 We now introduce a very important example of a stack which can be associated with any geometric morphism. So the idea is really to refer to what we have seen earlier concerning the way of inducing a comorphism of sites from the given geometric morphism regarding this morphism of sites via its inverse image. So you see here we have a domain of this comorphism of sites a comma category and in fact the projection factor from this comma category 2e is actually a stack which we call the canonical stack of the morphism and we can easily describe this from an indexed point of view because it simply sends an object of the base topos to the relative topos to the inverse image of the object under the morphism and the level of arrows it acts as the convex factor. So you see in particular that we indeed have a pseudo factor and not a strict factor. So it's natural to call the growth index topology on the comma category which is induced by the other projection, the one to F, because we want actually this to induce an equivalence at the topos level with the topos F. So this is why we have to induce the topology from F. So this topology it's natural to call it the relative topology of F because in fact this gives rise a first example of a relative site. You see we have a stack and a topology, a growth index topology on this stack and by taking shifts on this relative site we get the source topos of our morphism. So it's interesting to look at what happens if we take F to be the identity. Of course we can take the identical morphism on a topos and if we choose a site of definition for this topos we obtain this so-called canonical stack on the given site which sends any object of the category underlying the site to the corresponding slice topos. And in fact this relative site if we regard it in the context of this canonical site of the canonical site of this identical morphism well it gives us you see a sort of categorical thickening of the usual representation of the topos as the shifts on itself for the canonical topology. So it's quite interesting because it's something richer in terms of structure but the idea is the same, like regarding a topos as shifts over itself and here we see that we can not just regard as normal shifts but also relative shifts which are richer. Okay so now I would like to describe what happens at the level of operations on stacks that we might want to perform. You see I mentioned that stacks are generalizations of shifts so with shifts we are able to operate by taking for instance direct and inverse images so can we do similar things for stacks? Well yes in fact most of the theory generalizes quite smoothly. You see first of all you can observe that whenever you have a functor F from C to D composing with F yields a two functor from the category of index category over D to the category of index category over C. So this is just a pre-composition but what is interesting is that as in the pre-shift case we have two adjoins, one on the right and one on the left which we can call the pseudon can extensions. Indeed they are defined in a similar way to the usual can extensions except that here you see we replace the category of sets with the category of small categories. Of course sometimes it's possible also to take values in bigger categories by using some co-finality arguments. I mean these co-limits can make sense also more generally but to be technically rigorous here we restrict to index categories taking values in small ones and we also restrict to small generated sites or even small sites if necessary. Okay so they are defined like this so similarly as you define left and right can extensions you see what the changes here is that we take values in cut and therefore we take the appropriate notion of co-limit there which is the notion of pseudo-co-limit so it's a kind of two categorical co-limit and it is the right one to take because indeed one can prove that we have such two adjunctions. Okay so this is the starting point for understanding the action of morphisms and co-morphisms of sites on categories of sets because remember so I just come back to that for you to fix the ideas we had seen that we had characterised the action of morphisms and co-morphisms of sites at the level of the corresponding categories of sheets in terms of can extensions so you see direct image was for instance for a morphism was given simply by composition with the functor but the inverse image involved the left can extension and in a sort of dual way for co-morphisms the direct image involved the right can extension which restricted to sheets and the inverse image instead was just given by composition and so we are going to have an analogue of that in the context of stacks so let's come back to where we were okay so you see we have these fundamental ingredients the pseudo-can extensions then we are now make use of them in a formally similar way as we have done for sheets so a preliminary remark that is very interesting is that in fact I mean a priori you might wonder is the notion of continuous functor also the right one for stacks because remember that Grotendick defined the continuity in terms of the condition that composing with that would preserve the shift condition in fact you can prove that composing with any continuous functor also preserve the stack condition so in fact you don't need to introduce any other notion the continuity notion you have for sheets equally works well for stacks which is very good and so of course every morphism of sites in particular is continuous and therefore it induces a two-adjunction which in fact represents the analogue of the geometric morphism induced by morphisms so what is interesting about this is that in fact both the direct and inverse image of this morphism between categories of stacks can be described very nicely in vibrational terms so concerning the direct image in fact it can be described vibrationally just by taking the pullback along the found and concerning the inverse image while it is a bit more complicated but still we have a nice vibrational description obtained by considering the first of all the vibration of generalized elements of the composite functor so we take the composite of the functor with the given vibration so then we get a vibration out of that by taking the generalized elements of that and in fact in order to so this will give a lax co-limit but since we want a pseudo co-limit we will localize this vibration and this will give the required description for the pseudo left hand so I'm not going to enter into details in any case this slides will be available after the course we will be able to check everything and also of course for even more details there is a word for coming text which you will be able to read from next week okay so we have talked about morphisms of sites but things work very smoothly also for co-morphisms so when you have a co-morphism of sites similarly eating uses a two-adjunction the level of the corresponding categories of stocks so the right joint as in the case of sheets is given by the restriction of the right pseudo kind extension while the left joint is related just to composition with f of course you have to stackify in order to make sure that you end up in the category of stocks then it's interesting also to remark that if the co-morphism is more over-continuous and we have already said that in the context of vibrations we actually have many such panthers and co-morphisms then we have a further joint a left joint to the inverse image of the morphism between categories of stocks which is given by the pseudo left can extension again with the composition and now what is nice about continuous co-morphisms of sites is that you see we have a result in the pre-shift bundle adjunction which I stressed as a very interesting result the fact that one could compute the inverse image of sheets in terms of the corresponding etal bundles as a pullback which was geometrically very pleasing so in fact we had an analog of that here so I mean you can describe the effect of the inverse image of stocks along a geometric morphisms in terms of pullbacks in the category of toposes involving zero toposes but this happens at the level of toposes and okay it's nice but here we have something more I mean if f is a continuous co-morphism of sites we can understand this pullback already at the site level in the sense that by considering a pseudo pullback in cut of the corresponding fibrations I mean for any stock we consider the corresponding fibrations sorry here is a typo we consider the pseudo pullback and we know that this pseudo pullback will be sent by the fact or inducing morphisms from co-morphisms of sites to a pullback in toposes so you see this is really an analog of this vibrational way of understanding the action of inverse images on the chips and in this case stocks so it's quite pleasing okay so now I would like to focus on the main topic of this last lecture namely relative toposes how relative toposes are formally defined well the analog of relative pre-ship toposes was essentially defined by Giro in his paper classifying toposes and in fact so the analog of pre-ship toposes in the stock setting is given by toposes of chips on the given stock or more generally the given fibrations with respect to the Giro topology so this actually is the right notion of relative pre-ship toposes well because we have this proposition which in fact generalizes a proposition already present in the paper of Giro which ensures us that these toposes as built by Giro satisfy the desired property because you see they are equivalent to categories of indexed from the opposite of D because here I have an index category D and I take the opposite of it it's just defined by the fiber one no problem so I have an opposite notion of index category and I consider the index function from the opposite of that to what is our analog for the topos of sets you see here we are in a relative topos theory so we change we replace the topos of sets with an arbitrary base topos and so we replace the topos of sets with the canonical stack of the topos and so you see that this is really what we want so this ensures that the Giro's construction is really the right one and we are going to say something interesting about the construction of such Giro toposes which will yield our fundamental adjunction as mentioned in the previous slide so in fact the main result is that we can represent a Giro topos in a very natural way so let's explain this so first we have to recall the notion of weighted co-limit so this is a notion of two categorical co-limit that is useful in many situations it's a very nice notion geometrically very appealing and it is really what we need in order to have a very nice construction for the Giro toposes so if you have two categories two week two categories C and D C and K and suppose you have a pseudo fan tour D on C op this will play the role of our weight and a pseudo fan tour R so this will be the diagram over which we are going to consider the co-limit so it will be a co-limit of R indexed by D so the de-weighted pseudo co-limit of R is defined as an object of the target category which satisfies this universal property that the category of arrows from this weighted pseudo co-limit to an arbitrary object should be equivalent to the category of weighted pseudo co-cones on the given R so what is a pseudo co-cone well we can visualize that as follows so what the changes with respect to the usual co-cones is that you don't have just one leg you can have multiple legs related to each other by two more reasons so you see the classical co-limits the conical co-limits are obtained by taking the weight to be the trivial one in which you have just one object category and you can, it's very natural to authorize having categories that are bigger than just the one object one the one object and one arrow one so here we authorize arbitrary categories as provided by a pseudo functor and so this is how a pseudo co-cone looks like and so our theorem is that a total topos can be realized as a weighted pseudo co-limit of a diagram of a total topos so here is the picture so you see our total topos are really those which come from the bases the base site so they correspond to the objects of C you see that these are actually slice toposes I mean sheaves on C or X and J X this can also be represented as the etal topos on the image of X under the canonical functor and similarly for Y so you see these are the etal toposes which arise in the co-limit representation and the legs of these weighted co-cones are provided by morphisms of fibrations which are in turn induced by the fiber the yonida lemma so basically if you want to get an object of a fibrations in a certain fiber well it suffices to take a morphism of fibrations from C over that object to the fibrations and of course since you can have several objects in one fiber this is why you have a weighted co-limit because you have several such legs which also can be related of course by morphisms because a fibrations is a category so you have morphism okay so this is the picture now the universal property of the weighted co-limit is precisely what gives us the fundamental junction I mentioned in the previous slides because we have two pseudo functors one pseudo functor lambda and another one gamma so lambda is what takes the Xerotopos construction which of course is functorial and in the converse direction the gamma functor is a sort of home functor from the canonical stack on the site to the given relative focus and indeed this junction can be seen as induced by a schizo-frenic object which is precisely given by the canonical stack so you see you home into that in different categories so sorry I cannot give a lot of details because of time constraints but of course if you are interested you will be able to study all of this carefully through the references I should give okay so now I would like just to briefly say what happens in the discrete setting because of course so far we have talked about arbitrary vibrations and arbitrary index categories of course pre-shifts can be seen as index categories and they correspond to discrete vibrations so it's natural to wonder what happens if we restrict this fundamental adjunction in the discrete setting so in order to get something which really generalizes the usual pre-shift bundle adjunction for topological spaces in the context of an arbitrary site and for arbitrary pre-shifts so for this there is just a size issue to be fixed so we are going to restrict to the geometric morphisms that are small relatively to the base in the sense that when we consider geometric morphisms from metal to opposes to them they should form a set because we want our pre-shifts to take values in sets so this is a necessary requirement but that's the only thing one has to care about because everything works perfectly well so we have an adjunction of one category where on the one hand we have pre-shifts on three and on the other hand we have relative so the two factors work like this so a pre-shift is sent to the corresponding localomia morphism but look this localomia morphism can be presented in terms of comorphisms of sites because you have the canonical vibration associated with P which is a comorphism of size and in fact the localomia morphism is precisely induced by such comorphism of sites so you can understand this in terms of comorphism in the converse direction the gamma functor acts like a home functor remember that in the topological setting the gamma functor was the functor of cross sections you see over open sets of the given bundle here what we have is a functor a home functor where instead of taking the open sets we take the categorification of that namely arbitrary objects of the base category you see identified with the corresponding discrete vibrations through the canonical stack so you see all of this is very natural very satisfying and in fact as in the topological setting this adjunction restricts to an equivalence between shifts on the one end and ethalomorphisms on the other now a pleasing property that the pre-shift bundle adjunction led to is the possibility of describing the shiftification functor as the composite of the two functors in the adjunction and here we have exactly the same thing so we can understand the shiftification as the composite of gamma and lambda this is very nice from a geometric viewpoint it allows us to really understand concretely all the elements of the shiftification you see so they are not formal things they are morphisms so an element of the shiftification is a geometric morphism which can be locally represented as being induced by morphisms of vibrations and in fact this relates strictly to the reconstruction of the shiftification functor in terms of locally matching families of elements of the pre-shift which I have already talked about in the first part of the course so you see that really it's nice to have such an adjunction because it really allows us to reason geometrically about all of this so to regard the toposes as they were topological states so just a few remarks about the descriptions for direct and inverse images of shifts in terms of vibrations that we have in the discrete setting because things simplify to a certain extent when we restrict to the discrete setting well we have already said that the direct image corresponds to taking a pullback so here there is no difference but concerning the action of the inverse image here we can refer to the relative comprehensive factorization to get a very nice description of that so more precisely the inverse image of a shift P under the geometric morphism induced by a morphism of size F is given by the discrete part of the K comprehensive factorization of the composite factor F composed with the vibration associated with P so you see this is a universal characterization of that and so you don't need to make your hands dirty to understand what it is and in fact we have also natural analog of in particular this last result in the context of starts but they are a little bit more sophisticated so I am not presenting them in this course but you see how natural is the language of vibration in this context ok so now we are ready to talk about relative shift opposites we have already identified the right analog of pre-shift opposites in the relative setting now it remains to consider the sub-toposes of such relative pre-shift opposites we know that really the right notion to consider is the notion of sub-topos because it is the invariant notion which is able to capture the notion of growth and decline so we are using that to define a relative site should be so starting from a small-generated site we consider first of all the pre-shift opposites so which is the shifts on the growth and decline of the given index category with the gerotopology and so for us a relative site will be just an index category in particular and a growth and decline topology on the corresponding growth and decline construction which contains the gerotopology because remember that sub-toposes of shifts on a certain site correspond precisely it was recalled yesterday in particular by Charles Redskin in his course so they correspond precisely to the growth and decline topologies on the category which contains the given topology so here this is how we define things so a relative topology is actually a topology on the growth and decline category so a growth and decline category is actually something because you see it takes into account what happens in the base so we can think of this as the horizontal component but it also takes into account what happens in the fibres and this is the vertical component so it's really a two-dimensional entity and so as you can imagine in terms of expressiveness we get really a lot by passing from ordinary sites to relative sites in particular we might wonder if it's possible to classify in some way these relative topologies for instance by using horizontal or vertical data so one can show that in fact not every growth and decline topology as you might naturally expect on the growth and decline category can be generated by just horizontal or vertical data or both of them because basically the idea is to get the diagonal it's not enough to get the two sites so you cannot reduce two dimensionality to one dimensionality but in practice it's a matter of fact that relative topologies actually are induced by sets of horizontal and vertical data so here is a list of examples so of course we have already encountered the relative topology on the canonical stack of geometric morphism and you can see easily that there are no chances in general that it's going to be generated by horizontal or vertical data on the other hand the girot topology is an example of a relative topology generated by horizontal data you see because it's something which entirely comes from the base in another direction growth and decline introduced the notion of a fiber site without putting any growth and decline topology on the base category so in that setting you have just vertical data and so the total topology on the growth and decline category that he defines is an example of a topology generated by vertical data of course as you can imagine there are also nice topologies that are generated by a mixture of horizontal and vertical data one such example is a topology we have introduced in a work with Axel Osman so this work is actually going to be presented in a contributed talk at the conference next week by Axel so this topology we have introduced for constructing the over topos at a model and it is a topology defined on the stack of its generalized elements it is an example of a mixed relative topology so it's a very interesting topology in fact is the join of two pure topologies and in fact we were able to get an explicit presentation for such a topology in terms of the basis so then with Ricardo we have been wondering whether one could get a more general result like I mean whenever we present a topology by using horizontal and vertical data it satisfies some nice compatibility condition between themselves is it possible to calculate a basis an explicit basis for the generated topology well in fact we have a very nice result about that showing that under some very natural assumptions it's possible to get explicit descriptions of basis of such topologies in particular I would like to point out that common toposis as introduced by the linear in the context of vanishing cycles etc in fact can be presented by using such kind of topologies and well I have a forecoming paper on that in which I provide explicit basis for this topologies that in general are quite hard to describe because they are presented by different sets of horizontal and vertical data so you see in general calculating a groten topology generated by the family of sieves is a non-trivial problem it can be very easy in some situations but in general certainly it's not a trivial so in that case it's not a trivial but we are in a situation like this where we we have horizontal and vertical generators that we can tell with respect to each other and therefore in such a situation in fact we have basis for describing these topologies which are given by multi compositions of horizontal covering families with vertical ones but again you will find the details in our work okay so finally I would like to mention that the kind of work in progress we are pursuing with Ricardo is to identify the logical counterpart of this geometric approach to relative purposes in order to introduce a sort of relative geometric logic so just a few remarks about this to illustrate the spirit in which we are working so in fact if you take the classical formulation of geometric logic there are no parameters there so it's because it was designed starting from from classical model fear you see it was inspired by first order logic in classical formulations so you have the sorts that are going to be interpreted as objects but then you have variables and you see for instance you don't have the parameters embedded in the formalism for instance suppose you want to formalize the notion of a vector space over a given field where the field for you is you want to think of it as a parameter so you want it to be fixed so there is no syntactic gadget embedded in the geometric logic formalism for doing that but you can introduce that of course without changing its level of expressivity and in fact it's very important to do that because if you stay just with the base you don't need to talk about parameters it is not so important but when you make changes of base and you want to reason in a vibrational way parameters become really fundamental and therefore it is important that they are fully integrated in the logical formalism if we want to have a very nice correspondence between logic and geometry as we have in the case of geometric logic so in fact the logical formalism of developing is a kind of vibrational higher order parametric logic in which the parameters basically are what happens in the base and by using these parameters actually one can increase the degree of expressivity of the logic in order to accommodate a certain number of higher order constructions so you see by changing the universe you can formulate things in a simple way relative to that universe but then the universe itself can be quite rich and so externally you will get something more expressive so that's the basic idea okay so that's all I wanted to say in this course so I just leave you with a list of references so some of the results I have presented come from the monograph I have already mentioned several times in the course of the talk so this is available on the archive already from last year then there is this very big monograph we are writing with Ricardo on relative opposites from which the results I have presented in the last part of the course are taken from so this should be available very soon on the archive hopefully already next week and in fact I would also like to mention that Ricardo is going to give a talk on the discrete fundamental adjunction at the conference next week so today I was very short on that deliberately because I would like to know more detail about if you are interested and then of course for all the things which concern the bridge technique and how it can be applied in different contexts also the general philosophy you can refer to my book and also another natural references my habilitation thesis which is written as an overview of different areas of mathematics which have been discovered in the last years and this can be downloaded from my website okay so I guess that's all so thank you very much for your attention