 So it's my pleasure to reintroduce Lorent Le Fogg, who will continue his course on classifying toposes of geometric theories. Please, Lorent. OK, thank you very much. So I just want to remind the participants that yesterday, I introduced the statement of the existence theorem of classifying toposes. How to associate to any first-order geometric theory a topos which represents the function of models of this theory. So the first part of the lecture was devoted to the meaning of the theorem, of everything which appears in the theorem, and the second part, which we had begun, was devoted to sketching the proof of this theorem, which means to construct from a given theory T category, the so-called syntactic category, and owed with topology, GT, JT, the so-called syntactic topology, such that the associated topos answers the question raised in the theorem. So we had begun the second part. So saying first that we wanted to construct such syntactic category, CT, together with the topology, JT, and it should be endowed with canonical model, MT, which should be universal with respect to the construction of all models in all geometric categories. So then here, we had introduced the notion of geometric category, which is a type of category where geometric formulas in any signature can always be interpreted. And there is not only a notion of geometric category, but also a notion of geometric function. A geometric function is a function between geometric categories, which respects the interpretations of geometric formulas in sigma structures. OK. As a statement of the characterizing property of syntactic categories, endowed with their canonical model, the fact that these categories, that the syntactic category, represents a function of models on geometric categories related by geometric formulas. And then we had proposed the construction of the syntactic category. So by definition, the object of this category are formulas, or geometric formulas, in some variable. And these formulas are considered up to substitution of variables. Then there is a definition of morphism. So morphism are geometric formulas, which are provably functional. And in fact, yesterday, I had forgotten to say that these provably functional geometric formulas should be considered up to three provable equivalents. In fact, one participant made me remark that I had forgotten to say that. So here, it is added in the last line of this slide. So of course, this notion of equivalence, two provable equivalents, refers to what it means to be provable in the theory, just as the notion of provably functional formulas. So this heavily depends on the theory you are considering, not only on the signature. And then there is a formula for the composite of two morphisms. And of course, it has to be verified that this definition defines a category, and this category is a geometric category. OK, as this definition refers to the notion of tip probability, we need it to make precise what it means to be provable. So to be provable in a theory means that it can be deduced from the axioms of the theory using the inference rule of geometric logic. And we had given a full list of the inference rule which can be used. And of course, this list is just the list of deductions, rules which we use all the time when we do mathematics. Even without thinking about that. OK, and by now, as we have this notion of provability, there is an induced notion, which is a notion about the relationship between different theories. So let's consider two theories, T and T prime, with the same signature sigma. So the same language, but different theories considered with different axioms. And then there are elements of vocabulary we introduced. So first, we say that the T prime is a quotient of the T, if any geometric sequence in the signature, so a sequence relating two formulas, phi and psi. So any such sequence, which is provable in T prime, is also provable in T. So you understand that T prime is a quotient of T, when there are everything which is provable in T prime, excuse me, it is the opposite. Here I have made a mistake. I wanted to say that T prime is a quotient of T, when anything which is provable in T is also provable in T prime. So this means that in T prime, there are more provable properties than in T. So for instance, if you think about the theory of rings and the theory of commutative rings, so of course, the theory of commutative rings has more provable properties than the theory of rings. So it means it is a quotient. And we say that two theories are equivalent. If anything which is provable in one of these theories is also provable in the other, so each one is a quotient of the other. Then there are obvious remarks to be made. If two theories are equivalent, of course, the associated syntactic categories are the same. And if T prime is a quotient of T, which means if there are no properties, more sequence provable in T prime than in T, then CT is subcategory of CT prime. And they have the same objects because the objects only depend on the signature, as we have already remarked. OK, so this will be important for later. By now, what we immediately remark is that if two theories are equivalent, associated syntactic categories are exactly the same. OK, so by now, it is important to consider subobjects in syntactic categories. So the notion of subobject makes sense in any category. So in particular, we may wonder what is a subobject of an object in the syntactic category. So let's consider a geometric theory T and its associated syntactic category CT. And let's consider an object of CT, which by definition is a geometric formula, phi of x in some variables, in some context x. And of course, this is considered up to substitution of variables. Then the statement of the proposition tells us that the subobjects of such a formula are exactly the formulas phi 1 in the same context, in the same familiar variables, such that the second phi 1 implies phi is provable in the theory T. OK, so we see that in other words, in this category, the notion of subobject exactly corresponds to the notion of provability between geometric formulas written in the same variables. So this is a categorical translation of the notion of provability. And by now, if we have two subobjects of phi, phi 1 and phi 2, then the inclusion relation of one of these subobjects in the other corresponds exactly to the relation of provability between the formulas. OK, so the second part, of course, is the consequence of the first part because a subobject is included in some other subobject phi. If and only if A is a subobject of phi. OK, so this is very important to have such a translation between the categorical notion of subobject and the logical notion of provability. OK, in fact, this whole theory of syntactic categories is a way to make logic categorical. And when I say logic, once again, you should not understand this word as a particular part of mathematics, but rather as a way to present any mathematical theory in the linguistic way, which we always do when we do mathematics. OK, so then we have to make precise what is the canonical model empty in such a syntactic category. How is it defined? So once again, let's consider such geometric theory T in some signature sigma together with associated syntactic category. And of course, we suppose we are verified that Ct is a geometric category. So this, of course, requires a proof. We don't have enough time to give the proof, but it could be checked. And if you don't know the proof, it is a very good exercise to check it that it is really a geometric category. In particular, it has arbitrary finite limits and so on. OK, so as it is geometric, as this category is geometric, it has a meaning to talk about the models of the theory T in such a category. And then we are to introduce a particular model, which we shall call the canonical model. So a model consists in associating an object to any sort of A, a morphism to any function symbol F, and a sub-object to any relation symbol R. OK, so let's do it. So first, for any sort of A, we decide that the associated object is just the formula code in any variable XA associated to the sort F. Of course, it does not depend on the choice of variable because objects are formulas considered up to substitution of variables. Then if we consider a function symbol F going from a list of sorts A1, An to a sort of B, then by definition, the associated morphism in the model we are constructing is given by the formula, which is written there. So it is just the formula is just that the variable XB in the codomain is equal to F of the variables X1, Xn in the domain. OK, so this is just the formula we are used to when we want to define a function. But here, by definition, such a formula is a morphism. Here, we just have to check that this formula is probably functional, which of course is completely of use. And lastly, for any relation symbol R in the context of a family of sorts A1, An, we decide that the associated subject of true in the variables X1, Xn is just the formula R in X1, Xn. So R in X1, Xn is, of course, a subformula of true of X1, Xn. And here, we are using the fact that in the syntactic category, the formula true in the variables X1, Xn is the product of the objects represented by the formulas true in X1, true in X2, true in Xn. OK, so this, of course, defined a sigma structure. And here, there is something to be verified, which is the fact that this is a model of the theory T. So in order to do that, we have to consider the interpretations of geometric formulas in this model. So let's consider such a geometric formula phi in some variables X. And by now, we have a lemma that the interpretation of the model of such a formula phi of X is just the subobject defined by this formula. So just remember, here, we had said that the subobject of formulas consist in formulas in the same variables, which probably imply the ambient formula. OK, so here, the ambient formula, the context is the formula true in the variables X1, Xn. And it contains the subobjects formula phi X. And so the statement of the lemma is that this formula phi of X, considered as a subobject of the formula true in X, is exactly the interpretation of the formula phi of X in our canonical model empty, in our canonical sigma structure empty. And if we combine this lemma with the previous proposition here on the interpretation of the relationship between provability and inclusion of subobjects, we get as a corollary that a second is a geometric sequence, a sequence relating two geometric formulas phi on psi. Is t provable? Is provable in the theory t? If and only if, considered as a subobject of the object true in X, the first one is included in the second one. And this, by definition of a model, means that such a model, it means that the sigma structure empty verifies the second phi implies psi. So you see that here we have an equivalence. We are seeing that the second is provable in t. If and only if it is verified by the sigma structure empty. So an equivalence, of course, means we have an implication in two directions. So in the first direction, we have that if a second is provable, in particular if it is an axial, then it is verified by the sigma structure empty. So this means that empty is a model. But here we see that it also works in the other direction. If the canonical model empty verifies this second, then verifies a second. It means that this second is provable in the theory t. OK? And by now, we have a theorem, which is, so this theorem exactly tells us that the category we have constructed is an answer to the question we addressed. So let's consider the syntactic category just constructed of a geometric theory t together with its canonical model empty. So then, as for any model, there is for any geometric category c, from the category of geometric functors from city to city, to the category of t models in c. And this functor associates to any functor from city to city, the transform of the model empty by the functor f. OK? And the statement of the theorem is that for any such geometric category c, this functor is an equivalence. So the category of geometric functors from city to city is equivalence through this functor to the category of models of city. And not only it is an equivalence, but we can also construct the reverse equivalence. So the reverse equivalence has to associate to any model of the theory t in c, a geometric functor from the syntactic category city to city. So how is defined this reverse equivalence? So we have to associate to any object of city an object of c. And we have to associate to any morphism of city a morphism of c. So what is an object of city? By definition, it is a formula. It is a geometric formula up to substitution of variables. And what we do is just to consider its interpretation in the model m. And this, by definition, will be the image of this formula considered as an object of city by the functor fm. So the image of such a formula considered as an object of city by the functor fm is just defined as the interpretation of the formula phi of x in the model m. And by now, let's consider a morphism of the syntactic category. So it is a provably functional formula, theta in two families of variables, x and y. And this is such a provably functional formula up to three provable equivalents. And so what you have to do is abuse. You consider the interpretation of this formula. So the interpretation of this formula is a sub-object in the product of the interpretations of the formula phi of x on psi of y. But because the formula theta is provably functional, you get that this sub-object in the product is the graph of a morphism. And so you define the image of this morphism theta by the functor fm to be the unique morphism whose graph is the interpretation of theta. Everything has been defined so that all of this makes sense. OK, so the theorem tells you that these are two equivalences which are in this one to the other. OK, so of course, it has to be verified. But the verification is, in fact, is quite easy. It is more intricate to verify that the category city is geometric. Once you know this, everything goes through very easy. OK, by now, we have constructed the syntactic category together with its canonical model. And we know that the syntactic category represents the functor of models on geometric categories. But by now, we remember that what we want, in fact, is not a geometric category. We want a topos, which is a much more geometric object. And in order to get a topos, we need to define topology on the syntactic category. So we need to define when a sieve on an object in the syntactic category will be considered as a covering. And here is the definition we propose. We decide that the sieve on an object's psi of y is a covering. If it contains a family of morphisms, you see some domains phi i to the co-domain psi. And these morphisms are defined by probably functional formulas, theta i's. And so we decide that such a sieve has to contain such a family of morphisms, which is globally epimorphic. So this means globally epimorphic is, once again, a categorical notion. But because images and unions of sub-objects are well-defined in the geometric category city, the property to be globally epimorphic just means that the union of the images of this morphism is equal to everything, is equal to psi. So this is an inclusion property. And the inclusion property, I remind you, is equivalent to property of t probability. So in fact, it means exactly that the following sequence psi implies the union over the indices i of the images. The images are just given by the existential quantifier on the variable x i's. So this sequence has to be t provable in the theory. So this exactly means, so this is a logical translation, a translation in the language of probability of the categorical property for a family of morphisms to be t provable. And so we decide to define in this way the notion of covering sieve. And of course, it can be checked that this is indeed grotendic topology, which means it verifies the three grotendic axioms of maximality, stability, and transitivity. So maximality is obvious. Stability comes from the fact that both arbitrary unions and images, which means existential factors, commute with best change. So for unions, it is a so-called distributivity rule. And for the existential quantifiers, it is the so-called Frobenius rule. OK, so this means these are parts of the usual inference rules, deduction rules of the object. A remark we can make here, an important remark, is the fact that this topology is defined by the categorical structure of the syntactic category. You see a family of morphisms is covering when it is globally epimorphic. So this property is purely categorical. So it means, in particular, that if you have two syntactic categories which are equivalent, then they have the same topology. Because the topology is deduced from the categorical structure in that case. OK, so let's just think where we are. So we have constructed a syntactic category. And we have proved that it is universal with respect to models of the theory T in geometric categories. But by now, we want to move to a universality property with respect to toposies and to morphism of toposies. So this means we have to move from geometric factors between geometric categories to morphism of toposies between toposies. And these two things are different. And the relationship between the way to go from geometric morphism to morphism of toposies is given to us by Diakonescu's equivalence, which was already stated yesterday afternoon by Olivia. So here it is, so I repeat it again, just in the case when the category we are considering has finite limits, which, of course, is a simple case. In fact, yesterday, Olivia stated Diakonescu's equivalence in general. But for today, we only need Diakonescu's equivalence in that particular case. So let's consider such a small category with finite limits together with topology. And let's consider the canonical function from the category C to the associated topos. So the canonical function L is just the composite of the Yoneda embedding followed by the Shiffication function, J upper star. And let's consider an arbitrary topos. Then here is a statement of Diakonescu's equivalence. First, for any morphism of toposies from E, from this arbitrary topos E to the topos defined by the site C, J, then the composite of the canonical function from C to the topos C at J with the point-back component F upper star of the toposmorphism from E to C at J. So this component verifies the two following properties. So first, it is a flat functor, which means here that it respects finite limits. So the notion of flat functor exists in a much more general setting without any hypothesis on C, as Olivia explained in her lectures yesterday afternoon. But in that particular case where C has arbitrary finite limits, it just means that this functor from C to E respects finite limits. And the second property verifies is the fact it is J continuous. So this means that it transforms any J covering family of C into a family of morphism of E, which is globally epimorphic. So this is the first part of the statement. So the second part of the statement is the fact that the functor we have just defined from the category of morphism of toposies from E to our toposm is an equivalent to the category of functors from C to E, which are both flat and J continuous. So here, I remind you, flat just means that it respects finite limits. And J continuous means it transforms the coverings into globally epimorphic families. OK, so this is a general statement of Diacones Cruz equivalence. And by now, we have to apply it to syntactic categories. But for this, we need a lemma. So let's consider geometric theory, the associated syntactic category on an arbitrary toposm E. And let's consider a functor from C to E. And then the statement is that such a functor is geometric. If and only if it is flat, which means it respects finite limits. And it is JT continuous, which means it transforms globally epimorphic families of Ct into globally epimorphic families of E. I remind you that the topology of Ct, the syntactic topology of the syntactic category, was defined by deciding that a family of morphism is covering when it is globally epimorphic. So here, JT continuity just means that such a functor has to transform globally epimorphic families into globally epimorphic families. OK, on the earth, lemma, which is easy, tells us that such a functor from the syntactic category to E is geometric if and only if it is flat on JT continuous. But according to Diacones Cruz equivalence, it just means that such a functor defines morphism of toposis in the reverse direction. OK, also, we get the following corollary for any geometric theory. And if we denote by E, the cushion topos of the topos of pre-shifts on Ct by the topology JT. So we consider this topos on an arbitrary topos E. And then we have a composite functor which goes from the category of morphism of toposis from E to ET to the category of flat JT continuous functors from Ct to E. OK, so this first functor is Diacones Cruz equivalence. So it is an equivalence. But by now, we just said by the previous lemma that a flat JT continuous functor is the same thing as a geometric functor from Ct to E. And then we have already proved that the category of geometric functors from Ct to E is equivalent to the category of models of E. So here, we take the composite of two equivalences of categories. So it is an equivalence of categories. And we have proved that our topos is a classifying topos for the theory. For any topos E, to consider a toposis morphism from E to ET is the same thing as to consider a model of the theory T in the topos E. In particular, if you take for E the topos of sets, you get that the point of the classifying topos ET is the same thing as a model of T as a set theoretic model of T. OK, so here, of course, we have, as for any topos defined by your site, of course, we have a canonical functor from the syntactic category to the topos ET. But yesterday, Olivia introduced in her lecture a notion, a property which can be verified by these canonical functors. It is a property to be a fully-faced full embedding. This property is not always verified, but it is verified by some sites. And here, so the definition is the following. You say that topology on a category C is subcanonical. If the canonical functor from C to the associative topos, to the quotient topos by J, so if this canonical functor is fully-faced full. So it is a definition. And in fact, you can prove that this is equivalent to say that the unedited embedding of C into C at factorizes through the topos considered as a subcategory. In other words, it means that any representable functor on C is a shift for the topology J. OK, so it is a definition, a general definition, which was given yesterday in Olivia's lectures. And then we have a lemma, which is the fact that the topology GT on the syntactic category Ct is always subcanonical. Whatever the theory T. And so this means that Ct is embedded into ET as a fully-faced full category. And of course, this fully-faced full embedding also respects finite limits. So this is always the case because the unedited embedding respects arbitrary limits. And the shiftification functor respects finite limits. So the composites, the canonical functor, always respect finite limits. And so this functor respects finite limits. In particular, it respects subobjects. And it is fully-faced full. And as a corollary of this lemma and what we already proved, we get the following that a geometric sequence of the theory T, so an implication between two geometric formulas, phi and psi, is provable in the theory T if and only if it is verified by the universal model of T. So the universal model of T is the image of the canonical model in CT by the canonical functor L from Ct to ET. So here we really have, you see, a wonderful relationship between syntax and semantics. So we have that any property is provable in the theory if and only if it is verified by the canonical model of the theory in its classifying topos. OK. And I already mentioned as a side remark that if you apply that to so-called coherent theories, so here it corresponds exactly to theories whose classifying topos is coherent in the sense of topos theory as already introduced in FGA4, then what we get is that Godel's completeness theorem is the same thing as the theorem of the linear on coherent opposites having enough points. So this is a really wonderful fact because it tells us that a logical, most important theorem of logic is, in fact, the same thing as this purely geometric theorem of the linear. OK. So by now, an important remark we have made is it is a reverse direction, which is the fact that if we start with an arbitrary topos, then there are infinitely many geometric theories whose classifying topos is equivalent to the given topos. And here I just write a sketch on the idea of the proof of this proposition. So you start from an arbitrary topos is, so by definition, a topos is a category which is equivalent to the category of sheaths on some site, Cj. OK, so by definition, a topos is a category equivalent to some Cj hat, where C is a small category. And we can even suppose that C has arbitrary finite limits. And J is a topology on C. And by now, of course, there is a notion of a flat G continuous functor from C to an arbitrary topos. And this notion, of course, defines a theory. There is a theory of flat G continuous functors. And you can check that this theory is, in fact, a geometric theory. And then when you consider the geometric theory T, so you see the geometric theory of flat G continuous functors. In other words, you can say that this geometric theory is really the theory of points on the topos associated to Cj, according to Diaconix whose lemma. This theory is a theory of points of the associated topos. So it is a geometric theory. And then you prove that its classifying topos is equivalent to the topos of sheaths on Cj you started with. So here you see that any presentation of the topos in terms of a site gives rise to a presentation of the topos as a classifying topos of some geometric theory. So here you already see that there is an incredible diversity of presentations of topos by geometric theories. And so the fact for here there is an induced definition, you say that two geometric theories are so-called morita equivalents when they have the same equivalent opposites. So maybe it could be suggested to call this equivalent semantic equivalents. In fact, we have already there is also a notion of syntactic equivalents. It was mentioned yesterday when I answered the question. You say that two theories are syntactically equivalent when their syntactic categories are equivalent. But here there is another notion of equivalents, which is induced by the abuse one, but which is much more rich. It is a notion of semantic equivalents or morita equivalents. It means that two theories are the same associated topos. And it really means that two theories, which possibly are completely different languages, have the same mathematical content. So it gives rise to, it is the beginning of a theory of, I would say, of relationship between the contents. It is a theory of relations between the mathematical contents of different mathematical theories. And this theory, in fact, was introduced and developed by Olivia from her PhD thesis. And she has called that the theory of topos theoretic bridges. And she already talked about it yesterday. And today I will introduce some consequences of some consequences, some applications, some bridges, and some consequences of some concrete bridges. So here is the definition. Two theories are called morita equivalents if they're associated, the classifying topos are equivalent. And then, as a remark, we have the fact that the syntactic equivalents between two theories which means the existence of an equivalent of syntactic categories implies an equivalence between the associated topos. This is because the topologies, the syntactic topologies and syntactic categories are induced by the categorical structure. And here, a very important remark is that the converse is not true. Some theories are semantically equivalent or morita equivalent without being syntactically equivalent. So the relationship of semantic equivalence is much more subtle. OK. And by now, what I want to do in the third and last part of the lecture is to present some basic results which allow to get some first applications of the theory of classifying toposes. So these results I am to present are from the PhD thesis of Olivia on their book Theories, Sites, Toposes. OK. So a general principle for getting these results will be to use bridges. So a bridge, once again, is an equivalence between two toposes presented in two different ways. And the general principle of bridges, of toposteritic bridges, is to consider invariance of toposes, which means some informations which can be extracted or informations or structures or whatever type of properties of geometric or mathematical objects which can be constructed from toposes using the language of categories and which are invariant under equivalences of categories. And when we consider such an invariant, general principle is to try to express these invariants in terms of presentations of the toposes under consideration. Here's the first invariant we are to consider is the category, in fact, the ordered set of sub-toposes of given topos. So the way any topos, it is possible to associate an ordered set, a set with an ordering, consisting of its sub-toposes. And then we may wonder how this invariant can be expressed in terms of different presentations of the toposes we consider. So first, we need a definition. In fact, the first part of the definition was already introduced. So what is an embedding of toposes? It is a morphism of toposes consisting in a pair of adjoint factors, j upper star, j lower star, such that the right adjoint component, j lower star, is fully faithful. And this is equivalent as these factors are adjoint. It is equivalent to requiring that the composite factor, j upper star composed with j lower star for this composite factor, identify with the identity factor of E prime. So you see it really corresponds to E prime being a sub really embedded in E. And the pullback component from E to E prime has to act on objects of E prime without changing them. So in some sense, there are fixed points of this factor. So this is the definition of an embedding. And then a sub-topos is an equivalence class of embeddings. You decide that two toposes, two embeddings of toposes E1 and E2 into an ambient topos are equivalent if they can be related by an equivalence of categories which transform the first embedding into the second embedding up to isomorphism of functions. So this is the notion of sub-topos. And then there is a first result which, in fact, it is already in SGA form. So it is the fact that if we consider a sub-topos of a topos E, which is presented as a topos of sheaves on some site, then to consider a sub-topos of this topos is the same thing as to consider another topology, J prime on the same category C, which is bigger than the given topology, J. So of course, in one direction, it is abuse. If you have on C topology J prime, which is bigger than J, of course, it induces, it defines a sub-topos C at G prime of C at. And the embedding of C at J prime into C at factorizes through the sub-topos C at J because J prime contains J. So this means that any topology G prime bigger than J induces a sub-topos of our topos E equal C at J. But the proposition is that this map is one to one. So to consider a sub-topos is the same thing as to consider a topology. And here, as we can take for C, a small category, it means that as a consequence, it implies that the sub-toposis of a given topos E together with inclusion relation of sub-toposis is a set. It is a set ordered by inclusion. OK, so it is a corollary of this proposition. And by now, we can wonder about the translation of this property of this invariant, the invariant of consisting in the ordered set of sub-toposis in terms of theories. So by now, we consider a geometric theory of signature sigma. And we suppose that we are interested in sub-toposis of the classifying topos of T, in sub-toposis of E T. And here is a statement. So first, start from a theory T, which is a quotient T prime of T. So this means it has the same signature, the same language, and it has more provable sequence. There are more properties. There are more axioms in T prime than in T. And then the first part of the statement is that in that case, E T prime, the classifying topos of T prime, identity is a sub-topos of the classifying topos of T. And the second part of the statement is that this map, which goes from quotient theories T prime of T, considered, of course, up to equivalence, the map from these quotient theories to the order set of sub-toposis of E T. So this map is one to one. So to consider a quotient theory of T is the same thing as to consider a sub-topos of E T. So this means that this invariant, consisting in the order set of sub-toposis of a given topos, has an extremely nice expression, both on the sides of topologies and on the side of theories. OK. So maybe we can sketch or at least give an idea of the proof. So on one side, we have the quotients. And the other side, we have the topologies. But what we need to do is to define a map in one direction and then in the other direction. So start from quotient T prime of T and consider an axiom of T prime. So it is a second phi implied psi. And of course, this sequence defines a monomorphism in the category C T. So the monomorphism just corresponds to the object phi conjuncted with psi on the object phi. Of course, phi on psi is a sub-object of phi in the syntactic category. And then you decide that this monomorphism has to be a covering for the topology J associated to T prime. And you define J as the smallest topology of C T, which contains a canonical topology, J T, on these coverings. OK, so it is a topology generated by G T and by this family of coverings associated with all axioms of T prime. And in the other direction, when we have a topology J on C T, which contains G T, we want to associate to it a quotient T prime of T. So in order to do that, what you do is to consider an arbitrary J covering family of morphism of C T. So such a family of morphisms, of course, is a family of T-provable functionals. OK, and you can consider, of course, the images of this morphism. So this corresponds to taking an existential quantification. And then we consider the union of the images, which corresponds to the disjunction sign we have written there. And by now, what we just introduced this second five implies this property. And what we really want is this family of morphism to become a covering. And so, excuse me, it is a covering. So we want this family of morphism is a covering. And we want to translate that in the language of provability. So we decide that any segment of this form gotten from a covering has to be an axiom of T prime. And we define T prime as a quotient of T defined by this family of axioms. And so the theorem is that these two maps are inverse one to the other. So you see that it is constructive in both directions. OK, and before we make a break, I want to state the following consequence, which is the equivalence between the language of provability, between the notion of provability, and the notion of grotendic topology. So here, we combine the two translations of this invariance that had been computed on the one hand in terms of topologies, and on the other hand in terms of theories. So consider a geometric theory. And suppose it's classifying topos, ET is presented as a topos of sheaves on some site. So this site may be the syntactic site, but it also may be any site, any representation which you know. And then there is a correspondence, a one to one correspondence between on the one hand the theories T prime, which are quotients of T, and on the other hand, the topology is J prime on C, which contains J. So here, this is, as Olivia says here, we have a bridge, so the invariance is the invariance consisting in the sub-toposis of the considered topos. And we are considering two representations of this topos at the same time, on the one hand in terms of a theory P, and on the other hand in terms of a site Cj. And we express the same invariant on the two sides. And in this way, we get a correspondence. And so by now, this corollary has very striking consequences. So the first one is written there. Of course, when you have a theory given by your family of axioms, the first thing you want to know about this theory is whether it is contradictory or not, whether it is contradictory or consistent. And in general, it is a very subtle and difficult question to answer that is, in general, it is very difficult. But here, consequence of this corollary is that when the classifying topos of ET is presented from a site Cj, and j prime is the topology corresponding to t prime, then t prime is contradictory. If only if the topology j prime is maximal. And maximal means that the NTC is covering for any object. So this is a geometric translation of the property to be contradictory. And so this means that if, in the category C, there is at least one object which is not covered by the NTC, then the theory t prime is not contradictory. OK. And here, there is the last slide before the break, which is also, I think, very, very interesting. So it is just an obvious consequence of the previous result. So it is the following. Consider an arbitrary geometric theory of signatory sigma. And then consider an arbitrary geometric sequence. Of course, you want to know, you are a mathematician, so you want to know whether this geometric sequence is provable in the theory. Is it a consequence of the axioms or not? So in mathematics, usually, you want to prove results. So a result is always an implication, which has to be a consequence of the axioms of the theory. So you want to prove that you are a mathematician. This is exactly what you want to do. But then, formally, you can do the following. You just introduce t prime, the Cushion theory of t, which is defined by adding this axiom. And so the question is, is t prime equivalent to t or not? T prime equivalent to t just means that this sequence is a consequence of the axioms of t. It is provable in t. So on by now, let's consider the classifying topos of t, so ET, written, we suppose, as the topos of shifts on some sites. And let's consider the topology, j prime on j, which is defined by ET prime. In fact, we know how this topology is defined. We just consider the sequence phi implies psi. We consider the associated monomorphism, phi on psi implies phi. And j prime is just the topology generated by j and by this morphism considered as a covering. So the monomorphism from phi on psi implies psi. Okay, so this means j prime is constructed by just generation from j by one covering. And by now we have the following equivalences. We have that our sequence is provable in the theory. If and only if the caution theory t prime of t is in fact equivalent to t, and this is equivalent to j prime equals g. So in other words, is some topology generated in some way, is it equal to g or is it bigger? This is a question. So this is really wonderful because it means that all the inference rules, you remember there were maybe 10 or 12 inference rules which were the deduction rules of geometric logic. On the earth, they are just replaced by the rules, by the axioms of growth and dik topologies. And there are three axioms, so the maximality axiom, which is trivial, and then the stability and transitivity axioms. So in fact, the axiom of growth and dik topologies here appear as completely equivalent to the deduction rule, the deduction rules of geometric logic. And any problem of probability in a geometric theory can be translated in a problem of knowing whether some family of morphism in a category belongs to the topology generated by family of covariance. Okay, so of course it doesn't make the problem trivial, but it is a translation of one part of mathematics. It is really here in some sense an embedding of logic into topology. Okay, and so here in fact, I think something very, which has to be done is to try to implement that in the computer systems. I really think here there is an important subject to be explored. Okay, so I stop there for the first part. We make a break. Okay, thank you.