 So I am going to review some basic results on Topozis. So of course Topozis, the Nation of Topos, was introduced by Rotendik in the late 50s or around 1960. And they were first introduced as generalized categories of sheaves on so-called sites, which means categories endowed with topology. But it was discovered a little later that Topozis can also be presented in a very different way as classifying objects for first-order theories in the sense of logic. So it is a very different point of view on exactly the same objects. So the original presentation of Topozis by sites can be seen as a geometric way to present Topozis. And here what I'm going to talk about is the linguistic way to present Topozis from a mathematical theory, linguistically presented in some way. So my lectures will consist in three parts. So first I will introduce a sheer notion of classifying Topozis. I will immediately state the theorem of existence of classifying Topozis. And then I will spend some time explaining the meaning of this theorem. The second part of the lectures will sketch the proof of this theorem. And I shall especially present so-called syntactic categories endowed with their syntactic topology, which allowed to construct classifying Topozis as Topozis of sheaves on some sites associated to the theories we consider. So these two parts of the lectures date back to the 1950s. I will immediately say the names of the people who were involved in this theorem and its proof. And the last part of the lecture will also consist in basic results, but which are more recent. So this is taken from the Piaget thesis of Olivia Caramello 12 years ago, and her book Theories, Sites, Topozis. So these basics results allow to begin to make the theory of classifying Topozis efficient for establishing concrete results in mathematics. Okay, so by now I begin with the notion of classifying Topozis. And I immediately state the following theorem. So this theorem, I think, was proved in full generality in 1975. And quite many people contributed to this theorem. So here I have written the names of MacGyne and Reyes, because in fact this theorem was stated on proof for the first time in a book by MacGyne and Reyes. But this work certainly depended on some very important earlier work by Lawyer. And I think some other mathematicians or categoricians of the same circle played a role in establishing this theorem. So I also wrote the name of Joial, André Joial. And I should add that this theorem systematizes or generalizes some ideas which are already present in particular cases in the Piaget thesis of Monique Akim, who was a student of Grotendi. So this means that at least part of the stop, in fact, was certainly already present in the mind of Grotendi. So here is a statement, but maybe at that moment you will not fully understand the statement of the theorem. And the purpose of the first part of this lecture will be to explain, to fully explain the meaning of this theorem. So, yeah, you consider the so-called first order theory, which is geometric. So geometric is some kind of technical condition which will be explained later. But in fact it is not restrictive, which means that for any first order theory, in fact, there is a way to make it geometric without changing the set theoretic models of the theory. So this technical condition is not restrictive, is not very restrictive. And this is the first thing we can have in mind about this notion. Then, for any such theory, there exists a topos. So I shall explain a little later what is a topos. A topos, endowed with a universal model of the theory, such that for any other topos, there is a natural equivalence of categories. Between the category of models of the theory in E, and the category of morphism, of toposis morphism, from E to the classifying topos ET. So this statement looks familiar for anybody who knows the notion of classifying object, of representable from top. Here, what we are saying is that any theory, any first order geometric theory, defines a functor of so-called models, and this functor is representable by a particular topos. So this theorem is extremely general. It applies to any first order geometric theory, and it allows to associate to any such theory, so which means to any linguistic presentation of a first order theory. It allows to associate to it a topos, which, in fact, is a geometric object. The toposes were invented by Gotendic as the most general notion of space, of geometric object, which could exist in mathematics. This is really what Gotendic thought. Okay, so even if you don't fully understand the statement of this theorem at the present moment, it is not important because we shall explain that with more details in the coming hours. Okay, so before I begin with explanation, I want to make a few remarks. So the first remark is that, given such a theory, is classifying topos, endowed with a universal model of T in this topos, is uniquely determined up to equivalence. The second remark is that T belongs to the world of logic. So here, logic should not be understood at some very particular part of mathematics. Here what it really means is formalization of mathematics, linguistic presentation of mathematics. Anytime we do mathematics, we begin by presenting one or several theories. And even if we don't think about it, we are using the language of logic to present any theory we want to study. So this is really how things have to be seen as respect with this theorem. Okay, so on one side of the theorem, you have formalized mathematics, a linguistic presentation of some theory. And on the other side of the theorem, you have an associated topos, the so-called classifying topos of T, which belongs to the world of geometry and topology. So for growth and dig, I repeat, the notion of topos was the most general notion of space. Yet another very important remark is the particular case of this theorem, when we take for E, the simplest non-trivial topos, which is a category of sets. And if we just specialize the theorem to this particular case, we get an equivalence between the category of points of the classifying topos and the category of set theoretic models of the theory. So this particular case is especially important, because in general, when we study theory in mathematics, we are looking for its set theoretic models. For instance, if we think about the theory of growth, what do we mean by your group? We mean a set and doves with a structure of group. What do we mean by your ring? We mean a set and doves with a structure of ring. So this means we are considering set theoretic models of the theory of groups or set theoretic models of the theory of rings, and so on. And here, what the theorem says is that there is a geometric interpretation for that. The set theoretic models of any such first order geometric theory can be seen as the points of some associated geometric objects, the classifying topos of this theory. Okay, so here this is, so this means that in some sense, this theory of classifying topos is could also be called a functorial theory of models. By now remark five, which is also extremely important. It is a fact that, of course, as we said, for any first order geometric theory, there is an associated classifying topos, which is uniquely determined up to equivalence. But in the other direction, given any topos, there are infinitely many theories with the same classifying topos up to equivalence. And this remark gives rise to the theory introduced by Olivia Caramello of toposis as bridges, because what Olivia has begun to use in a systematic way is a relation between theories, which is defined by the property to have equivalent classifying toposis. So this can be seen as a theory as a general theory of translations, because, as I said, theories have to be understood as linguistic presentation of mathematics. Any theory is a choice of a language and a grammar to present some mathematical content. And the associated classifying topos is a geometric object, which incarnates the meaning of that theory. So when there are two theories with the same associated topos, it is exactly as having two languages, possibly very different, which have exactly the same mathematical content, which means which exactly tells the same story. And as we already remarked, for any given content, which means for any given topos, there are infinitely many theories, or if you prefer infinitely many languages, which allow to present this content. So this remark is the beginning of the possibility to use toposis as bridges between theories, which possibly could be completely different. Okay, so by now I want to explain the different ingredients in the theorem. And so the first ingredient, of course, is the word topos. So I want to explain a little or at least to recall what is the topos. So in order to have an idea of what is the topos, we need to begin with examples. The first example is consistent toposis associated to topological spaces. In fact, this is really like that, that toposis were born. So starting from any topological space, it is possible to associate to it the category of sheaths. And here I mean sheaths of sets on the topological space X. So I will not recall here what is a sheath. In fact, Olivia in her own lectures this afternoon will do it. But immediately I state the following proposition, which tells us that it is almost the same thing to know a topological space or to know the category of sheaths on this topological space, which means to know the associated topos. First, there is a general notion of points for topos and the first basic fact is the fact that any point of the topological space X we consider is a usual sense in uses in use points of the associated topos. And this map is one to one, if X verifies an elementary properties, a property to be sober, which in fact is verified in most cases, it is verified by a whole or topological concrete topological spaces. And in the same way, open subset of X correspond to open sub-toposis of the topos EX. So I don't define here the notion of open sub-topos or the notion of point of a topos. But what I stress is the fact that in the world of toposis, there are categorical notions, such as the notion of points and the notion of open sub-topos, which allow to recover not only the underlying set X, but also its topology, so the full topological space. And so this means that the notion of topos generalizes the notion of topological space. And it also means that this notion realizes an embedding of topology into category theory. So this is really amazing because it means that in this way, we go from one world to another world, from the world of topology to the world of category. The world of topology is a world of continuous structures, whereas the world of categories is a discrete world. So it is very surprising that such an embedding exists. And the sheer fact to go from one world to such a different world is already extremely deep in itself. And not only it is deep, but by going from topology to category theory through the construction of the associated toposis. In fact, we enlarge a lot the world of topology. And this also is very important. This becomes immediately clear when we come to the second type of example of toposis, which is given by groups. So for any group, we can consider the category of sets endowed with an action of the group. And this also is a topos. In fact, here we have the two basic examples of toposis. On the one hand, categories of shifts on a topological space. On the other hand, categories of sets endowed with an action of a given group. And the notion of topos is pertinent, in particular, because these categories of shifts on a space or of sets endowed with an action of the group, have very similar categorical properties. And as Grotendick had remarked. Okay, and the example of groups can immediately be generalized in the following way. If we start with any small category, then we can consider the category of pressures on this small category C. This is by definition the category of contravariant factors from C to the category of sets. So this is a generalization of representations of groups. And this again is an example of a topos. So all these categories, which I have mentioned, shifts on a space, actions of a group on a set, or pressures, which means contravariant factors from a small category to sets. All these categories have similar categorical properties. And in fact, I will come to that a little later. What are the main general properties of topos? So when we come to these categories of pressures, it is very important to state the following proposition, which of course is very well known. So start from a small category C, then consider the category of pressures. And the content of the proposition is that this category of pressures C at can really be seen as a completion of C. So the first point is that C is embedded in C at. For any object of X, you consider the associated contravariant representable hash if. So which consists in all rows from an arbitrary object to X. And so this function, which any object of C associated the function it represents, so this function is fully faithful. So we have a fully faithful embedding of C into C at. And on the other hand, any object of C at, so any pressure, can be written as a core limit, or if you prefer an inductive limit of such representable factors. So here, in order to realize that the indexing category is the so called category of elements of the pressure if you consider. So this means that the objects of this category of elements consist in object X of C. Endowed with an element of the image of X by the pressure P you consider. So this is an indexing category, but the first important factor is that any object in the category C at is an inductive limit of representable object. So this is why I say that C at can be really be considered as a completion of C. Okay, by now, I propose a general definition of topazis. But once again, only if you are with an impact point this afternoon. So here I define a topazis as a category, which is both a quotient on the subcategory of some category of pressures. So a topazis is a category such that for some small categories C at. There exists a pair of adjoint functors from C at to C on from E from, excuse me, from C at to E on from E to C at such that so first this pair of functors is adjoint. So this more precisely, the functor from C at to E is left adjoint to the functor from E to C at. Secondly, the right adjoint functor from E to C at is fully faithful. So it is a fully faithful embelling. It can be seen as through this functor as a subcategory of C at. And as this pair of functors is adjoint, the fact that this functor is an embelling is the equivalent to say that the composite functor of the embelling by compose with its left adjoint. Is isomorphic to the identity functor of E. So E is a subcategory of C at. It is also a quotient of C at through this left adjoint functor, which is called the shiftification functor. And the composite of these two functors is naturally isomorphic to the identity functor of E. And the last point in the definition is that we want the left adjoint functor to respect finite limits, finite projective limits. Okay, so this you see is a very categorical definition. Then we can make a few remarks. So first, the fact that the G lower star component from E to C at respects arbitrary limits. This is because it is right adjoint to some functor. And the J upper star component from C at to E, the shiftification functor respects arbitrary co limits arbitrary inductive limits. This is because it is left adjoint to some functor. As a consequence of this remark is the fact that if we consider the canonical functor from C to E, which is a composite of the Yoneda embedding into C at with a shiftification functor. So this composite can be denoted L. Then we get that any object of E can be written as a co limit of images of objects of C by this canonical functor. So this is just a consequence of the fact that this is already true for C at and the functor G upper star respects the limit. Okay. By now, I want to recall that such presented in a very abstract way can be defined in a more concrete way by a so-called grottendic topology. So here we can define topologies in the following way, consider an arbitrary object X of C, and let's consider sub-object of the representable pressure associated to X. So what is such a sub-object in the category C at it is just a family of morphism from arbitrary object X prime to X, which is stable under any composition with any morphism from some X prime prime to X prime. It is stable under right composition. So such a family of morphism stable under right composition and which all go to X is called a sieve of X. And by now, if we have a topos presented from Seattle as in the previous page, a sieve on X, which means a sub-object of the representable pressure associated to X, is called covering if this embedding from S to Y of X becomes an isomorphism after Shifififififififififififif. Okay, so here we have a notion of covering sieve for any object X, and we can denote GX, the family of all covering sieves of such an object X, and then we can consider not only one X but all possible objects X of C. Also, we have an index family of covering sieves, and this is called a grotendic topology on sieve. Okay, and by now it is easily checked that such topology defined in this way verifies some axioms. So in fact, there are three axioms, which are very easy to verify. There's a maximality axiom, which says that when we consider an arbitrary object X, then the full sieve of X consisting of all rows to X is always covering. Secondly, the stability axiom, if we consider any row from X prime to X in the category C, and any sieve S on X, which is covering, then its spread back through the morphism we are considering is still a covering sieve of X prime. And lastly, there is a slightly more subtle transitivity axioms, which says that if we are considering a covering sieve S of some object X, and a sieve S prime of X, would spread back by all elements of S are coverings, then it is a covering sieve of X. So this is a little more subtle. The important point is that it is not difficult to get that any topology defined as in the previous page verifies these three properties. And now the second part of the theorem is the fact that conversely, any index family of sieves which verifies these three axioms defines a topos endowed with two factors, which allow to recover the topology jet. So to consider a topos presented from Seattle through these two factors is the same thing as to consider a growth endic topology on X. Okay. Okay. And by now, I want to state the main categorical properties, which are verified by two poses. So two poses are particular types of categories. They are categories, which can be constructed from categories of precious by the type of construction we presented in the previous pages, but it happens that because they are deduced from categories of precious in this way. They inherit from categories of precious some very special properties. So first, any topos is a locally small category. So these things that given two objects in a topos, the morphism between these two objects make up a set. Secondly, in an arbitrary topos, there are in any topos, there are arbitrary co limits on arbitrary limits. Thirdly, for any morphism in a topos, the associated pullback factor defined by considering the fiber product over this morphism, not only respects arbitrary limits, but it also respects arbitrary co limits. Another important properties is that in a topos, sums are designed. So these things that, for instance, if we consider two objects on the topos and we make up their sum, the limit of the diagram consisting in these two objects without any morphism. Then, of course, the sum contains as sub-objects the two objects we started with. And the intersection of these two objects inside the sum is equal to the initial object of the category, or if you prefer to the empty object of the category. So this, of course, is familiar to us in the case of sets, but here we say that it is also verified in an arbitrary topos. Another extremely important property, which is also familiar to us in the case of sets, but which is verified in any topos, is the fact that to consider a quotient object of sum object E, which means to consider an epimorphism from E to sum object Q, is the same thing as to consider an equivalence relation on E, which means a sum object of E cross E, verifying the usual axioms of equivalence relations. Contains a diagonal, it is symmetric, and it is transitive. And the link between these two presentations is the fact that the equivalence relation is just the fiber product of E with itself over the quotient. And the quotient is recovered from the equivalence relation by considering the co-limit of the diagram consisting of R, the equivalence relation, E, the ambient object, together with the two morphisms, the two projections from R to E. Okay, another important property is that in a topos, the sub-object of any object and the quotient object of any object make up sets. And the last property I want to state is the fact that in a topos, a morphism which is both a monomorphism and an epimorphism is an isomorphism. Of course, this is also familiar to us in the case of sets. If you have a map which is both injective and onto, then it is objective. In other words, it is invertible. But this is true, this is still true in an arbitrary topos. And here, there is a very remarkable theorem of Giro. So Giro was a student of Autendic. The theorem tells us that these properties characterize toposis. For this, we need an extra condition, which is the fact that the category we are considering is not too big. So here is a precise statement. We consider a category E, verifying all properties of the previous page. And then we suppose that in this category E, it is possible to identify a small full sub-category, such that for any object of the E, there is an epimorphic family of morphism from objects of C to X. So in some sense, this means that E can be considered as generated by C. And then, for such a C, there is an induced jet. You decide that a sieve of an object X of C, which means a sub-object of Y of X in SEAT, is covering if it contains an epimorphic family. So here epimorphic refers to the categorical structure of E. So this is a definition. And then the statement is that the category E is the coefficient of SEAT defined by the topology G. In other words, it is a category of sieves on the category C, endowed with a Grottendig topology J. So this means that by now we already have a double characterization of toposites. So the first characterization was constructive. Toposites were categories which could be constructed in some way. On the earth, thanks to this theorem, we have a second characterization. Toposites are categories which verify the properties of the previous patch, plus the existence of such a generated small subcategory. Another remark we can make here, which is very important, is the fact that this theorem means that a given toposite can be presented in infinitely many ways from small categories C, endowed with topology. Because here we can take in this theorem, we can choose for C any small category, any small subcategory, full subcategory of E, which is big enough to generate E. And there are infinitely many choices for that. And this also is part of these extremely interesting properties of toposites that they can be presented in infinitely many ways. So I already mentioned that they can be presented in infinitely many ways from theories, as we shall see later with more detail. But here, thanks to this theorem, we also know that a toposite can be presented in infinitely many ways from categories endowed with topologies. So this presentation, of course, is more geometric, whereas the other one is linguistic. And so here I mentioned again that the full theory of that Olivia Caramello has proposed to use these basic facts to make toposites bridges between theories and also between categories endowed with topologies. So I will come back to that later because in fact we shall make use of several such bridges to establish present. Okay, so by now I can just recall the basic examples I had already talked about. So the first basic example is the example of topological spaces. So we said that for any topological space, there is an induced toposite, the toposite achieves this topological space. But of course, when we talk about topological spaces, we also have to consider continuous maps. So topological spaces make up a category. And here, the basic fact is that whenever we have a continuous map between two topological spaces, then this map induces a pair of functors between the associated toposites of sheaves on X and Y. So a pair of functors which are right joined to one to use the other. So these things that they go in the two reverse directions. So there is a so-called push forward functor from which transforms sheaves on X into sheaves on Y. And there is a so-called pullback functor which goes in the reverse direction. It transforms sheaves on Y into sheaves on X. And it is a fact that these two functors are adjoint. So the pullback component is left adjoint to the push forward component. So this means the push forward component respects arbitrary limits, the pullback component respects arbitrary co-limits. And as a matter of fact, the pullback component also respects finite limits. Okay. By now, we remark that if we have a topos which is defined by a topology J on some categories C, then the topos C-hat and the topos C-hat and J are related by a pair of functors. And for us, it was a definition in the way I presented toposes. So they are related by a pair of functors which verify exactly the same properties as the functors induced by a continuous map between topological spaces. So this pair of functors are adjoint. And the component J upper star respects not only co-limits, but also finite limits. Find out whether we know other situations with such pairs of functors. So in fact, there is one which is very important. So let's consider an arbitrary functor between two small categories. So of course, this functor defines an associated functor in the reverse direction between the associated categories of pressures, because a pressure on D is a contravariant functor from D to set. So if you composite with row, you get of course a contravariant functor from C to set. In this way, you define a functor which goes from D-hat to C-hat. It transforms pressures on D into pressures on C. And here, a very important fact is that this functor, this functor of composition with row has two adjoints, one left adjoint and one right adjoint. The right adjoint is denoted row, lower star. And the right adjoint is denoted in this way with such a right of dot of mark. And as the functor of composition row, upper star has two adjoints, of course, it means that not only it respects arbitrary limits, but also arbitrary limits. Also, we have an adjoint pair, row, upper star, row, lower star, such that the left adjoint component, row, upper star, respects in fact not only finite limits, but arbitrary limits. Okay, so this example, some, especially the first one, lead to the following definition. Morphism of toposys by definition is a pair of adjoint functors, the so-called pullback functor and the so-called push forward functor, such that the pullback component, which means the left adjoint component also respects finite limits. So here it is just a definition. Okay. And here, of course, we remark that as we are talking about functors or pair of functors, they naturally make up a category. Given two toposys, E prime on E, and two morphisms of toposys from E prime to E, so this morphism we call F and G. And by definition, a transform of such morphism of toposys is a transform of functors between the pullback components. So by definition, a morphism from FG is a morphism of you if you prefer a natural transformation from F upper star to G upper star. And because they are adjoint, this is the same thing as a morphism from G lower star to F lower star. So here you have to be careful that taking the adjoint, the arrow has to go in the other direction. So to give a morphism from F upper star to G upper star is the same thing as to give a morphism from G lower star to F lower star. So in particular, the category of points of a toposys is by definition the category of morphism of toposys from set to E. So of course, the category of set is a toposys, it is just the category of pressures on the category consisting in just one object and one morphism. Or if you prefer, it is also the category of shields on the space of the topological space with one element. So a point of a topos by definition is a morphism of toposys from set to E and points defined in this way make up a category. It is not only a set, it is a category. In fact, in general, it will not be a set because it can be bigger, but it is a set in quite many cases, but not always. And the last part of the definition is a notion of embedding of toposys. If you prefer the notion of subtopos. So what it is, it is a morphism of toposys consisting in a pair of adjoint factors, J upper star, J lower star, such that the push forward component J upper star is fully faithful. And this is equivalent to saying, because of adjectness, that J upper star composed with J upper star is isomorphic to the identity factor of E prep. This is the definition of subtopos. So you see that in this definition, everything is categorical. Everything is phrased in the language of categories, which is very natural because toposys are just particular categories. But we see that in the language of categories, we have expressed geometrical notions, especially the notion of point on the notion of sub space, the notion of subtopos. And also the notion of a relative toposys, which means a toposys considered over another topos through a morphism. And in fact, this will be studied much more in Olivia's lecture. So, of course, this definition was given in order to have the following examples. So first, any topological map map, any continuous map between topological spaces in uses a morphism of toposys. And in fact, this is one to one is why is a sober topological space. So this means that if why is sober, to consider a continuous map from some topological space X to Y is the same thing as to consider a toposmorphism from the topos associated to X so EX to the topos associated to Y. So this of course applies in particular, if X is a topological space with only one point. And so this means in particular that the points of why identify with the points of the associated topos, if why is a sober topological space. Another remark is that if we have topology J on a small category C, then the topos defined by J is a sub-topos of the topos of species of C. And the last example is that any factor between two small categories in uses a morphism of toposys from C-at to D-at, so in the same direction. Okay, so by now I have given a review of the theory of toposys. I hope I have given to you an idea of what a topos can be. In fact, I have given full definitions together with the basic most important examples. And by now I want to explain the other alpha of the ingredients necessary to understand the theorem of existence of classifying toposys, which is a linguistic presentation of mathematics. Here we have to use some language which is familiar to logicians and just as a warning we should not be afraid by this language because it is just a way to make ourselves aware of what we are doing all the time when we write mathematics. So the first order theory consists by definition in language, in a language plus grammar rules. So if you prefer in a vocabulary in a list of names on the one hand and on the other on grammar rules. And the grammar rules in mathematics are called axioms. So first we begin with the language. So the first order language consists in three types of names. First there are names of objects, but which logicians call sorts. So, for instance, if we want to consider group theory, here there is only one name of object, which is just the name group, or if you prefer, the letter g. If we consider the theory of rings, there is only one name of object, which is the name ring, or if you prefer the letter r. If we consider the theory of fields, there is only one name, k, or one letter, k, the name field, the letter k. If we consider the theory of vector fields over vector spaces over fields. So then there are two names, the name vector space and the name field. And we can choose letter for that. So for example v for vector space and k for field. If we consider the theory of modules over a ring, and here there are two names, modules and rings. Or we can write them as letters, m and r. Okay, this is just the names of objects we are considering. Then the second type of names are names of morphisms, or names of functions, or function symbols, or they are called. The function symbol consists in a letter f, going from a collection of, from a finite family of names of objects to a name of object. So for instance, if we want to talk about addition, so addition is certainly a name of morphism. So addition is a binary operation. So this thing, it is a function symbol from, so in the case for example ring, from f, f, because addition is on two variables, both in f, from f to f. And we can consider outer multiplication in a module over a ring. So this is the function from r, m to m, because one variable has to be in r, one variable has to be in m, and the function goes to m. And lastly, there are families of, there is a family of names of relations. So for instance, if we want to talk about the theory of equivalence relation, so what is an equivalence relation? It is a sub-object in the product of some object. So here, the equivalence relation will be some sub-object r of, for instance, e, where e, for instance, can be a set, or it can be an object of a topos. You see you have a name of object, e, and an equivalence relation, r, which has to be a sub-object of e, e. Here an important remark is that it is possible in this definition to take n equals 0. So a function symbol from 0 to b is called a constant symbol. And the relation inside the empty family of names is called a proposition symbol. So let's immediately take examples. So for instance, the language of the theory of rules. So in the theory of rules, you have one name of object, which is g, and there are three function symbols. So first for multiplication from g, g to g, then going to the inverse, this is to g from g to g. And lastly, there is a constant one, which goes from the empty family of names to the name g. It corresponds to choosing one element of g, one constant in g. And in that case, there are no relation symbols. If we want to talk about the language of the theory of equivalence relations, we have to choose one sort e, which will be the underlying object, no function symbol, and the relation symbol are inside e, because we want to consider a sub-object of the product of an object with itself. Okay, so this is the language of a theory. And by now, I want to spell out what it means to interpret such a language. So technologicians don't use the name language, they use the name signature. So for such a signature, which means a first order language, and for any topos, or more generally for any category with finite products, including a terminal object, then we can talk about interpretations of the language of the signature sigma. So what is a sigma structure m in such a topos e, or in such a category e? It is a map. A map where it shows the associates to any sort, which means to any name of object a, an object, which is called m a. And it has to associate to any function symbol f going to a1 a n to b, a morphism, which goes from the product m a1 cross m a n to m b. And here we are, of course, when we write this m a1 cross m a n, this is a product of the object m a1 m a n in the category e, and m b is also an object of e. So it has a meaning to talk about morphism from m1 cross cross cross cross m n to m b. In this case, n equals zero. So we have a constant symbol, and then we structure, sigma structure consists in associating to such a constant symbol. A map going from the morphism going from the terminal object to the object of name b, which is m b. Any relation symbol are inside a1 a n. We have to associate sub-object m r of m a1 cross m a. The sigma structure is a way to associate to any name of object, an object of e, to any name of morphism, morphism in e, between the corresponding objects. For any relation symbol, a corresponding sub-object of the product of objects associated to the names of objects, which are the context of the relation symbol. And by now, it is also possible to talk about morphism between two such sigma structures. So let's consider two such sigma structures m and n, which means two maps, which are associated, I repeat, to any name of object, an object with the name of function morphism, to any name of relation sub-object. Also, a morphism of sigma structure is a family map, which associates to any name of object a, to any sort a, a morphism in the category e, from m a to n a. So morphism between the corresponding objects with the name a, associated to a, through the two sigma structures m and n. And they have to verify two properties. So first, they have to be compatible with the function symbols. So this means that for any function symbol f, from a1 a n to b, the associated square, which is drawn on the edge, is commutative. And for any relation symbol r, inside some a1 a n, there is an associated commutative square as at the bottom of the page. Okay. And so this means by now that given a signature, which means given first order language, we can associate to any topos, or more generally to any category with finite products. The category of the sigma structures in this category e are not only that, but if we have a functor between two such categories, which respect finite limits, or more generally which respect finite products on monomorphism, there is an in use factor between the associated categories of sigma structure. So in particular, for any morphism of toposis consisting of a pair of adjoint factors, f lower star and f upper star, there are two induced factors, which in fact are adjoint between the categories of sigma structures on e on e prime. So the pullback from top and the push forward from top. They are well defined because both f upper star and f lower star respect finite limits. So in particular, they respect finite products, they respect the terminal objects, and they respect sub-objects. Okay, so I just look, yeah, so by now I am going to Excuse me, this is what you wanted to say.