 we just explained that this notion of growth endic topology is so rich that it gives another formulation for the problems of provability in geometric logic. So this is really an incredible result and it makes all the more important the following question. For any geometric theory you are interested in, how to present its classifying topos. So you have the associated classifying topos, ET, and you wonder how to present it in terms of sites. Because any presentation in terms of sites, in terms of topology j in some categories, we'll give you a new translation of the problems of provability in t in terms of topologies on c which contain a j. So the first remark we can make is that for any topos there are so many presentations in terms of sites that in fact it is not even a set. This is just because E itself is not a small category because except the trivial topos with only one object and one rule, all other toposes are not small categories and I remind you that any full subcategory of E which is smaller and generating gives rise to a presentation. So there are incredibly, the way to present it, it is not even a set. What we have done so far is to say that of course there is the presentation of the classifying topos in terms of the syntactic category endowed with a syntactic topology. But this presentation of course is not is not always the best. In particular it is a presentation which is syntactic in nature. So it is very close to logic and it would be probably more telling in many instances to have presentations in terms of sites cj which are not so closely related to the linguistic presentation of t. Here I should also mention that in fact the classifying topos of the theory in general as several ways as several presentations define even in syntactic terms there is a syntactic geometric category we already define but there are also exist other syntactic presentations for instance if the theory is algebraic there is a partition there is a partition syntactic categories the regular the coherent sort of different types of presentation and maybe I can also mention that next week there will be a talk by Joshua Bradley. So first year PhD student of Olivia who will present in particular a new way of constructing the classifying topos of theories which he has found. This also is very interesting but by now we consider in general the question of presentation of a theory in terms of sites. So the first remark we can make is that of course if we have such a presentation then the topos e equivalent to c at j will be a sub-topos of the topos c at the topos of pre-shifts on c and the topos of pre-shifts on c as any topos can be presented as the classifying topos of some theory t prime and then because e is a sub-topos of c at t will appear as a caution theory of t prime so we can try to play in this direction. So for that we need a definition where we you see that the consideration of pre-shift opposites and the fact that any topos by definition is a sub-topos of a pre-shift topos so this consideration leads to the following definition. A geometric theory t is called of pre-shift type when it's classifying topos can be written as a topos of pre-shifts on a small category. Okay so of course not all toposies verify this property only parts of the only so this is a particular class of toposies and maybe I should also mention that when you have a pre-shift topos like that c at it can be equivalent to some other topos some other presentation so c at c prime at j prime with some non trivial j prime so you see you can define a topos through a non-discreet site and nevertheless get the properties that this topos is equivalent to the topos of species on some category. It is a quite subtle question. So there are examples of theories which are of pre-shift type in fact which I'll come back to that later but I immediately state a few examples first up in any signature sigma you can consider the so-called empty theory which is a theory without axioms and then it is a general fact that this theory is of pre-shift type. Another example of another very important class of pre-shift type theories is a class of algebraic theories so what is an algebraic theory it is a theory whose signature has only source on function symbols no relation symbol and whose axioms all have the form true implies inequality between terms okay so if you think for instance about the theory of groups what are the axioms there is associativity of multiplication the fact that the the element one is unit the fact that g is a product of g and g minus one in indian direction is equal to one so all of these are equalities between terms okay and it is the same for the theory of rings of modules over rings of vector spaces over fields and so on so here I am saying that all algebraic theories have pre-shift type in fact it is not an obvious fact it requires a proof okay and then what I want to do in the following minutes is to study in general pre-shift type theories so for this first I consider two poses of pressures so let's consider a topos e which is a topos of pressures on some categories c and in order to understand better the relationship between e and c we shall consider an invariant of the topos under consideration which is the category of its points so let's consider the category of points of seat and of course the point of seat is given by djakonescu equivalence and in that way you prove that points of the category of points of seat is equivalent to the category of functors from c to set which verifies two equivalent properties so first the category of its elements which means the category of objects of x endowed with an element of p of x so this category is filtering and the second equivalent property is the fact that this functor can be written as a filtering co-limit of representable functors so in particular we get that any representable functor defines the point of seat so as a consequence of that we get that the category of points of seat is the end completion the end category built on the opposite category of c so you see that in this way the category of points of seat is computed from from c in a quite concrete way this is the end category the end completion of the opposite of c okay this theorem can be proved and of course there is a corollary by now if we have a pre-shift type theory who's classifying topos can be written as a topos of pre-shifts on seat then we get that the category of models of set theoretic models of t is equivalent to the end category of c opposite to the end completion of c opposite sorry in order to make that more clear we just draw the bridge we are considering so here the invariant under consideration is the category of points of the classifying topos and the classifying topos is presented in two different ways on one side from a theory t on the other side from the category c engulfed with a discrete topology and so you compute the category of points on the two sides on one side by definition of the classifying topos it is a category of set theoretic models on on the other side according to our previous theorem we get the end completion of the opposite category of c so this means in particular that the end category of seop is determined by t up to equivalence so by now we can ask whether to which extent it is possible to recover c from the topos seat so far what we have recovered is the end completion of seop we did not recover c so we may wonder whether it is possible to fully recover c or not so in order to answer this question we have to introduce the following definition so let's consider a pre-shift type theory and let's consider a set theoretic model m an object of the category of set theoretic models and then we say that this model is finally presentable if the function the co-variant function associated on the category of set theoretic models respect filtering co-limits okay when you can check that for instance if you think about models of algebraic theories such as groups rings and things like that then the definition of finitely presentable really corresponds to finitely presented groups finite which means groups which are defined by your finite family of generators and relations or rings which are defined by your finite family of generators and relations and the same for all type of algebraic structures so this definition really corresponds to what we have in mind but you see that this is a categorical definition on the category of set theoretic models and this definition singles out part of the models okay so we can introduce in this category of set theoretic models the full subcategory of models which are finitely presentable and then we have the following proposition so let's consider a theory t which is of pre-shift type so that the category of its set theoretic models is uncompleted of co- yes here i have made you see i have written and of co-opt i could have removed removed uh okay uh on my now let's consider a model uh and so the the proposition tells us that such a model is presentable if and only if it is a splitting of the representable uh object of c so if and only if there is an object x of c opposite on the pair of morphism p and i whose composite is the identity of m and you see this is a pair of morphism from between m on the representable function associated to x so the this proposition tells us that the finitely presentable models are exactly objects in the uncompleted which are uh retracts of representable objects but uh this has a name in uh in category theory when you start from an arbitrary category and you add to these categories all the retracts which means you add an image of p for any idempotent element p so this is the the caroby complication this one this construction was introduced by caroby and so by now this means that we that the set that the the category of synthetic models of t which are finitely presentable is the caroby complication of the category c opposite uh and if uh we denote by m this caroby complication so this is the category of finitely presentable models we get that the classifying topos of t is equivalent to um uh the topos of pre-shifts on the opposite category of m and we also get that any synthetic model is a filtering co-limit of finitely presentable models so for instance any group is a filtering co-limit of finitely presentable groups of groups which are defined by a finite family of generators and relationships on the same for rings for vector spaces for any type of algebraic structure so uh so yeah we see that uh uh when uh pre-shift type theory t uh as it's associated classifying topos uh presented as a topos of pre-shifts of term c then c is almost determined by t it is determined by t up to caroby complication okay and uh by now uh uh we can uh apply the begin to apply that to the problem of constructing uh class of presenting uh the classifying topos of a theory so it starts from a theory geometric theory t on a signature sigma what we want to do is to construct a pre-shift type theory t prime such that t is a coefficient of t prime uh so suppose we are able to do that we just uh add them to uh t uh possibly some other sorts uh some other functions or some other uh no we no no we don't excuse me we don't change the language but we remove some of the axioms okay we want uh t to appear as a coefficient of t prime so t prime will have the same unlanguage but less axioms than t and we would want to remove enough axioms so that t primes becomes of pre-shift type and suppose by now that it is done then we can consider the category m of finitely presentable satiric models of t prime and then we get that the classifying topos of t prime would be equivalent to uh the topos of pre-shifts on the opposite of m and then uh according to the correspondence uh between uh cushion theories and topologies there would be a unique topology j on the opposite of m such that the classifying topos of t is equivalent to the topos associated to the opposite of m endowed with the topology j okay uh so if we want to realize this program uh we see uh that uh we have to we need criteria for deciding whether a geometric theory t prime is of pre-shift type or not so here i am going to present uh in fact two different uh criteria uh uh first criterion which is purely syntactic uh which means uh which is a criterion uh written in the language of logic on the second criterion uh which is a criterion about the relationship the special relationship of syntax and semantics in the case of pre-shift theories so here i begin with uh first uh syntactic purely syntactic criterion so for that i consider an arbitrary geometric theory t together with its classifying topos associated to the syntactic category endowed with the syntactic topology and let's consider uh canonical the canonical function from the syntactic category to the classifying topos and we already said that this canonical function n is fully phase four by now uh we uh recall that the object of ct by definition are geometric formulas the morphisms of ct are geometric formulas which are tip-probably functional up to provable equivalence and lastly the gt uh governing families are families uh whose families of morphism whose unions of images is equal to everything okay so i just uh something recalls that and by now i introduce the following definition uh an object of a topos is called irreducible if any globally epimorphic family from from a family of objects to this object are the splitting so these things that there should exist an index i not in the list of index of the family uh such that the projection from e i not to e uh are the splitting which means uh there is a morphism from e to e i not whose composite with the projection is the identity of e so if any covering family if any globally epimorphic family in the topos uh verifies this property we say that the object e is irreducible so by now uh in uh uh we can translate this property in the language of sites so uh if we have a small category c undone with a topology j we shall say an object x of c is irreducible with respect to the topology j if it's only j covering c is a maximal c and uh lastly if we apply this general property uh to uh the particular case of the syntactic category of a geometric theory uh we get the following definition a geometric formula phi of x is called irreducible if it is irreducible as an object of the category ct with respect to the topology j t okay and here we can remark that of course this definition we have just given in geometric uh language in categorical language uh can be translated uh into uh into the the language of probability because we just we just uh make explicit what it means uh to be a covering family in the syntactic category uh what it means to be a morphism and so uh here i have written uh the meaning in terms of uh probability uh so uh you will see see that in the slides uh so here the remark is complete there is a complete statement of what it means so it is quite uh it is a little intricate i don't say that it is a property which is easy to verify when you have a theory but at least it is well defined and uh here um uh we have uh uh the following lemma uh which once again is uh an illustration of the bridge technique so this times you consider the classifying topos of some theory uh t uh and uh uh you suppose uh this topos is a pre-shift topos so it is equivalent to the category of pre-shifts on some um category c and the invariant you consider is uh the full subcategory of irreducible objects and by now you represent your topos in two ways so on one side from the syntactic category ct gt on on the other side uh from uh the category c uh endowed with a discrete topology so on the side c art we get that the irreducible objects of it are splitings of representable objects of the category c so you see uh in fact here we get the same result this is the same thing as uh um as um uh finitely presentable models you see uh so here it is this is a result that in fact we get the same thing and on the other side uh computing the irreducible object of the topos we just get the irreducible objects of uh of the irreducible formulas in the syntactic category so which is a purely uh uh logical uh notion so uh the consequence of this lemma uh there are some remarks to be made so first uh the fact that the caruby complation of c uh is uh the full subcategory of the topos and irreducible objects so this is just a rephrasing of the of the first characterization and secondly uh we get for such pre-shift type theory that there is a one-to-one correspondence between irreducible formulas and finitely presentable set-based models of the theory so you see this is really very remarkable because on the one on one side you have the notion of irreducible formula which is a logical notion and on the other side you have the notion of finitely presentable model which was defined as a purely semantic notion it is defined purely in terms of satiric models okay uh on cell here we have the following uh first criterion which is a purely uh syntactic criterion consider a geometric theory t then t is pre-shift type if and only if any geometric formula considered another object of the syntactic category as a covering admits a covering by formulas phi i which are all irreducible so you see the criterion the syntactic criterion is that a theory of pre-shift type if and only if any formula can be covered for the syntactic topology by irreducible formulas okay so i don't give the proof of this theorem but what i want to do now is to give the other criterion for a theory to be pre-shift type so let's consider a geometric theory on signature sigma and then the criterion is that this theory is pre-shift type if and only if the three following conditions are verified before stating them i want to stress that all three conditions amount to saying that something which is defined on the syntactic sites correspond to some other thing which is defined purely in semantic terms so the the conclusion is really that a theory is of pre-shift type if and only if syntax and set-based semantics correspond to each other okay so the first condition is the following a geometric sequence of sigma is provable in the theory if and only if it is verified by all set-based models so remember that we have already said that for a geometric theory a sequence is provable if and only if it is verified by the universal model in the classifying topos but you see this is a model in a topos which of course is almost never the topos set here we are saying that the first condition is that t provable which is a syntactic notion is equivalent to the fact that this sequence is verified by all set theoretic models so this is the first condition the second condition is the fact that any set theoretic model which is finitely presentable so I remind you that finitely presentable means that the associated covariant function respects filtering co-limits so you see it is a categorical notion and so the condition here is that finitely presentable models are finitely presented and finitely presented by some irreducible geometric formula phi in some variables x1 xn so what does it mean finitely presented it means that for any model n the model morphism from m to n corresponds to families of elements in the sets n a1 n a n you see the sets associated to the sorts a1 a n in the model n so families of elements which verify this formula so you see for instance if you have a group generated let's say by two elements j on h verifying some formula for instance g to the square equal h to the power 3 then what is a group morphism from this group generated by this element with these relations to an arbitrary group it is just it corresponds to the choice of two elements in this group which verifies an event formula okay and here this makes sense in an arbitrary setting and so if a formula is pre-shift type in fact any finitely presentable model is finitely presented by such a formula so this is a second condition and together with the cell condition we will immediately give it is equivalent for the theory to be of pre-shift type so here is the third and last condition for any sorts in the signature and for any map which associates to any finitely presentable model a subset of the product set m a1 cross m a n in a factorial way so this means that for any morphism of models from m to n the in use map from m a1 from m a n to n a1 cross m a n sends the subset m p to the subset m p so here you see p is any choice of subsets indexed by set theoretic by finitely presented set based models and which are factorial and the third condition is that if we have such a factorial map then it is defined by a formula in the variables of source a1 a n of course in the reverse direction it is obvious that if you have a family of subsets of the finitely presentable models which is defined by your geometric formula then it is factorial but here we have the reverse condition that anything which is factorial is defined by your formula and so the criterion for a theory for a geometric theory to be pre-shift type is that these three conditions are very fact it is unequivalent so I remind you the first condition the first condition is for the fact that you can check probability on set theoretic models the second condition is that finitely presentable models are finitely presented by formulas which in fact happen to be irreducible and the third criterion is that factorial families of subsets in models are necessarily defined by formulas okay so we can say a few words about the proof in the direct sense so when we have a theory of pre-shift type why are these properties verified so in fact it is a consequence of three bridges so for the first it was already done we just have a classifying purpose which by hypothesis is on the one hand the purpose of shifts on the syntactic site on the other hand it is the purpose of issues on the opposite category of the category of set based models of a finitely presented fed based models okay so as a consequence of that it is an immediate corollary is that it is enough to check probability on set based models the second property comes from the bridge which consists in computing the invariance of irreducible objects both in the syntactic site and on the presentation sites by finitely presented finitely presentable set based models and for the third property you just consider the universal model of the universal model ut of the theory and of course ut associates to the sort a one a n objects of the topos and we can consider the product of these subjects and by now we are interested in sub-objects of this product so on the on the syntactic site what we get is that sub-object of this product is just they just are formulas because in the syntactic site sub-objects are always formulas okay and on the semantic side the presentation by the opposite category of the category of set of finitely presented set based models so you are in a pre-shift of us and you have to just to understand what it means to be a sub-object of a certain pre-shift and you get exactly what is written in condition three okay so you get one direction of the theorem just by considering three bridges okay so by now if we together what we have said we have the following scene which has a following process we can try to implement uh suppose we are considering uh theory which is a quotient of another theory so here i do not T prime the quotient of T and suppose we have proven that T is a pre-shift type theory so for instance by using uh uh two the one of the two last criteria uh so in fact in the book uh theory sets of the opposites by olivia there are yet other criteria in fact there are many criterias for many criteria for a theory to be a pre-shift type so suppose you have applied one of these criteria and you know that T is a pre-shift type theory and as it is a pre-shift type theory you know it's classifying topos it is a topos of precious on the opposite of the category of uh uh finitely presentable set base models or you can also say that this opposite category uh is just the full subcategory of the syntactic category on irreducible formulas so you see your this category m can be seen either as a semantic thing or as a syntactic thing okay but any you see because pre-shift type theories are reality theories for which syntax on the set theoretic semantics correspond to each other and so by now uh according to the general characterization of caution theories in terms of topologies we know that the classifying topos of the caution theory T prime uh is equivalent to the topos of pre of shields on this opposite category m up endowed with some topology J and in fact uh the topology J here can be made explicit so here it is you will say that the family of morphism between irreducible formulas so you see a family of uh this means a family of probably functional of T probably functional formulas uh between irreducible formulas uh is covering if and only if the corresponding sequence from fire to the disunction of the images which means the existential quantifier applied to the formula state eyes so if this sequence is provable in the theory T prime so be careful uh uh you see in this statement when I say that the theta eyes are provable are probably functional from phi i to phi here I am talking about T probability okay but then I uh he says that the J is defined in the following way so the second from fire to the disunction of the existential quantifier applied to the theta eyes so this sequence uh is T prime provable and so in fact this defines topology J on this category of irreducible formulas on the classifying topos of T prime is uh is the topos of shields uh defined by this topology on this category so this gives you a way to present uh the classifying topos of T prime but here I should stress that this representation is not unique in fact um uh there are um uh as I already said infinitely uh many theories infinitely uh many presentations of the classifying topos so this means that uh uh if I replace if I allow to replace T prime by a syntactically equivalent uh category I will be able uh to present T prime in infinitely many ways as a cut short of pressure type theories okay on each presentation of T prime as a cut short of a pressure type theory will give rise to a presentation of the associated classifying topos okay and as soon as we have uh you see every presentation can be useful there in general there is not uh a best presentation uh the in fact the interest of the presentations depend on what you do and uh in general it is extremely uh useful to have many different presentations and these different presentations are related through their associated toposis and this is exactly so you see the idea of the bridge such as here anytime you have good different presentations of a topos and you consider an invariant you can try to compute the invariant in the different presentations and this will create a correspondence and uh so uh here uh just because toposis are so general we have seen that any geometric theory defines a topos so this means that toposis are incredibly general it is not uh toposis this is not only for algebraic geometry it is for uh for it can be defined everywhere in mathematics so anytime you have a topos on anytime you have different presentations of the topos and for any invariant of the any toposterity invariant you consider this generates correspondences equivalences so this is really uh a principle uh whose uh generating power is uh absolutely uh huge under so Olivier says that this is uh some type of morphogenesis uh of mathematical morphogenesis so this means that you have a unifying principle the principle of bridges which depending on the the toposis you consider on the presentation of the toposis you consider on the the the invariant you consider will generate extremely diverse results okay and uh so uh I think uh I hope at least that you have understood that the opposites are extremely diverse so you see the this theorem of existence of classifying toposis uh tells you that any geometric theory as a classifying topos so you you don't you you don't have to wonder whether it does exist or not you know it always exists but then the problem is to get more information on the topos and the topos is a huge object it is a category which is not small so it is not possible to to to fully know a topos but uh what we can hope to do is to extract some partial presented some partial information on a topos by choosing concrete presentations and considering invariance which are meaningful for the type of problems you are considering so this gives huge flexibility okay and uh so you will never exhaust all the topos you will never know all the information contained in a topos but you will be able to extract some information and as soon as a topos is presented in different ways which means through different sites or through different or as a classifying topos of different theories so usually what you will get is that this topos with in some sense is mysterious which you don't know which is too big to be known this topos relates it is really a bridge between things which are concrete and meaningful and then you will be able to relate two concrete things through an object which in fact you is too big to be known but this can be done as soon as you are able to compute or to express some invariance so here in what I have explained in my lectures I think I have made clear that when we consider practical concrete theories it is possible to produce concrete presentations of topos of toposis if only by uh syntactic sites but in other parts of mathematics it also different presentations of a topos may also appear so for instance I mentioned that next week there will be the talks by uh clausan on the show so just in clausan on the better shoulder uh in fact what they do in their talks is to introduce a particular topos they call that the topos of condensed spaces and they they they introduce this topos as the right context uh to study uh functional analysis in a better way this is a new foundation for functional analysis and in fact when you begin to read the definitions one of the first reasons why you understand uh this topos is a fine topos is the fact that it has two natural presentations so one of these presentation is by the site of compact topological spaces and the other presentation is by the site of co-finite sets so these are very two very different sites with the same associated topos and the fact that these two very natural sites generates the same topos is a very good reason to believe that this topos is especially interesting here you see I repeat the names on the one hand you have the site of topological of compact topological spaces so you see this site belongs to topology and on the other hand you have the site of profinite sets and this site is of combinatorial nature so here you have two different parts of mathematics on the one on topology and the other on combinatorics which are related through a joint topos they both define okay so this is an example but once again there are infinitely many topos this and for each topos infinitely many presentations and on topos this infinitely many invariants you may be interested in. This is really a principle for producing a huge diversity of results in mathematics okay so thank you for your attention I think this is the end of my lecture. Thank you Laurent.