 Thank you so much to the organizers and anyone in particular for this invitation. I'm very happy to give this course, even though unfortunately through video conferencing because of the coronavirus emergency. So today I would like to start my course, which is going to be an introduction to the theory of grotendic toposis from a metamathematical point of view. So what do I mean by that? I mean that I would like to present to you the basic notions of grotendic toposphere with the specific aim of giving you an idea of the sense in which grotendic toposis can play the role of concepts capable of shedding light on a greater variety of different mathematical subjects. So I will present things in such a way that it will become clear why toposphere can be regarded as a sort of unifying metamathematical theory which can be used in a greater variety of different ways for studying mathematical theories, mathematical problems, for connecting them with each other, for multiplying the points of view on a given subject, etc. So I have decided to start my course with this striking citation by grotendic about the unifying nature of toposis. Grotendic says it is the topospeem, which is this bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the word of the continuous and that of this continuous or discrete structures. And then it has, it is what I had conceived of most broad to perceive with the finest by the same language, rich of geometric resonances in essence which is common to situations most distant from each other coming from one region or another of the vast universe of mathematical things. It is of course a very beautiful extract which illustrates the way grotendic used to conceive toposis not just as mathematical objects that are interesting to study for their own sake but as mathematical objects which can play a metamathematical role in the sense of studying mathematics from a higher point of view. And so this is the point of view which I will try to illustrate in the course. So I will try to present you the basic elements of the theory of grotendictoposis starting from scratch. So I will not suppose you already know what a topospeem is. We will introduce the notion of Schiff on a topological space. First then we shall go to the categorical generalization, the notion of Schiff on site. Then I will present the definition of a grotendictoposis as originally introduced by Grotendick in the early 60s. Then I will present you the classical points of view that have emerged throughout the last decades on the notion of topos. So in particular I will focus on three classical aspects of toposis which play a crucial role in the development of the subject. The point of view of toposis as generalized spaces. The logical point of view of toposis as mathematical universes and their own internal logic in which one can develop mathematics in a similar way as one is used to do in the classical set theoretical foundation. Then third point of view which is the point of view of classifying toposis. So toposis is viewed as the classifiers of the models of certain kinds of the first order theories which should be considered up to the so-called notion of Morita equivalence which is the relation which identifies two theories of how they have equivalent the classifying toposis. So these are quite the classical viewpoints of toposis that we shall first illustrate in the course. And then the final aim is to give an introduction to a more recent perspective which has emerged in the past 10 years I would say which is the perspective of toposis as unified bridges for connecting different mathematical contexts with each other and transferring notions, ideas, results across different theories. So going to the history of the subject which is directly related to this three classical points of view that I have mentioned the toposes were originally introduced by Glotendick in his quest for a general setting for developing homology theory. So in fact that was a problem raised by the Bayes conjectures the problem was to define a new homology theory that couldn't be a classical homology theory of topological spaces in the sense which was already known. And so it became clear that it was necessary to enlarge the realm of general topology and so to enlarge the notion of topological space in order to define more general homology theories. So this was the original motivation that pushed Glotendick to introduce his notion of toposis as a setting for developing homology. And in fact this program was successful because in the end all the Bayes conjectures were proved in particular thanks to these new Bayes homologies which were introduced. So the original conception that led Glotendick to introduce his toposes was that of a notion of generalized space so the idea of understanding a space through the category of sheets of sets on it and defining more general notions of sheets so that the notion of sight was introduced with that purpose to clarify the general categorical context in which it makes sense to define sheets and then a topos as we shall see was defined as any category which is equivalent to the category of sheets on it. So through this duality between the language of topology and that of category theory which is given by the construction of sheets on a topological space one is able to revisit general topology through the lenses of category theory and this as we shall see is very interesting because it leads to a great re-foundation of topology itself. So this was the original point of view so toposes as objects which provide the notion of generalized space. Then I mentioned in the previous slide that there is a second point of view of more logical nature on toposes which emerged actually in the same decade in which toposes were introduced. Category theories such as Villo Vier and Miles Tierni decided to undertake an axiomatic study of grotendic toposes which led them to the identification of a crucial object the object omega inside the topos also called its sub-object classifier which in fact encodes a great part of the internal logic of the topos as regarded as a sort of mathematical universe in which you can develop mathematics. So this point of view was exploited in particular to understand set theory in a more flexible way for instance one of the first applications was to reinterpret the forcing construction in terms of toposes and of course this perspective made it clear that it is important to develop mathematics not just in the classical set theoretic foundation but in arbitrary toposes and so from the very beginning there was this idea of categorical semantics of interpreting theories in arbitrary toposes and also studying their models in all these toposes aiming also for a classification of these models and this brings us to the third perspective on the notion of topos which is the perspective of classifying toposes which emerged in the 70s thanks to the work of many people but most importantly the Montreal School of Categorical Logic which was active in the 70s, so Makayre, Yes, Juallet So this point of view is that of toposes as the classifiers of certain kind of structures which can be formalized within a certain logic we shall see that this logic is precisely what is now known under the name of geometric logic So in fact this point of view of classifying toposes was already introduced by Grotendick in the specific context which he was addressing which was theories of rings and so special kinds of rings like local rings, strictly in c and rings etc but in fact it was already clear to Grotendick that this idea of understanding a toposes from the point of view of the morphisms to it from arbitrary toposes so this is the unique paradigm of course was potentially quite interesting because one could describe these morphisms in quite elementary terms in fact by using first order logic and in fact this task of identifying the right logical framework to formalize all of this was, as I said, identified by the categorical logicians of the Montreal School So this introduced actually a noble point of view on toposes because you see since you need any object in a category is determined up to isomorphism by the collection of its generalized elements we can present toposes by specifying the structures which they classify and then of course if we want to take this point of view we have to investigate the relationship which might exist between theories which might be classified by the same toposes so this relation is called mojite equivalence we shall define all of this formally in the course of the lectures and so of course we have to understand toposes as mojite equivalent classes of theories written within geometric logic just because the different theories may have the same classification of the toposes what is quite important about this point of view is that it is complete in the sense that every dotted topos actually can be seen as the classifying topos of some theory of course very interestingly this theory is not unique in fact this notion of mojite equivalence as we shall explain in the course is far from being trivial it is actually quite interesting and deeper because it expresses precisely the idea of describing the same structures in different languages and so it can in particular relate the theories that are syntactically quite different from each other but which share the same semantics ok so then this perspective of toposes as unifying the bridges to a certain extent builds on all these classical approaches in actually exploiting the fundamental ambiguity which is present in topos theory which is given by the fact that a topos can be presented in many different ways actually infinitely many different ways so toposes are presentable categories so they by definition as we shall see and so they can be built from a great variety of different presentations and these presentations can in general belong to different areas of mathematics and so it makes sense to try to use toposes as bridges because connecting these different presentations with each other and hence for transferring knowledge across different areas of mathematics so actually the key idea behind this use of toposes as bridges is really this fundamental ambiguity that a given topos is associated in general with infinitely many different presentations technically speaking sides but not just sides because toposes can also be presented from other mathematical objects but still the idea is that we can build toposes from different areas of mathematics which might turn out to be equivalent or strictly related and then in such a situation one can consider invariance of these toposes and try to understand these invariance from the points of view provided by the different presentations and this actually is a quite effective technique which can lead to deep transfers of knowledge across the areas to which these presentations belong so in fact what I would like to illustrate in the course is how also this bridge technique works and the fact that it represents in a way a methodology for investigating the mathematical theories in a quite dynamical way in the sense that you try to understand theories by multiplying the points of view of them by relating them with other theories and so all of this becomes really a way not just for relating the necessarily different theories with each other but just to study a given theory in a given domain but in a dynamical spirit ok so now I have I would conclude my conceptual introduction and I would start the technical part of the course by recalling the basic notions that we will need for introducing formally the notion of glottic topos ok so let's first talk about ships on a topological space so before introducing ships we have to talk about pre-ships so what is a pre-ship on a topological space well it is a way of assigning to each open set of the space a set and to each inclusion of open sets a function between the corresponding sets going in the other direction so if you have an inclusion p into u there corresponds a function going from the set corresponding to u to the set corresponding to v and one of course requires these assignments to be factorial in the sense that we want the identity inclusion of an open set into itself to be sent to the identity map of course it's a very natural requirement and we also want this compatibility relation between the maps which correspond to the inclusions in case we have a chain of three open sets one included in the other like this w included in v we want the map corresponding to the inclusion of w into u to be the composite of the two intermediate maps so these maps which correspond to the inclusions of open sets are called restriction maps because one has in mind the classical example of a ship on a topological space which is the idea of taking continuous functions on open sets of the space with values in a fixed space such as r so this is a classical example of a free ship then we will see it's also a ship because when we have an inclusion of an open set v into an open set u of course here we have a canonical induced map which is just given by the restriction of our continuous function to the open set v and of course you see immediately that these properties which give the definition of a free ship are really satisfied so this is why the terminology restriction maps of course the notion is completely axiomatic so these maps shouldn't be general restrictions in fact even the sets of this form shouldn't necessarily look like sets of functions but this is an intuition that you can use in a number of situations ok so all of you are familiar with the basic language of category theories so you will have immediately realized that the free ship is just a fact defined on the opposite of the category of open sets of the space so of course if you have x topological space you can define a category which is denoted o of x those objects are the open sets of x and those arrows are precisely the inclusions between these open sets so between two given open sets regarded as objects of this category there is at most one arrow between them so there is one precisely when the first is included in the second so it is a very poor in terms of structure category but it is still a category and so one can consider functors defined on it and if you take the opposite of it the opposite you take because you want functors defined on it to be contravariant because here you see any inclusion is sent to a map going in the other direction well you realize that the funcals on o of x of two sets are precisely the pre-ships then of course we have also a natural notion of morphism between pre-ships which is just the notion of natural transformation between these functors so concretely you see the notion of morphism of pre-ships is what I have written here so we want for each open set view a map connecting the set F of U with the set G of U and we want this family of maps to be naturally in the sense that it should be compatible with respect to the restriction maps ok so now we can talk about sheeps so the notion of sheep is quite important because it relates in a very natural way the global and the local behaviour so in a sense the idea is to consider pre-ships whose behaviour on a given open set is determined in a natural way by the behaviour on a family of open subsets which cover the given open set and so we give this definition we say that a pre-ship on a topological space X is a sheep if for any open set U and any open covering of this open set so by open covering I just mean a family of open subsets such that the union of all is given to the given open set so in such a situation if we have two elements of F of U whose restriction to all the elements of the covering family are equal then the two elements should be equal so this is a condition which expresses the fact that elements of the pre-ships are determined by the restriction but then there is the other fundamental condition which represents a converse to that I mean is it possible to reconstruct or to uniquely determine a given element of F of U starting from an arbitrary family of elements of the F of Bi provided that this family satisfies a national condition of course you realise that you should have a compatibility condition in order for the existence of an element defined on the big open set which generates all the restrictions because you see if you have an element S whose restriction to all the Bi is given Si then we should have this compatibility relation on the intersections but in fact this is the only obstruction that we put in defining ships so we want that whenever we have a family of elements of the values of the pre-ship at the open sets in a covering family which are compatible in that sense we want there to be an element of the value of the pre-ship at the big open set which restricts to the elements of this family so in fact we understand the significance of this condition when we revisited the example of continuous maps because suppose that you can cover U by the family of open sets Bi and suppose you have for each Bi a function which is continuous we pay using R and suppose that these functions are compatible with each other in the sense that they agree on all pairs of intersections then you certainly can define the continuous function defined on the world U which will restrict the world this at Bi and so you understand that in fact we already know many ships we have already encountered many many ships in different fields of mathematics in fact you can not only consider the ships of continuous functions you can consider if you have a variety the ship of regular functions on this variety and if you work in differential geometry you have of course the ship of differentiable functions and differentiable manifolds and similarly you have the ship of all of the functions on a complex manifold I mean the idea is always the same that of gluing things that are compatible with each other in order to get a globally defined object from locally defined data okay then an observation we can make is that many ships in mathematics actually are not just the ships of sets but they are ships of of algebraic structures such as modules, wings, abelian groups already this we can resume so yes I was saying that many ships which arise in mathematics are actually ships of richer structures than mere sets so in particular we have ships of modules, ships of wings ships of abelian groups etc already you see the example we gave here you see we can these sets of functions defined on the open sets of our topological space this is the structure of a ring because you can add and multiply such functions etc so one might be tempted to consider just the ships of these richer structures but in fact Grotendig never wanted to do that even though of course he was interested in particular in ships of wings because all the ships that are existing in algebraic geometry in particular in the structure shift which is associated with a scheme is a ship of rings but in fact he understood that it was good to privilege just the pure notion of ship of sets and try to understand these more refined types of ships in terms of them in fact what we will see is that thanks to categorical semantics one can understand these ships of algebraic structures as models of the given algebraic theory inside the category of ships of sets and so in fact it is a good idea to consider as the primitive notion of ships of sets because then you can understand also more complicated kinds of ships in terms of ships of sets and on the other hand the category of ships of sets is good because this category has as we shall see all the present categorical properties that we are used to in the familiar context of sets in fact these categories inherit all these present properties such as the existence of all small limits all small coordinates etc while it wouldn't be like that if we consider categories of ships of a more complicated kind so in fact as we shall see Grottingdijk defined his toposies as general categories of ships of sets not just a methodological space more generally honest art but still the ships were considered that we were using as sets ok so what can we do in order to understand ships from a categorical viewpoint well a first observation that we can make is that in order to define a ship on topological space we actually didn't use the points of the space at all we had just reduced the open sets of the space and the inclusions between these open sets and so in categorical terms in order to define pre-ships we have just used this category because pre-ships were defined as funtals or opposite of this category towards the category of concerning ships the situation is is a little more subtle because of course if you want to express the ship condition you have to do it with reference to a notion of covering of an open set of the space by a family of open subsets so you need to have a notion of covering the family in your category in order to define a notion of ship on this category with respect to this covering family and as we shall see this notion of covering family is what is provided by the notion of growth and pictopology on a category so why does it make sense to understand topological spaces from the perspective of the category of ships on these spaces well because in fact many important topological properties of spaces can be naturally understood and formulated as invariant properties of the associated categories of ships of subsets on these spaces for instance you can immediately see these four properties which do not involve the consideration of points of the space think for instance about compactness or connectedness these are properties that you can formulate in terms of the category of open sets of the space and then you can expect them to be expressible in terms of the category of ships on these open sets I will tell you exactly how this is done later but for the moment I just wanted to remark that in fact these are the two basic remarks which led Grotendicker to introduce his categorical generalization of the notion of ship and then the notion of a topos as a category which is equivalent to the category of ships on a site which is just a category equipped with a natural notion of covering family before introducing the notion of site let's just remark how nice these categories are so for any topological space X the category of ships on X actually has all small limits why? because in fact this category sits inside the category of pre ships on the space and this category of pre ships is a category of fanters with various insets insets you have all limits and so by defining them point wise you can get limits in the fanter category and in fact when you take limits of diagrams which take values in ships you still get a ship just because if you think a little bit about it you realize that the ship condition can be expressed as a limit condition because the ship condition can be expressed as saying that a certain arrow should be the equalizer you see if you have such covering then here of course you have a canonical map and then here maps so you can understand the ship condition as the property that this canonical map is the equalizer of these two maps and so you see you can express the ship condition in terms of limits and since limits commute to be limits you understand that this category is closed in the category of pre ships under arbitrary small limits in particular it has all small limits concerning co-limits the situation is more interesting because they are not computed in the same way as in the category of pre ships in general but there is a fundamental construction which is that of the associated ship factor which is just defined as a left adjoined to the inclusion factor of ships into pre ships and which allow us to compute the co-limits because in fact since this factor is left adjoined to a certain factor it will preserve our co-limits and so we know how we can calculate them because we take given a diagram of ships we compute its co-limit point wise in the category of pre ships and then by applying to the result the associated ship factor we get a co-limit of the diagram in the category of ships and so thanks to this we understand that actually all the categorical properties that we appreciate in the classical category of ships are inherited by the categories of ships of sets on any topological space. Now I would like to explain how revolutionary is this idea of understanding topological spaces through the associated categories of ships by revisiting the classical notion of ponds because in fact one of the great innovations which was brought by Topos Fury is being able to treat the ponds in a richer way in particular define a notion of point which admits automorphisms. So you see in topology we have a notion of point which is quite poor because well the topological space by definition is a set of points with a collection of open sets but then the points by themselves don't have a richer structure. While thanks to toposes one can define points in a way which is much richer because points become functions between toposes satisfying certain natural properties as we should see. And in fact more precisely the way we get such funtals from points in the classical topological setting is provided by the construction of the skyscraper ship on one end and the construction of storks of ships at a given point on the other end. So these constructions actually define funtals which are joined to each other. So the stork funtals is actually left joined to the skyscraper funtals. And so what we see from this is that any point of the topological space actually induces an adjunction. An adjunction between the category of sets and the category of ships on the space. In particular of course when you have adjoined funtals it suffices to give one to uniquely determine the other up to four pieces. And so here the fundamental funtals is actually the stork funtals because this funtals not only preserves arbitrary limits since it has a right adjoined but it also preserves finite limits. And in fact these funtals which preserve finite limits and arbitrary limits are the fundamental funtals that we want to consider between growth and ectoposis. Because in fact they provided the right categorical generalization of the notion of continuous map between topological space. You see when you have a continuous map when you take the and you try to understand this from the point of view of the open sets you realize that the inverse image of this map which of course by continuing this sense open sets and open sets actually preserves finite intersections and arbitrary limits. So what is the categorical analog of finite insertion? Well it is finite limits. What is the categorical analog of arbitrary limits? And so you understand we shall define this later in the course the notion of geometric morphism which actually is a pair of adjoint funtals such that the left adjoint preserves finite limits. And so we shall see in particular that every and the point of topological space defines a geometric morphism from the topos of sets to the given topos of ships on the space and in fact geometric morphisms from the topos of sets to a given topos are what we call points, points of the topos. And so you understand how powerful is this factorial reinterpretation of topology. In fact Grotendig referred to that as a metamorphosis of the notion of space. So a metamorphosis in the language of category theory. So it is not strictly a generalization because in fact the different topological spaces may have equivalent categories of sheets even though if the space is satisfied good separation properties this is not the case. Technically speaking if the spaces are so bad then they can be recovered from the corresponding topos of sheets. So it is not strictly a generalization but almost because I mean all the spaces that you encounter for instance in functional analysis or algebraic geometry satisfy this sovereignty. But most interestingly it is a reinterpretation of the notion of topological space in the language of category theory. And finally for today I would like just to explain how we understand open sets from the point of view of sheets. In fact open sets of topological space correspond precisely to certain objects of the category of sheets on the space. These objects are precisely the sub-terminal objects. So what is a sub-terminal object? Well it is an object whose unique approach to the terminal object is a monomorphism. We have seen that we always have the terminal object in any category of sheets on space because if we take the three sheets which sends every open set to the singleton of course we realize that the sheet condition is satisfied. So this is the terminal object and if you try to understand what are the sub-objects of this terminal sheet you realize that in fact these are the sub-sheets. So the notion of sub-sheet is what I have written here and if you look at this notion in the particular case where f is the terminal sheet you understand that in fact all these sub-sheets are actually determined by open sets which can be attached to them in a way to get actually an isomorphism between the sub-objects of one and the open sets of the space. So this is quite interesting as well because in fact you see that open sets of the space sits inside the topos as objects as particular objects of the topos while the points correspond to suitable kind of factors relating the topos to the topos of sets. So in both ways I mean both these ways are extremely important and you also understand that they are quite quite different from each other because I mean speaking of an object of the topos means that you are in a sense objects are what the topos is made of so they had a certain substance while the factors are a way of relating the topos with another topos so they are a less fundamental notion if you take the point of view of the topos because they induce a certain deformation of what is in the topos along a certain morphism and so in fact this could already explain the fact that toposphere really should be seen as a generalization of pointless topology rather than general topology because the fundamental notion here becomes that of open set of a topological space and in a sense the notion of point is a secondary more derived notion so I still have four minutes so we can start talking about the notion of seed which will allow us to define sites and then ships on a site so first we define what a pre-seed is in a category it is simply a collection of arrows with a fixed co-domain and then we say that a pre-seed is a seed if this collection of arrows is closed under composition of the art of course any pre-seed generates a seed because I just had to take all the arrows in the pre-seed and I can consider an operation of pullback of seeds along arbitrary arrows which is defined by this formula so seeds are important because they allow us to define a notion of ship without having to refer to the intersections of open sets you see here apparently you might be led to think that you need your category to have finite limits at least in order to talk about the ships on this category for a certain topology because you might think that you need some replacement for these intersections which you have to consider for expressing the notion of a ship but in fact the notion of of a sieve allows you to get around this problem because in fact you can make this observation which is here in the remark giving if you have a three ship on a topological space X and a covering family in of a given open set of this space then you realize that giving a family of elements of the three ship which agree on the intersections is actually the same as giving a family of elements indexed by the open sets of the sieve generated by the three sieve in the covering family which satisfies this compatibility condition and so you understand that actually this idea of generating sieves from three sieves by taking all the arrows which factor through an arrow in the pre sieve can solve the technical problem of referring to the intersections because the idea is the intersection is of course a universal way of connecting the two open sets with each other but more generally you could consider all the things which lie above the two open sets without necessarily the need of the existence of a universal object with this property and so by using this remark actually one can define ships on in the general categorical setting so what gives the replacement for the notion of covering of an open set by a family of open sub set is as I already anticipated the notion of grotendic topology on a category so for today I stop here and tomorrow we shall start again from grotendic topologies