 Matthias Utsler from the University of Augsburg is going to talk about gluing classifying toposes. Thank you. Thank you. So first of all, it's a great honor to give a talk here at this conference and thank you to the organizers, to the organizers and especially to Olivia for organizing this conference. Yes, my talk will be about gluing classifying toposes and let me say right away, sorry. So by classifying toposes, I always mean four geometric theories. So actually I could also call the talk gluing geometric theories. And by gluing, I do not mean the classical art in gluing construction, but I mean gluing along open subsets. So like gluing where I have overlaps of the toposes I want to do, and these overlaps are open subtropuses. So yeah, I'll take the perspective in this talk that the only nice way to present a quote in the topos is by a geometric theory as opposed to the previous talk. Of course, it's not the only nice way, but I take the perspective that I want to know a geometric theory for epicotentic topos that I need. And of course, if I have a geometric theory, then the topos is presented by the theory in the sense that it is the classifying topos for the theory T. And if I have this perspective, then there's a natural question how to calculate with these presentations of topos is how to calculate with theories. Okay, glad I mean what constructions of toposes do we know. And in particular, I'm thinking of toposes as generalized spaces or constructions of spaces to know and how to implement them in this presentations as theories. Perspective. And of course, for every topos there is a theory which is justified. We can always take the theory of continuous factors. But we want concise geometric theories, otherwise it knows a nice presentation of my topos. So it's known that for some toposes there's a really nice concise geometric theory, which is much shorter to write down than the definition of the topos by a site. And it feels much more elementary. And I think this is a good thing to have. So we will always look for concise geometric theories, at least as concise as can be. So here's the first example. What if I start with two toposes, classifying theories T1 and T2, and they take their product. Product in the sense of toposes or spaces, not in the sense of categories. The product category would be the co-product topos, but I mean the product topos here. So I really imagine this as the product space. And in this case, we can be the answer quite easily. This is the classifying topos for a new theory, namely the theory where we have just took all the stuff from T1 and all the stuff from T2 and put it together. So all the sorts of T1 and T2, all the symbols of the axis. So the disjoint union of theories is the product of toposes. And to see this, it's not hard. If we start from the right, then we have to see that geometric morphisms, which I just write wrong now, geometric morphisms from the test topos T2, the classifying topos of T1 plus T2. Well, these are models of the theory T1 plus T2. But the model of the theory T1 plus T2, since T1 and T2 have nothing in common, there's no connection. It's just a model of T1 and a model of T2 because they have to interpret every sort of T1, every sort of T2 and so on. So this is just a pair of models, T1 model in T and T2 model in T. So to be a bit more precise, I should write equivalency. And yeah, of course, this is the universal property of the product. So this is quite obvious. And so we've seen our first construction with theories. But now let's turn to the co-product of toposes, which is the simplest case of bluing, right? If they have two toposes and they glue them trivially, not at all. That means they take the co-product. And well, in this case, what is the new classified theory? I would say it's not so obvious. So it has something to do with T1 and T2. But a model of this new theory is a morphism into this disjoint union. So it does not contain a model of T1 or a model of T2, but only models of T1 and T2 on some parts of the domain topos. So this is not as obvious as before. And the more general question, which would be the central question in this talk is if I have a topos E and I can cover it by toposes where I know a classified theory, where I know a syntactic presentation. And I should say that these are supposed to be open subtoposes. So if I can cover a topos by open subtoposes and a syntactic presentation for these open subtoposes, how do we construct a syntactic presentation for the whole topos E? This is our central question and I will give an answer in the end. But before we have to we have to see some constructions with our theories because, well, it won't be so simple but what's written here, but it will still be of a kind where I would say that in many applications, if you really calculate the construction, then what comes out is again a really concise theory if the TVIs you started with were concise genetic theories. And here's a natural example, which was the motivation for this question. Namely, if I take a scheme, a scheme X, always admits a cover by open sub schemes, which are fine. So let's fix such a cover. X is the union of the affine schemes spec AI. Then we would like to know what the big risky topos of X classifies. And if in the in the fine case, spec AI, the big risky topos of spec AI, and they put finitely presented here because we have to use the side work on the steeps of locally final presentation over spec AI. This big risky topos classifies local AI algebras. And I mean, it depends on the ring AI, but if I know a nice maybe finite presentation of the ring AI, maybe not finite, then I have a nice, concise formulation of this geometric theory. But what about the general non-northine case? And it is true that this cover by affine schemes induces a cover of the big risky topos. But we need we need some machinery to construct the theory classified by the basis of X. Okay, so we'll have to ask next the question, what is our preferred syntactic presentation of geometric morphisms? Because you see, as you might already have guessed, this construction here will not only depend on the theories TI, but there has to be some gluing data of the TI in the sense that we should also look at the intersections of these open subtropuses. And we have to need we need syntactic presentations of these intersections and of the inclusions of the embeddings of these intersections into the set TI. So let's just ask more generally, what about a general geometric morphism, how to present it? And my answer here in this talk is, we should present it by extensions of geometric theories. And by these extensions, I mean, that we're not only allowed to add axioms, but we can add sorts, relation and function symbols, and also axioms. Of course, all these are optional. And this is the definition I want to work with. The same thing is called expansions in Olivia's book, The Resides Topos. I think for me, the name extension is a bit more natural. Okay, and I wrote about notation. So when I can add arbitrary things, but I have to keep the things that were present in the base theory TI, then I could write TI subset, sense of sorts symbols and axioms are subsets of those of some theory TI prime. But actually, I want to look at the extension as a first class object. So the extension is something I really care about, not just the theory TI and TI prime. So it will rather write this as TI prime can be decomposed into TI plus some extension. So I give the extension itself, which contains some sorts and axioms, but it does not have to be a theory in itself, because the symbols and axioms can also use sorts from TI. So this extension gets a symbol E. And I also think about these extensions as kind of morphisms, of course. So I sometimes write an arrow from T to T plus E. And this corresponds to the fact that, of course, if I have a T plus E model in a topos, in any topos E, then I can forget the E part of the model. So I have a function to T models. And this, we could call this U, we could sort of get the answer in the sense. And of course, this is induced by a geometric morphism, which we might want to call pi, because it's kind of a protection from the generalized space of all T plus E models to the generalized space of all T models. Okay. And while I do sometimes write an arrow, I will stress that I do not define a notion of morphism of theories here. As opposed to two other things, which are more general. Firstly, morphisms of theories in the sense of figures, which here you are allowed to give a map from the sorts of the first theory to the sorts of the second theory. And similarly for the symbols and axioms, which is still completely syntactical, one might say, but it allows to identify two sorts, for example. And I do not want to allow this in my setup here, because I want to be really, really explicit about my theories. I really want to think about them as syntactical objects. And I do not want to identify things if I am in a situation where I want to identify two sorts, then I will instead introduce a bijection between these sorts, which is something I can do, it's an extension. Okay, so this is a more general notion. And another more general notion is interpretations in the sense of a little terminal. So it's described in the book, theory science toposys. And their interpretation of one theory in another theory is defined to be so for geometric theories, geometric factor between the classifying, between the syntactic sets. Geometric. So these are totally fine notions, of course. And the only reason I look at it in a different way is because I want to be really explicit, how I can build up these. So these extensions are about building up complex theories from simple theories. And in particular, these notions here are appropriate to turn the collection of theories into a category. It's both kind of morphism of categories, while I do not want to talk about the all extensions that turn one T into some T prime. This is not, in my view, here, a useful question. Of course, I can ask about all extensions of T, which happen to make T equivalent to T prime. But usually you will have a T and add something to it and then give the result to name T prime. Okay. So I think I really have to hurry up. Sorry. Here are examples of extensions. For example, a quotient of a theory is always an extension. I write Q for a quotient, where Q is just some axioms. So new axioms is an extension and it's called a quotient. And a concrete example to maybe keep in mind is that we start with the theory of rings and add an R algebra structure to it for some external ring R, this typical example of an extension of theories for me. And of course I can always add two theories that have nothing to do with each other. I can also become UT2 as an extension of T1. And here's an important theory about these extensions, namely it says that they actually suffice to present any geometric morphism. So every geometric morphism over some topos with a given classified theory is of the form of such a forgetful geometric morphism for some extension. And almost even more important, this geometric morphism might be localic, but might even be an embedding. And in these cases, I get that E is a localic extension, localic, or respectively a quotient. So quotient means only add axioms, so no new sorts and no new symbols. And localic just means no new sorts. And I think that this three-step generalization or specialization from general extensions to localic extensions to quotients, this seems quite important to me. Maybe we'll show up again in the talk. Okay, let's now be optimistic and just start to build our theory T for the topos E, which is covered by these classifying toposes. Yeah, let's see what we have. So these are open. And let's give them the name EI. And that they're open means it is the open subtopos corresponding to some UI where UI are subterminal objects and opens of E are subterminal objects. And so we have that the terminal object of E is the join of these UI as subobjects of the German object in E. And from this, we can see that our theory T must have propositions for these UI for these subterminal objects, because these are internal truth values. So a classified theory for E must have known that it must be able to express these these truth values by some closed geometric formulas. And we can just as well assume that they are actual proposition symbols. So we have these proposition symbols PI here. And since the join of the UI is the whole topos, we know that this has to be derivable. So the disfunction of the PI has to be true in our theory. Okay, this is something we can start with with our theory T. But what else? Maybe is it true that the theory T also contains all the theories TI? And here's the reason why this is not true in general. Namely, there is no canonical TI model in all of E, only in a part of E, namely the open part EI. So TI cannot contain no, T cannot contain all of the theory TI. Instead, it should contain something. It should only contain something which we might want to call TI, if TI is true, right? So wherever or internally speaking, if TI is true, we want to have a model of TI. And this informal notion is made precise here. I call it conditional extension. If we have an extension E of some theory T plus PI, where PI, it was a proposition symbol above. Now I say it can be closed geometric formula. So any formula of TI is good enough. So if we have an extension E of this, then we can form T plus E over PI, which we should read as T plus E if PI is true. So a model of this should be a model of T plus a model extension along the theory extension E, but only wherever the T model satisfies the formula. E model if PI is true. And here's the construction E over PI, this conditionalized extension has the source of E. We can actually add the source, but we have to make sure that the sorts are no superfluous data. So if PI is false, then we actually don't want to add the sort. And for this, we force the sort to be empty in case PI is true. That is, we add also an axiom. If there exists something in the sort A with no special condition, then PI must be true. And quite similarly for relation symbols, if there exists something which makes the relation true, then this can only be the case if PI is true. And for the axioms of E, it's quite simple. If we would want to add an axiom, psi entails chi in some context, then we instead add the axiom psi and phi. And a short remark. I didn't mention function symbols here. That's because they need a bit more work. But keep in mind that function symbols are, in a sense, not really needed in geometric theories. You can always transform them into relation symbols. You need a few more axioms to say that your relation symbols are functional, but they're not really needed. But it can be done. It's just a bit more work. Okay. So here are examples. And we can now write down a theory just specified by the district union of two classifying topics. Namely, we start with two proposition symbols for these two open parts, which are also closed. So we take proposition symbols P1 and P2. Now, we ask that one of them is true. Then we also ask that they are not both true because there's no intersection. And now, up to now, we have only presented a discrete two point space. But now we can add T1 over the one proposition symbol and T2 over the other. This is an example which is also included in the elephant by Tronson. But it is not with this construction. So this is just a first example how to apply this construction here. And the second example, the Sipinski cone, or scone for short, is you take a topos and you add one closed point to this topos, i.e., you make your topos in an open part of a new topos where you have one extra point. And this is even simpler. We only need one proposition symbol P. And then we add T over this P. Okay. Yeah, there will be a bonus question. What does this thing in general actually present? What does the topos look like? But I don't have time now because we want to come to this general situation where E is covered by open subtoposis EI. And there we need a presentation of this whole system of subtoposis. They are iterated intersections. And for this, well, you could think we start with theories T1, T2, T3 to present these basic open subtoposis here. But instead, let's start with one theory T0 and already take extensions of this T0 to present the open subtoposis. And this is just to be able to take a common part of all the theories I want to talk about out. For example, in the case of Siriski topos, I want to take out of the whole construct in the common part that I want to talk about the local ring. And I only want to say that the extra structure of the local ring has to be viewed in the proper way. So this goes on to the right somehow. So something like E12 will appear here, which is the extension that turns the previous things into a presentation of E1 intersected with E2. So in general, we need a definition of a system of theory extensions. And because I don't think of theories as objects of a category, I don't want to call diagram of the extensions, but it feels like a diagram of theory extensions. So a system of theory extensions is something that looks like this. I have a base theory T0. And then I add the sum of a lot of other things to it. And these things can have complicated interdependencies. So they are indexed by not necessarily a natural number or something, but they are indexed by any partial order. And to be clear on this EI, this EI is an extension of everything before. So the base theory plus for all j smaller than i dj. So this is a notion of a system of theory extensions. And then we can define a notion of a system of presentations of such subtopuses. And here's the theorem. I think you can say that in one or two minutes. Yes. So the theorem starts with we have a base theory T0. And we take any model of the base theory in our topos. And of course, we assume as before that topos is covered by open subtopuses. And that we have such a system of presentations of the intersections of these open subtopsis. So this index S here is supposed to be a finite subset of the index z i. And I denote it like this. So this is supposed to just mean S is a finite inhabited subset. So for all these intersections, we need a presentation. Excuse me, this is wrong. I wanted to write the presentation consists of a theory extension and a model extension, which turns all the previous parts of models into a classifier and the universal model in this interactive topos. Okay. And then you classify something which I can write down like this. I can be written tiny bit shorter if you put the proposition symbols here in the base theory, but I can't talk about that now. So we take the base theory, we add proposition symbols, which we knew we had to add. We add that one of them is true and no two different ones can be true together. And then we can add all these extensions over the appropriate conjunction of these proposition symbols. So the principle is that we have to take our system of presentations and then conditionalize all these theory extensions that appear here. We conditionalize them over these proposition symbols. And then we can add this. And there's also a similar looking description of the universal model. And yes, this is then our classified theory, which presents the topos E. The only thing that, so no time for a proof sketch, but application to the risky topos. Let me just say it in one sentence. So if I have the scheme X, then the big seriously topos is actually covered by these big seriously toposes of the fine schemes. And then the big risky topos classify something. So this might still look somehow a bit scary because I have to take all finite intersections of these open subtoposes, but it turns out since I can here take local extensions of the base theory of local rings, because I only have to extend it to the theory of local AI algebras. And this AI algebra structure is a local, is a local extension of the theory of local rings. And this means that they only have to look at the intersections of every two open subtoposes. And yeah, it turns out that the big first seriously topos of non defined scheme justifies local rings, which are an AI algebra for some I. So with AI algebra structure, and we write for some I, which is supposed to mean that it has to be a satisfaction for some I. And if it has a satisfaction for two I and J, then they have to be compatible and a bit more. But there's no really short way to say it's a theory of something very short. Okay, thanks for your attention. Thanks a lot.