 Thank you very much. I just wanted to present the notion of language or signature of first order theory and the interpretations they may have in the topos or more generally in the category with finite products and these interpretations are called sigma structures. So by now I want to come to the notion of first order theory. So first order theory consists first in a language or signature sigma as before and secondly collection of axioms and these axioms have to be formalized and they are formalized as so-called sequence so this is denoted in this strange way which is familiar to logicians. So you have two letters phi and psi and these two letters are related by this strange symbol which just means an implication and phi and psi have to be formulas so more precisely geometric formulas. So the meaning of such an axiom is that one formula phi has to imply another formula psi. So for instance if you think about the theory of order relations one of the axioms is that if A is lesser than B and B is lesser than C then A is lesser than C. So here phi is the formula A lesser than B and B lesser than C and psi is the formula A lesser than C and in fact all the axioms of first order theories can be phrased in this way implications between formulas. So the formulas which appear in such theories are geometric formulas and they have to be formulas with variables so you see that in the example like just gave there were variables A, B and C. So here to write formulas and implications between formulas we need variables and each variable has to take values in some sort. So this means that any sequence has a context consisting in a finite family of variables and each one of them is associated to some sort and of course we may have several variables with the same associated sort. Okay so by now we want to make precise how formulas phi and psi are built. So geometric formulas are built from more elementary formulas which are called atomic formulas. So at the present moment I don't say what is an atomic formula it will be on the next page. I just say that the general geometric formulas are used from atomic formulas using three types of symbols. So first a symbol of finite conjunction. So for instance in the example I just said if you remember there was the formula L, S, R, and B and B, S, and C. So here you see there is and okay and this is this symbol of finite conjunction but we also allow arbitrary disjunction. When I say arbitrary I mean that it can be infinite but of course indexed by some set. And of course there is also the empty conjunction which is a true symbol and the empty disjunction which is a false symbol. So they have been noted in this way. And lastly there is also the existential quantifier in part of the variable. Okay so we are used to do that all the time when we write mathematics. Okay so by now what is an atomic formula? So an atomic formula is deduced from a relation formula. So a relation formula is a relation symbol half in the context of some sorts a1, an, but it is considered as a formula in some variables x1, xn affected associated with the source a1, an. And we also allow equality formulas between variables with the same associated source. Okay so an atomic formula is either deduce either it has either the form of inequality or of a relation associated with a relation symbol. But we also allow substitution of variables by terms. So what is a term? A term is just a function symbol in some variables or a composition of such function symbols. So the notion of term is defined in an inductive way. You start from a function symbol f in some variables x1, xn associated to the source a1, an of the function symbol. And you allow to replace one of the variables by another function symbol so g in some variables. But of course you have to respect the fact that the function symbol g takes values in a sort ci which is the sort of the variable you have decided to substitute. Okay so here you see it is just a formalization. We want to make precise what we do all the time. In mathematics we are always using variables and we are used to substitute variables with some more complicated expressions. And these more complicated expressions ultimately have to come from some function symbols. Okay so here we are just trying to write in a precise way what we are used to do all the time in mathematics. Okay so here is a remark which is the fact that of course in mathematics even in first order mathematics we may use some other symbols than the symbols which were listed in this definition. So these other symbols which we are used to employ are first arbitrary conjunctions not only finite conjunctions. So arbitrary conjunctions we already said existential quantifier we already said but we also use universal quantifier meaning for any variable. And also the implication symbol. And the negation symbol when we have a formula we sometimes we decide to take the negation of this formula. Or if we have two formulas phi one and phi two we can consider the formula phi one implies phi two. Okay so all of this as meaning it can be interpreted for instance is the context theory. On here in fact we have a full list of the symbols which are used in first order mathematics. So for people who don't know you may ask what is higher order mathematics. Higher order mathematics is when you also allow exponential operations such as replacing two sets by the set of maps from the first one to the second one or replacing a set by the set of subsets of this set. So these are exponential operations and in first order mathematics they are not alone. So here the theory we are talking with is a theory of classifying toposies for first order mathematics. In fact here I can mention that if you want to take into account higher order mathematics then you have to consider relative toposies. I mean a topos over one topos possibly over another topos and so on. And this is some work presently done by Olivia Caramello and her student Ricardo Zanfa and I think they will talk a little about it Olivia in her lectures and Ricardo in his talk next week. Okay but for my lectures I only talk about first order mathematics and here I want only to repeat what I already said is that if we have any first any first order theory including with this other symbol of universal quantifier then we can replace it with by a geometric first order theory which means with only finite conjunction arbitrary disjunction and existential quantifier without changing the set theoretic model. Okay so this means that this condition to be geometric is not restrictive at all and in fact to make a theory geometric it is as we shall see a way to make it topos friendly and in fact when we are doing mathematics it is a very good idea to formulate the theories you are interested in in in a way to make them topos friendly. Okay so by now I have explained what is first order geometric theory and I have to talk about the way such a theory can be interpreted. I have already explained how languages which mean signatures can be interpreted. So here let's consider a signature and first let's consider a term. A term is just a function symbol in some variables or a composition of such function symbols and let's consider a topos and let's consider a sigma structure m in this topos so what is the interpretation of this term so the interpretation is also defined in an inductive way it will be a morphism from the product of the objects associated with the source which are associated to the variables and it takes values in the object associated to the source where the function the term f takes values so if f is a function symbol the interpretation of f is just the associated morphism mf mf by definition is mf not okay when you have a sigma structure in particular the sigma structure consists in associating in associating to any function symbol a morphism and so this is the interpretation and by now if we have a term which is deduced from a simpler term by substitution of a variable by a function symbol then the morphism msk the interpretation msk is deduced from the interpretation msk minus one just by composition with the interpretation of the function symbol g so you see the definition is inductive and here we just use a categorical structure we just use the fact that it is possible to compose morphism in a category okay so by now we can move to the the interpretation of atomic formulas so let's consider once again a signature then a topos then a sigma structure m on an atomic formula so let's consider also the case of the formula true and the formula fails so if the formula is true in the variables x1 a1 xn an is interpretation by definition is the full sub-object f m1 cross m an if the formula we consider is false its interpretation is by definition the smallest sub-object of m a1 cross m an if you prefer it is the empty sub-object okay you see I am just seeing things which are obvious in the context of sets but in fact from the point of view of categorical structures a topos is just as good as the category of sets so everything which makes sense in the category of sets makes sense in the context of an arbitrary topos so by now if we consider an atomic formula which is just a relation symbol r in the context of some a1 an its interpretation by definition is a sub-object f m r of m a1 cross m an by now if we consider an equality relation between two families of variables then its interpretation is just a diagonal sub-object m a1 cross m an embedded diagonally in m a1 cross m an cross m a1 cross m an and by now if the term phi k excuse me the if the atomic formula phi k is deduced from a simpler atomic formula phi k minus phi k minus one by substitution by substitution of a variable by a function symbol then the sub-object m phi k is deduced from from the sub-object m phi k minus one by pullback by base change along the morphism associated with g the interpretation m g of g so and of course a pullback of a sub-object is a sub-object you see everything I have defined here is um is a sub-object if phi is a true formula the interpretation is a full sub-object if phi is a failed formula the interpretation is an empty sub-object if phi is a relation symbol the interpretation is associated sub-object just by definition of a sigma structure and if phi is an equality relation the interpretation is a diagonal sub-object and then when we substitute from these basic atomic formulas when we substitute variables with function symbols and do that repeatedly at each step we just take the pullback and the pullback of a sub-object is always a sub-object so these things atomic formulas are always interpreted as sub-objects okay and by now we can obviously interpret all atomic formulas so let's consider once again a signature a topos a sigma structure in this topos and a geometric formula a geometric formula in the context of some variables x a 1 x a n x 1 a 1 x n a n and then the the geometric formula was always been interpreted as a sub-object of m a 1 cross m a so let's consider the different cases so the first case is when we have a formula phi which is deduced from some simpler formulas phi 1 phi k by finite conjunction so we these simpler formulas phi 1 phi k are interpreted as sub-object m phi 1 m phi i of m a 1 cross m a n and in order to interpret phi we just take the intersection of this sub-object so the intersection is just a pullback it is a finite limit operation so of course it it perfectly makes sense in the context of the topos e then if phi is defined by arbitrary union of some family of formulas phi i so each phi i is interpreted as a sub-object m phi i of m a 1 cross m a n and we just take the union the union is by definition the biggest the smallest sub-object which contains all of them and here we remark that this union for this union we have a formula and the formula is a co-limit formula so the the the co-limit of the diagram i have written the diagram consisting of the interpretations m phi i and their fiber products by pairs their pairwise fiber products so you see that in this formula there are only finite limits and co-limits it only makes use of finite limits and co-limits okay and lastly we need to interpret the existential quantifier so then we have a formula psi in two families of variables x 1 the x i a i's and y j b j and then of course m psi is interpreted as a sub-object of m a 1 cross m a n cross m b 1 cross m b n and this big object has a natural projection on m a 1 cross m a n and here the interpretation of the existential quantifier is the fact that we replace this sub-object n psi by its image in m a 1 cross m a n and here we so the image by definition is the smallest sub-object of m a 1 cross m a n who spelled back contains m psi so this is the definition but because we are in autopos this image is given by your formula the formula is i have written it is once again the co-limit of the diagram consisting of m psi and the fiber product of m psi with itself over m a 1 cross m a n so this formula is obviously true in the context of sets and it remains true in the context of an arbitrary topos and this is very nice because you see for the interpretation of the universal the existential quantifier we once again we only need finite limits and arbitrary co-limits okay so this is a remark at the bottom of this page because we are in topos this we remarked that the all these formulas which only make use of finite limits and arbitrary co-limits are always preserved by base champ this is because in a topos base change not only respects limits but also arbitrary co-limits and they are also preserved by pullback functions by the pullback components of morphism of toposes because by definition of a morphism of toposes f from e prime to e the pullback component respects arbitrary co-limits on finite limits so this means that all the interpretations of formulas are respected by pullbacks and they are also respected by the pullback component of morphism of toposes so here we can remark that in fact the other symbols of first order mathematics so arbitrary conjunction universal quantifier implication negation they are also interpretable in any topos just as they are in any set in the context of set because once again from the point of view of categorical structures a topos a grottendig topos is just as good as the category of sets and the interpretations of these formulas are always respected by base change but if we have a topos morphism from f f from e prime to e the pullback function between the categories of sigma does not respect in general the interpretations of these symbols they do not respect arbitrary intersections they do not respect universal quantifier they only respect the symbols which are allowed in geometric first order theories so this is why when you have a first order mathematical theory if you phrase it in a geometric way you can say that you have made it topos friendly okay so by now at last we can talk about models so let's consider a signature let's consider a geometric first order theory and let's consider a topos so by definition a sigma structure in this topos m is called the model of t so here i have made a mistake it is t is here it is theory it is not the the symbol of cool okay so it is a model of t if for any axiom consisting in a second between two formulas phi and psi the sub-object which is the interpretation of phi in m and the sub-object which is the interpretation of psi in m so this of course has two sub-objects of the same object so they have to verify the inclusion relation the interpretation of phi is included in the interpretation of psi so when this is verified for any axiom of the theory you say your sigma structure is a model of the theory and a morphism of models of the theory is just the morphism of the underlying sigma structures so this is a definition so of course in particular it applies to the category of sets so in this way you get the notion of set theoretic model or set base model of a theory but it makes sense in the contest of an arbitrary topos okay and so with this definition we realize that the models of a theory t in a topos make up a category this is by definition the full category or the full sub-category of the category of sigma structures on the t models so the morphisms are the same they are just morphisms of sigma structures but the objects are the sigma structures which are models of t which means where all the axioms of t are verified and because the theory you consider is geometric and so the pullback components of topos morphism respect the interpretation of geometric formulas then we get that the pullback function f per star between the categories of sigma structures induces a function between the categories of models and so by now if we consider an arbitrary geometric theory t and an arbitrary model of t in in a topos e there is for any topos e prime an induced function from the category of topos this morphism from e prime to e to the category of models of t in e prime so the function just associated to any topos to any topos this morphism f the pullback of m by f okay and then the definition of the pacifying is what you require for a topos to be classifying the theory t is the fact that this function has to be an equivalence of categories for any topos e prime okay so what we want is that for any topos e and for any topos e the function which associates to any topos morphism from e to e t the pullback of the model ut of t in et so this we want this function to be an equivalence of categories and here it is a formal consequence of everything of the just the meaning of the statement that if such a couple ut exists it is unique up to equivalence but of course the existence of such a topos is a theorem and it takes some time to prove it okay and so by now i want to begin the proof of this i want to begin the proof of this theorem so i repeat what we want to do we start from first order theory which is presented in a geometric way we start from a geometric first order theory t and we want to associate with the topos but toposists are always constructed from a pair consisting of a category c endowed with a grotendic topology j so grotendic has called that a site a site is a pair consisting of a small or a socially small category endowed with a grotend with a topology and so what we need is in order to construct a classifying topos et is to define a category ct together with a topology jt on this category so that the associated topos et so the quotient topos of the category of the topos of issues on ct by the topology jt so this topos has to answer the question so more precisely we shall get the following first we will construct a category ct with enough categorical properties for the category of models of t in ct to be defined so ct be careful ct will not be a topos it will be much smaller in fact it will be an essentially small category but it will have enough categorical properties for geometric formulas to be interpretable into this category and so for the models to be defined and in these categories there will be a canonical model empty and and as i already said a topology gt such that denoting et is a classifying topos and ut the image of mt through the canonical functor you see the canonical functor l which goes from ct to the category of pre-shifts of ct so to ct at through yoneda embedding composed with the shiftification functor and so with this definition we will get that the functor from the for any topos e of the category of topos morphism from e to et to the category of t models in e so this functor to be unequivalent for any topos e so this is what we want to do okay and i just said that this category ct has to be good from it we need for this category enough categorical properties for geometric formulas to be interpretable in this category so here we give a definition of what we want uh so we call that a geometric category so a geometric category is a category where geometric formulas can be interpreted so what do we need first we need this category to have all finite limits in particular this means that for any morphism in the category p from x to y there is an induced functor on the associated categories of sub-objects so here you see i have denoted omega of y and omega of x the categories of sub-objects of x and y because in any category with finite limits if you consider a sub-object of y let's say y prime then the fiber product of y prime with x over y is a sub-object of x so in this way we define a pullback functor from the category of sub-objects of y to the category of sub-objects of x and what we want is this pullback functor to have a left adjoint and this left adjoint will be the interpretation of the existential quantifier so we denoted in this way with the existential quantifier on p as an index of course it has to go in the reverse direction from omega of x to omega of y okay and by definition it has to be an adjoint of the pullback functor okay and of course this uniquely determines this functor and we also want this image this functor of image this functor there exists in xp to be compatible with best charge so this means that for any cartesian square as i have written in the at the bottom of the page the associated square with you see at the horizontal level you have the pullback functors on sub-objects and the vertical rows are just the existential functors and so we want this square to be commutative as soon as the square on the left is cartesian okay so this is part of what we want for a category to be geometric but we also want the arbitrary adjunctions to be interpreted so this means that for any family of sub-objects as i of some object y then this family of sub-objects as a well-defined union so what is a union it is a sub-object such that for any subject s of y the union is lesser than s if and only if each si is lesser than s so we want this to exist for any family fsi of sub-objects so when it does exist for any such family we say that unions are well defined in our category only one not only unions to be well defined but also to be compatible with best charge so this means that for any morphism from x to y we want the pullback of the union to be equal to the union of the pullbacks okay so a category will be called geometric when it verifies all these properties and in fact we have already remarked that any topos is a geometric category and by now when we have a functor between two geometric categories we say that this functor is geometric if it respects first finite limits then the existential functions so this means that for any morphism from some x to y for any morphism p and for any subject x of x then the transform of the image of s by the functor f is equal to the image by the morphism f of p of the transform of the subject s by f okay so we want this property to be verified and we also want f to respect finite unions so if these three properties are verified we say the functor is geometric okay and by now we remark that if we have an arbitrary signature and the geometric category and the sigma structure m in this category then any geometric formula in family of variable phi with associated source a1 and n is interpretable as a sub-object m phi of x of m a1 cross m a n in fact we have already done that in the case of topos and here in the previous definition we have really defined the notion of geometric category in order for the interpretation process to be possible so there is such an interpretation and as the geometric formulas are interpretable it is possible to talk about the models of a geometric theory t in such a geometric category we say that a sigma structure m is a model of t if for any axiom of the theory phi implies phi then the sub-object m phi on m psi verify the inclusion relation m phi included into m psi and so the notion of model is well defined and we can define the category of models of t in c as the full category of the category of sigma structures c on models of t on the remark that if we have a geometric function between two geometric categories it induces a function between the associated categories of models in fact all the definitions were written to in order for this to be verified okay so by now we have the following theorem which is true for any geometric first order theory and the system says that we have what we want so we are able to associate to the theory phi a geometric category c t endowed with a canonical model m t such that the following property is verified for any category c if we associate to a j any geometric factor f from c t to c it's the image of the model m t by f which is well defined as a model of t in c then this is an equivalence of categories for any geometric category c on the second part of the statement is that this couple c t m t is well defined is uniquely determined up to equivalence okay so this means that at the level of geometric functions and of geometric categories on geometric functions the function of models of theory t is representable it is representable by a special category called the syntactic category endowed with a special model which can be called the universal model of the theory okay so this is a theorem and by now we can construct it so let's start with such a geometric theory and yeah before I say before I begin to construct it there is a remark which is the fact that if this theory endowed with this canonical model exists then it is unique up to equivalence so this is the part two of the theorem and this is former okay this is exactly as this is a categorical version of the yoneda lemma if a functor is representable by some object this object is unique up to unique isomorphism so here the functors don't take values in sets they take values in categories so for this reason the usual notion of isomorphism in the context of the yoneda lemma has to be replaced with the notion of categorical equivalence but the proof is exactly the same so this is okay for uniqueness but the difficult point is existence and in order to prove existence we are going to construct this category in a concrete way we shall actually construct such a category so these things we are going to list to give a list of objects a list of morphisms and a definition for the composition law of morphisms okay so let's do it let's start with such a geometric theory tip and we have to define the objects what are the objects of city what are the morphisms of city and what is the composition law of morphisms of city so here we decide that the objects of the syntactic category are just geometric formulas okay so geometric formulas were defined in the first lecture this morning so we consider all geometric formulas up to substitution of variables so we which means that we consider two geometric formulas as equivalents when one is deduced from the other by just a change of notations of the variables okay so this is of course is very natural so then we have to decide what is a morphism the idea is to define morphisms through the graphs so if we want to define a morphism from a formula to another formula from an object to another object in a category where of course the products will be well defined to define a morphism is the same thing to define a graph as a sub-object of the product object of phi of x and psi of y and the product object is of course as to be a formulas in the union of the variables x and y so this means that the morphism from phi of x to psi of y have to be formulas in the union of the of families of variables x and y and when you have such a formula yeah of course a formula will define sub-objects and what you will need in order for a sub-object to be a graph are three in fact three properties first you want the projections of this sub-object to go into phi of x on psi of y so this requirement this request is represented by an implication between formulas we want the formula theta which has to define the morphism to imply both the formula phi in the variable x and the formula psi in the variable y secondly we want the projection on the first part on phi of x to be in order to be a graph and this is represented by the following sequence if the formula phi is verified then there should exist some y such that the formula theta of x y is verified so you see this sequence will correspond to the property for the projection morphism to be on two in order to define a graph and lastly we want this projection on the first component phi to be an isomorphism so it has not only to be on two but also to be a monomorphism and so what does it mean it means that if the property theta is verified both by x on y on by x on y prime it has to imply y equal y prime okay so this we put as definition of formula theta to be provably functional so we want these three sequence to be provable in the theory t yes here i see i see that i have written at the last line are provable in theta in fact it is not theta this is are provable in t provable you want in your theories that these three sequence are provable so by now what will be the the composite of two morphism so here you you just have to think about the following question when you have two morphisms which are given by two graphs what is the graph of the composite morphism and the graph of the composite morphism is given by the formula i have written there at the last line of the part three we want we are considering here three variables x y and z and the definition of the graph of the composite is the image of the intersection of the graphs of the first and two morphisms you see so you take the image so this means you use the existential quantifier on the variable y apply to the conjunction of the formulas theta on theta prime and this defines the graph of the composite okay so here we have defined some things of course we have to verify it is a category and not only it is a category but it is geometric and it represents the filter of models so before we do that we remarked that first that in the definition sorry come back you see in the definition of objects objects are geometric formulas up to substitution of variables so you see that the list of objects does not really depend on the theory it only depends on the signature sigma okay it does not depend at all on the actions of the theory okay this is the first remark the objects only depend on the signature but of course the morphism depends on the notion of probability in the theory t so in order for our definition to make sense we need to make precise what it means for a sequence for an implication between formulas to be provable in the theory so this is another definition a sequence or if you prefer an implication between two geometric formulas it's called provable in a theory of signature sigma it if it can be deduced from the actions of the theory by the rules of logic okay which we are used to to use when we do mathematics when we do mathematics we are always making deductions and in order to make deductions we use implication rules and in fact is this definition we have given the full list of implication rules which are implied so these are all basic rules and every one of us is used to them but it is interesting to to to write them without forgetting any rule in fact I think the first one in history to do that to just list the usual rules of logic is Aristotle and in the context of mathematics of course it was done again by Hilbert so here are the deduction rules so first there is a second cut rule if phi one implies phi two and phi two implies phi three then phi one implies phi three okay then there is the identity rule for any term f true implies f equals f so this means that f equals very f has to be verified all the time then there are the identity rules if f one is equal to f two then f two is equal to f one if f one is equal to f two and f two is equal to f three then f one is equal to f three okay so then there are the substitution rules if we have two terms f one and f two and we consider another term f and suppose that the terms f prime one f prime two are deduced from f one f two by substitution of f to some variable you see we we suppose that f one and f two are in the same variables and we decide to replace one variable by f or we suppose that f prime one f prime two are deduced from f by substitution of f one and f two to some variable of f then of course we have that if f one is equal to f two then f prime one is equal to f prime two on the same goes for relations suppose we have two terms and a relation and by now we have two atomic formulas r one and r two which are deduced from r by substitution of the terms f one f two to some variable of r then if f one is equal to f two it implies that the atomic formulas r one and r two are equivalent each one of them implies the other okay so then there are the rules which define finitary conjunction first any formula phi implies true so this anything implies truth and if we have formulas phi and phi one phi k phi implies the conjunction of the phi i's if and only if phi implies each of the phi i's and there are the corresponding rules for in for arbitrary disjunction so first first uh felicity implies anything and secondly for any phi and phi i's possibly infinite the union of the phi i's the disjunction of the phi i's implies phi means that each phi i implies phi okay uh then uh there is the so-called distributive rules which is a factor that uh um um if we apply a conjunction with some phi to a formula obtained by disjunction we can commute the two operations so phi conjuncted with the disjunction of the phi i's is the disjunction of the phi i of the of the phi i's conjuncted with phi this is unequivalent so in one direction it is the consequence of the previous rules and in the other direction it is a rule we introduced by now there is a rule which defines existential quantification if we have a formula phi in the context x y on the formula psi in the context x then phi implies psi means that uh the existential quantifier and the variable y apply to phi so this considered as a formula in x implies psi so this is the definition of the meaning of the symbol of existential quantification and lastly there is a so-called Frobenius rule which is a fact that existential quantification commutes with conjunction okay so it is written there and so in one direction it has been introduced it has to be introduced as a new rule and in the other direction it is a consequence of the previous rules okay and by now we have the the meaning of provable something is provable in the theory when it can be deduced from the axioms by applying this list of rules so what is remarkable here is that this list of rules is finite it is finite it is completely explicit and so we know with utmost precision what it means to be provable so by now the syntactic category ct is well defined i remind you that in the definition of the category the notion of provability does not appear in the definition of objects which only depend on the signature but they appear in the definition of morphism morphism are formulas which are provably functional in the theory t okay um okay on the earth of course we have to verify that this is indeed a category that this category is geometric and that we can define a model in this category which is universal which means which represents all models in all geometric categories so this we shall do tomorrow and we shall continue with the study of classifying the opposites so for there is top here