 Okay, so yesterday we finished our presentation of the geometric side of the toposphering, and today we shall begin talking about the logical approach to toposis. So we shall start by renewing the approach to toposis as mathematical universities, which emerged in the 60s and 70s thanks to the work of categorical logicians, in particular Billovir and Mike Stearney. So what logicians undertook at that time was an axiomatic study of glottendift toposis from the point of view of regarding them as places where one could develop mathematics. So alternative mathematical universities with respect to the classical set theoretic setting to which everyone is used. So one can wonder, given the fact that a topos has a very rich categorical structure, whether one can really do mathematics inside a topos. And in particular one wonders whether the classical logical rules of deduction we are used to apply become sound in toposis when we develop mathematics inside a topos. Investigating these issues amounted to focus the attention on a particular object inside a topos, which we already introduced yesterday, the sub-object classifier, the omega object, because in fact this object captures a great amount of the internal logic of a topos as a mathematical universe. So in fact yesterday we already talked about hiding algebras and their relation with inclusionistic logic. I remarked that hiding algebras provide the analog for inclusionistic logic of Boolean algebras for classical logic. So in fact they provide a sound and complete semantics for propositional inclusionistic logic. And hiding algebras arise naturally in toposis because in fact as we shall see, all the sub-object lattices in a growth-indict topos have the structure of a hiding algebras. So this means that really the internal logic of a topos is intuitionistic. So of course this has a lot of consequences because it ensures us that if we develop mathematics in a constructive way, then what we establish will be sound not just in the classical set theoretic setting but in every topos. So as you could imagine this point of view had a lot of applications in fact it paved the world for the construction of new mathematical words satisfying particular properties because of course thanks to the generality of the notion of sight one can construct toposis adapted to the needs of the particular situation that one wants to investigate and so one can profit from this flexibility of being able to change the universe according to one's mathematical needs. So in particular it is possible as we shall see in a moment to consider models of any kind of first order mathematical theory in a topos. So it is possible also to try to classify models of theories in arbitrary toposes and we shall see that if the theory has a particular form namely if it is expressed within the framework of geometric logic its models can be classified in a very satisfactory way by a growth-indict topos called the classifying topos of the theory. So you see this possibility of considering models of theories in arbitrary toposes the possibility also of changing the base topos over which work and work all of this is extremely important. Think for instance about growth-indict refoundation of algebraic geometry a very important principle which applied throughout this work was the relativity principle so the idea of considering schemes over bases which could vary according to the needs of the specific situation so here we can do the same with toposes so we can not just consider toposes of sheets of sets which means working over the topos of sets but we can develop topos theory relatively to a base topos so consider if S is a growth-indict topos S value the sheets on it which will again be a growth-indict topos and so all of this of course makes the theory extremely powerful both conceptually and technically of course things become even more interesting when one gets the duality between toposes and the structures that they classify a duality which is well established for toposes defined over sets and which is not quite completely understood at least from a user-friendly point of view for toposes over an arbitrary base but of course it works equally well but it is quite powerful to be able to dispose of this point of view of toposes as the classifiers of certain objects and not just for toposes over sets but for toposes over an arbitrary base but before going to this let's be a bit more precise about these height-ing algebras which I have mentioned so I have told you that in any growth-indict topos for any object the collection of its sub-objects has the structure of an height-ing algebra in fact a complete height-ing algebra these are all frames and in fact we also have another quite important construction which is the second point of the theorem which is the construction of left and right adjoins to the pull-back factor between sub-object lattices induced by an arrow in a topos so you see given an arrow f in a topos you have the pull-back factor along f which sends sub-objects of b into sub-objects of a and what is quite interesting and was discovered by Lomir at the time of this exploration of the logical side of toposes is the fact that this factor has these two adjoins which in fact can be used to interpret the existential and universal quantifications along f in the sense that they can be used in categorical semantics to interpret these quantifiers appearing in first-order formulas so let's give a complete description of all of these so first the height-ing algebra structure so what does it look like so here I have reported the operations that you have on sub-objects in a glottin topos so the sub-objects of a sheaf they are just sub-ships in the sense of this definition and the operations are defined in this way so notice that there is no surprise concerning the myth because it is defined point-wise but things become interesting when you consider the other operations in particular the orb operation, the disjunction which is interpreted by referring as you can see to the covering sieves for the topology so you see that here the logic becomes much richer than the usual binary logic because it is able to capture the idea of local truth so the truth relative to a covering and you see this precisely by looking at these formulas so of course since you have this complete height-ing algebra structure on each sub-object lattice and the object omega classifies the sub-objects you can conclude from the yonida lemma that the sub-object classifier has the structure of an internal height-ing algebra in the topos now the interpretation of quantifiers is also quite interesting so these are the formulas which I have reported for you so of course the pull-back factor is defined in the obvious way because pull-backs are particular kinds of limits and so as we saw yesterday the limits are computed as in the category of pre-ships so they are computed point-wise but the left adjoint and the right adjoint admit interesting descriptions as you can see in the formula which gives the left adjoint you see the existential quantifier appearing so see here and in the description of the right adjoint you see this condition, this inclusion which again involves quantification this time a universal quantification and so you understand how this really represents a way of interpreting the quantifiers in the topos theoretic setting ok so now let's go to the point-of-view of classifying the topos so the point-of-view of developing model theory in topos and then aiming for a classification of models of theories so what is possible to do is to interpret any kind of first order theory in a topos in fact it is possible also to interpret higher order theories because of the existence of the exponentials in a topos but we shall not be concerned with this in the course so here we just limit ourselves to first order theories for which there is a well-developed classical model theory and we shall see what topos is bringing to this subject so as you know Tarski introduced the notion of model of the first order theory in the classical set theoretic setting so first you have the notion of first order signature over which you write the axioms of your theory and how are first order signatures interpreted in the classical Tarski and Semantics well in a first order signature you have sorts, function symbols and relation symbols so sorts are interpreted as sets function symbols as functions relation symbols as subsets what is going on in categorical semantics is a very natural generalization of all of this because sorts get interpreted as objects of the category function symbols as arrows relation symbols as sub-objects and then of course when you have to define the interpretation of the formula in a given context recall that a context is any finite set of variables which contains all the variables which appear free in the formula so whenever you have a formula in a given context it will be interpreted as a sub-object of the corresponding Cartesian product or the interpretation of the sorts which correspond to the variables in the context and of course as you can imagine as in the classical case the interpretation of the formula is defined recursively on the structure of the formula and of course this definition involves precisely the categorical structure present on the topos for instance as you can imagine to interpret conjunctions you will need to consider certain limits to interpret these junctions you will have to take unions of sub-objects to interpret existential quantifications you will have to take images of certain arrows but we know that we can do all of this because of the original categorical structure present on the topos and so everything is defined in a very straightforward way and then in this way you get the notion of the structure of the first order signature and then a model of a theory will be just a structure in which all the axioms of the theory are satisfied ok now we respect our attention to a class of theories which is very important because the theories in this class have classifying toposes so this framework of geometric logic is a framework of first order logic but where the axioms of the theories need to have a particular form so a geometric theory is a theory of a first order signature whose axioms can be presented in this form so this is the sequence notation and the meaning of this notation is for all the variables in the context the first formula entails the second and we require these two formulas to be geometric which means built up from atomic formulas over the given signature but by only using finite and even junctions infinity junctions and existential quantifications so this definition might seem at first sight quite straightly but in fact it is really not because on the one hand most of the first theories naturally arising in mathematics are already geometric order signatures so you don't need to make any operation to make them geometric but if this is not the case there is always a canonical procedure that you can apply to turn any finite and even standard theory into a geometric one this procedure is called more regularization so you see that thanks to this basically you are able to investigate any kind of finite and even standard theory from the point of view of geometric logic because this process of more linearization doesn't essentially change the set base the markets of the theory so as far as classical model theory is concerned there is no difference now why do we need this restriction to geometric logic well because we want to develop a factorial model theory so in particular we expect the morphisms that we have defined between toposes mainly the geometric morphisms to send models of a theory into models of the theory and this will not in general be the case for an arbitrary classical theory but if the theory is geometric then it is too so it means that when you have a geometric morphism between growth and if toposes so the fundamental the one which preserves the greatest amount of structure is in the seamage because it preserves the finite limits as well as arbitrary limits so this is the one we are going to use so what we want is that if we have a model of the theory inside f this is transported to a model of the theory inside so we want such a pseudo-factor defined to be induced such such a contour if the theory is geometric if the theory is not geometric this will not in general be the case so the reason for this is that when you interpret arbitrary first order formulas in toposes you use these constructions present in the topos for instance the universal quantification the the hiding pseudo-complement the sorry the hiding negation the hiding pseudo-complement etc. and in fact these operations are not in general preserved by in this matter fact also geometric morphisms while if you stick to geometric formulas you see geometric formulas are built from atomic formulas by only using finite conjunctions possibly infinitely disjunctions and existential quantification so you see these finite conjunctions are interpreted in terms of finite limits these are interpreted as arbitrary unions of sub-objects and these are interpreted as images of arrows so you see all these constructions actually involve finite limits and arbitrary co-limits which are all preserved by functors of this form and so you see that if the theory is geometric then we indeed have a functor making the capitalism models of the the theory induced by the geometric morphisms between the toposes and so since we have such a pseudo-functor we can wonder whether this functor is representable and now we get to this fantastic result which was discovered in the 70s which says that every geometric theory has a classified topos by this universal property that the geometric morphisms from an arbitrary topos to the classified topos are in natural categorical equivalence with the models of the theory inside that topos and what is also very interesting is that classified toposes not only exist for an arbitrary geometric theory but can be built in a canonical way by taking into the category of sheets on a particular site attached to the theory called its syntactic site which in fact embodies a great amount of the syntax and the proof theory of the theory so this is the definition of the syntactic category so it is a category whose objects are geometric formulas in a given context over the signature of the theory considered up to renaming equivalence and whose arrows are the formulas which are the geometric formulas always over the the signature of the theory considered up to provable equivalence which are provably functional from the domain to the column one requires this proof-theoretic condition so that when you pass the syntactic interpretation of such an arrow becomes the graph of an arrow from the interpretation of the domain to the interpretation of the column then you realize immediately that this is a category because you have a very naturally defined the notion of composition of arrows you have also the identity arrow on each object and so you have a well-defined category which in fact has the structure of a geometric category ok so on the syntactic category of a geometric theory one can put a naturally defined glottendictopology called the syntactic topology in fact this is precisely the so-called geometric topology on that category which actually has the structure of a geometric category so concretely this syntactic topology is defined in this way so a family of arrows to a given formula in the syntactic category is governing if and only if the sequence which expresses the fact that this family is public is provable in the theory so you see how beautifully this relates the geometry of the syntactic category with the provability in the theory so why do we get the classifying topos when we consider ships on this side well we just have to recall the theory which we presented yesterday which gave the equivalence between geometric morphisms to the topos of ships on the side the connection with the plate path also so let's recall that what we saw yesterday was that we had equivalence between the geometric morphisms from the topos e to the topos of ships on the side and the flat j continuous path also from c2 now if we apply this to the topos of ships on the syntactic side what we get is this so we have to consider the flat jt continuous path on the syntactic category now this category is geometric in particular it has finite limits and yesterday acted that a factor defined on a Cartesian category is flat even only if it is Cartesian so it means that here we get the Cartesian jt continuous path from ct to tablish a factor equivalence between these factors and precisely the models of the theory in e why this is an equivalence because we can define a factor going in one and in the other direction I will just give one direction so given a model here the factor which corresponds to it very naturally defined because it is the factor which sends every formula to its interpretation in the model in fact the factor going in the other direction uses the existence inside the syntactic category of a distinguished model called the universal model of the theory inside it and it will denote this model by so universal model a factor defining this equivalence which sends a given Cartesian jt continuous path f to the image of this universal model and you can show that you get an equivalence of categories and so by putting these two equivalences together you can conclude that the category of sheets on the syntactic side a classifying topos for the theory ok so it may happen of course that two distinct geometric theories have equivalent classifying toposes when this happens the theories are said to be more equivalent as we shall see in a moment this equivalence relation is extremely deep and interesting because it formalizes in many situations the feeling of describing the same structures in different languages or constructing the same object in different ways so just having different linguistic points of view on a given field so for the moment let's just remark that by definition of classifying topos two geometric theories are more equivalent if and only if they have equivalent categories of models in every growth index topos naturally in the topos so it means that if you change the topos by means of geometric morphism you should get a cognitive square connecting the equivalences existing for each of the toposes so is it true that every growth index topos is the classifying topos of some theory? yes because by definition of growth index topos any growth index topos can be presented by using a site and you can attach to each site a geometric theory which will be classified by the associated categories so given a site cj you can write down a theory a geometric theory the theory of j continues show that the classifying topos for is precisely the c that infect the language of sites and the language of theories are strictly related because to any geometric theory you can canonically attach a site it's a syntactic site and conversely to any site you can naturally attach a theory which will be classified by the corresponding topos so in fact we really have two levels here at play the level of toposes and the level of their presentations which can be sites which can be theories or more generally any object from which you can construct a topos and in fact this is by no means limited to sites or theories because toposes can be associated with many other mathematical objects for instance they can be built from two points either topological or localic or they can be built from quantals or more generally quantaloids so in fact it is possible to construct the toposes from many different kind of mathematical objects but all of these objects will play the role of presentations so actually it is most important to distinguish these two levels the level of presentations and the level of the toposes and to investigate the interplay between these two levels and this will be really the art of the the bridge technique that we shall present in a moment so for now let's just conclude that this construction of the classifying topos of a geometry theory allows us to think of a growth in the topos as a canonical representative of equivalence classes of theories, mobulo, morita thank you very much ok so now we we can start presenting our perspective of toposes as bridges by making a few remarks about this notion of morita equivalence how should we think about this so apparently this notion is quite sophisticated because you require the theories to have equivalent categories of models in every growth in the topos which in general is a condition much stronger than simply the condition of having equivalent categories for instance of set-based models so you might think that this is a very strong condition which is rarely found in practice, in fact the opposite is too my experience with the topos theoretic study of theories have led me to conclude that actually the notion of morita equivalence is ubiquitous in mathematics whenever you find yourself in a situation where you can talk about the same structures in different ways where you can describe a given mathematical content in different languages it's very, very likely that you find yourself in presence of a morita equivalence and it is worth to try to capture this feeling of describing the same things in different ways to capture it through this notion because then the feeling becomes a definition and a theorem on which you can build and from which you can extract information so in fact throughout the work of last few years I have undertaken the task of formalizing many dualities equivalences and correspondences that I could find in different fields of mathematics in terms of this notion and in general I have to say that it has turned out to be quite effective so of course many more dualities and equivalences still wait to be investigated from this point of view but the results that have been obtained so far indicate that indeed morita equivalence provides a very general notion of equivalence of theories which is naturally related both to the classical notion of categorical equivalence and to the categorical notion of duality and also to the notion of bi-interpretability in logic but before I go into that let's remark that in fact topos theory by itself is a primary source of morita equivalences because in fact having different theories classified by the same topos corresponds actually to having different presentations for this topos talking about theories or about sites is essentially the same and so actually the existence of different theories classified by the same topos translates in particular into the existence of different sites which present the same topos and conversely if you discover that a certain topos presented in different ways this can be interpreted from a logical point of view as the existence of different theories which are morita equivalent to each other and so all the representation results for topos that you can prove by using topos theoretic tools actually can help you generate morita equivalences that can be relevant in different mathematical context I have really used in my work such a point of view to generate new equivalences new dualities in different mathematical fields really using the tools of topos theory so in fact normally topos theory allows you both to shed new light on classical subjects in particular on classical dualities but also it allows you to generate many new insights many new dualities or equivalences or correspondences that would be difficult to obtain otherwise now as I mentioned the relation of morita equivalences strictly related to by interpretability so what does it mean that the two theories are by interpretable well we have the notion of the syntactic category available to express this so intuitively two theories are said to be by interpretable when there exist a sort of dictionary relating them to each other so by a dictionary I mean a procedure which will assign every formula written in the language of the trans theory a formula written in the language of the other and conversely so as to induce an equivalence at the level of the categories of models of the theories so by using the language of syntactic categories you could define by interpretability as an equivalence between the syntactic categories of the two theories because you want every formula the language of the trans to correspond to the formula in the language of the second and of course if you add that the two theories we have the same classification because the classification is built by equipping this geometric categories and so such an equivalence of course entails morita equivalence but what is most interesting is that in fact most of the morita equivalences are not of this form so it means that they do not come from dictionaries and this is a very important remark to make because it shows the relevance of topos theory to understand theories from multiple points of view and also relating different theories with each other because what this is saying is that in fact most of the correspondences which arise in mathematics are not induced by concrete dictionaries they are much deeper and so thanks to toposes you can identify them if you don't have toposes you will never realize about them or realize about them but with great difficulty and without most natural point of view so to realize about this difference between dictionaries and morita equivalence you can think again about the example I gave yesterday of the basis for a topological space so what I said yesterday is that if you have a topological space so if you have d1 and b2 bases and you can on the one hand represent the topos of shifts on x in terms of the first base with respect to a certain growth in the topology which will be induced by the canonical topology on the category of open sets of x and on the other hand you can represent it in terms of the other base so in particular you can conclude that the two toposes are equivalent but look you don't necessarily have an equivalence between d1 and d2 so in general there will not even be a factor going from d1 and d2 so you see that the connection between the bases cannot be understood in a linear way so you need to complete the two bases to the whole space in order to be able to link them and this is what the topos is able to achieve so of course this case of the basis is very simple because in this case the topos basically can be identified with the frame of open sets so you don't actually need the topos you just need this to connect the two bases but this is a very simple example in general when you go from the propositional setting to the setting of of first order theories then you indeed need the full topos to connect the theories ok so I will continue in the second part of the lecture to illustrate the theory of topos theoretic bridges so how to build bridges and I will illustrate this by analyzing a number of examples so for the moment thanks again for your attention