 nezaj. Včešče in vse, da se je vse zespečen, z tem, da je vse zespečen, od vsez, v korišu, in vsez, da je vsez, in vsez, da je vsez, koriš, in vsez, da je vsez, ta vse svojstje koukaj skupaj je sezdano. Početno štef, da je to vsi zelo, da je to tako zelo začostil. Našli vse se je zelo, da so možemo pokazati, da je to vsezdano v konfliktim kanonijstvom, zelo pošli, da je to vsezdano vsezdano. zitebenي, kajavingi dva vsi, in potem topoti z vsi o tudi z ateptikimi zesefati, zalojte to dorge v universalnih properti in v glasbi proprtiti vsi, počusi imam vsi basen, tudi počuti vsačega. In tudi, ko je decitivno v dopovi, glasbi proprtiti, to bese na kategoriji te pike, z modelami tega v tega, v toto spi, način, v tega. Zato to tudi ima, da imamo nekaj matematik, nekaj vsega, to je tega in toga, kaj je tega modela tega, kaj je tudi tudi tudi tudi, ki se nalazi do identitiju tudi, ki se nekaj je zelo v tudi v klasicjih vljautih, ki se nalazi do identitiju, ki se nalazi vljauti tudi v klasicjih vljautih. Tudi se nalazi vljauti v klasicjih vljautih, in tudi model, ki se nalazi uti, je ta zelo, da tudi vljauti vljauti vljauti v klasicji Vseh toboz, ovo je naredno, ki bomo pošličili. Vsi nekaj modeli se občajati vse unijersalno modeli, z kajom vsečenem vsečenem vsečenem vsečenem vsečenem, kaj je unijerno vsečenem. Tudi je unijda vsečenja. Unijda je argument, ker to bude rezentabilitiv, kar naredzana kaj je to prostavi, ki je to doj rača, JT gleda domovesti. Na pé veliko stvari, ne igre taj sem sprejden, tako moment strana, kako ta je, bo izgvorega včet04 način je to blizna, na častrojno Efrije, compt忌, stvar. In zelo, da se nesetak je zelo, da ne zelo, da je nezelo, da se tako je zelo. Zelo je, da je dobro. Zelo, da se nezelo, da se nezelo, da se nezelo. To je po vseh vseh opozitivnih. To je nezelo. Tak, tak. Zato sem bilo vzelo v tih tih početek, zato pa vse objezujem, da bo vzelo, da se vzelo, da se je zelo v početek in v tem, v tom početek, pa so vzelo, da se je uvršen, in neko... Uvršen je, mt. Tudi mt je vzelo prejderzaj taktiko in pr carry buljaca, imaš zelo, da také, kako Ingrediente mežije na topolo, je to, da dogažujš, v kratku zrani v posledu. Tuk,,, tudno je take, ki je vse tako kona, kot se najmega je, kako je tega, ki ne je, kaj je to? Tako, je to, če je, da razprvamo in je, če je, gaj, si je občasno this model there, nared nared, in so what you get is that this angle will be azo-mortific to angle will be azo-mortific to. Eštečne. So this picture shows you that, basically, this is sort of a center of symmetry of the theory because everything that happens elsewhere, in some sense is deformation včoč, da se pravno prišljamo v točko, če pričo seznačnih vsemačnjih, kot vseznačnjih, zato se zelo, že v twentvoj modelu, katera je večo objavna, katera je vseznačnja. Jeste tako vseznačne kompletnosti, da je zelo izredalja iz toj model. OK, zato je je vseznačna pristim vseznačnja, In vseč, naredite je mnogo, da ne jste ne zelo. To je modem, je modem od... V moji modu, na različenju, majske mode, ne bo modem, ne bo modem. Zato je, da bo, da bo, da bo, da bo, da bo modem. Tako, da bo, da bo, da bo, da bo, da bo modem, in da bo, da bo, da bo modem, da bo modem, da bo modem. Zato, da so posledili toga vsemo, da bi bomo zelo tudi sega, da so tudi zelo, da bomo zelo, da so tudi zelo, da bomo zelo. Zato, da smo pričeli ti način, da se obšihamo, da vseh je tudi vsega vsega vsega vsega tega vsega. vsega trijna z vsem. In na toga nekaj to je ta, kdaj se je ovo počutila, kar je vsega vsega vsega, ko je vsega, kaj je začel, kar je z vsega z vsega, kar je vsega z vsega... da je vsega vsega, ali, kaj z vsega, da se je zelo, da se je zelo, in, če je vsega, da je več vsega, v delilj, je, da se ustačil odlična, in lahko da se pripozvanje, zelo, da je to obličke odlogi Kronend, nače ne museli tem nebej zimne, da je prvej, in nekako je oblik, zači se razljega je prvej, ispečne, nače je odličnije v obliku matematickim kontekšnima, matematickimi tebe, nače se smarnost politej, Proto, če sem nečalout tako, da jim je inžiativ, kako je to vsega, če se dači vsega, kako je to, da je, da je inžiativa, zelo, če je to, da jo je pribalo se vsega vsega, če je vsega vsega vsega vsega vsega. Da se počasno, da vsega vsega vsega vsega vsega vsega, je vsega vsega vsega inžiativa, tudi nekaj verj, ki je vsega nekaj sprednje, ali nekaj, ki je vsega nekaj pri teba ta vsega, če pa je, da je to jaz tudi izgleda v svoj klasik. In to, da se pridrečno sem, da se je prijezena pri teba na vsega klasik. Vse je tudi vsega vsega klasik. Vsega je vsega klasik. Tudi je to vsi kategorije, a zato je to vsega zelo. Ovo, kako je to prijezena? Tukaj, da je ta vsega? Nelj, kako je je zelo? Asositi štefnike razičiljajoveda o dogrom construjnji, dogrom studiarcije, navr Address, so izgledamo to, da je tega. T and t prime are set morita equivalent in observacije. in ki sem tukaj v equivalentnih klasitarnoj, to je tukaj pravda definitiva. To je nekaj in nekaj, da je tega vzgleda, in zelo, da je vzgleda na to, ker je tukaj, klasitarnoj, in tukaj vzgleda na vzgleda, da se poču, da se poču, da je vzgleda na vzgleda na vzgleda, in je zelo, T gay, shouldn't see it later. But I just to say that this is really a strict generalization of the notion of War-Clinker and targ of the classical one. Okay, so, this means that we can identify growth and ectoposis with geometric theories considered up to war with equivalence. And so the bridges that I would like to talk about vseč potrebno o časneje, da se može izgleda, da imajo dva teori, ki se izgleda, tudi tako greba, po transparati vsi z odku in odkorti. Vse transparat vsečo zpravimo, da bi se pristajila v barji, vzniče v barji, ki je vznat na to, vzniče tudi, in izgleda, da se vseznega vzrednja taj način, v tem, da se je vzrednje z vseznačenjem. Kaj je tega nekaj vzvečenja, da vseznačenje je nekaj nekaj vzvečenja, zelo je nekaj nekaj nekaj vzvečenja vzvečenju bih komersi, da je dobrozne. Zelo bih komersi ima, da bom je vzvečenja, the usual formula is in the language of the force theory to to translates in the language of the second in a way that this induces an equivalence of the semantic level. So there are many, many examples, of theories that are not Certified but that are equally equivalent. So in those situations you don't have additionally so you cannot perform a direct translation between the two theories but the fact that the two theories share the same classification of those means pa nekaj ne bo, da sem se zelo početila. Zato, da je pa različila, da se početila, da se zelo početila, in zelo početila, da je to več načinčno. Sve, da se početila, da se početila, da se početila, da se početila, da se početila, da se početila, da se priča vsih vrvih, v svoju vrvih vzduči, s veče včešče, vzdučnje vzdučnje, da vzdučnje vzduči vzduči, zaradi vzdučnje v vzdučnih zeljegitavih operacijov, ki vi svoj imeli vzdučnih svej. Začelim, da se istradi, da zelo veče več vzdučnje, ker, zato vzdučnje, zelo vzdučnje in vzdučnje, vzdučnje vzdučnje, became, then when you look at these operations from the point of view of fear, you often find things that are very interesting, I will start with a very simple invariant notion, which is the notion of subtropos and I will describe how this notion expresses in terms of logic, which is logical countert art of the notion of subtropos. Tukaj je to rezultat, ko sem vzela v moj tizijskih. Če sem tukaj vzela geometričnih teori, je tukaj klasika in opos, sem tukaj na definicijs, ker sem tukaj nekaj... Zato, nekaj je tukaj vzela o težko, zelo, da bi se počuče definicije. Kaj je to tukaj, tukaj nekaj, je je počut, nekaj je tukaj, tukaj je tukaj geometrična inkluzja. So it is a geometric morphism, such that it's a direct image is full and taken. What is a quotient of a geometric theory while it is simply a theory which is obtained from the given theory just by adding more axioms while remaining over the same signature. So you don't change the signature, you just add more axioms. And what is a syntactic equivalence while it is the reaction that identify the two theories over the same signature when they prove exactly the same geometric sequence. To change the set by so much, the theory remains the same. Yes, exactly. I mean, it is the obvious thing that you want. Because, of course, I mean, if you add the things that are provable, the models do not change, so you certainly cannot expect this to hold. And so this bijection works in the following way. So when you have a quotient of t prime, a quotient of t, so here you will have the universal property of the classified topos of t, which gives this. And then you can write something analogous for t prime and t diagram, where here you compose with this inclusion and given by the subtopos. So in particular, the domain of the subtopos will be a classified topos of the quotient. So you really, you can localize everything in a most natural way. OK, so how does this work? Well, the proof is not particularly difficult. I will just give the idea. We just used the representation, the syntactic representation that we gave of the classified topos of the theory. So let's do this. You take shifts on the syntactic side and you use the fact that the notion of subtopos admits a very natural side characterization. When I say side characterization, I mean a way to describe this invariant in terms of the given side of the definition for the topos. So this means that when you consider subtopos of shifts in a certain side, this corresponds exactly to the grotes of topologies t prime on c, which contains to refine the topology j. Very natural thing. Yes, it means that to obtain j prime, starting from j, you just add more. And in fact already from this remark you can be more or less convinced about the truth of this result because here on one hand you add more shifts and on the side of logic you add more axioms. And in fact we shall see how to add little shifts on the other side. So more precisely here what we have is that we have a direct bijection between the grotes and topologies j which contain the syntactic topology and the quotients. So how does it work? So imagine that you have a quotient t prime so here it means that you have added axioms of this form for instance to your theory. You can go on the other side so here you have to know what to add to the syntactic topology and so here you just have to recall that flat, I mean here we just have a Cartesian Cartesian jt continues counters which yesterday remark did the same as the geometric counters this to any toposki the same thing as the models and this word in the following way a model was sent to the factor fm which sent any formula any geometric formula to this interpretation in n. So you have said that this is the same thing this of course and then you can consider this monomorphism in the syntactic category and basically requiring that this is sent by to an isomorphism by the factor fm precisely amounts to the condition that m should be a model for this additional sequence and so from this you get the idea of considering the growth in the topology generated by all the the sieves generated by things of this form and so this is how one direction is defined and the other direction well, yesterday we observed that we gave the definition of this syntactic topology you have a situation of this kind so if you have a we describe our sieves on this syntactic category well so let's just replace this and so starting from a sieve here you can be the sequence in the following way just take and so by adding all the sequence obtained in this way you will get your portion so I think it is pretty clear how it works of course there are some technical things to make the tool completely it goes back I think you should make the remark that it is not completely straight forward application of what you did yesterday because when you go from t to t prime their syntactic categories as defined yesterday are not the same in the sense that here we represent no, of course here basically I change the topology in the sense that I represent the classifying tools of t prime as she is on on this but we also know by the general construction of classifying tools that this is also equivalent to to this so you see that here we have a different category so I mean that you come all the time with multiple ways of representing the toposis in the sense of multiple size of definition for the same toposis it is very, very easy to generate different possibilities for describing the same tools OK, yeah describing the same group exactly, exactly I mean already Grotendick made this comparison he said that if we we think about this analogy group toposis then basically different sets different sizes of definition for one toposis can be in some sense a sort of analog of different presentations for the same group OK, so so quotient of t just means that you keep the same signature we just add more things over the signature and the formulas are course in geometric yes, always, always so if you do it with other fragments it's a very good question it's a very good question that I asked myself actually some years ago and this is very interesting because even if you start for instance with a theory that lies in a smaller fragment of geometric logic there is no intrinsic way of identifying only the sub-toposis that correspond to them so I mean there are some invariants for instance you can say that a toposis is coherent so the notion of coherence was already present in SGA4 and you can prove that a toposis is coherent if it is the classifying topos of some coherent theory but this is the only thing you have and this doesn't allow you to restrict the equivalence even by this duality to those fragments so this is interesting because it makes appear the fact that really geometric logic is the fragment that corresponds to the geometry of growth and decomposition so it means that if you decide for instance to restrict your attention to coherent toposis because maybe you like the assumption of choice and you are assured that they always have enough points when you perform operations which are as natural as the sub-topos construction you come out of that so you see this is an illustration of the fact that really the logic that corresponds to growth and decomposition is geometric logic and nothing else ok so now that we have this duality we can start playing with it because in the sense that we have another question at the site of 50 prime essentially you are just setting more atoms, the objects are the same but what I think yes the objects are the same what changes is the probability because probability in t prime or probability in t so you are less and essentially the sites will be the same because you are adding the sieve on a certain object if that sieve is a sequence derivable no you have a different notion of arrow because so it's a different category and you have a different notion of probability but what we see in the old one in the c t j for those in the old one should we say you get sieve I thought there could be new sieves that essentially are covered already by the one in the next what happens if you add contradictory actions well you can always do so in this way you get a contradictory theory and this is how can you have a model no it has no models you have a topos which has no models no points at all and which is trivial as a topos so no it's a very important feature of logic the fact of being able to talk about contradiction because I mean it's not I mean no no it's important it's like when one works with imagineries I mean you can add and then you will see whether it is contradictory or not but you add so the message of logic is don't worry add things and think like that yeah you say it has no model I might object because the degenerate topos will be a model in model yes I mean no at least no set that's the when usually in logic they say something has no model the logician means non-empty model yeah that's what it means but the case of the you have always been the degenerate topos no no you have this degenerate topos and you can really study it as an object and you can also prove for instance that the fear is not contradictory by showing that the topos is non degenerate I did this for instance in one of my papers so I really proved that the topos is non degenerate I knew for instance that this topos always had enough points and I was able to deduce that there was at least one point for instance so I proved the existence of one set based model of a theory using the degenerate topos for which set? well it was a theory of homogeneous models for certain theory of ancient type so it was a theories arising in the context of crisis construction so maybe I will say a bit about this later ok now let's come back to this duality so in this duality one last question if the theories maximum is complete what happens to the classifying topos well if it is complete in the sense of geometric logic it means that the classifying topos is too valuable and if this topos is also atomic which is something important to have because if you have atomicity or at least bovianess you can prove that the notion of completeness for geometric logic which means that any geometric assertion is either true or false in the theory this notion becomes equivalent to the first of the notion of completeness which is not generally the case for any geometric theory and for those theories like that these are maximal with respect to the natural order of theories that it is induced by this duality but I will come back to this because here of course the duality gives a lot of things because at the level of subtoposis you have many structures that you can consider in particular you have the order between subtoposes you can say one subtoposis including the other and so you can wonder what corresponds to the level of theories and so duality in order between theories you say two theories one is well, let's try it like this one is smaller than the other if this is a portion of that it means that everything is provable in tipa exactly exactly another way of saying it so we have an order in the level of geometric theories which are quotient of the given theory and knowledge of elementary topos theory applied to the lattice of subtoposis of the given topos tells us that this order is actually a niting algebra structure because at the level of subtoposis you have a co-hiting algebra structure so since you reverse the inclusion we get a niting algebra structure here so it means in particular that you have a join that you have a meet etc between theories and it sounds that you can even give a very explicit description of what these operations are you get an atomic height you get an atomic height because there are maximal theories yes you can almost it is almost it is not exactly but almost in the sense that for most theories you can find a maximal one that extends it for most so which operations are available on the collection of subtoposis you say that you have a join and a meet do you have infinite meet yeah you can also take that of course and do you have some kind of complement or this doesn't make sense yes you have this sub pseudo complement if you want this you always have when you have in a niting algebra do you have infinite joins also yes of course if you wanted to take here what you do at the level of logic you just put the actions of how do you define the join of two theories when you take the actions of the first the actions of the second and you put all of them together so you get certainly a new theory provided that it is a set so yes you can define is there a simple example of a geometric theory that does not admit maximal extension no but maybe I didn't think about this problem but I proved that in many situations I showed how you can get something maximal out of a theory with some properties but it depends also on which kind of set theoretic assumptions you want to make many choices yeah that's the point if you want to do things completely constructively then it can be a problem but with choice do you get the mathematical point? no, not even sure in many cases yes but I mean I repeat I haven't really addressed this completely but I can tell you that in most cases yes so you get this I think algebra structure so of course we have talked about the join you can take the myth so this is you take the things that are crucible in both and and from the fact that you have a co-writing algebra structure on the lattice of some topos you get also some nice properties of this like you have automatically distributivity for these operations etc so it's a nice thing to have and now I don't have the time to write down the specific externalizations for these operations but if you are interested you can find them in my paper so this is called the lattice for the moment my aim is not really to talk so much about logic but mostly about mathematics I mean the kind of insight on concrete mathematical structures that this duality can give so for instance well, just remaining on briefly on the subject of logic on the topospheric side you have the notion of open closed sub-topos and so these correspond to extensions of these kind so open sub-topos correspond to adding geometric sentences like that and the closed sub-toposis which are complements of these are given by sequence of these kind always in the empty context because these are geometric sentences so you can see it's very natural of the topospheric side and at the logical level another important construction that you can make in toposphere is the subjection-inclusion factorization of geometric morphism so what you can do is the following thing if you have a geometric morphism of toposis you can always factorize it as a subjection followed by a geometric inclusion of toposis so a subjection is a geometric morphism such that its inverse energy is painful so you always have this factorization it is a general result so you can look at what it is in the setting of the classifying topos of a theory because here you have sub-topos now it tells out that these sub-topos can be identified as the classifying topos of some theory which is related of course to this geometric morphism by the universal property of the classifying topos is the universal it is the geometric morphism corresponding to a model of the theory t in the topos here f and it tells out that this is the classifying topos of the theory of this model so you see it's very natural in logic to take all the things because we made an application of that even for non-remotives we used this kind of construction to take the theory of the model even though we did this in the context of regular logic just if you are interested this is the paper look I will probably mention this in this talk consider the theories of this in those paper and we showed that an appropriate version of the regular syntactic category of this theory yielded something that is equivalent to classical non-remotives so you see this is just to show that logic is really not a part from mathematics so it interacts very well with classical mathematical context sorry here I know what to say so here we have talked about open sub-topos and here about closed sub-topos we have also talked about the logical interpretation of this construction there are of course many other things to say for instance there are many constructions that you can make local operators that you can consider on a topos to produce sub-toposis so when you have a topos and you have a universal closure operation sub-objects so this means that you have a thing like this and there are some notable universal closure operations that one might want to consider for instance you can consider the double negation closure operator this gives what is called the booleanization of a topos so this is called booleanization it is a universal way of making a topos boolean and you had many other operations that you can construct sorry, have you well, you take for any sub-object you consider the double negation of that here the negation is the pseudo complement in the hiding algebra in which the sub-object is so the sub-object you had a hiding algebra of sub-object so you just take this and you take it two times and so you get something bigger so you define it in this way you operate in the lattice of sub-object of sub-object sub-object so so this is one example but of course there are many more and I mentioned this just because of the relevance that it has in the context of various national mathematical theories there is another construction that I introduced which apparently is very eccentric so basically I proved that given a topos there is always a large dense sub-topos satisfying the Morgan's law now I am not going to explain exactly what it means but just to show that apparently you see here we are talking about the Morgan's law on a topos so this suggests that in this hiding algebra sub-object of this law also and the demorganization is a sort of universal way of the topos demorgan and basically as you can see here you are doing completely topos theoretical considerations so apparently you don't expect that when you look at this thing under the mirror of the duality you will find meaningful examples in the context of particular mathematical theories but actually the reality is the opposite because for instance we proved with Peter Johnson that if we take the classifying topos for the theory of fields so a very simple and natural theory and we consider the demorganization of it then this corresponds under the duality to a very natural theory which is the theory of fields of finite characteristics which are algebraic over the prime field so you see this is I mentioned this just as an illustration of the fact that apparently weird and remote constructions that you can make here might give rise to naturally defined mathematical notions and here since I have talked about dualenization, what does it give in the context of the same theory when it gives the fields that are more over algebraically closed so you see you get really natural mathematical examples if you change the theory because you might think what is missing from the demorganization to dualenization no, the demorganization is bigger so when you take the dualenization you get smaller topos and this corresponds to taking the fields that are finite characteristics which are algebraic over the prime field and if you want to get the dualenization you have to add more over the condition that they are algebraic. but logically speaking the rule that is not satisfied by the demorganization but the law of excluded middle because here this is weaker than the law of excluded middle the law of excluded middle is this so this is a weakening it is an intermediate logic in whitenistic logic in classical logic but this law does not hold in general the first law does not hold in general no, no, not atrope this is why it really defines an intermediate class of toposis which are called demorganized toposis and so let's just change the theory to show that this fruitfulness for classical mathematical situations is just not completely by chance is not peculiar to that particular theory if you take the theory of linear orders for instance the dualenization gives the theory of dense linear orders without 10 points which is again a very natural theory ok, so this is just to give you an idea of the fact that this correspondence given by the theory of classifying toposis between geometric theory of one hand and growth in the toposis on the other is actually very, very natural so now let me move excuse me when you say that the theory of you consider the theory of fields as a geometric theory what do you mean? the usual first order? no, it is a coil you can formalize the theory of fields within coil and logic you can do it without changing the signature you do it in the signature of commutative rings within units so it becomes a portion of the theory of commutative rings within units ok, so now let us talk about theories let us talk about a class of theories that is particularly important in topos theory at least concerning the logical understanding of classifying toposis is the class of theories of pre-shift type the theories that are classified by pre-shift topos why are they interesting? well, simply because since every growth in the topos is a sub-topos of the pre-shift topos so you can expect every geometric theory to be a portion of a theory of pre-shift type this is actually true so basically theories of pre-shift type you can regard them as the basic logs from which you can construct any kind of geometric theory so in some sense these are the simplest theories that you might want to consider and all the others will be given by an addition of axiom to them so the definition is theory classified by a pre-shift topos so in principle you don't specify the kind of pre-shift topos that you might want but it turns out that this category here is essentially determined by the theory in the sense that if imagine that the tree is classified by such a topo then let's look at what it says at the level of the set-base models of the theory well it means that the flat topos like that but this is precisely the in the completion of the category C why? because flat functions are just the functions that are filtered co-limits of representables so this is basically the filtered limit filtered co-limit completion of the category C and it turns out that you can recover a scene from the in the completion by taking so scene up to C completion is equivalent to the category finally presentable objects of this category and so it's also finally presentable objects of yes so what is a finally present, yes to say so definition so well I will give it for a model but it is a general definition you say that it is finally presentable when in a category which has filtered co-limit when you consider the home function, the covariant home function and you require this to preserve filtered co-limit ok so this is so here we have so this means that when a theory of pre-shift type it is always classified by this because if you just take the co-ship completion of a category, the resulting category of pre-shifts does not change so you can directly take this and this gives the sort of canonical representation of the classifying purpose notice that yesterday we have already identified a subclass of the class of theories of pre-shift type namely the Cartesian theories because we showed remember that if t is Cartesian then you have that the classifying purpose is just take the Cartesian syntactic category of the theory and this is this classified by a pre-shift purpose so it is a theory of pre-shift type now the interesting thing is that this class is much much broader than the class of Cartesian theories because it includes many theories that are even infinitary whose syntactic form is completely different from that of Cartesian theories so I will give just a few examples so you can consider the theory of linear arms and for instance to consider to formalize the notion of linear order you need a disjunction because you have to specify that the order is total so you see we are no longer in the context of Cartesian logic or you can also consider the theory of intervals by interval I mean a linear order with bottom and top and this theory is interesting because it is classified by the topos of a simplicial set which is in fact a pre-shift topos we have many other examples so the theory of algebraic extensions on a given field this is an infinitary theory because if you want to express the algebraicity condition then you need an infinitary disjunction you have the theory of lattice order and a billion groups with a strong unit for instance the notion of strong unit also requires an infinitary disjunction over the natural numbers you also have Amin Alankone and Katja Konstani have used both the cyclic topos that Kon had introduced in the 80s and the more recent 80 cyclic topos these are two pre-shift topos and one can describe few result pre-shift type classified by these topos in a paper entitled cited theories if you are just curious to see what it gives they are relatively simple theories that one can describe but simple but not cartesian which are classified by these important topos and so I mean this is just to illustrate the fact that in some sense these theories form a huge class and so even though apparently in terms of syntax they look quite different from each other they share many properties because actually you can prove various reports about these class theories for instance a very natural question that you can ask about the theories of pre-shift type is what about the classical syntactic notion of finite presentability because I can define you see normally in universal algebra one defines a model to be finitely presented if there is actually a formula that presents this model while the definition that I gave here is completely semantic you just categorical you work in a category and then you require a normal function to wonder whether there is a relationship between this syntactic notion of finite presentability and the semantic one but let's give a definition so m is said to be is considered in the context geometric formula in context interpretation of this formula in any model n are in natural correspondence in the homomorphisms so you see this is for those of you who are familiar with the free structures this is a generalization because in the case of where you have two you just find the notion of free model and here we allow any possible geometric formula so you see we have this definition a significant difference between the two and actually you can realize that if a model is finitely presented with syntactic in this sense then it is also finitely presentable this is quite easy to show but the converse is by no means straight forward because imagine that you start from something that is finitely presentable in this semantic sense it is not a priori clear at all how to find a formula but we have a sort of special method to address this kind of problems a bridge technique which suggest us to work with multiple representations of the specificity of our feeling and so in the case of fields of pre-shift type this is very nice because we have on one end the classical syntactic representation and on the other end we have this semantic representation I say semantic because you see the notion of finite representability that I use here is the semantic one so I use this and I know that it is correct because we obtained this description of the classical entobos by using the semantic definition rather than the semantic one so I have that and now I try to play with and I put off if I want to embed this into here I can do this but of course it is going to be co-variant only or to sort of pass from here to here because you see we start with something we start with a finite representable model in the semantic sense and what we want to do is to find something here you see here is the formula so we want to go from there to there but it doesn't seem feasible to do it directly because we have no key so how to do so we start from M and we have our immediate embedding that allows us to regard M inside the classical entobos now we use the fact that because this is a pre-shift entobos the objects of this entobos of this form they are very special they are objects that are irreducible because it means an irreducible object in entobos so it means that whenever you have anything more thickening so you say that A is irreducible if whenever you have anything more thickening like this then the identity on A necessarily factors through one of these arrows so you see it is a very very strong condition it is not something that in fact it is a condition that characterizes exactly in entobos up to Koshi completion of course if the category is not Koshi complete then you could have a retract of something that also is irreducible without coming from the side but here you see we have a Koshi complete category so we know that this is irreducible then we can transfer it here so let's call the object that we get here A and so what we can do is we know that A is irreducible as well because of course irreducibility is invariant so I can look at it from the other point of view and what I am very tempted to do since I really want a formula well I don't yet know that A is a formula but I can certainly cover it with formulas because in every object in entobos of sheep this is a general fact I mean if you have totes of sheep on a site then if you take all the arrows from the representables when I say representables of course you have to apply the associative sheep kind to this representables but so I denoted like that A, J or C you make a C vary give a few more feet do you assume it's a chemical? no no no not at all no because you have this is a consequence of the fact that the level of the pre-ship totes every object is a collimit of so you just apply and you get no no you don't need so it means that I can certainly the things that come from the site in these formulas and and since we know that A is reducible well then we have such a factorization but this implies that this factorization is monique but it means therefore that A is a sub-object inside the distopos of something coming from the site and you can easily see that it must come from the site itself because this category is closed under sub-objects in the totes so we are done we have found the formula so you see in a completely indirect way and we would have never been able to do it in a direct way because you see we are really exploiting the geometry of the totes what if I try to do it by contradiction I say there is no such formula then I keep going and I contradict the fact I don't think I tried myself a bit before and then for me it was a big surprise to find this result just by playing with this I didn't believe at first but then I understood that in some sense this is really the right point of view you see it's very simple and sometimes you get in a simple way you get different results that even in particular cases you could not prove so this shows that the two notions are equivalent and this is going to be useful now for describing the quotient the classifying toposes of quotient theories of which type and this is useful and of course it gives us a way to get more semantic representations of those classifying toposes because so far we have just syntactic ways of building classifying toposes which in practice are not very useful unless you are just interested in logic but the advantage of working with theories of bishop type is that you can pretend in some sense this category of penalty presentable models is semantic even though it is not actually because I mean this correspondence you can formalize it as an equivalence between this and the full subcategory of the syntactic category on the irreducible formulas what do I mean by irreducible formula when it is a completely syntactic notion that derives just by the expression of the property of the object corresponding to the formula irreducible in the topos so basically it means that every j covering sieve on the formula is trivial so you see this is a completely syntactic thing but ah sorry I forgot the op and so you see here you can really switch from the syntactic thing to the syntac and of course it works in this way so here you have formula and on the other hand you have the model presented by this formula and and conversely and this is a categorical equivalence so anyway you can pretend this could be semantic and this is important because you see you can identify finitely presentable models just by using the semantic definition usually this is what one does one doesn't necessarily care about finding the formula but for our purposes it's important to know that there are such formulas because for making certain computations it is useful so now we consider portions of pre-shift type so, portions of theories of pre-shift type this is a natural something to consider because as we have remarked as any as any glotending topos is the sum topos of sum pre-shift topos so any geometry theory is a portion of sum theory of pre-shift type so we want we often to be able to obtain classic pre-topos of these portions by starting from the representation the double representation that we have for the theory of pre-shift type so we start from this and then what do we do? so we know by the theory that we proved above that if I have a quotient t prime t prime quotient t then it means that I have a sub topos of the classifying topos content but now I move here and I recall that sub topos is of such a topos are just the localizations of these so it means that I have a glotending topology that I would like to be able to describe to describe explicitly in terms of t prime and t basically I mean talking about t prime is the same thing as talking about j but you see that it is not completely clear how to go from one to the other because you see here the formulas are so imagine if you have a discipline like that then you add to t in order to form a t prime how can you find the glotending topology j starting from this you even in the following way you suppose that these axios have a particular form because after all this assures you that there should be a way of presenting t prime in this way since all the sequence that come from j will have a particular form that will involve these irreducible formulas necessarily because you see we are working here and we are precisely working on the opposite of that but the opposite of that is precisely that and maybe we can just write like that emphasize this point expressible in terms of irreducible formulas but this is a bit a bit strange because why should this be the case well, the reason is that you can cover every formula in the classifying topos of the fluid t which is appreciate type you can cover it by irreducible formulas this is the logical translation of the fact that you can cover every every shift in every object of the classifying topos with things that come from the side so what does it mean concretely, but it means that I can cover in this way so this is my geometric formula it means that there exists a jt common in sieve which is generated by formulas that are irreducible so I will have my cover means of course that this and this the disjunction of all this and this gives me the key because in some sense you see, thanks to this decomposition what I can suppose is that I can suppose this to be irreducible because you see we are working inside the fluid t anyway because t is a quotient and I can suppose that my sequence is something of the form because here of course I can always add that so I can suppose that I can cover this formula here by irreducible but then this formula I can send it to this formula so it will give a family of irreducible that go to my formula so I can suppose that I have a situation like that where all these things are arrows from an irreducible formula to our formula which also is irreducible and this is the way to make the translation to calculate j because you see we can exploit precisely this formula and so I mean concretely it as you see there I adjust a sieve basically in in the category c t here and by using that duality this gives me a sieve on the opposite of finally presentable models of the theory so I have got a sieve starting from each axiom and then so this will be so you see also in this case it is not straightforward you cannot do things directly you have to pass through this sort of the composition by irreducible formulas and I want to remark that it is by no means clear even if you take a particular theory of pre-shift type that you should have this way of the composing you see you don't see magic in the sense that you don't even in particular cases these are not trivial things at all and it's full of these situations toposphere is full of these situations so I mean Grotendik was really right in thinking that the notion of topos has an immense potential and in particular the notion of classifying topos adds a lot to it because the logic is of Grotendik's toposis in fact even small changes that you can make the level of the signature of a theory can have a huge impact the level of the corresponding toposis so in some sense toposis are very sensitive to logic and so I mean we shall see in some examples after a break and then I will give more examples of bridges and more sophisticated nature to convince you of the power of this general methodology and of the fact that the toposis can indeed as Grotendik was claiming and indeed after significant places ok so see you later