 ima se vse matematiko in v kratičnih unijegov, v kratičnih unijegov, ali imamo vse vse kratičnih unijegov, vse kratičnih unijegov, in tako, nekaj je vse zelo, če se je vse, kaj je vse, ko je zelo, kako je zelo, in kako je vse. Zato, da je hljel, kaj je vse, kaj je vse, kaj je vse, kaj je vse, kaj je stručaj M v C. Zelo si imaš imenje, da, kaj je F predstavila finačne produkce, zelo se prišlo s sigma stručaja v C, in s sigma stručaja, ko sem zelo tako, f m v d. I this is clear, because the only thing that I had for asynchronous structure is the interpretation of sort, functional symbols and relation symbols, and this only equals finite products. So if I preserve them, for sure, I get a schematic structure here. But what is interesting is to wonder which kind of conditions we have to put on the functor Isprav double transfer models, of a certain theory. Here in models of the same theory, there. And here, you realize of course, that the presentation of finite products junior exist. Poor teacher, whether monops are. Ah maybe. Yeah. Yes yes, because you want. Yeah, okay. And wait. You need the friend version, Let's say, in v fina Boosty presez Mulna? Se je potrebno bolj. Tak, ne, je je potrebno je genulite. Fina Boosty presez19 In, če processing je v modu, tak, kateri mi je, tko z tem modul aleski teori se oblada, Se, da je to nekako, da je to nekaj model od nekaj, da se četakči objezajte, da se je vse tega, da četakči je, da je tudi nekaj, da je tudi f, da je nekaj, dači prezervej vse, da je tudi nekaj na kategori, tudi, ko se pravno odložilo, da je zelo, kaj je počet, in da je tudi v tukaj, in tukaj lahko nekako se vsega vsega. Zato, kako se najbolj izrobijo, nekaj tega srednje vsega, tudi sem zelo, mu bo, da početem vsega vsega. Tudi, da sem zelo, prezervovali franče. Tukaj bomo vzivati regulje franče. Zato vse je regulje franče. To je franč, ki je vse kratižije in imač prezervovali. To je franč, ki je vzivovali, in imač je vzivovali. Kojernje kategori, nekaj nekaj nekaj franč. Kojernje franče, je regulje franč, ki nekaj prezervovali vzivovali franče. In initalija nočnja, in initalija nočnja, je nočnja geometrična. To je več vzivovali, ki je vzivovali, ki je vzivovali franče nekaj nekaj nekaj geometrični. Koj? To je več vzivovali, ki je vzivovali vzivovali, ki je vzivovali vzivovali, in nekaj vzivovali. Če ste vzivovali, ki je vse franče, nekaj prezervovali Tudi je to pomembno početnja. Vse malo pa vse nekaj, ker je zelo tako, da se predjeljajo, ali naredajte, da je vse zelo načinje imegi in tudi in vzelo vzelo na načinje imegi. Če za imegi, da vzelo, nekaj početi pa vzelo v zelo. Zato imegi zelo, da se nekaj posleda nekaj vzelo, Zdaj zanimaramo, da je in se je, da je učniko, nekaj je zanimaramo. Vse bo, da je, da je, da je, da je, da je, da je, da je, da je. Vespoči, da je, da je, da je, da je, da je, da je. in cosnjenje tudi, da sodatek ozok je, tako, da vse zelo, da idea iz entreprisesu imati prezert, zelo, da probatilo, zelo, da su zelo od vso dolične povezate domovina. In zelo, da je obozrečeno zelo je, nekaj ne je odpovr 20. Zelo, da to podrečijo, nekaj ne bo iz 좋아, da tukaj ne bolj tudi, nekaj ne? Spolite, da njistila, da se naje to, da ima se, ta ne, ne, skupaj, ne, ne, ne, ne, ne, ne! Tukaj, da se pozor, da je zelo, da se pozor, da se pozor, da se pozor, da se pozor, vzgleda vzgleda vzgleda vzgleda. Tako, ki je vzgleda, odgleda na prvstvenih vzgleda in imaš najske imaš, kako priježujem nekaj. Tako, ki smo preddajali kajto je geometri. In taj je to vso vzgleda kar so priježijo, da je kot in dikt, kar so priježijo vzgleda način način vzgleda priježijo, ker nekaj, da smo preddajali modelj, in način, ali vzgleda način, da smo priježijo, ki je modelj, Moj seìnho. In? Od začustju. So? Kaj je dobročne poskupovanie? Mi je vse začustje, da ne mojemu... Zelo, da vse igrali. In začusti, da je noted, da vse so dobročne poskupovanie. Pa vse, ta je dobročnja poskupovanie vsisak. Začusti, da je dobročnja poskupovanie vsisak. … ne, da je dobročnja poskupovanie... ... in tudi, da je tukaj začel, ki je zelo skupil, ... in je to začel vsega. In to je tukaj, da smo vsega vsega vsega vsega. Zato, da je, ... ... zato, da smo ... ... in tukaj, da smo vsega vsega vsega vsega vsega vsega vsega. ... in tukaj, da smo vsega kratizija vsega.행, je bilo, da je candidatesi z vrčestovanih moriasnih, prizavljamo tudi, da imam počkora. Sreč? Zvon, da, zato tako, da sem počkora, da je bilo. Paj ne bi njegi zrčet, da bi bilo, da bi bilo, da bi bilo, da bi bilo, da bi bilo. Sreč? Tako, ker je bilo, da bi bilo, da so, da bi bilo, da bi bilo. Nišli si bilo, da bi bilo, da bi bilo, da bi bilo. The only thing you need to take equal access to have something different from... Yeah, okay, so for formulating axioms this subject, we want to formulate arbitrary sequences, right? No, but you might want, there is actually an intermediate level here, that is called Orn logic. And in Orn logic you are allowed to consider this finite directions and basically the right environment to interpret the dc partition category. bo to je, da je to vse zrpončeno kathesijanom logiku. Zelo tudi sem tudi prikazala, da je to vse zrpončeno kathesijanom logiku. Ok, zelo, da sem počkala, da bi se vse zrpončeno, če je bilo, da je vse vse kathesijan. Zelo, da je vse kathesijan, zelo, da je vse kathesijan. Now it's interesting to wonder if you can go in the opposite direction, in the sense that imagine that you have a signal structure sent by F to a model, then under which conditions on the fan core F you can go back and say that the original structure was a model of the theory. If F is conservative, conservative means that it reflects biomotivza, then you Kaj je to pravda, da je je občasne, da je je občasne, da je je občasne. Zato da je bilo vse, da je bilo vse vse način, da je bilo vse način. početki. S njom je pejno druh horko, če očeraj cezeni, če. So, zelo pa početki je je izvahnovati. Če je to… in tudi, ko ti svaši in presence prihleriti horko, inčo jest to se vse zelo nekako ležimo, tu je horko reprezent http. Tako, je to početno, da je početno, da je reprezentable, ker smo imeli nekaj kategori, ki smo početni v nostrih teorijah. da sem vzgledaj vzgleda vzgleda. Zato smo vzgleda, da se počem se vzgleda kaj je vzgleda. Vzgleda, kako smo imeli, je to nekaj kategori, ki je to, izgledaj, ešte se, ko je četil zrčen. Presezno je to, kaj je što se modez, ki se zrčen v početri, po vsej češte početnih, je pravno prišel, ki so prišel iz vsej četil, da bomo prišel iz vsej četil prišel iz izstavnih, z vsej četil pristim, in z vsej četil z rupim. in tudi način, način način v zi. Zato način način v zi izgledaj, je to, da je konec, da je konec. Zato tudi vidim, da je to več stranje, da se modelje zelo vse. Vse je univerzal model, ? ? ? ? ? ? ? ki se je zelo od njihoj pragem zelo, je to skupno, je to je vžetka. I sih se ne bo ozorit. I sih se ne bo ozorit. Sih se ne bo ozorit. Daj, moj pa bo ne. Vši se so koncijno v vse. Ah, OK, so. Mi je počo. Včešči. Tako, češči. Načinje všakvačne transformacije. Zato vse je vse kaj je zelo dva kategorij, da je tudi... Tako, tako. To je začala, tako. Tako, tako. Dobro. Tako, sve, sem tudi, kako je svetljena modelov. Modelsi korrespondi na vse. Now we have to see how it works, both the construction of the syntactic category and the definition of the significance. But before doing that, I just wanted to give a few examples because I promised them in the abstract of my course. Otherwise, this seems very much in the sky. Since we have given many abstract definitions, let's see that they correspond to something very natural in practice. So Alankone talked about the theory of groups at the beginning. So let's consider, for instance, the theory of groups. So notion of group, the axiomatic notion of group, by design the theory, formalizing the groups that we had talked about before. And here, you see, we have at our disposal many different categories in which we can interpret this notion. This is an algebraic field, so we can interpret it in any category we find at products. So we can, for instance, choose the category of topological spaces, or we can choose the category of algebraic varieties, or we can, of course, always choose the category of sets, or we can choose the category of differential variables. And we might wonder what does it like the notion of structure of model of this theory in these particular categories. And you see that you recover very well known notions. So let's start from sets. Here you recover the notion of group. Here you recover the notion of the group. Here you recover the notion of algebraic group. And here you recover the notion of topological group. Here you see, you have one abstract notion that incarnates in different forms in the context of different semantics. And, well, apart from being nice to have such an abstract understanding of these notions, what is useful is in connection with soundness, because, you see, by soundness, whatever that you prove here will automatically descend in all these different directions and so it will give results about each of these particular structures. Of course, the most interesting things about these structures do not come from here. So I'm not saying that because, I mean, the things that you can prove here are very general things, effects about groups, and I mean, you cannot expect the implementation, the incarnations of these into these different concepts would be particularly deep results. But still it's something that can be useful in some occasions in the sense that instead of proving the same theory four times, you just prove it once, and then you have these canonical interpretations in all these different attributes. Yeah, for instance, I mean, this kind of trivialities. I mean, of course that... No, but I mean, things that do not exploit the specific combinatorics because, of course, you see results about topological groups that are really deep. You cannot really expect them to come. But anyway, I just wanted to point this out because it can anyway be of some usefulness. At least it is, I mean, a clear picture and it is an application of this soundness that we describe. OK, so, other examples. How does syntactic category in this case? The category is finite products? Well, in this case, I mean, basically, the syntactic category is you want them to be at least the Cartesian, so even if you had an algebraic theory, you treat it as a Cartesian theory and then you take the syntactic category of that and it will be a Cartesian category. So it will be a category with finite limits, but, OK, we... It is more or less your problem corresponding to these two groups. Well... I think it's more the opposite of the category of finite represented... Yes, it is equivalent, yes. We shall see this tomorrow in the more general context of theories of preceptive type because of theories classified by a preceptive topic. We shall see that all Cartesian theories are classified by preceptive topics, but there are many interesting examples of theories classified by preceptive topics, which are not at all Cartesian. So, in some sense, it is more natural from a toposkoretive viewpoint to study the whole class of theories classified by preceptive topics, which are called theories of preceptive type. OK, other examples. Pierre Cartier mentioned the fact that in the past one didn't just consider ships of sets, but rather ships of rings and other kinds of... Before, even before, yes. And Groden Dick instead said, no, let's be purists, let's just... And there is a reason for that, because by considering ships of sets then you get a toposkoretive and if you want to recover ships of rings, then what you just have to take is a model of the theory of rings inside that toposkoret. And it also works not just for algebraic theories like rings, but also if you want, say, to consider ships of, say, local rings in the sense that the stocks are local rings, then it's the same thing. You consider models of your theory inside your topos of sheets. And this will give you exactly the notion you want. You will have that the stocks of your sheets are models of teeth. In the case of an algebraic theory is equivalent to the stalls or the sections, is the same. For... Because you have... Or... Yeah, you can, of course. You can. And so... But in the case of a general geometric theory the right notion of shift of models is a shift such that its stocks are... Because each stalks gives a point of the topos and if you consider all the points points are just geometric morphisms from the topos of sets to the given topos. So each point if you consider the inverse, so let's call it like that, if you consider the inverse images of these points these are jointly conservative. And so to have a model here of a model still. Ok, so you see that in fact what Enrique was quite right to say let's just take shifts of sets because all the others basically you can get out of that in this way. Similarly, you can... You might want to formalize the notion of a bunch of models of geometric theory indexed by a certain set. And then here you consider just the set as a discrete category. You take the corresponding topos of the theory inside this course. So, I mean, these are just simple considerations but just to give you an idea of the usefulness of being able to consider models of theories not just in sets but in arbitrary categories. But of course... Jointly conservative? Jointly conservative means that all together they reflect isomorphisms. All together. So not just one single but to consider all of them so it means that if the image of an arrow is a nice morphism through all of them then they are amazing. Excuse me, in this long and impressive list there is a very important notion which isn't in the list. Namely, how shall I think? Using your terminology Cartesian theories. No, Cartesian theories are here. Cartesian theories. So, things which are definable finite limits. Yeah, yeah. That's... Oh, yeah, oh, because also finite limits. Final limits. Yeah. I have these ideas. Cartesian theories are theories. I... My French is the stickter. No, it's okay. It's terrible. So, these formulas are required to be Cartesian relative to the theory. It means that they are built from atomic formulas by only using finite conjunctions and existential quantifications to prove of the unique. Prove of the unique. I told it, but I did it right. Prove of the unique. Prove of the unique in the theory. It doesn't fit in this... So, once you have... It's not syntactical. No, it is proof theory. Oh, that's why I said it... No, I mean it is proof theory. You need to refer to... I mean you don't have a completely syntactic... I mean... No, Orn, of course, is intrinsic. I agree. Yes, it's weaker, of course. For instance, the theory of categories, the theory of small categories, if you want to formalize it, you will have the Cartesian theory, because if you want to formalize the composition of two arrows, since the composition is not always defined, because it's always defined when the co-domain of the first coincides with the domain of the second, the way to formalize is to use a ternary predicate. And so a ternary predicate for composition, which is something like that, predicate is a synonym for relation symbol. So, this is a ternary relation symbol and which expresses composition, so the meaning of that will be that h is given by this composition. And, of course, you will have that if co-dom of g then there exists an h, such that this. One moment. No, because you cannot write it. Because you cannot write here the unique. You have to write another axiom that says that it is unique. No, no. You have to write this. See, c f g h prime. No, no, you cannot write together. So, you see, this is a Cartesian theory. Now, I have given the examples, so now we can give the description of syntactic category. Yes. No, it is not an atomic formula, but it is incoherent logic. It is incoherent logic, because you have the false, and you can write 0 equal 1 implies false. So, it is incoherent logic. It is a coherent field. You can't formalize it as a coherent field. Because you can say that x plus 1 invertible implies x invertible or y invertible. So, and this, you see, you can do in incoherent logic. So, this is the notion of localism. You can formalize it as a coherent field. Ok. In which Yeah, I mean, it is a theory, in my sense, provided that every time that you have no, it is lightly, I mean, it's not defined exactly in the same way as the others. Yeah, but I mean, it's a standard. It's like that. Yeah, yeah, yeah. But what he meant was that it was not the form of the formulas were not enough to specify the fragment. That is what he was saying. So, I take this remark. It's correct. But, I mean, in practice it's not a problem. Ok. So, here probably we can leave. Ok. So, notion of syntactic category. So, as I have said, you have one notion of syntactic category for basically each of the fragments that I have written there. Now, just for the purpose of illustration I will give the definition for the geometric fragment and then I will say how to modify it to get the one for regular Cartesian and coherent. Ok. So, syntactic category for a geometric theory because this as we have said there it gives representability. And so it's very important because basically it shows that there is one place where the syntax and the semantics of the theory meet together and which generates all the other models because as we shall see from the way this equivalence is defined in this universal model under a functor preserving the appropriate structure. Ok. So, let's give this definition syntactic category for a geometric theory. So, suppose that t is geometric over a signature sigma. So, to define this syntactic category we have to specify objects and the arrows. Now, the objects are formulas, geometric formulas over the signature of the theory in a given context. So, we consider both the context and the formula together to form an object. So, in particular if I change the context I will have a different object. It's very important to remark this and I want to consider these things up to renaming equivalence. Just change the name of the variables I don't want the object to change. But, ok. These are my objects and what are the arrows? Well, the arrows are this is lightly more sophisticated. You have to take all formulas geometric formulas in the context here you suppose for simplicity, thanks to renaming you can suppose that x and y are disjoint. So, you take all the formulas here which are provably functional from the domain to the co-domain. So, what does it mean? It's provably functional and you don't actually you take equivalence classes of such formulas for technical reasons mod you know the relationship that identifies two formulas over the same context if they are provably equivalent in the theory. Ok, so you will take equivalence classes of such formulas. These formulas as I was saying they are required to be to this. What does it mean? Well, in order to understand what's going on here you have to think at the semantic level. What you want is that when you take a certain model m here you will be able to interpret this as certain sub-objects. So, let's take the domain of this sub-object. It will be that. And in the same way you will have sub-object on the other side and what you want is an arrow here. So, you want basically to put some conditions at the proof theoretic level at the syntactic level which ensure that when you take the interpretation of this formula which will be you see this formula is in these two variables. So, its interpretation will be sub-object of the product of this and this. So, it will be a relation in general. But you want this relation to come from an arrow. And here are the conditions that one puts in order for this to happen. So, first two conditions is the fact that if you have something in the domain there should be associated to it through this function. Let's say. So, it means that you require this. First axiom. You put the functionality axioms just written in in sequence forms. So, this is the first. Then of course you want that if something is in this relation then it should project on one hand to the domain on the other hand to the co-domain and so it should this should be true. And then the third axiom is the one that characterize the real functionality. The fact that I cannot assign to an element of the domain two different elements. And so it means that if I consider this and at the same time here I don't change the X I just make a substitution I want that this falls. So, I require that all these sequences should be provable in T, in my theory. That's it. By soundness the famous soundness that we talked about earlier in fact you can easily prove that this, the probability of this sequence ensure that when you take the interpretation you really have an arrow here in your category. So it is completely straightforward to define identities and composition in this category so I will not do it I think it's pretty clear that you have a category. And basically it's also not hard to see that this category is geometric because basically you see all the here you really have the unification between logic and geometry that Grotendik was between logic and category theorem that Grotendik was alluding in his text called SMI because you see the logical operations transform into categorical operations you see for instance the disjunctions become unions of sub-objects the intersection the conjunctions become intersections of sub-objects so you see that there is this unification between the logic and the category theory that is magnificently realized by this notion of syntactic category which is due to the Montreal school of categorical logic which was active in the 70s. So syntactic categories of course are important also for building classifying opposites probably the Joyal will tell me maybe it was the motivation for constructing classifying opposites to define syntactic categories or maybe not maybe one did that I mean the people that did this were mostly Makai and Reyes that published this in their book First Order Categorical Logic 1977 even though of course they have not been the only one to work on this subject back in the 70s ok my category see where it will be geometric I mean I build it and then no no it's over there yeah it is independent yes so it means category of t that's the meaning so it's like saying I take the category of t yeah exactly ok so yeah ah yes of course because you need the probability in the theory ah no no it's completely intrinsic yeah of course you can also do ah yes you can always take the empty theory if you want you will probably talk about that tomorrow no axioms yes the empty theory that's how I call it no empty in the sense that of course you always have the axioms in the logic this of course you always have empty in the sense that you don't add anything yeah yeah of course that is always present you cannot eliminate for sure yeah ok yeah no this is interesting in fact yes because the objects you see for defining you don't need to invoke the notion of probability in the theory while so a syntactic category can be seen as really a structure presented by generators and relations if you want so you see generators can be seen as the things that the language over the signature of the theory is able to express and the relations that can be seen as the axioms of the theory or more generally whatever is provable in the theory from the from the definition you can see it easily ok so now yes yeah I change that I make a substitution I introduce the farther so here I actually have over these three variables and yeah the uniqueness of of the element we shouldn't speak of elements but just to understand the element that corresponds to a given x through this function ok so now we come back to this equivalence now in our context we are working in the geometric context here we have geom so geometric functions here and so now you will wonder and now that we have defined this you might wonder how this is defined now it is defined in a most natural way well let's start in one direction because we can do it immediately for the other direction I will need to give a description of the universal model of the theory inside city now in one direction you can go in this way you start with a model and how would you define the functor is very simple what would you associate to this formula well the most obvious thing to do is to associate just the interpretation of the formula in the given model right so and it makes sense to think that all the information about the model is actually contained in the functor associated to it we shall see that this is the case through the the identification of the universal model of the theory lying in this syntactic category how to define it well we call it m t universal model of t well these are the arrows the equivalence classes of formulas that are probably functional these are the arrows it is straight forward if I have an arrow here by what I have said you see you will get an arrow here so it is you see if I have an arrow in the syntactic category it will be this and this I mean we defined arrows in the syntactic category in order for this to be to give us a functor we define them in order to obtain this ok of course it is well you get an essentially small category so it is fine yes it is not small actually but it is essentially small so it is enough it is equivalent to a small category so it is the notion of well powered category so all the sub-object lattices are essentially small and so it is you didn't hear yes yes yes it means that it is equivalent to a small category a small means that it is based on a set of course this you only need when you have geometric theories because of the infinitary nature in the other cases it is always straight forward the problem doesn't pose ok doesn't pose ok so we have this functor and now to go in the other direction you see I would like starting from a functor to produce a model a natural way to do this is to apply the functor to a certain model we shall apply the functor precisely to this model so we are now going to define this model so in order to define a sigma structure what we have to do is we have to specify where the sorts function symbols and relation symbols go so as sort A I am going to interpret it as I take a variable of sort A and I take just the formula true function symbols well imagine that you have a function symbol which takes as input sorts of this kind and so here I take this and this is isomorphic and I have to define to like that and what do I do? well I simply take this so you see this is certainly functional because I need more functional than that I take a function symbol so it is certainly functional so it is an arrow it is an arrow in the syntactic category and this is the interpretation of my function symbol relation symbols it is straight yes completely completely so you see this result about this representability is at the same time trivial and extremely important so sometimes totalogical things can be yeah is an example so I would say that it is a miracle in the style that you were referring to because to have a representability of a function is always a very good news it means that all the information about the function is contained in just one is condensed in one object, one element and so you see the function in general can contain a lot of information imagine here we have the semantics to function of the theory so you see it contains a lot of information the fact that you know that everything is actually condensed in this universal model and the syntactic category is really striking and it brings symmetry to the whole theory because it shows that basically the center of symmetry from which you should look at the theory is not the set base models as one is used to in classical logic but it is really these syntactic models ok so just to complete relation symbols well it will be a sub-object of this thing so if you consider just same context and you just put R you realize easily that this is a sub-object and so this will give the interpretation of relation symbols and now it is equally tautological to verify that a geometric sequence is valid in this model if and only if it is provable in the theory I've write it here because it's very important valid in nc and only if it is provable sigma I mean geometric sequence over the language of the theory provable in t and here you can immediately see it because of the way mt is constructed it is really a generalization for those of you who know for a propositional theory ok so now that we have the definition of this universal model we can complete the description of this equivalence of categories because we can also specify how it works in the other direction and in the other direction what it does is simply to take f and to send it to f applied to this universal model and here we use the fact that since f is geometric and this is a model by applying f to this model we get a model of the theory so we are done and we have already at the level of syntactic categories a representative result a factorial understanding of models and now it remains to bring all of this at the level of toposis because this will add additional tools that will be very interesting to use that are not available for just classical syntactic categories so just a word about the variance of all of this for the other fragments of logic this is interesting because they will give us alternative ways of creating the classifying topos of our theory now how do you modify well just in terms of notation so if you have a regular theory t well let's start with Cartesian theories you denote the Cartesian syntactic category of the theory like this with cart it means so it's exactly the same construction except that instead of taking geometric formulas you take exactly the same nothing changes same thing for regular theories you have reg and here just to specify for coherent I will write like that and for geometric I will simply since this is the fragment that is of most interest to us I will not put any adjective here I just write city ok now classifying toposis so far we have considered regular coherent geometric functions but if we want to work with growth and diktoposis as it has already been said the right notion of morphism to consider is the notion of geometric morphism and we already remarked that inverse image functions of geometric morphisms are geometric we we work in the fragment of geometric logic ok so so what we can do is we can consider this pseudo factor that assigns to each growth and diktoposis the category of models of the theory inside it and if we change the topos by a morphism we will have a functor going in the other direction given by the action of this and the fundamental theorem due to the Montreal School of Categorical Logic in the 70s is that every geometric theories every geometric theory has a classifying topos which is defined as a topos that represents this pseudo factor so it means that the geometric morphisms from any growth and diktopos to it so I will call it ET the classifying topos are in categorical equivalence with the models of the theory in E naturally in E so this is the theorem and what is interesting about this theorem is that the construction as the construction of universal models and the syntactic categories is completely straight forward in the sense that it is completely canonical what you do I mean you need a site because growth and diktoposis you have to define them or systems of generators but we shall do it with sites basically this classifying topos is built by equipping this syntactic category that we have just built with a particular growth and diktopology which is basically the canonical topology on this category and which admits a completely explicit description so you basically you can give the description of this topology so suppose that you have a seed in your syntactic category you have to decide when it is covering for the topology so you suppose that you have things like this so you say that this is covering on this object if and only if the sequence this sequence here which says basically intuitively says that any element here should that this should be epimorphic I mean that every element here should come from some element that is either here either here or here this sequence provable in the theory so you see it is very very plural definition of topology and you can prove that this is the classifying topos and in order to prove this one uses a result that probably Andrei will mention tomorrow is the fact that the geometric morphisms from any topos to a a topos of sheets on a site the same thing as the flat j continuous factors on c with values t so this result is really fundamental and sometimes I mean the way people refer to it is diakonescu equivalence because diakonescu proved the most general version of this I think it's essentially the SGA yeah it is but then it proved I think a relative version of that so it has been if you look at the literature they use this term so I have stick to it but yes I know that it is already there and now if we apply this to our context now I'm not going to tell you what the flat factors are because we don't even need and in the case of a Cartesian category flat is the same thing as Cartesian so we are going to apply this to that and so we get that the geometric morphisms from a topos e to this are the same thing as the Cartesian j t continuous factors from c t to e in j t continuous it means that it sends j t covering sieves to etymorphic families and you realize easily that this is the same thing as the geometric factor exactly the same thing and so from this you get to the models as we saw about and so this is how it works and interestingly there is a converse direction to this theorem is the fact that any groten diktopus is the classifying topos of something and the way it works is you have to choose a site so so any groten diktopus we can represent it like this for a site cj and basically you can attach to this a theory a geometric theory which informatizes precisely the j continuous flat factors on c and so because of this result the the topos we started with will be the classifying topos of this theory you might wonder over which kind of signature this theory is formulated while it is formulated over another kind of a total logical construction I mean the internal signature of a category in our case the category c it is a signature which contains one sort for each object of c and one function symbol for each arrow and one relation symbol for each sub-object in c so this kind of signature is something very useful because it allows to argue in a set theoretic way about things happening in your category so this is not relevant for this particular purposes but it is just something which is useful to know when people say that toposes behave as generalized universes of sets well it means that essentially when you consider this internal language of a topos defined in this way you can reformulate many properties of your topos in terms of sequence formulated by using this formal language which basically you can prove by using a set theoretic intuition a simple example is imagine you want to prove that f is a monomorphism if you were in set how would you write the condition you would write this condition injectivity you would write that and it turns out that using this language just do like that and you have a sequence and you can formulate the condition of your arrow to be a monomorphism by saying that this sequence is satisfied in a sort of topological structure which consists precisely in interpreting this signature in the obvious way so you interpret these sorts as the corresponding objects the function symbols and similarly for and so is satisfied in the internal in the topological structure and so you see this is just one example in general you can write very complicated things here you can argue set theoretically imagining that you are in a set theoretic foundation and you will be able to say a lot of things to argue inside your topos as you were working in a set theoretic context so of course you have to pay attention to some things you only have bounded the quantifications you cannot use lower excluded middle or axiom of choice of these things but provided that you stick to these good rules you will not have any problems so this is the essence of this construction we minute ok yes yeah yeah for sure it's very practical ok so now I don't have much more time I would like just to mention that as we had different versions of this syntactic categories we also have this give us different ways of building the classifying topos so I gave this standard construction but you can also I mean for ok let's just erase this or maybe that t Cartesian then you will have that the classifying topos will be represented just like that off if t is regular you have an alternative way to build j, t, reg this is called the regular topology so it is a topology defined by saying that a covering c is one that contains a cover a cover is an arrow that has the identity as its image so it's defined in this way so it's not the same thing as this topology that we defined so you see we have genuinely different sites of definition for the same topos let's conclude coherent then you have and here you will have a finite type topology which is called the coherent topology on our syntactic category this is characterized by the fact that a c is covering if and only if it contains a finite covering family covering in the sense of this category here ok so as you can see you see you have already imagine if you have a Cartesian category you have four different ways of building the classifying topos because of course you always have the this is the same thing this is shift with trivial topology because here you don't need any topology because the topology becomes essential from regular on but for Cartesian you only need that the functor preserves finite limits and this you have it automatically because for this theorem you see here if I have a Cartesian category and I take the trivial topology I just get Cartesian functor well algebraic theories there is an alternative construction of the classifying topos that you can do but the most natural thing to do is still to consider it as a Cartesian theory because otherwise you will not end up with a Cartesian category and with a syntactic category that are a bit more difficult to compute but Blasen Shedrov they have they have given an alternative construction which was not just for algebraic theories but for universal orn theories but anyway ok so and just to conclude for today I would like to make just a couple of remarks about the usefulness of having different sites of definition for the same topos so here we have seen that we have this different possibilities of building the classifying topos and there is a way to make a good use of this to extract information about this theory and a natural question that one could pose is that we raised at the very beginning is imagine that for instance I have a regular theory but I decide to consider it as a coherent theory or I decide to consider it as a geometric theory so of course I will have in principle different notions of probability for sequence which are regular because when I have a regular sequence I can also consider it as a coherent sequence and so I will have a notion of probability of it relative to this fragment and also relative to this fragment and of course relative to this fragment as well so in principle these things will be different and in fact they are not and there is a very simple way to see this independence by just comparing these different representations of the classifying topos you know that basically when you decide to consider a theory in a fragment or in another there is something invariant that you can immediately point out the models are clearly the same and so the classifying topos will be the same but it will have two different representations so imagine for instance that you consider a regular theory both in regular logic and in coherent logic this and you can exploit this to deduce that the two notions of probability are the same how you do? Well you observe that here you have a universal model of the theory built in the fragment of regular logic and here you have a universal model of it written in the fragment of coherent logic but these two models correspond to each other under this equivalence now since they correspond to each other under this equivalence validity in one will be the same thing as validity in the other but validity in one as we have seen corresponds to probability in the corresponding fragment so here we shall have probability in regular logic so you see they will be necessarily equivalent with each other because they are connected by this bridge so you see it's very easy you don't even need to look at how the rules of inference are defined you rely on soundness of course but you see it's very simple yeah yes yes exactly exactly exactly exactly that's precisely the meaning yes, exactly yes, precisely and you can see it very simply like that other things that you can see so you see this is just considering an invariant which is the notion of universal model of a theory from the two different points of view basically just to write you see the complete bridge you have so you see here you have one invariant which is the notion of universal model and you look at this from the two different points of view here you find your empty because it lives in the side actually and here also and no, no, no here you of course you can say this refers to sequence written in regular logic because otherwise it would only be here and not there so you want to compare things this is bigger than that so you take things that come from there now just another example so we talked about a coherent logic a coherent logic is something that musicians are particularly familiar with because it is the finite fragment of geometric logic and it is characterized by the fact that it satisfies a form of compactness compactness theorem plays an important role in classical model theory and it is certainly not satisfied by a general geometric theory but if the theory is coherent then you have this compactness theorem which amounts precisely to saying that if you have a coherent formula in the language of theory and if this entails in a given context a family of a possibly infinitary family of coherent formulas here then there exists a finite subset subset of i and much that this entails a finite sub disjunction of them you see this is a compactness compactness of course it doesn't hold for a geometric theorem but it holds for and now the interesting thing is that this also you can read basically in one line from the form of the classifying topos how to do? Well here I replace this with I decide to consider my theory both in coherent logic and in geometric logic so to exploit the existence of these two different representations so I do this and I will have my two sides and now no longer consider this as an invariant I change my invariant I consider the notion of compact object in a topos a compact object in a topos is just an object such that every covering of it contains a finite sub covering so if I look at this notion so imagine I start with something that comes from this side so and notice that this condition here it is not a condition that I can formulate in this side because here I only have this finite type topology so I cannot formulate that but I can do it here and how to connect? Well through the topos how to do? Well I take that I send it here through the you need embedding and it comes out by a very simple calculation on sites that since this topology is a finetary then this object is compact it's just a very trivial thing that you can verify so it is the geometry of the site that implies the capacity there and once you have that of course this is an invariant so you transfer it here and then you read it off there so you see another very simple way to deduce this and in this way you can you see this is just to illustrate the fact that even innocent variations like this can lead to straight forward proofs of results that would not be otherwise trivial anyway now I think I had to stop so I today I wanted to talk about the duality for sub-toposis of the classifying topos unfortunately I had too many questions and I couldn't but tomorrow we will start with that ok thanks