the fact that we have a very simple formula for describing the closure of a sub-object induced by a grotendictobol. So in fact, this closure operation is useful because in many situations one can reformulate things involving the shitification operations in terms of closure operations. So, and whenever you can do this, you can get rid of the shitification and replace them with closure operation and this can simplify your expressions for site characterizations of invariants. So for instance, very simple example, suppose you want to characterize the sub-objects of the shitification of a certain pre-chip. Well, there is a natural objective correspondence between this and the closed sub-objects of the given pre-chip. So you see, you have eliminated shitification and you can do this in many situations. So in fact, I have written a paper called site characterizations for geometric invariance of toposis in which I apply this technique for several significant invariance. So in order to get rid of shitifications by expressing things entirely in terms of the closure operation. So this is one very nice tool, one has a disposal for trying to understand the things happening at the topos level in terms of sites. So I will now give some illustrations of some description of things happening in the shift topos directly at the site theoretically. Just to show you that indeed one can describe everything which happens inside the topos as well just in the combinatorial terms in terms of the generators of the topos and in the objects of the site of definition. So for instance, one natural question that one might want to address is describing the arrows between two objects in the image of the canonical functor going from the site to the topos. Well, if the site is sub-canonical, then of course the canonical functor to the topos is just the unit embedding which is full and faithful. And so all the arrows between any two sub-objects actually come from a unique arrow at the site. But this also for sub-canonical sites. And of course, there are many interesting sites that one might want to consider that are not sub-canonical. So you might wonder what does it happen in the non-sub-canonical case. Still, you can get a site theoretic description of such arrows. In fact, instead of having just one arrow which induces that arrow, you have a local representation of your arrow by means of arrows in the category defined over objects which cover in the sense of the topology the domain of your arrow. So this is what is expressed by point one of this proposition. So as you can see, things are a bit involved here. They involve this notion of local equality with respect to the given growth in the topology for arrows in the site. But you see all of this is explicit. You see, you say that two arrows are locally equal if there is a covering sieve such that when you compose the two arrows with all the arrows in this sieve, they become equal. So you see it's still a quite intuitive notion to give. And you see that in fact saying that two arrows are sent to the same arrow, two parallel arrows are sent to the same arrow by the canonical function to the topos, this can be expressed purely elementary at the site theoretic level as the local equality of this two arrows. So already this is an illustration, you see of this back and forth between the site level and the topos level. And of course, this is basic remark, then of course there are some more complicated developments, but so here are the statements, I'm not really going to read them. Of course, you will have the slides after the course, so you will be able to read everything in detail. I have projected them just to show you that such characterizations exist and they are fully explicit in terms of the site with no reference at all to she. So you see, you really have a nice back and forth between the site level and the topos level. So in fact, every arrow in the topos admits a local representation and conversely, when you have such a local representation, this will present arrow and then you can compare also different local families and try to compare the arrows that they induce and characterize those which induce the same arrow at the topos level, you can also do that. You can show that they do induce the same arrow if and only if they are locally equal on the common refinement, which again is something that is fully explicit at the site level. And you can also describe the composition operation for arrows in the topos in terms of this combinatorial data of site theoretic nature that you have used for presenting these arrows. Now, let's come back to shiftification. Well, we can apply these techniques of local representations of arrows in the topos to get a more geometric understanding of the shiftification process. In fact, calculating the elements of the shiftification of a pre-shift amounts to describing the arrows from some object L of C. Coming from the site to the given shift. And of course, we want to describe these arrows in terms of the site and we can indeed do it. And this yields description of the elements of the shiftification factor of the shiftified pre-shift as locally matching families as equivalence classes of locally matching families of elements of the pre-shift, which personally I find a much more satisfactory understanding of the shiftification process. You see, you don't have this double plus construction, which is not very inspired. Here you really understand geometrically what is going on. And you will understand even more once we have developed the analog of the pre-shift etal adjunction for topological spaces for an arbitrary site. Because in fact, we shall get a description of the shiftification also as a composite of the two factors giving the adjunction. And so we will have also a concrete understanding of the elements of the shiftification as particular geometric morphisms of toposis, which can be locally represented as morphisms induced by some morphisms of the vibrations. This will be, so you see, if you say things this way, you very well understand what is going on. You, at least from a geometrical view point. Okay, so now more generally you can wonder if you can describe the arrows between the shiftifications of two pre-shifts. Sometimes this is something useful in practice if you want to understand certain constructions happening in a topos at the site theoretic level. So you can do this in terms of this notion of J-functional relation from one pre-shift to the other. And again, you see this notion of functional relation is fully explicit in terms of the given pre-shifts. So if you apply this to pre-shifts, which have a combinatorial nature, namely that are built from the site in some definable way, you will get a combinatorial description of your arrows in the topos. Okay, so now we turn to the main topic of the second part of the course, which is morphisms of toposis, which we shall consider in particular as relative toposis over the base, the target. So what is a geometric morphism of toposis? Logan has already talked about them. So let me quickly review what they are. So we put the objective geometric just to emphasize the difference between this kind of morphisms, which are the topologically motivated notion of morphism between toposis and another notion of morphism between toposis that is sometimes useful, which is the notion of logical counter, but it's a much more rigid notion. So for geometrical purposes, this is the right notion. And so we use the objective geometric for emphasizing that. So a geometric morphism is just a pair of adjoint functors, one direct image functor, which is the right adjoint, another functor called the inverse image, which is the left adjoint, which preserves, which satisfies the property of preserving finite elements. Then given a pair of toposis, we can define a natural notion of geometric transformation between geometric morphisms. So this is defined conventionally to be a natural transformation between the inverse image functors. Then if we take the domain topos to be the topos of sets, so the topos of sheets on a point, if you prefer, we get the notion of point of a topos. So of course, grotendictoposis and geometric morphisms form a two category with, of course, the notion of geometric transformations between them. And now here are a number of examples of geometric morphisms. As I said, this notion of morphism between toposis is motivated by topology. In fact, every continuous function between topological spaces gives rise to a geometric morphism between the associated shift of poses. The direct image is defined in the obvious way, just by composing with the inverse image of the continuous map. And the inverse image functor acts just by taking the pullback along the given continuous map when you regard the shift as a dull bundle. Then you also have a very important morphism which relates every grotendictopos with the topos of sets, not in the directional points, in the other direction. So it's a structural morphism. So this structural morphism has a direct image functor, the global section functor, and as inverse image, the functor sending a set to the coproduct of the terminal object index by the elements of this. Then, of course, when you have a site, the pair of functors formed by the inclusion and the shiftification functor, well, we have remarked that these are adjoined, and moreover, the shiftification functor preserves a finite limit, so we indeed have a geometric morphism. So such geometric morphisms considered up to aeomorphism, they are called the subtoposis, subtoposis of, well, in this case, pre-shiftopos. Then we have change of base morphisms induced by an arbitrary arrow between two objects in aeomorphos. So because the pullback functor has between the slice toposis has both a left joint and the right joint. And so we indeed have a geometric morphism going in the same direction as the morphism between the object. Okay, so now I think, how much time do I have? I think I have still 15 minutes, right? Yes, yes, Olivia. Ah, okay, okay, good. So I can go on with the classification of geometric morphisms in terms of flat functors. So this is very important also in relation with the theory of classifying topos, isn't that Loran treats in his course? So it's important to review this as well. So in fact, it's important to classify geometric morphisms from an arbitrary topos to an arbitrary shiftopos. So a topos presented as topos of shifts on a certain side. So let's first focus on preshiftoposis and then we shall introduce the topology. So a little of preshiftoposis, something that we can remark is that whenever we have a functor a from a category, a small category c, to say a topos, a grotendictopos e, well, it works more generally for a locally small, co-complete category, but for simplicity and for our part, let's suppose that e is a grotendictopos, then such a functor induces an adjunction between the category of preships on c and the topos e. So this adjunction can really be thought as a general home tensor adjunction because the right adjoint, the functor ra, is actually a sort of home functor. You can see this from the description. While the left adjoint is defined by a co-limit of a particular form, so it's a co-limit of the composite of the functor a with the canonical projection functor from the category of elements of p to c. And in fact, given this particular form of this co-limit, when we rewrite this co-limit in terms of co-equalizers and co-products, we realize that what is going on really is a sort of general tensor product. So we can think really of l of a as a sort of general tensor product functor. And so what we have here is really a general kind of home tensor adjunction in this categorical setting. So we have such an adjunction. Now, if we want to build a geometric morphism starting from such an adjunction, there is a condition we need to have, namely that the left adjoint should preserve a finite limits. So in general, this will not be the case for an arbitrary a. So we decide to focus on those functors a, such that this left adjoint l a preserves finite limits. So we call such a functors the flat functors. By analogy, what happens with modules? You see the modules that are flat are those such that the tensor product functor is exact. So in fact, this really generalizes that notion. Even if here we are in a non-linear setting, but it's just the analog of that. So if the category C has a finite limits, flatness corresponds exactly to finite limit preservation. This is very important to remark. So if you just deal with Cartesian sites, flatness simplifies a lot. It becomes just finite limit preservation. In general, the notion is much more subtle. It can be characterized explicitly. You can find these characterizations, for instance, in McLean and Mordex book. In fact, they have proved an equivalence between the notion of flat functor and the notion of filtering functor, which can be made completely explicit. So you can refer to that for such a characterization. In any case, we shall not be concerned with that in this course. So what is important for us instead is to talk about this equivalence between geometric morphisms from a topos E to the pre-shaped topos on C and the category of flat functors on C with values in E. So how does this work? When in one direction you can imagine because you see if you start with a flat functor, we have seen that by using this home tensor adjunction, because of flatness, we indeed get a geometric morphism. In the converse direction, if you want to build a flat functor starting from a geometric morphism, what you do is simply to take the inverse image of the morphism and you compose it with the unit embedding. And you can show that this is a flat functor. And in fact, one can see without a lot of work that these two correspondences are factorial and give rise to an equivalence of boundaries as stated by the theorem. So this theorem is called Diakonescu's equivalence. Also in the more general setting with sides. Because here we have not introduced grotendic topologies, now we are going to do that. So we are going to focus on the flat functors which are j-continuous in the sense that they send j-covering seeds to epimorphic. In fact, these are the functors which correspond precisely to those such that the corresponding geometric morphism factors to the subtopos of shifts for the topology. And so basically what happens is that it restricts to an equivalence between geometric morphisms to the shift topos and flat j-continuous functions. This is very nice, very useful because in fact it reduces the complexity of the description of a geometric morphism because you see a geometric morphism is a pair of adjoint functors defined between toposes. Toposes as we have said, they are quite big. And on the other hand, we have just flat functors defined on something small. So they are more tractable. And this actually is a starting point for understanding toposes as classifiers of structures from a geometric point of view. So it is a very fundamental result. Okay, so now I have still some minutes. So let me introduce another important topic that we are going to treat in detail in the next part of the course, the topic of functors inducing geometric morphism. So there are two main classes of functors which we are going to focus on morphism subsites and comorphism subsites. They will induce respectively geometric morphisms between the corresponding shift toposes in a contravariant way and in a covariant way. So in fact, morphisms subsites, as I have anticipated, can be thought as a combinatorial algebraic way of presenting of understanding morphisms in toposes, while comorphisms subsites really embody a more geometric intuition about morphisms. So as you were talking about morphisms induced by continuous maps of topological spaces, so this point of view relates, as we shall see later, when talking about vibrations, it relates much more naturally with the original geometric perspective. Okay, so what is morphism of sites? Well, morphism of sites can be defined in terms of flat functors, as a functor, such that when I compose it with the canonical functor from the site to the shift topos, I get a flat and continuous functor. So in this way, by the result we have just mentioned, it will certainly induce a geometric morphism between the corresponding shift toposes. On the other hand, a comorphism of sites is defined as a functor satisfying the covering lifting problem. So the definition is quite different. What is very interesting is that, in fact, these two notions are in a sense dual to each other in the sense of bridges. This we shall explain next time. So in fact, we shall see that we can really unify these two notions, so to construct from any morphism of sites, comorphism of sites induce in the same geometric morphism up to equivalence and conversely, in such a way, has to really obtain a dual adjunction between morphisms of sites from a given site and comorphisms of sites towards that same site. So in this way, we really understand that we have a sort of algebra, geometry duality going on here. And one better understands the relationship between these two apparently quite different notions, at least from the point of view of the definition, they look quite different. So the main result, which is completely classical, is that of course, every morphism of sites induces a geometric morphism in the opposite direction. And every comorphism of sites induces a geometric morphism in the covariant direction. So let me just mention that morphisms of sites as flat functors can be characterized in a completely explicit way. So here I have a written down for you, I'm not going to read this, but you will find them in the slides, full characterization, explicit characterization of morphisms of sites in terms of the sites and all the covering sieves in the sites, et cetera. So you see we have an equivalence between the property of the functor to give rise to such a commutative diagram, where below we have the inverse image of some geometric morphism. So the existence of such a diagram is equivalent to the functor at satisfying a number of properties. So four properties, so the first is very easy to understand, it's just covered preservation. Of course, the functor should preserve covering the families. And then the other, so it corresponds to continuity. And then the other three condition, two, three and four, they correspond to flatness. So in any case, I think simplify when we have Cartesian categories, because in such a situation, a functor which preserves finite limits and which is covered preserving, it will be automatically a morphism of sites. So if you deal just with Cartesian sites, you can forget about this more refined notion of flatness and just to work with Cartesian with finite limits. Okay, so just finally before stopping, let me mention that in fact, one can describe explicitly the direct and inverse image functors of geometric morphisms induced by morphisms or comorphisms of sites in terms of common extensions. This is quite nice, because already it will show some sort of duality between the two. So first we have to recall what the right can extension is, what the left can extension is. Well, they are characterized universally as right adjoint to the composition functor for a given functor F between three shifts. So right adjoint for the right can extension, left adjoint for the left can extension. So the right can extension is computed through a projective limit indexed over a common category. The left can extension is computed in dual way by taking a co-limitem, still indexed by some kind of a common category. And what I was referring to concerning morphisms induced by morphisms and comorphisms of sites is this result. So as you can see, you see the direct image of a geometric morphism induced by a morphism of sites is simply given by the restriction to shifts of F up or star. While the inverse image involves the left can extension. While what happens for comorphisms of sites is that the direct image of the morphism involves the right can extension while the inverse image is given by the F up or star. Okay, so, and so this will not, sorry, F up or star, but composed with the shiftification and the inclusion. So, but you see the similarity between the two treatments. So this already gives you an idea that there might be a duality between the two notions, but we shall say more about this next time. So we shall start on this slide, which talks about how to unify morphisms and comorphisms of sites through bridges. Okay, so I'll stop here. Thank you.