 zabijam, da je počkina vzelo posloženja v te ležne. Urtečno se je napravil datorizča energetiak 가čenja in tezna stacije,ny odpovi iz grimódov, ležno se je tako prop kad je način plansarje Honor consider, tako vzelo. Zelo tukaj, da se napravila posloženja v klasicajih radiopost, je prej vzelo veklu stojno, o čez način je sljela, njih za klasicajh posloženja. Pazisli, da zelo sem požegljala, da je zelo vzelo, da je zelo vzelo, da je začeljno objezal, da je začeljno objezal in da je vzelo, da je vzelo vzelo, da je vzelo in da je vzelo, integral domains, just to give an example of an illustration of how the actions transform into covenin states. So in the case of the Zalisti topos, the relevant theory is the theory of local rings. And even though the notion of local ring is apparently a second order notion, there exists a third order formalization of it, which is in following. If x plus y is invertible, then either it is invertible or y is invertible. And of course, you also require non-prediality, so you see what is there. So you have some following. So yes, the language is the language of commutative rings with unit. And what are the covenin states? So in this case, we can take as a theory of appreciate type the theory of commutative rings with unit. So the finitely presentable models of these are just the finitely generated rings, which is the same thing as the finitely presented rings. In this particular case, not in general. And so how do the covenin states? Because here we are taking states on the opposite of the category of finitely presentable models, where basically we have these localizations. The states that cover are those which contain these families of localizations, such that the ideal generated by these elements is one in the ring. So you can recognize, you see how the course one needs to carry out a little calculation to calculate the topology starting from the axiom, but you recognize where they come from. So for integral domains, what you have is where the axioms are again. And then we have the product equal to. And how the covenin states are made? Well, they are all of the form. Notice that we consider these families. You see, here apparently you start with just two things, but in order to have the transitivity property of growth and topology, we have to repeat these things a finite number of times. And this is why we end up with these things here. So here the condition is that the product of all these elements is zero. So you see very well in this situation that you see this corresponds to this condition and taking the portions. You see, each formula gives rise to a model presented by and then by pulling back these basic states along the arbitrariados, because of course when you consider a growth and topology generated, what you have to do is to take all the possible pullbacks and all the possible multi-compositions. So you can see that you end up with these things. So I just wanted to give this simple and classical example. And of course it would be interesting to take more to oppose these elements for algebraic geometry and try to describe specific theories that are classified by them. The subject has been made to a very, very little extent because even though the theory of classified hypothesis was born, as I said yesterday in the 70s, the subject has remained quite adornment, unexpectedly, because it was clear from the beginning that it was a very promising subject and got indik himself very convinced about it. But then research took a different direction and so these theories could essentially undeveloped. It is partly due to the fact that some people from the old generation of category theory sort of promoted a site-free ideology. I mean, they thought that sites were not really good objects to consider because of the fact that they are not canonical, they are not intrinsic, so they tried to replace toposes with concepts that didn't need sites to be studied or investigated. And the problem with that is that, of course, you really need sites in order to connect with mathematical practice, because without sites you can do axiomatic considerations, which might be very interesting as well, but you don't realize the kind of unification of which got indik was talking. Got indik, when it refers to unification, it really refers to the fact that one can build the toposes really from the ground, from objects such as sites or theories or group oids or quantals or whatever. I mean, from the fact that you can really build the toposes from the ground, and then this allows you to discover links between different theories by comparing the toposes that you can attach to them. I think that now classical toposes have actually come back to life, so we don't have to worry anymore, but this is just to explain why, even for very, very classical questions or even for very classical results, there is not much literature available. OK, so now I would like to explain in general in what consists this bridge technique, which is what allows actually to perform this invitation. So how does a bridge work in the abstract? We have already encountered, I think, four or five examples of bridges so far, but it's worth to systematize this in a more abstract way to know how to orient oneself in a variety of situations. OK, so as I have already said, the basic idea is to have different theories which are classified by toposes which might be equivalent or sometimes you might also be interested to compare theories whose classical toposes are not equivalent, but between which there is a strong kind of relation but for simplicity I will just treat the case in which you have an equivalence because in this way you have more invariance at your disposal to perform translations because essentially by an obvious metaphor in which you can formulate a categorical language with the automatically equivalent and so this gives you a great amount of possible invariance to consider and of course for each such invariant you will be able to try to make a translation see what this invariant tells you in the context of one theory and then what tells you in the context of another and now this is in some sense the logical outlook on that but since toposes are built out of sites most of the time this has a technical translation the situation where you have different sites of definitions for the same toposes so most of the time you work actually with sites so it's worth to consider situations like this and so how do you do well basically you consider an invariant I so by invariant I mean any kind of property or construction or notion that applies to toposes and that you can transfer across this equivalence so a property that you can decide to look indifferently from the point of view of the first representation or from the point of view of the second ok so this is sort of the deck of the bridge and then how do you build the arches of the bridge well what you do is to try to find properties in the context of the two different sites so here I write property p, cj here I write properties q, dk and I want essentially these properties to represent sort of invariant links in terms of the specific sites of definition so what I want essentially is imagine that i is a property ideally I want equivalence of this kind the site satisfies this property if and only if the corresponding topos satisfies the invariant and of course I want the same thing for the other site because this will give me the first arch and the other one will give me the second arch so imagine we have such a situation we can conclude that these two properties are equivalent and this can be interesting because even though at this level there is just one property when you look at the way this property manifests in the context of different sites you can have huge surprises in the sense that you can find that in the context of these sites this expresses in a certain way and you can find that in a context of a completely different site this expresses in a way that you would have never imagined without the aid of toposis to be related to the original property to convince you about this I will briefly mention a few simple invariants that one might want to consider so as invariants are you can take for instance the property of the topos the bullion the mordian equivalent to the pre-shift topos located and connected so you see people who practice algebraic geometry of course are used to particular invariants or the homological invariants that of course are very important and that they can attach to toposis and homology was actually the main motivation of the concept of topos but it does sound that these are by no means the only invariants that it is meaningful to consider on the topos there are many, many invariants here I have just put some of them but in fact there are infinitely many such invariants and you can even fabricate your own ones in a very simple way basically whatever a thing crosses your mind you write it down and automatically in fact it might seem a weird thing to two but in fact one has really to change the way one usually thinks in the sense that even a very weird invariant might give when translated into context of different styles might give properties that are unexpected there is one that should add which is a category of points yes, of course points so you see, we consider that the organization, you see this is a slightly weird invariant and nobody could expect a priori that it could be relevant to find that characteristic that are algebraic over the prime field I mean, it's something that certainly you would have not come out with but in fact the reality shows that so you see it has been to take this view seriously in the sense that even apparently weird invariants can possibly give deep and interesting results anyway just as an example let's take this invariant which is really a logically motivated invariant de Morgan's law so it means that for any subject and Boolean means that the lower excluded middle in the topos, in the internal logic of the topos and let's look at how these invariants express when you apply them to different sizes such as this Boolean, here, what you get is almost discrete almost discrete it means that every open is closed if you look at it here you get that C is a loophole the other invariant de Morgan here you get that X is extremely disconnected and here you get that C satisfies the amalgamation property which is this property so you see we have just considered two simple invariants and already you see the power of this it means that the closure of any open set is open so you see on the two sides you obtain very natural things but if I had asked you before do you think that there is a relationship between the property of the space to be extremely disconnected and the amalgamation property in the category what would you have enhanced? probably no relation at all while we have toposes, I mean you have many situations in which you have equivalences between toposes of this kind and toposes of that kind and so you can really use these things to build bridges which allow you to translate for example image you are interested in understanding if a space is extremely disconnected if you discover that the corresponding topos can be represented in an alternative way this will give you a way to an alternative how to look on your property and depending on the cases we will find something one has to really work with a sort of open spirit in the sense that you cannot control you cannot control a priori what you will get because it is a top down way of doing things what is almost this? it means that every open is closed in the you see the link with the boolean as you see the need of the property is contributed so there are many examples important ones where there is a theory such that for every topos E the models of T in E are again toposes can they be characterized? this I don't know because then you could repeat the operation and take models of T prime into the topos models of T E in terms of product I know that in the context of pardon? anyway we shall also give an illustration in the case of the Tuber-Eliutnes this is also interesting because Matteo has asked me a question about completeness now here is a simple application this using this environment Tuber-Eliutnes now Tuber-Eliutnes if you look at it from the point of view of the scientific site it means that there are just two sub-objects of one in the topos and they are distinct from each other and so you have you have just these two and they are distinct this is Tuber-Eliutnes so if you look at what it means in terms of the classifying topos of the theory then it means that T is say, geometrically in the sense that every geometric sentence over the language of the theory is either true or true no, no now, if you imagine that you represent this in another way so you consider a category C which satisfies the amalgamation property and you put on the opposite of it the atomic topology which is the drotenic topology in which the covering six are exactly the non-empty ones then you get a topos that what Tuber-Eliutnes means for this it means precisely joint and getting property on C no, no, it is on C no, no, no it is on C so you see again that here we have a property of completeness which is in general a very hard property to establish in logic which becomes equivalent to joint and getting property on a category which is something much, much simpler in general at least in terms of the statement because it tells you that any two objects can be mapped to a third one so you see it is much more tractable in many cases than completeness but it is equivalent to it if you have an equivalence like that and there are indeed the situations in which you have equivalences like that for instance in the context of the priceless construction in model theory you have basically a situation exactly like that in which you start from a theory of appreciate type t then what do you do when you start with its classifying topos and then you proceed to posing that here you have the atomic topology sorry here you have the amalgamation property so the fact that you have the amalgamation property allows you this is an hypothesis that you suppose it allows you to take sheets on the opposite of this with respect to the atomic topology since this is a sub-topos of that it corresponds on this side to a sub-topos of that but sub-topos of the classifying topos of the theory are classifying topos of the quotient of the theory so this means that I have quotient that I still don't know t prime such that I can make this commute now let's work on that what is atomic topos this is an atomic topos what is it? in atomic topos means that the sub-object lattices in the topos are complete atomic Boolean algebras so it means that you can cover every object as a disjoint co-product of atoms and what is the atomic topology ah it is the atomic topology so it means that the sieves which cover are exactly the non-empty ones so you have that and then you can start playing in variants and ridges so you immediately observe so here let's write the remaining part so here we have t prime let's say so you discover that the fact that the geometric property of amalgamation on this category implies that this topos is atomic so this is one arch of our bridge so we know that atomic that amalgamation property here implies atomicity here now we move here since it is an invariant and we descend there we discover that t prime is atomic as a theory so it means that every formula is very composed as a disjunction of formulas which behave as atoms with respect to probability in the sense that you don't have I mean these are formulas that make even context such that for any other formula which I can for any other formula in the same context when I consider this conjunction I know that this is between and false and we say that the formula is say atomic in the terminology is complete if either this is isomorphic to that or it is isomorphic to that so you don't have anything intermediate so as you can see again you have started with something geometric here and you end up with atomicity here which apparently was completely unrelated so you see how many surprises one gets and then as I have already mentioned if I take joint emitting property here then here I get two values and then I descend there and I get completeness of the theory but what ensures that the category is non empty no you have to suppose it sorry I can't say but suppose this category here to be non empty but in fact it suffices to suppose that this is non empty to have a non degenerate purpose it suffices so of course if it is empty it's not interesting ok so if I understand correctly you also need because you see as the final object joint emitting property is trivial yes but here I am stating things in full generality so I am not supposing any property of initial or terminal object but it's true that of course if you have an initial object then joint emitting follows from amalgamation but I am not supposing it ok so you see here from these two geometric properties we have arrived in syntactic properties but it is not finished yet because we have not talked about the points of these purposes so this is our third invariant you see the scheme is always the same I am just changing the invariant so here I put the notional point and here of course this is by definition the classifying scope of t prime I find the t prime models in set here I find the so called t models so what is an homogeneous model well it is a model m such that when you have finitely presentable models a and b with an arrow like that and an arrow like that you can so you see what we can conclude from all of that is that starting from a theory of pre-shaped type such that the category prezentable models is non-empty and satisfies the imagination property then the theory t prime which axiomatizes the homogeneous models is atomic and complete if and only if this category satisfies the joint embedding property this represents a broad generalization of a classical theorem due to Freisse in model theorem so I decided to present it just to illustrate the fact that you see how this works apparently one would not expect this very abstract invariance to yield completely concrete properties when you unravel them in terms of size but this is what you get and just an example then of course you can go on and you can wonder if you still want this in an alternative way for instance you might wonder when is it the case that I can represent my topos as this precise topos when can I represent it as a Galois topos I mean Galois type topos I mean topos of continuous actions of a certain topological group of automorphisms of a certain structure that of course I want to be a model of my theory and possibly of the portion that you try me as well so I can wonder when it is the case that I have equivalences of this kind because you see this is a sort of general framework for analyzing the Galois theories of Galois here as a theory of pre-shift type you can take you start to be the Galois extension and you take all the intermediate extensions which are filtered for limits of finite extensions so this will be just the finite extensions and this will be the Galois group so as you can see if you have a classical Galois context this rises to representation of this kind if you want to build the Galois theories in a more general setting you can ask yourself the question when it is the case that I can represent toposis like this in this way now of course you would not be satisfied just with an equivalence and the level of toposis you would like this to restrict to an equivalence and the level of sides in some sense because you want something more complete Olivia, just as a preliminary of course it is abuse for you but you should say that for any topological group yeah you have a toposis so for any topological group set the discrete set under with a continuous section and it is atomic with two values yes always for any topological group you take all the actions where here you consider this as a discrete set not necessarily finite and this was with this Galois category and you require this to be continuous by endowing this with the discrete topology and I presume you have very few points for this toposis but two isomorphism probably well yes exactly because this will be things like algebraic closure so you see for instance it's pretty rigid so you wonder whether you have equivalences of this kind so you would like of course these equivalences to be induced as in classical Galois theory so basically here what would you have by the unit endaging you go there and you would like actually this to take value in the non-empty transitive actions of the same group here I have not said that normally there is no topology to put on the automorphism group of a certain model it is the so-called point wise conversions topology so you wonder also when it is the case that this restricts to a counter like that and of course one would like this counter to be an equivalence in order because this is the case in classical Galois theory you really have an equivalence already in the development size now it turns out that the fact of embedding everything in a topos allows you to discover conditions necessary and sufficient conditions for the existence of these Galois-type theories why? why do you think the topology is the atomic topology? well it is forced by the fact that when you consider such a topos this is always a valued atomic topos so I have said that in atomic topos you can always decompose the object as a disjoint union of atoms which are the atoms in this situation where they are exactly the non-empty transitive actions so you see you decompose transitive actions just as a union of all the orbits ok so you see you understand that this always you can represent by taking sheets on just the transitive action with respect to the atomic topology because this is the topology that you get when you induce the canonical topology that you have here on this full subcategory so it is determined by that in the sense that when you induce given the fact that this is an atom you just get the atomic topology so you see it looks very similar to what you get in the other framework and so it is natural to wonder when you have that now if you realize again topos is help you a lot to solve this kind of problems because you see starting from so you would like this to be a functor that sends you would like this to be the functor that sends C and finally present a role model to of the C to N and order with the action of the automorphisms of N this is what you would like now you realize that basically every object here is sent to an atom here but then it is sent because of the fact that the this topology is atomic and basically I have an atom here then I go here and so it means that I have an atom here but an atom here means transitive non-empty action so necessarily this will take place in the transitive non-empty actions by purely abstract considerations the problem is first when does such an equivalence exist and if a delivery of toposis and then such a restriction is a full and fake punter or even an equivalence so how to answer this question well first of all we care about the equivalence of the of the toposis so so you realize that essentially that it is natural to put on M so M is universal because I have observed that I want my function to take values in the transitive non-empty actions of that but I want this function to be given by taking M from C to M so I want this M to be non-empty this is universality so I want that for each M for each C I have an arrow from C to M I want that in response to say that the corresponding action is non-empty so this is non-empty and then I have the condition of transitivity so transitivity what does it mean that if I have two different ways to send C into M I have a way to send one to the other through an automotivism and this corresponds to transitivity necessary conditions but a theory that can be proved by using logical techniques and I have not been able to find a proof which is concrete only in particular phases but in full generality I only have logical proof for that these conditions are sufficient as well so basically if you have that M is universal and ultramusinus now the only thing that remains to be seen is if this factor here which I call F why is for even an equivalence because already full and faithful would be quite good because it means that at least you can embed your category as subcategory of this category of transitivity non-empty actions but of course equivalence would be even better because basically it would mean that every open subgroup of this automorphism group come from something, from some C from some object C in your category I mean in the case of classical galva theory you really have an equivalence but you might be content already with something like that because if you are in a situation like that you can wonder what you have to do to complete your category in order to get something that is equivalent to the category on that side and in fact I have described a procedure to complete a category which satisfies the first condition to a full galva theory this is a procedure that consists in a sort of addition of imaginaries basically if you have the first condition but not the second some quotient by natural equivalence relations that you can take in the first context they are not realized by something by the category itself and so in order for them to be realized you have to enlarge the category but it is a completely explicit you can give a way to complete it by it's a very calculatory activity I mean I should stress that the toposies when you look at them from the point of view of sites they become very computational devices in the sense that you can really compute very, very well not just with sites but even with other way other objects that you can use to represent toposies such as groupos sites are the main way to represent them and I can tell you that for many of those invariants that I mentioned they made extremely well in terms of sites in the sense that you can find site characterizations that give you the arches of your bridges in many, many situations in some cases you can even find them by purely mechanical calculations for instance the results about Boolean or the Morgan all the logical conditions that you can put on the internal sub-object classifier of the toposies this you can do in a completely mechanical way this means that it is extremely well beheld but let's come back to this so how to characterize when f is full and faithful well you realize that full and faithfulness of f corresponds precisely you can look at f as taking values here in the topos and it corresponds precisely to saying that this growth in the topology is sub-canonical and this in turn you can express by saying that all the arrows here are strict one more reasons which is a completely concrete condition so we have an answer to the first question and so thanks to this for instance you can discover many new Galois type theories in different areas for instance we have Galois purely for finite groups in that sense for Boolean algebras for graphs etc these are all situations in which the first condition is satisfied for the second condition for instance all the context that grew in little bit in superior Galois category provide actually equivalence as well but you might not have an equivalence in general due to the fact that there might be atoms of this topos which are not representable in the sense that you see you know that all the objects here give rise to atoms here but in principle there could be atoms here that do not come from there so you see you can formulate a completely explicit condition on this category for it to be atomically closed in the sense that every atom that you consider here comes from something there so again it is completely explicit I do not write it down if you are interested all of this is described in this paper that is called topological theory and again what I want to stress here is that really it is topospheric invariance that guide you in this sort of otherwise very concrete investigation because after all you are really looking for concrete correspondence between open subgroups on one end and your finitely presented objects on the other so it is a completely concrete problem but in some sense it is really fruit opposites that you get the right notions that you will get the right conditions because all I have said here is also necessary so you see in some sense it is the optimal framework that you can use to address this kind of problem if you want and so as you can see it is always a game of invariance and how to express this invariance on one side or the other ok, now you could mention that this general setup in this sort of yes, this I have said you were in fact I was a bit upset did you one shot the example of the Samuel Topos? no, the Samuel Topos not but this is really the simplest possible one so if you take as a theory of pre-shift type the theory of it is called the theory of decidable objects in which you have just one sort and you have this difference predicate with the axioms that I brought down yesterday at the request of Goyal so this will be the basic theory of pre-shift type I and basically what you have is that so of course you have a category I category of finite sets and injections of finite representable models of this theory so this is the Topos and you can represent it as continuous actions of in automorphism group of whatever infinite set you might want but I cannot replace any with any S and this is very interesting because you see whatever cardinality and that is very interesting because imagine that you want to put into so in particular you have that continuous actions of a certain S is the same thing as of the automorphism group of S I suppose that this Topos has only one point no, it has every infinite set as a point it's continuous action of a group yes so let's take the group to be finite yes no, but it's not a finite because here I don't have a finite group it is auto-infinite set no, no, I understand that what I don't understand is that when we take so this is it's not the same thing as taking the group itself and taking the dual of it as a small category with a single object it's not the same thing no, these are each of the things in which you only have no no, this is continuous the topology in place of all no, of course so now you see here it is interesting because you cannot put S and S prime in relation to which are directly we have no way but they are unified through the Topos so you see even in a very simple situation like this I mean even at the level of automorphism groups you cannot compare very often but still when you pass to the Topos they are unified so you see all of these illustrate the fact that if you really want to perform unification Toposes are in some sense the best way to go because no, it is infinite you take S infinite and you say for any infinite S any infinite S so you see in fact I wanted to apply this framework about this both the algorithm and generalize to the study of the independence of L in eladic Comology because I have written a paper about that recently which is called Motivic Toposis and I suggest that this eladic Comology should be homogeneous for a certain appropriate choice of theory of bishop type and this of course is a longer term project and I don't expect to come up with an answer I don't know whether they are actually homogeneous or not but if they were in some sense it would explain the fact that we cannot really compare these different homologes in a direct way because they are defined over different coefficient fields so it is not natural to don't have a dictionary a natural dictionary for comparing them but this could provide a dictionary if it works so I'm not claiming it will work there are several indications that show that it is not a completely unreasonable point of view on the subject especially because the exactness condition for Comology theories follows as a particular case of homogeneity so in some sense it is an indication of the fact that at least the approach is relevant it is not completely maybe you could also at the remark that not depending on the fact that this approach works or not anyway this general setup of specifying opposites which are totally counter-valued encompass in the same framework the Galois approach and the Tanaka approach which means we don't have to distinguish anymore between what is linear and what is non-linear yes, if you want this is also another aspect ok, anyway how much time do I have? ah, perfect so I would like to conclude with more general considerations and a few other examples and these were the main ones but of this technical bridge because I have pointed out that the two basic ingredients of bridges are tags of bridges which are given by mojite equivalences which are given by site characterization I say a few words more about each of these topics we have already told a lot about the site characterization that total theoretic invariance express often in very natural ways in terms of sites but we haven't really said too much about the first topic mojite equivalences because apparently the notion of mojite equivalences is a very restricted one I mean at first site because you have that two theories are mojite equivalent when they have equivalent categories of models in every topos naturally in the topos so this by definition of classifying topos of course so you have that for all e naturally in e so apparently if you look at this definition it might seem very rare very rarely occurring in mathematics because after all you require that the categories of models should be equivalent not just in sites in particular you also require them to be equivalent in sites but you require them in some sense to be equivalent from the point of view of whatever topos you might choose and since we have infinitely many topos they resemble to one and other for many aspects but there are also differences between them so a priority is not clear whether there are many mojite equivalences around or not so I would like to make just a few general considerations about this especially because Ancon has asked about the relationship between equivalence of categories of models in sets and now what we can say is that suppose that you have a practice that first you discover equivalence like that and then you wonder is it a mojite equivalence because of course imagine that you can lift a mojite equivalence then you can apply intelligence and this as we have seen allows you to translate a lot of things in surprising ways so do you have a high likelihood that your equivalence lifts or not? Well it depends on how you have built this equivalence of course it is not in general true that you can lift because for instance there are geometrical theories that have no points that have no set based models so you see from the point of view of sets they could be perfectly equivalent to the contradictory theory but without bringing the contradictory themselves so for sure it is not always true but if when you have established this equivalence you have only used constructive principles because you need this in order to be able to generalize the definition of the two punctors forming the equivalence you want to lift it so you want not to have exploited anything inherently set theoretical when you have done this so if you so for instance you cannot use the law of expression admitted you cannot use the axon of choice you have to but if you have been careful to avoid this and if the punctors that you have used to define the equivalence are in some sense geometric in the sense that you have only used geometric constructions such as finite limits and arbitrary coordinates in defining this then you can reasonably expect this to lift so this is a sort of realistic principle of course but we have applied it with Anna Karla Rousse that will give a talk on this in the contributed session on Wednesday we have applied this to two known equivalences in the context of the algebra and we were able to by making a careful analysis of how these punctors were defined we were able to lift them we were able to discover new results that the specialists of the area couldn't attain with their meters for instance we had an equivalence which was an interesting one between on one end the category of Boolean of Nd algebras on the other end the category of L of a billion which is an infinity theory so you see you have on one end a finite algebraic theory on the other end an infinity theory but Moviči proved that they are equivalent but the two theories are not by interpretable in the sense that you can prove that the scientific categories of the two are not equivalent we proved that it was an interesting question because of that it could be all the more interesting to find the more equivalent because for instance the duality theorem tells you that if you have more equivalence even though you don't have by interpretability you have a bijective correspondence between the quotients of the two theories and this is by no means trivial because imagine if you have a dictionary of course you know how to do because you take each formula and you send it on the other end so it is completely trivial but imagine if you don't have still we managed to show that you have a bijective correspondence and we were able also to describe how this correspondence works but it doesn't work in a linear way of course and this is the kind of things that specialists in that area couldn't see because one really needed the double-spirited viewpoint because the first thing I wanted to say the link between between equivalence in sets and equivalence in arbitrary topos so I have already talked about by interpretability I should also mention that all the classical notions of more equivalent that have been considered for many structures such as I have already mentioned associative rings before in topological groups in versus semi groups small categories so all the classical notions of more equivalence fit into this in the sense that you can attach to those structure toposes and then the more equivalences become equivalent to equivalences of these classifying toposes so in some sense this general notion of more equivalence encompasses most of the notions of more equivalences in the literature now there is something that I would like also to say it's about duality theory you might wonder what is the link between more equivalences and dualities because you see here we have talked about covariance equivalences but not about dualities I mean equivalences between a concrete category and the opposite of another concrete category of course this is will not be a more equivalence in that sense but there is a nice link between the two in the sense that dualities this is a sort of slogan if you want so it means that so when you have duality you can consider each pair of objects related by the duality and then what happens very frequently is that this pair of object is related the two objects of the pair are related to each other through a topos that you can construct from each of them separately and so you have in some sense a bridge for each pair and then by focalizing this bridge you get duality just an example will be clearer than many words so take duality very simple duality for instance for a Boolean algebra so you have a Boolean algebra and the corresponding in stone space that is complete or not complete or not no later I will also discuss the case of complete atomic Boolean algebra for those where you have another duality you have Lindenbaum here duality this is for complete atomic Boolean algebra discuss later so here stone duality so of course it is duality between the category of Boolean algebras and the category of stone spaces and here basically you see that for each Boolean algebra you have a topos that you obtain by equipping this algebra with the coherent topology which is this is a distributive lattice so it is a coherent category it is a coherent full set category so you can certainly put the coherent topology on it and what you get is that it is precisely equivalent to the topos of sheets on the space so you see the way you get the duality out of that is as follows so you imagine you want to make a b variable so here you have b and here you have that so if you take a morphism of Boolean algebra here what you get there is a is a geometric morphism going here then you transfer the geometric morphism here and since these spaces are outside then you can exit the bridge there and so you get so it was going there and so it is going here so you see that j is coherent coherent it means that finite zones so the cobering sheets are those which contain finite zones so I mean things like that d d1 so you require this and for linear boundary duality it is also cute in the sense that if you have such an atomic Boolean algebra complete atomic Boolean algebra a then you know that you can reconstruct it from the set of its atoms just by taking the power step so it is starting very trivial at least at the level of objects it is slightly more interesting kind of understanding that the topos is provided on that because in the other direction it works like that so you see this is the duality and topos theoretically it is like that so you have on one end the topos of sheets on A we respect to the canonical topology here you take all joints this is equivalent to atoms you see when you take the arrows between this complete Boolean algebra you don't just in this reality you don't just take the prime arrows you take all the complete arrows the ones that preserve also infima, arbitrary infima so conceptually it is not completely clear why you take those now the topos theoretically explains you this as well because you see when you imagine you have just a frame morphism here then for sure you have a geometric morphism here but then you wonder when it is the case that this morphism comes from a function you see between this not all geometric morphisms here come from geometric morphisms induced by this they are exactly the essential geometric morphisms essential means that the inverse image has a left adjoint having a left adjoint in the context of means exactly so you understand exactly why you need this coefficient anyway this was just we illustrate that can you do something similar in the same thing for local? well yes of course I mean you might take care about this which is called the topos theoretic approach to strong type dualities all with local so you need to work with local because it is more I mean when you work with topos as Jojad said they are strict generalizations of local I mean for the duality between the frames and local does it fall into this pattern? yes if you want but it is sort of quite trivial adjunct so I mean these are a bit more content content full dualities than that one anyway so this is just to explain this slogan now of course I mean these are maybe elementary examples that we will illustrate in the point how much should we have now? 5 ok so 8 if you want sorry? if you want you can have 8 minutes oh excellent we will see what the topos is this side ok so finally I would like to say this is actually formalizes in this situation the feeling of looking at a certain object in different ways so when you feel that you can look at something when I say look at something I might mean you can construct something or you can represent something in a certain way or in another it is very likely that you are in presence of a more equivalent this I tell you because even in presence it might seem incredible but it's like that because a mathematical theory is not like a dead object I mean it is something that starting from the axioms it starts developing so whatever new result that you find in the theory allows you to split the theory into the past and the future in some sense I mean what you already know to be true in the theory and what is potentially true in the sense of approval about your theory will give rise to a different representation of its classifying topos by the duality theory about the subtosis because you see if you look at a certain theory t as a quotient of a certain theory s then you will get a representation of its classifying topos as a subtropos of this so it will be you see it will be the scientific category of this with a certain topology so you see it will depend on s so if I change s I will get many different and it is not innocent as it might seem for instance, because as I said the classical topos are very sensitive to logic so even if you make a small change at the level of logic you can have huge differences in the difficulty or easiness of calculations that you might want to perform for instance I mentioned the paper that I brought with Peter Johnson about the Morgan's law in the theory of fields first we tried to compute the invariant that we wanted to compute by using the representation of the theory of fields as a quotient of the theory of rings just commutative rings and we didn't manage to calculate then we changed the theory we chose the theory of von Neumann regular rings which is just a little variation because after all of course I mean the only thing you have to observe in which field is a von Neumann regular ring which is completely trivial well it means that you have assumed the inverse so I mean instead of having just an inverse you require you see to have something like that you see something like that so you see it is completely trivial but for us the calculations became much easier because we got that the topology was sub-canonical in that case but not in the other case and so we managed to calculate much better so you see even a small difference can make a huge difference at the level of calculation Did you try the skew fields? No, no I didn't try anyway just to illustrate so we have talked about the big zariski spectrum of a ring let's talk about the small the Pt topos so of a ring so we have just skews on the zariski spectrum of a and here it's a very simple object but even if it is simple you can come up with different points of view on it and each of these points of view translates into a different way of representing these topos this is the classifying of a certain theory which axiomatizes the prime filters of the ring the prime filters are exactly the compliments of prime ideas so you have a logical theory so this you can see as a topological representation and this is a logical representation and then you can also represent these as skews on a distributive lattice with respect to the coherent topology because it turns out that this theory is actually coherent and so by using the coherent way of building classifying the way of building classifying topos for coherent theories here by taking the syntactic the coherent syntactic category of this theory you get an alternative representation of the distributive lattice so if you want to work with distributive lattices instead of topological spaces you mean the theory of the prime filters and the ring is fixed and you axiomatize it these are propositional theories propositional theory is a theory over a signature which has no sorts and propositional so you only have zero re relation symbols and propositional theories correspond exactly to localic toposis in the sense that a localic topos is always classified by propositional theory and conversely any propositional theory has a localic but you don't see how can you axiomatize it as a first order theorem it is first order but you have to quantify on these subsets that we no no no because you take one symbol for each element so in this way so but then it's not finished because you also have the possibility of cutting this site down to a mixed semi lattice because it sounds that actually you don't need this whole thing it suffices to work with a mixed semi lattice which is given by you first look at a as a monoid yes and just by taking the multiplication and then you quotient by the smallest equivalent relation that identifies a with a square and then you get this monoid and notice that here you have only told about multiplication they wrote a diptopology that you will put, will talk about addition and so the unification of these two things will give the same result and of course it doesn't finish here so you can always consider the frame of radical ideas of A and then you have the canonical support on that so you see, this was just to illustrate that every different point of view you have translated in something like that this is useful because for instance imagine that instead of considering just spectra made by prime ideals you want to consider spectra made by maximal ideals for instance what you want so well there are situations so in these situations what do you do what you observe that here you have a subtopos if you take the induced topology and then you can localize everything else in particular and this can be surprising because you are sure that you can axiomatize the complements of prime ideals in first of the logic but it is not a priori clear that the notion of maximal ideal allows you to do that as well and sorry, by the surprise you know that by the duality theory for subtoposis you know that there exists a theory there exists a theory and you don't know what it is but you know that there exists a geometry theory over the same language how do you handle the morphisms for this max the morphisms A is fixed I am fixing A so you see here I know that there exists a theory a geometric theory over the same signature as the one that I had used for describing the prime filters that works for the maximal things this can be surprising sorry, you don't expect there to be a first of the thing but you know for instance that if this is a sober and I mean normally this is Ausdorf so it is necessary sober you know that the set based models of this will be exactly they will be in correspondence let's say with the maximal ideas of A and it is actually possible to calculate this theory so I have just as an exercise I have calculated it because you see the duality theory is not just something abstract in disguise something that allows you really to calculate and so it is actually possible to calculate this and so you get a theory that actually axomatizes the complements of maximal ideas and well it is useful for something but anyway it is just to say that you have a guide to work you see everything is very continuous I can make a change and this change has consequences everywhere you see everything is connected so I mean to conclude I would just like to say that well I hope to have suggested to you that you did act very effectively as unifying spaces across different situations and even inside even mathematical fields not necessarily for connecting things belonging to different mathematical fields but just working you see as in this situation with a specific structure it can be very useful so it is a very technically and of course as you can see in all this story sites are fundamental so I mean by working with sites I have attracted a lot of obstinity from the people that I was mentioning earlier in the talk it is very unfortunate because I mean sites as you can see are really central so I hope that this controversy will be done and as beautiful mathematicians that have been reproduced.