 No, da bojte. Nisem, da se kaj sem odmah vse pravite. Tako je, da se nekaj vse prekazam odmah. Da si dobro vse, da sekaj sem izgledam. Da sem izgleda, da je to, da imamo tukaj, NEkaj, da je tukaj, nekaj poskupen. Nekej poskupen, da pa je zelo, da je tukaj, da je tega. To je tukaj, da se nekaj poskupen. si neko nekaj tričenih tih komandov, ki je počesnotov na načinitvi q.a. počesnotov na načinitvi q.a. Všem da ne da bo početno zpečiti z Grozendikov na koncerti tih komandov ta pripovodil, kaj se je zpromečnja izgleda tih komandov, Gorod endik se presented, ki je tozda, ki je tozda, načinj robo, ki sem kromsi, geometri in algbara ne bo, tukameneci, in rythmeti. Matematika, logika in kategorij. Vrk the wife of the continuous is then he has, je to, da sem jasne to Potom, asi jasne, pa izprav. Kjer smo ne zamo dojevali iz ležnosti geometriku, vse znači, zato je zelo zelo svečen, ko je vse možno izvečen, z nebezavršenimi zvom, vsakimi vizivnih vzivnih. Zato mi zelo lahko spredal vse, da izgledeš dob to pos, ker to posi je, da nekaj zelo se razgardez, nekaj zelo se zelo razgardez, participated by Grotendik, as sort of unifying spaces in mathematics. So what Grotendik is saying there is essentially that one can build that opposite starting from the most disperate mathematical context. And so, topposes are unifying in the sense that one can build in some sense the same kind of entity, the same kind of mathematical object. z kompletnih čelvih kontekstov. In, vseč, da je težka v težkjih toposfjeri, je se vsečo včasila o vsej všim zrekeljih, da je srednje vsečje, da se betreboje vseče vseče in vseči vseči vseči vseči vseči. in je vseča tukaj vseči vseči vseči vseči vseči vseči vseči vseči. Proti, to je zelo več dinamijne svoje bovno, ki ne različi vse vse zvori, ki bine zelo izgleda, ko se ne bo. Tato je površenje z tem, kaj je odrečen, površenje z drugi teoriji, in je dovolj, da je je zelo hvaljena vsobečen, je vzivno, pričo vsev tega matematika, o čas na izgledaj, površena površenja, izgledaj z nočenem topu. Zelo, da sem bilo predstavljena na svojo tudi, v kursu tudi. Zelo, da sem početila tudi pravi konceptučne remače o zelo, zelo, in kako tudi zelo tudi spesivne matematice v tudi, metod in vse, vzelo, da je tudi da kratične v nekaj pravdu. Po zelo je tudi tudi vrč interdisciplinarne vsočne. Se vzelo, da vzelo je toga in zelo vsega in vsega. In vsega je transversena v mene za vse nočnje matematika. In za tojo razljete, jel je to pričo komplementar, da tukaj nih vse njeli v nekaj pravdu. Sreč je, da najbolj je, da je prihledaj do vsepojstvenke vzvrče vetradi, je, da se prihledaj do vsepojstvenke vzvrče, in da se razkovno zelo n sei prihledaj, kaj bi je instaj, da vsepojstvenke vzvrče se in vsi vzvrče, ki je nema neč nezavrčen, neč nezavrčen, neč nezavrčen, neč nezavrčen, neč nezavrčen. mostly the interesting results are obtained when there is a combination between the concrete expertise available. In this new perspective, because really it opposes make you think in a rather different way about mathematical reality. So it is really thanks to this interplay that one can get the most interesting results. that one should go from the abstract to the concrete and really work on different levels. In in fact what it turns out is that reallyroprimiş techniques things are liable to make you see things that would be almost invisible otherwise or at least very hard to attain And using alternative methods. In in spite of the generality of the notion of topos, the kind of insights that toposes can generate are quite deep. In particular, the connections that toposes can establish can be highly non-trivial, from a technical viewpoint. To je zelo vse več, da je vse izgleda, da je neskrišno zelo, boš je objev, ko je in vsega, ali, da je hvalo in nekaj matematik, zelo je to nekaj počke, na matematika, je bilo nekaj objev, ki je vse od njega počke, ali je je izgleda. A z nami je tudi nekaj zelo vsočen. Zdaj se zelo vizivajo na detelj, da so vzivne objev Jopoz se vzivova po tudi unifajenih spasih, kako je. Tudi, in vse vzivno, je vzivno objev je vzivnja o nekaj zvržanjama vzovu. V temneče toplji so občosti, o kjer bi je tudi svoj poslednjami in z njeljavim projčaj v počke. To je, da je tudi je najklasne počke, kako je vznala iz konceptu toplji na jah se z nekaj 40, 50-ci ljudi, zato je izvedena z groznjima. Petro deposit would be in the context of Algebraic geometry. So, I wrote this in the first appearance of the topics in the context of Algebraic geometry. Toposes were, in some sense invented by Grotendick to use his own words as a metamorphosis of the notion of space. I will say more about this in a minute. In fact, this gives the first perspective on the notion of the topos. Zavvisi se ja. David Ocho vpraša je bilo, ljudi se razstavila ja. Oko se ga iz odručila, naredilo lahko, da bih vse laboratory pošli na mene, da je pomečena in pošlo se koncerjala v sez Afremije o fakti. Nerečijo zame na pomečenje, začal tega na uročaj vse. To se predstavila vse. kot nekaj, na različenem različenem. Pogledaj je tudi vrštno obježenje, ki je to tudi vzostaj tudi vzostajvali v nekaj priklju formu, vzostajvali je zvršenje, vzostajvali je tudi modulo, to, ko je početnja, ko se nekaj teželj, da teželj, nekaj semanticoj. da vse tebi tebi tebi tebi tebi vseh ljubov. Zato vse tebi je to posačaj, kjer je prijevila občas, kaj je semeničnja tebi, v semeničnih semeničnih semeničnih semeničnih. So now let's say more about each of these different points of view. So I already mentioned that the inventor of the notion of topos is Grotendik, and in fact the original motivation was to enlarge the realm of applicability of the topological intuition well beyond the classical setting of topological spaces. Grotendik realize that there were many situations in particular with respect to homology in which in some sense topological spaces were a true restricted environment to work in and that it would make sense to try to enlarge the realm of applicability of topological intuition and also in other contexts. And the key idea was to pass from a topological space X to the category of sheaves of sets on that space. So this is how the metamorphosis took place, because you see we are going from a concept to a quite different one. So the concept of a topological space is not at all a categorical concept. On the other hand the category of sheaves on this space is, I would say, a maximally structured category in the sense that it is a very rich category in terms of structure. It has all small limits, co-limits, it has exponentials, it has even a sub-object classifier. So you see you start from a topological space and you build out of it an extremely rich category which nonetheless is strict related with the space you have started with because in fact many important properties of topological spaces can be naturally formulated as invariant properties of these categories of sheaves. When I say invariant properties I mean really categorical properties. I mean properties that are invariant under the notion of identity for toposis given by categorical equivalence. So this was the first remark which justified in some sense this metamorphosis of the notion of space. And then of course what is very interesting about this idea is that it can be generalized. So one cannot define sheaves not only on a topological space, but on something much more general than that. One can take sheaves on a site. So a site is basically just a pair consisting of a small category C and a so-called grotendic topology on C which is something which defines a notion of covering of objects of the category by families of arrows going to it and which is of course required to satisfy some natural conditions. And so basically grotendic show that one can define sheaves not just on topological spaces which basically amounts to define sheaves on the associated canonical site which is given by the category of open sets of the space with the canonical grotendic topology on that which is given by the classical notion of covering of an open set by a family of open subsets. So basically sheaves are defined as particular pre-sheves, so pre-sheves are just contravariant functors with values in sets defined on the category and one among these pre-sheves selects those which satisfy the gluing condition corresponding to the sheaves in the topology. And in this way one gets a category satisfying the same remarkable formal properties that the category of sheaves on a topological space. So all the categories of this kind have this very rich structure which is extremely pleasant to work with because you have everything you want essentially, you have all small limits, co-limits, exponentials and sub-object classifiers so you can make computations very, very effectively in a category of this kind. While on a site you might find yourself in trouble if you want to perform certain operations because in general a site will be just a small category and so for instance this category might not be closed with respect to certain operations such as I don't know quotients by equivalence relations or products or co-products and so on. So you see all these problems disappear when you pass to the topos. In the topos you can perform all these computations and this is I think a quite striking aspect of toposis which accounts for their great relevance in mathematics. So originally toposis were in fact introduced to define the homology theories so this was the original aim, I mean to define the veil type homology theories and in fact the homological invariants are still I could say the most used and invariance of toposis that are used in mathematics even though of course they are by no means the only invariance that one can consider on toposis but what one can say is that they have certainly had a tremendous impact in the development of modern algebraic geometry and the refoundation and generalization of homology through toposis that was made by Grotendik has really allowed to think homology in a much more powerful way and in particular it has led Grotendik to introduce this six-operation formalism which by the way still awaits for a fully unified topospheric treatment because there are many instances of these six operations in different mathematical context but what is still missing is a general topospheric theorem which would imply all these formalisms in different settings and Grotendik explicitly asked for that in fact he recalled the same idea but this is something which has not yet been done but nonetheless I mean homology and viewed through the lenses of toposphere has had a tremendous impact in the development of mathematics not just algebraic geometry and in fact this is also due to the greater flexibility of the notion of topos because in fact Grotendik thanks to the topospheric definition of homology was able to go well beyond the classical homology theories that were considered for topological spaces or for groups and was able to define also pretty exotic homology theories that turned out to be quite useful in algebraic geometry situations and in particular for solving the veil conjectures so I mean these homological invariants are of course very important but in fact there are many other invariants that one can consider on toposis in particular theomotopy theoretic invariants in which the experimental group can be defined at the toposporetic level and there is also recent development of the theory of higher toposis which allows to treat theomotopy even in a more richer and most effective way now of course these are quite deep invariants in fact if one thinks about toposporetic invariants in general one realizes that in fact there are infinitely many of them because you see what is an invariant well it is an ocean or a construction which you can transfer across an equivalence of toposis but what is an equivalence of toposis is just the equivalence of the underlying categories so you see by an obvious metaphorium whatever you formulate in categorical language will be invariant under categorical equivalence so this means that you dispose of infinitely many invariants of toposis and so this is also quite striking aspect of the theory because you have these very very rich categories on which you can consider infinitely many invariants of whatever nature essentially of algebraic, logical, geometric nature and of course you can always introduce new invariants and in fact finding invariants for toposis is I would say quite a trivial subject in the sense that you see you don't really have to check that the invariance is satisfied if you work in the setting of category theory you know that whatever you will formulate will be automatically invariant then of course the interesting and difficult thing to do is to choose the right invariance for the concrete problems you want to investigate and this of course is by no means trivial but at least you know that this is a word in which invariants are naturally defined and one can consider really many of them and it turns out that even quite apparently weird topospioretic invariance can be quite relevant for the study of concrete questions when they are studied in relationship with concrete representations of toposis for instance given by sites or other objects used to construct them Ok, so now let's go to the second point of view the point of view of toposis as mathematical universes Well, you see we have seen that the topos by definition is a category which is equivalent to the category of sheaves on a site and I have already mentioned that these categories have this extremely rich categorical structure so as examples of toposis you can take for instance the topos of sets so of course you can take the trivial topology and one object category and this is what you get the topos of sets so the objects are sets and the arrows are functions between them then you have many other examples so for instance if you start with a group you can construct a topos out of it by taking the actions of this group on discrete sets you can also suppose the group to be topological in which case you take the continuous actions of this group on discrete sets and in this way you get a grotendic topos and then of course you have the toposis of pre-sheves so pre-sheves is what you get when you put the trivial grotendic topology in which the only covering sieves are the maximal ones then of course you have the toposis associated to topological spaces so as you can see you have a greater variety of different toposis of course there are infinitely many toposes there are infinitely many sites of course there is great variety here but I have just written these three important classes because they make you realize about really the generality of this notion and these are toposes that you can think of continuous in a sense coming from sites which are quite rich in the sense that you see here if you have a topological space which is quite interesting then this grotendic topology is pretty rich because it contains all these covering families and on the other hand the toposes of this kind can be thought of as combinatorial toposes in the sense that you see here you have just one category and no trivial grotendic topology on it so essentially it turns out that when you study in variants of toposes of this kind and you try to understand them from the point of view of the category then you get really some combinatorial conditions on the category conditions concerning the objects and arrows and so on so you can think of this as a sort of discrete kind of toposes and then these toposes are what allow you to generalize groups toposes generalize discrete groups for topological groups the situation is more subtle because you can have different and non-isomorphic topological groups which give rise to equivalent toposes but this is also a very interesting relation of more equivalent and in any case you see if you just want to understand a group through its actions the topospirative world is where you should do that so you see this means that you have many different universes in which you can you can work and in fact so the idea is we are used to do mathematics here in sets so for instance we are used to manipulate sets to consider I don't know Cartesian products of sets so to perform all sort of operations like this or to take the functions from a set B to a set A or to take the disjoint union of two sets so we are used to make these operations and then you might wonder if you can also make operations satisfying the same formal properties as this one in categories and in fact it turns out that that you can because of the very rich categorical structure that is present on a topos so for instance Cartesian products they are a particular case of well they are just finite products in a category so they are a particular kind of limit in a topos all small limits exist so you have that in a topos so this is the object of functions from one object to another in the category of sets and this is what is called an exponential and you have these things in a topos as well then here you have disjoint union and this is that and also you have it in a topos and so on so I mean all the usual constructions that you that you make and which involve limits or exponentials you can do in the same way in a topos so this means that you can think of a topos as a sort of alternative mathematical universe in which you can do mathematics because when you do mathematics actually you do this I mean you perform many operations but then a point comes when you try to understand what kind of logical principles are sound in a topos because you see when you work in mathematics you use some axioms and inference rules to carry out your reasoning and so you have to wonder do these axioms and inference rules are also sound in a general topos and not just in the topos subsets so for instance in many proofs one uses the lower excluded middle which corresponds to the fact that if you have a subset S of a given set A then there is a complementary subset such that the union with the given subset is the wall set and of course this is also true so for instance this is low of excluded middle so basically it logical it means that either phi is true or the negation of phi is true and for instance this rule will not be sound in general toposes so there will be toposes in which this rule is sound these are called the Boolean toposes and other toposes in which it is not sound so for instance if you take a preshift topos this rule is sound if and only if the category C is a groupoid so you see in general it will not work just for so on the other hand some other rules of logic such as this one for instance this will be sound in a general topos and so you understand that some logical rules have a status different from others so you might wonder which are the rules that are sound in all toposes and in fact the answer is that if you argue constructively which means if you renounce two principles such as the low of excluded middle or the axiom of choice or other non-constructive principles then all your arguments will be valid in general topos so this is a general metaphor which is not very hard to prove but it is quite useful because it shows that if you are constructive in your way of doing mathematics your mathematics will be in some sense more solid than the mathematics that relies on non-constructive principles because it will be valid not just in this setting but it will be valid more universally and so this also is a very good argument for the constructiveness of mathematics and for the importance of trying to prove even results that were originally prove non-constructive in a constructive way because if one is able to do so then the realm of applicability of the results which will get much larger constructive interpreting the usual way in tuitionistic logic so yes so this of course is quite relevant because it it shows that we no longer have one mathematical universe in which we can work the category of sets but now we dispose of many alternative mathematical universes each of which has its own peculiar properties because I have told you that as far as the formal properties formal general categorical properties are concerned the topos is shared the same because they all have limits, co-limits, exponentials and so on but each topos has its peculiarities its specific properties so when you study topos theoretic invariance you will see that you can by using them distinguish toposes one from one another and so there is really a whole variety of toposes and this gives a lot of flexibility for constructing new mathematical words with particular properties and in particular it allows one to relativize mathematics in the sense that one is no longer obliged to fit everything in one box one disposes of many different words and the idea then comes to choose a particular topos which best represents a certain theory for instance or a certain problem that one wants to investigate so this great technical flexibility is a crucial aspect of the theory which is in particular useful when it comes to problem of studying models of first-order theories because you see thanks to this very rich categorical structure present on toposes one can consider models of essentially any kind of first-order mathematical theory in a topos just generalizing the classical Tarskian definition and then the point is given a specific theory is there a topos which in some sense represents the most natural environment in which I can study my theory in which I can in which the symmetries of the theories are most naturally revealed or not and here comes the problem of building classifying toposes so in fact a very striking result which was obtained by a number of people in the 70s says that to any first-order theory presented in particular form it must be formulated within geometric logic one can associate canonically a topos a grotendic topos which is called its classifying topos and which in a sense represents this most natural environment for studying the theory in which sense well so so given a theory which is geometric so one requires the actions of this theory to be presented in a particular form for technical reasons so one requires this to be geometric formulae which means formulae obtained from atomic formulae by only using finite conjunctions possibly infinitary disjunctions and existential quantifications so if the axioms of a theory are all of this form we say that the theory is geometric so this might seem quite weird at first sight but in fact you can always starting from a finite first-order theory you can geometrize it in the sense that you can construct a theory over a bigger language whose set models can be identified with those of the original theory and this is a process which is called in the literature moralization so essentially by means of this process we can really restrict to geometric theories and geometric theories are important because they have classifying toposes so what is a classifying topos well it is a topos which represents the models of the theory in whatever topos so basically I have said that for any grotendic topos one can consider the category of models of the theory in it and when you take morphisms of toposes then the inverse image of this morphism will send models of the theory in this topos into models of the theory in the other topos so this will define a pseudo functor of models of the theory and the assertion is that for any geometric theory this functor is representable and the representing object is the so called classifying topos which is denoted so the key property is that the category of geometric morphisms from an arbitrary topos to the classifying topos are in equivalence with the category of models of the theory in the topos naturally in the topos so of course naturality is important as always when you deal with representability issues so it means that if you make the topos in which the models are taken very you get a commutative square and so what does it mean well it means that essentially all the essential semantic information about your theory is actually embodied in the classifying topos so in this picture the big star represents the classifying topos of the theory and the smaller shapes inside the bigger shapes are models of the theory so you see in the classifying topos I have identified one particular model which I have called U so what is U? U is the so called universal model of the theory so you see if here you take E equal to the classifying topos so I have the identity here and and the identity will correspond to a model which by naturality will generate all the other models of the theory in arbitrary toposis so this is what the picture shows you see these big shapes represent different toposis so inside them we have models of models M, N and P of the same theory and basically all these models will be images of this universal model under this structure preserving funtors which are precisely the images the inverse images of the geometric morphisms associated with the model so you see these pictures and you understand a lot of things actually because it makes you understand that if you really want to understand the symmetries, the invariance of a theory the right point of view is not that of the topos of sets which is just one of those shapes there so of course in particular I can take the topos of sets and I can consider models of theories in the classical set theoretic world actually if I do this what I am studying is just the deformations of an object which lies elsewhere which is precisely the universal model inside the classifying topos so you see this is a situation of the kind which often happens in mathematics when you start with a maybe you are just interested in a concrete problem and you work in a restricted setting but you find the symmetry only when you enlarge to a bigger setting you see here we have switched from the consideration of set based models of a theory to the consideration of all possible models of the theory in every possible grot and diktopos and by doing this we have got this very classification result which of course does not exist if you restrict to one particular topos such as in particular the topos of sets you see so this shows that really as I was saying in the previous slide topos is able to accommodate theories in the sense that they are able to provide very natural mathematical words in which the symmetry of theory can best be understood so examples of classifying topos is just a few to show that actually this is a point of view that is not just abstract but also pretty concrete and which can be used technically the simplicity of topos I mean the topos of simplicity sets classifies the so called abstract intervals for instance and this was shown by Zvajali in the 80s and the Zariski topos for instance classifies local rings, the theory of local rings so as you can see this gives you another understanding of topos through the structures that they classify this is the Uneda paradigm so to try to understand an object the arrows going to that object so here we understand the topos looking at the arrows in the category of toposes to the topos and in fact this paradigm was already introduced by Grotendik but which who identified in the thesis of his student Monika Kim they identified four toposes, useful in algebraic geometries as the classifying toposes for certain theories that are extensions of the theory of commutative rings with units so he introduced this point of view but then it was the logicians that defined the logical framework such that every theory formulated in it will admit a classifying topos and obviously any Grotendik topos can be seen as the classifying topos of a theory in that framework so it was the Montreal school of categorical logic in the 70s that led the foundations of the so called geometric logic ok, now the point comes when you want to understand the relationship between theories and classifying topos so in particular you can wonder when do different theories have the same classifying topos this is called the Morita equivalence and in fact as I mentioned the beginning of my talk it means essentially that the two theories talk about the same structures in different languages because having the same classifying topos by definition means that the two models of the two theories are equivalent in every Grotendik topos naturally in the topos so you mean at the semantic level you have an equivalence but syntactically you can have completely different presentations and this is something quite useful in fact so you can really exploit this ambiguity inherent to the notion of topos to actually multiply the point of view on a given theory by attaching to it other theories which are Morita equivalent to to it and in fact this duality between theories or sites and the corresponding topos is something very deep which can have great technical consequences in fact I would like to say a few words about this notion of Morita equivalence I would like to stress actually the fact that when you find yourself in a situation where you build the same mathematical object in different ways or you describe the same thing in different languages it is very likely that there is a equivalence and therefore that you can formalize it as having the same classifying topos associated with different theories or different sites and in fact many important dualities and equivalences in mathematics can be naturally interpreted in this way and in fact conversely topos theory provides itself a primary source of Morita equivalences because in fact having different theories classified by the same topos corresponds to having different representations for the same topos so actually whenever you are able to represent a topos in different ways it means that there are Morita equivalent theories behind what is also interesting to remark is that when you have a dictionary between two theories which allows one to transform the syntax of one theory into the syntax of the other in a sort of bijective way this is called biinterpretability in logic then of course you have Morita equivalence but what is interesting is that most of the Morita equivalences are not of this form so this means that topos theory can allow you to perform transfers of knowledge between theories even when you don't have a dictionary allowing you to pass from one theory to the other and actually you see the way you perform this transfer is really to use the topos as a bridge so basically what plays a crucial role here is invariance of topos so you see when you have different theories or different sites which present the same topos you consider invariance on these topos and you look at them from the points of view of the different theories or the different sites and basically this will lead to connections between properties or notions in the context of the two different theories or sites because they will be seen as different manifestations of a unique invariant line at the topos theoretic level so this is the general technique and technically this is how it works I mean normally you use sites for representing topos even though of course they are not the only means for constructing topos but they are one of the most commonly used so when you have different sites which present the same topos you take whatever invariant and I mentioned at the beginning of the talk that you have infinitely many invariants at your disposal so you can do this for any invariant then you try to understand how this invariant expresses on the two sites and what is interesting is that while abstractly you have just one invariant concretely you can have completely different manifestations of it so you can give a few examples so if you take for instance the invariant property of a topos to be two valued if you use this representation of the topos as the classifying topos of a theory two valuedness means that the theory T is geometrically complete so the notion of completeness in logic means that every assertion is either provably true or provably false in the theory geometrically complete means that you have this statement for any geometric sentence in the language of the theory so this is the notion of two valuedness now if you change the site and you take for instance an atomic site then this manifests as joint embedding property on the category C supposing that this is non-empty so joint embedding property means that for any two objects of the category there exist a third and two arrows from the two objects to it so you see if you compare these two properties they look completely different from each other but in fact they are just manifestations of the property at the toposporetic level and there are of course thousands of examples of this kind that are pretty striking because one would have very hardly imagined how to come up with these connections without the toposporetic viewpoint another example is for instance the property of the topos to satisfy the Morgan's law the Morgan's law so I mentioned that the logic of the topos in general so the Morgan's law will be satisfied only in some toposis so this gives a meaningful invariant and if you study this on a topos of pre-shifts what you get is this property on C the fact that any two pair of arrows with common co-domain can be completed to a commutative square so it's a dual of amalgamation property and if you look at it on the topos of sheets on a topological space X it gives you the property that the space is extremely disconnected which means that the closure of every open set is open so you see again you find a relationship between two apparently very different properties and so on so you see what is going on here is really a sort of mathematical morphogenesis which as you can imagine have many consequences so here I have just made a very brief list of applications of this bridge technique so for instance I got a generalization and extension of Freisse's theory in model theory by using atomic to valued toposis and the bridges of this kind and actually I managed to unify this theory of Freisse which comes from model theory to the theory of Galois categories of Grotendik through a more general unifying framework because in fact it turns out that the toposis that you use is to understand Galois and the ones that you use to understand Freisse are really the same kind of toposis more or less and so you can unify the two theories with each other there are of course many other applications so for instance in topology I have shown that all the classical stone type dualities dualities between preorder structures and post sets, locales or topological spaces they can be naturally recovered from topospheric bridges for instance stone duality is like this I mean when you have a Boolean algebra in the associated stone space they are connected through toposis in this way so the topos of shifts on the Boolean algebra with respect to the coherent topology is equivalent to the topos of shifts on the corresponding space then toposis turned out to be useful also in proof theory and they have made some applications also of toposis on that so basically I have introduced new proof systems based on grotendictophologies which turned out to be computationally better behaved than the classical proof systems because in fact instead of having all the rules of inference in the classical Hilbert style deduction system inference rules which correspond to the key axioms of grotendictophologies and for this reason the computations become much easier and in fact I was able to prove some syntactic theorems about geometric theories by using that point of view anyway there is much more on that of course and if you are interested to know more about the use of toposis as bridges there is my habilitation thesis that you can read in some more general remarks in fact all of this can be seen as a sort of mathematical morphogenesis in the sense that you see this essential ambiguity given by the fact that the toposis associated with an infinite number of different representations is something that really can be interpreted as morphogenesis because you see you have all these different forms which are just manifestations of a unique thing defined at the toposporetic level and so this suggests a way of doing mathematics which in some sense is reversed in the sense that it is guided by moret equivalences and toposporetic invariance so in a sense it is the invariance that tells you where to go which are the right notions which are the notions the concrete notions that in some sense are more fundamental than others so for instance is a very good question to ask if you have a concrete notion does it correspond to a toposporetic invariant or not because you see if it corresponds if you are able to build a topos out of your concrete problem and find an invariant on that topos which corresponds to the concrete property you are interested in then it means that your property is modular in the sense that it is transferable while if it doesn't correspond to a toposporetic invariant it means that probably it is quite concrete and so maybe it is useful for the concrete purposes one has but it will not be transferable according to this means ok so in fact another crucial aspect about toposis is the fact that they are completions in some sense of the theories or the sites from which they are built in fact if you think about the crucial passage from a site or a theory to the associated topos you actually understand that it is a sort of completion by addition of imaginaries in the model theoretic sense in the sense that you see you start from something real and quite small in some sense and quite unstructured because if you think about the axiomatization of a theory it is something quite unstructured just a list of axioms in a given language and you turn this into a sort of maximally structured entity a topos and this duality between the real and the imaginary is actually what is going on when you play with these bridges see this is a diagram which in some sense summarizes the way of proceeding when you want to make use of these morita equivalences of toposis so you see you start with a concrete point, a concrete fact a concrete problem that interests you you try to build the topos which in some sense captures the essence of your problem then you try to represent many ways in order to exit the bridge and go back to the concrete but following a different path so in some sense it is you see the idea is that the symmetries reveal themselves in the big environment in some sense you see think about a vase with a lot of beautiful symmetries and then you break this vase and then you lose the symmetries you are just with the fragments in some sense don't see symmetries maybe it is because you are working with a fragment and the topos theory allows you to complete your fragment and recover the unity and thanks to this you can then go to other words ok so final just to summarize these are some key features of toposis that one can can identify toposis are very general we have already remarked that in fact the level of generality of topospheric invariant is quite unique in mathematics because it allows one to compare effectively with each other theories of objects coming from different fields of mathematics on the other hand we have this aspect of expressiveness of invariance because in fact many of the invariance commonly used in mathematics can actually be seen as topos theoretic invariance this is the case for instance for the homological or homotopy theoretic invariance I mentioned in the beginning then of course there is also an aspect of centrality of topos theoretic invariance because imagine for any topos there are infinitely many theories or sites associated with that so this means that whatever happens at the level of toposis in mathematics so even if one doesn't necessarily care about that this is what actually happens so the fact that there are so many invariance defined at the topos theoretic level necessarily makes this concept centrally in mathematics and finally of course we have already talked about the technical flexibility of toposis this is due on the one hand to the very rich categorical structure that they have and on the other hand to the very well behaved representation theory so this means that when you try to understand invariance of toposis in terms of different sites you normally are able to by means of computations that can be harder or easier depending on the invariant but still you are able to establish to calculate in some sense manifestations of this invariance in the context of different representations so in the future of course what would be desirable to do is to continue the development of topos fury both at the theoretical and applied level in order to make the best use of this quite noble way of doing mathematics that topos fury provides so of course I think that the future of the subject is very very bright so I take this opportunity to invite you to this international event school and conference on toposis that we will have in coma at the end of June so this will be I hope a very good opportunity to push forward the development of this subject ok, so thank you very much for your attention any questions comments you cause me for each individual topos if I look at the first example topological space of group people look at continuous maps of group so they come themselves in the category where you have more than equivalence how this is integrated of course I mean I focused a lot on morita equivalences which are just equivalences of toposis just because in that case we have many more invariants at our disposal because as I mentioned whatever formulated in categorical language will be automatically invariant under categorical equivalence so in some sense for this reason it is good even when you have two toposis which have a relationship that is not necessarily an equivalence it is good to try to make it an equivalence in the sense that you see you have various ways to operate with toposis at the two categorical level so you can suppose for instance you build a topos which is not Boolean but the feeling that it is related to another topos which is Boolean then there is an operation that you can perform the Booleanization which makes an arbitrary topos Boolean in a sort of universal way and so you see you can by applying this operation you can try to to make the two toposis equivalent and then to make the transfers because what happens is that if you just have morphisms between toposis or more general kinds of course you can play the same game and you also have bridges but the point is that you will dispose of many less invariants because you will have to check for any property or notion whether it is transferable or not and so this will limit a lot the possibility of transfers so I think that because one can really calculate with toposis one can make a lot of operations with toposis to construct new toposis out of old ones I think that probably strategically it's a good thing to you see when you have the feeling that there might be a connection between two things of course very likely you will not immediately find an equivalence of toposis you will find some connections between these toposis but then I think it's worth starting from this connection to try to extract some equivalences in order to be able to transfer more otherwise one can just remain with the relationship and wonder about for any invariant that one wants to transfer one can wonder does it go through or not but you see if you consider how much can go through when you have an equivalence it encourages you to go at that level you also mentioned the contending pharmacological in this setting you don't only have one toposis you have a whole collection of toposis related by functors so of course it it is not always possible to obtain equivalences and what I stress is that if you are able to obtain them then of course in some sense you are in a perfect ideal situation because then you can basically transfer everything otherwise well in the logical setting you see since you can you can view very grotendic topos as the classifying topos of some theory so it means that any operation any two categorical operation that you can make on toposis will have a logical counterpart so I have described some of these things in my work in logical terms so for instance I proved that the sub toposis of the classifying topos of a theory correspond to the geometric theory extensions over the same language I proved this then I also described in logical terms the hyperconnected locality factorization of geometric morphism of toposis which really admits a very very nice logical description so what I can say is that really this duality between geometric theories and classifying toposis is very very good in the sense that you can really understand many things using the logical viewpoint I mean in fact the best reference for that would probably be my book because in my book there are many many such results really many, I mean I have mentioned too but there are maybe 10, 12 even if we think about classical comological formalism for example as built by grotendic and visital comology and things like that so you can interpret the toposis as classifying some theories so the theory of and salian local rings together with the structure morphism to the scheme you are considering but you can also say that if you are interested in only in comological invariance so there is a general result that the comological invariance of a small topos are the same as of the big topos so this means it has as a consequence that in fact you can consider all comological invariance let's say in algebraic geometry as invariance of one topos and its localization what is the localization of a topos is when you have a topos E and an object S in E it is a topos consisting of objects of E and odd with a morphism to S and in fact it is enough to so this is just a remark it is a kind of philosophical question but when do you know in classical mathematics when a property is of a topos the right of nature ha, this is but our reciprocity loves in number theory hey, it's in general quite hard because when you in some sense you see I said that the kind of way of doing mathematics that the topos theory seems to promote is being guided by invariance and then computing them now what turns out by computing invariance is that really you have to make a lot of computations and so it can be very lengthy computations at the end if you are lucky you find the property that you are interested in but to go backwards in general is quite hard I mean probably at this stage we are still in trouble about you see given a concrete problem to find a topos theoretic invariance which corresponds to that probably is there but we don't know so maybe what I think one should do is in some sense to make a sort of encyclopedia of invariance and their characterizations to try to document the behavior of invariance in terms of different sides one could even implement this on a computer because all of this I mean many invariance can be computed in terms of different sides in a quite mechanical way not necessarily trivial but it can be mechanized so you see one could imagine that one asks a computer to make all these calculations and then one ends up with a sort of encyclopedia of invariance and characterizations and ideally once one has such an encyclopedia you see look at that you would start with your concrete problem and you would in some sense develop a sensitivity for what can correspond to an invariant at what cannot correspond but in general it is really I mean quite hard it is easy on the other way you see it's like going down the river or going going up so if you try to force things it is always difficult then to follow the natural flow of the river in some sense in Topos theory the natural flow of the river is like Topos is at the center in a sense invariance are at the center and then you study how this invariance manifests in order to make the theory more user friendly of course there is time needed for getting to know these things better but in general there is quite a non-tribial no I mean what you can do for instance suppose you have a property of a theory so I can give you a simple example so suppose you have a property of a theory which says the signature of this theory has exactly two sorts for instance so I this is an example of a property well I can tell you immediately that it doesn't correspond to a Topos theoretic invariant because I can find a theory which is more equivalent to that one and doesn't satisfy this property and so I know that it is not so you see in some situations you can immediately understand that it is not invariant in relation with that particular Topos that you want to associate because of course this is if you want to associate the classifying Topos to your theory but if you want to associate another Topos maybe the same property might correspond to an invariant of that Topos for sure it will not correspond to an invariant of the classifying Topos but in general you see at the moment the theory is not really very user friendly there is more work to do at the theoretical level to understand the applications for instance for me it was like a surprise in a sense so for instance I started so during my PHD for instance I decided to investigate the Frisez construction which is a very important theory in model theory using Toposis and I had the feeling that the right kind of Topos to consider was atomic Toposis this was the initial intuition but then I started calculating with some invariants on this Topos and then for me it was sort of a surprise to find the concrete properties involved in Frisez theory just as manifestations of Topos theoretic invariants that I considered there but to be honest for me it was a surprise I mean I just I could have a feeling that I would get some properties that maybe could be near the ones I was interested in but actually I found that they were exactly what I needed and so there is an element of I wouldn't say luck but at the moment we have a very limited knowledge of the way invariants manifest and for this reason you see if you are quite if you have worked a lot with Toposis of course you develop a certain sensitivity so for instance I think I have a sort of sensitivity I can understand for many properties if there are invariants behind but by no means in a universal way I mean also it depends on the knowledge I have of the field in which I want to make the applications so for the moment because one of the feeling I have in terms of shortcomings is this try and error approach that you have to do which is, as you said, not user friendly because by the time you can construct and complete it's tedious Maybe the computer could be very useful about that I have never really worked myself on a possible computational implementation of all of this even though it must be said that it should be possible but here we are more like in the field of automated theorem improving and not just verifications of proofs which is, I mean conceptually much more straightforward you have the proof here you have to invent new results I think that the history of Toposis we already have teaches us a lot on what can happen for instance, so Olivia mentioned the first very important application of Toposis in relation with Comology and then, of course for instance, Kotelnik was confronted with key aspect of Comology which is point-caridability and so the question for him was really how to understand point-caridability in topostheoretic terms and then he came up with the formalism of the six operations but this was not automatic at all in fact it was really genius to design this formalism it is completely clear that it took him many months or years to design that and it made Comology much more powerful than before so the rewards were huge and in fact by now many results including extremely deep concrete results have been obtained using this unreached Comological formalism designed by Kotelnik but so these are the rewards of the effort Kotelnik to make in order for this key aspect point-caridability to become topostheoretic it was not at all automatic you really needed automation and subtlety so there is a huge element of automatism but on the other hand there is creative input which is still needed especially if one wants to starting from a concrete thing to find the right toposes and ok