 Thank you very much for the introduction. Also thanks for organizing this conference. I'm really having the time of my life here. It's really good. And thank you for giving me the opportunity to speak here. So I want to start with a question of Stephen Gubkin, who posted on my overflow. He said in the middle, it is convenient to reason about toposes and their own internal logic. Has there been much done about the internal logic of the gross or risky topos, or what the logic of the topos required too much communicative algebra to feel like logic? I think that Stephen has a very good point here. I mean, you see the basic objects of much of algebraic geometries are schemes. And schemes are locally ringed spaces, which are locally isomorphic to the spectrum of a communicative ring. So it shouldn't come as any surprise that many notions in algebraic geometry are like sheafy notions or globalized notions of ordinary notions in communicative algebra. And also, you know the business with the proofs in algebraic geometry. Often they go like blah, blah, blah. Therefore, without loss of generality, we may assume that the scheme is affine. And then it's a simple consequence of the following theme of a communicative algebra. Other proofs go like blah, blah, blah. Therefore, it suffices to check the condition of stocks. They are trivial because of communicative algebra. Okay, and I think topper's logic has a contribution to make there. The starting point is to realize that any scheme has its structure sheaf OX, and this is of course a sheaf, a sheaf of rings. But if you switch to the internal logic of the Piotr Zeliski toppers, the toppers of sheafs on X, this complicated sheaf of rings will look just like an ordinary plain old ring, non-sheafy, okay? So and if you have a sheaf of X modules on your scheme, then from the internal point of view, this will simply look like an ordinary module on that ordinary ring, okay? This is the starting point. This was already noticed in the 70s when the internal language was developed by several people. And so I quite want to see how far we can bridge, how far we can push the theory. So what we want to do is like build a dictionary between external notions and internal notions. Build this dictionary once and then we can use it as often as we want. So I just want to give you a few of those items. For instance, a sheaf of finite type, okay? So externally, a sheaf given on your space X, a sheaf of modules is a finite type if and only if from the internal point of view, it simply looks like an ordinary, finitely generated module, okay? Okay, we can push it forward. For instance, a sheaf which is externally finite locally free. This is precisely the same thing as from the internal point of view, an ordinary module, which really is finitely free, not locally but really globally from the internal point of view, okay? So the internal language is a device which allows you to pretend that the base scheme you're working over is in fact the point, yeah? To reduce to the situational point on a single point. Okay, let's just do a few more examples. So for instance, maybe when you're a beginner and then you're learning algebraic geometry, then maybe you are scared of the definition of a tensor product of sheafs, tensor product of sheafs of modules. I mean, there's nothing scary about them in fact but when you're just learning it, then you might be scared. The problem is like the naive definition only gives you a pre-sheaf and then you have to sheafify. And maybe you're still in the face where you have anxious feelings about sheafification, okay? Okay, anyway, from the internal point of view, the tensor product of sheafs of modules is simply the ordinary tensor product which you have learned in your first semester in linear algebra or something, yeah? Okay. A particular nice example is the sheaf of rational functions, Kx. So this is a sheaf of rational functions and so if you know its definition, then you also know that it's a little bit delicate. In fact, there's a short paper by Stephen Kleinman titled Misconceptions About Kx where he lists like three definitions about Kx which are very commonly found in the literature and which are wrong. Which do not even define a pre-sheaf, for instance, yeah? Okay, so the definition is a little bit delicate but not from the internal point of view. From the internal point of view, you obtain the sheaf of rational functions very simply, namely, take Ox. Ox is just an ordinary ring from the internal point of view, yeah? Okay, and then take its total quotient ring. But if you don't have elements, how do you take the quotient quotient ring? Yeah, I mean you have to use the topos magic, yeah? So you use the internal topos language and there Ox is simply an ordinary ring and it has its sets of regular elements and then you localize it then. This is exactly the beauty that it becomes so easy in the internal language. Actually, there are other kinds of quotient ring considered by algebraics which are less often used but I saw in some references that they have some other ways to define it. It's not interesting the regular elements. Yeah, I mean you can localize at many sets, yeah? In fact, at any subset if you want to, yeah? I mean, there will always be a quotient ring but if you localize at the set of regular elements, then you will obtain the thing commonly called Kx. And with this, for instance, you can give a beautiful internal account, purely internal account of the basics of the theory of Cartier divisors, okay? Let me make one more remark. So we had in a Booth lecture the situation that we had a theorem which hold it if and only if the topos was Boolean. We have a similar situation here, namely, from the internal point of view, you can ask yourself, what is the Krull dimension of the ring Ox? Any ring has a Krull dimension, yeah? I mean, you have to be a little bit, I mean, you have to use a constructively sensible definition of the Krull dimension but such a definition exists thanks to several people, some of which are in the audience here, okay? And then you can use this definition and ask yourself, what's the Krull dimension of Ox? The answer is the Krull dimension of Ox is precisely the dimension of the scheme, the base scheme, okay? And then you have the following observation. The dimension of Ox, the Krull dimension of Ox is zero if and only if the scheme is zero dimensional if and only if the internal language is Boolean, okay? So in some sense, the ring Ox, the structure sheath controls the logic of the topos. I want to take a minute to explain why in some sense this talk is like a praise for Mike Schulman, you see? So there have been several lectures on the internal languages of topos this year and most of the time geometric logic was stressed which is very fine, yeah? But in fact, the topos can interpret more higher fragments of logic. They can interpret full first order logic, they can interpret higher order logic, they can also interpret dependent types which some of you might not know about but I promise to you, you're using dependent types all the time, yeah? And if you really want to import all of constructive mathematics into the topos setting you also need dependent types. Okay, but the thing why I want to praise Mike Schulman is because he made it possible to use unbounded quantification in the internal language. So unbounded quantification is when you say for all rings it should hold that, for all modules it should hold that, yeah? This is unbounded quantification and of course you need it all the time because you want to formulate universal properties, okay? And Mike Schulman wrote this little paper which you can totally understand even if you don't know what a stack is. We just give a small addition to the usual cryptocryptial semantics of the topos to be able to speak about unbounded quantification and locally internal categories in contrast to only internal categories, okay? Okay, so with this small extension we can really import all of constructive mathematics into a topos setting. Okay, and I want to give examples for this. Are there any questions up to this point? Okay, then let's have a basic example. So in your first semester in university you'll learn that if you have a short exact sequence of ordinary non-sheafy modules and you know that the two outer ones are finally degenerated then solve the middle one. This is a basic theorem and this basic theorem has an obvious proof and this obvious proof is constructive. Therefore, you can interpret this statement in the internal language of any topos and then it will automatically give rise to a more advanced statement, the statement which you see on the board about sheaves. Okay, okay, so this is a first example where the topos magic allows you to like prove a theorem once and then interpret it in many, many contexts at once without further work. One remark, so you might object that the ordinary proof of this statement is not difficult at all. You might remark that it's simply routine and that it's very easy to do. Okay, then you're right. In fact, you can, if you're familiar with the notation then you will probably be able to proof the statement at the bottom in like a minute, okay? But I argue that this is not a minute where I spent. It's a minute spent with like saying, okay, we have this generator on these open subsets. Well, the open subset is too big. I have to shrink it if necessary. I have to shrink it again and mm-hmm, mm-hmm, mm-hmm. It's routine, but it's like 60 seconds where you could have done something different, okay? And with a topos language you can, you have this 60 seconds for free, they're given to you, yeah? Because you simply look at the statement, realize because of the dictionary that it's the interpretation of the well-known statement of linear algebra and then you are done, okay? You're going from 60 seconds to one second. This is, I think, a great idea. Also, you gain conceptual understanding because you now really know where this sheaf theory comes from. Before that, you only had like a feeling but now you have like a formalized rigorous proof. A more advanced example is the following. So, take a scheme X, assume that it's reduced. Take a sheaf of all the modules of finite type over it. Then the statement is that this sheaf will always automatically be locally free on a dense open subset, okay? This is statement in the sheaf theory. So, for instance, Ravi Vakil in his excellent lecture notes in algebraic geometry says that this is an important hard exercise. It's an exercise with like a hint which goes over like half a page, yeah? Okay, but in fact, this is not a hard exercise at all. It's a trivial consequence of the constructor theorem of linear algebra at the top. In constructor of algebra, you know that any finitely generated vector space is not not free, does not not possess a basis. Okay, so constructor, you cannot prove that any finitely generated vector space really is free. You can only prove the slightly weaker statement and the not not translates to on a dense open subset. Okay? I really like this example because otherwise you would have to like follow through this hint which just goes over half a page and then do Nakayama once and do Yakanama twice and so on and so on. And in fact, it's a trivial consequence. And you can also prove, for example, the level of generic flatness in this, in this, in this, I mean, it's true in this form. Yeah. And there you have to work a little bit more in the usual way. So if you're an algebraic geometry, then anyway you have to do the usual for me because. Yeah, well, at some point in time, you will have to work. But the point is that you can shift lots of this work into work which has already been done in constructively linear algebra. Yeah? And you can automatically import these kinds of things. Any more questions at this point? Okay. Then let's continue the tour of the putesia-risky-topos of the scheme. I want to show you a curious property which the internal universe of the putesia-risky-topos of any scheme always has. Namely, so from the internal point of view, or X is a ring, yeah? And this ring has the following curious property. Any element which is not invertible is nile potent. Any element which is not invertible is nile potent. So or X is almost a field, yeah? If you were to quotient by the nile potent elements, you would have a field in this sense. So this is a statement which holds in full generality on a scheme, even if your base scheme is a base ring, for instance, is really a ring and not a field, yeah? It always holds. So this was already noticed in a special case by Mulvey in the 70s. And here you see Tierney commenting on it. He says, this is surely important for its precise significance is still somewhat obscure, as at the case with many such non-geometric formulas, non-geometric in the sense as explained in Olivia's lectures, okay? So in the one purpose of this talk is to convince you that there is actually a real meaning behind this statement, that it's not at all obscure. This will come in a moment. First, let's have a purely Topos theoretical interlude. It's a thing which I stumbled upon. I presume that it's well known, but I have never seen a written reference to it. Okay, say you have a Topos E and you have a Sub-Topos given by a local operator, a modal operator, topology on it. There are several synonyms, yeah? Okay. So I have the Sub-Topos and you have the Super-Topos and now take a formula. And you might be questioning yourself, what's the relation between the formula holding on the Sub-Topos and the formula holding on the Super-Topos? Is there any relationship? Okay. And the answer is yes, there is. Namely, the relationship is exactly given by this observation. So a formula phi holds in the Sub-Topos if and only if its diamond translation holds in the Super-Topos. The diamond translation is given recursively by these rules. They are just the rules for the in logic, well-known double negation translation just with a diamond instead of the not-not, okay? Good, and this is the answer. This is a very useful observation because it allows you to use the internal language of the Super-Topos to speak about the Sub-Topos, okay? And we'll see why this is very useful in algebraic geometry on the next slide. The diamond translation, is it a geometric morphism? No, I mean the diamond translation is a purely syntactical operation on formulas. Given a formula, your pain is diamond translation by like putting diamonds everywhere in front of everything. And then in fact, you can prove that it surfaces for you to put the diamond in front of the exists and in front of the ore. So the top left, this is a geometric morphism on the top axis. And the diamond is an internal reflection on the universe, on the Sub-Topos. Maybe I should say that I'm a little bit envious of the philosophers. Re-Mathematicians, we only study what's true. But the philosophers are to study things what should be true, what do I know, what does he know, what can be true, what is necessarily true and so on. So the philosophers, they study model logic. And so with stopper theory, model logic is actually useful in like real world applications in algebraic geometry as well. We'll see it just in a second. The thing is, depending on which model operator you choose, you can incorporate like a whole host of statements in one. I just want to concentrate on, for instance, the first one. So if you choose for diamond, the double negation topology, then diamond phi will mean that phi holds on a dense open subset. Okay, you can check that with a Krippke-Jouill mathematics. Okay, and then you can also wonder what phi to the omega means, the diamond translation of phi. And if you do the calculation, you will see that, I'm assuming the scheme to be irreducible for the moment, that phi to the diamond means that this formula phi holds at the generic point. So in algebraic geometry, it's an important question when properties spread from the generic point to some dense open subset. This does not always occur, it's a good thing when it occurs. Okay, and you can analyze this question politically because you can put it in this form. When does phi to the diamond imply diamond phi? When does phi holding at the generic point imply, implies that it holds on a dense open subset? Okay, and the good thing about it is that you then can import well-known theorems of logic to tackle this. For instance, the theorem that for any geometric formula, phi, this always holds, okay? And in that way, you have like one meter theorem which gives you a whole host of individual statements about spreading from points to open neighborhoods. And in fact, if you vary the modal operator, you gain even more statements. Yeah? Should phi in this case have intuitively like one free variable which is a point of the scheme X or should it be have no free variables and when you say it holds on a dense open subset, you mean like? Yeah, it may have three variables. For instance, it may refer to a given sheeps or something, yeah? So... But not to points of the space X. Yeah, yeah, not the points of the space X. The confusion is some of these feel like quantifiers, but quantifiers eliminate variables. Yeah. Or bind variables, and this doesn't. Maybe I have time for one very short example for this. So you might know that the following does not hold. If you have a sheaf of modules and you know that it's stock vanishes, then it does not hold in general that it will vanish on a nape of the point. This is because the, so you know again a conceptual understanding for this non-theorem. This is because the formulation for a module to be zero is non-geometric. It's for all X in M, X should be zero. Because of the universal quantifier, this is not a geometric formula. Okay, but now presume that the sheaf is known to be a finite type, yeah? So internally, this means that the module is finally generated. So you have generators, X1, X2, Xn. And now you know that you can rewrite the condition that M is zero, simply as X1 is zero, X2 is zero, and so on, Xn is zero. And this is manifestly a geometric formula, okay? So now you have a conceptual understanding of why in general, spreading does not occur for being zero and why it does occur when the sheaf is a finite type. Let's talk about quasi-coherence. The condition for a sheaf of modules to be quasi-coherent is very important in algebraic geometry. In fact, there are arguments that sheaf of modules, which are not quasi-coherent, should never be studied because they're like not geometric, yeah? Okay, so of course you're wondering whether it's possible to characterize quasi-coherence in the internal language. And it turns out that yes, you can. The condition is on the board. And I quite like this condition because, so normally you're always reduced to well-known constructive algebra. But in this case, you create new constructive algebra because this is a condition which is not normally seen in constructive algebra. Okay, so the condition is that, finally F of OX, if you localize this module E away from F, then this new module should be a sheaf with respect to a certain internally defined model operator which we can see on the board, okay? So it's possible to give the whole of sheaf theory, separateness condition and so on and so on in the internal language. In fact, I'll do it for you. This is the separateness condition, just the separateness condition. And I quite like it because it's of our curious logical form. It says that if you are able to deduce that S is zero, given the assumption that F is invertible, then you can unconditionally deduce that some power of F times N really is zero, okay? So, and we can return to Mulvay. If you take for E, in a special case, simply O or X, the structure sheaf, and if you take for S one, then you exactly reduce to Mulvay's curious formula. And then now you know the deeper meaning of this formula. This was just like a little shadow of a greater picture than the picture of quasi-coherenceness, okay? The last few slides are for all fans of Munik Hakim and Peter Arndt, especially his really great answer on overflow on how to motivate schemes. Okay, so an abstract motivation of how to motivate schemes is the following. Take a ring A. You want to construct the free local ring of it. Okay. So this is the universal property. It should hold that, like you have a morphism from A to A prime, then to be constructed ring. And it should hold that for any map into a local ring R, this map should uniquely factor over a local map of rings from A prime to R, okay? And then you do the calculation and are disappointed because this optimization problem has a solution in set if and only if the ring A has exactly one prime idea. Okay, this is about the situation. Okay, and now we read this answer by Peter Arndt and I'm delighted to know that this optimization problem does have always a solution. You just have to allow yourself to change the topos. The solution is the structure sheet O-spec A in the scheme spec A, yeah? Of course you have to define what it means for what a morphism of rings living in different toposes is, but you can do it. It's very easy. And the side is the spectrum with the Zaris ketopology. Yeah, right, yeah. Okay, and then of course you want to go one step further because you're a fan of Munikakim. Now you want to construct the free local ring over any ring, not necessarily a ring in set. Okay, so what you do, you do. It's by the general theory of this topos magic, it suffices to give a constructive account of the theory of the usual spectrum inside the topos. So our first attempt might be to define the spectrum as the topological space of the prime ideals of A, but you know that prime ideals are elusive in constructive mathematics. This will not have the right universal property. Okay, so a better attempt might be to consider this topological space of all the prime filters of A. A prime filter is classically the same as a complement of the prime ideal, only that it's directly axiomaticized. It fulfills exactly the dual notions. Okay, this is much better, but also prime filters are elusive in constructive mathematics. There's a count example by Jo-Yal where he constructs a ring, which is not trivial, but doesn't have any prime filters, okay? What you instead have to do is build the locale of the prime filters of A. And in fact, this definition was already on the board. Remember Olivia's last lecture? She had this long chain of equivalences. Here was the shift topos of a spec A. This had a non-constructive definition, it doesn't work. And then there were three more topos, all of which you can constructively use. And I just kind of randomly decided for the final one, which is a locale, okay? Okay, and in this way, you obtain an internal characterization of Monique Harkim's very general spectrum functor, which gives an adjoint to the forgetful functor from local ring toposes to ring toposes. If you check her feeders, she does this like explicit site construction. So she has to build a site. And we know by Olivia's lectures that sites are good. But even better, it's to not have to construct them by hand, but use a general meta theorem about toposes over X and toposes internal to X, which does this kind of construction for you, okay? Okay, and if you're familiar with algebraic geometry, then you also know the relative spectrum. Given quasi-Korean chief of algebras, or X algebras on a scheme X, there's the relative spectrum, which has a scheme of this into X, yeah? Okay, and you might wonder how this can be described internally. So the obvious choice, the obvious guess, would be to just take Monique Harkim's spectrum functor, right? It's very general. It applies to the situation. But then you make the calculations and realize that Monique Harkim's spectrum functor gives a different result than the relative spectrum considered in algebraic geometry. In fact, you can check that Monique Harkim's spectrum gives the correct result if and only if the scheme, your base scheme you started with is zero-dimensional, which of course is not the general case, yeah? So it's interesting to see how you have to change, fix the definitions. And here's how you do it. You again define the relative spectrum as a local, so you give it frames. But you don't take as a frame the frame of all radical ideals, but only the quasi-Korean ones, okay? And you can do a few correlations to arrive at this short characterization of when radical ideal of a quasi-Korean algebra is itself quasi-coherent. Again, notice this curious-looking behavior. So this ideal E, i has to satisfy the condition that if an element s is an element of i under the assumption that f is invertible, then f times s really should be an element of i without any assumptions. Okay, and just want to end with a remark on the points. So this local is not spatial from the internal point of view, but you might still be interested in the points, just to have a better feeling for it, yeah? To know the theory which this spectrum classifies. And it turns out it's not the theory of all prime filters, but only of those which fulfill this additional condition. Okay, and then you can, of course, generalize it. X doesn't have to be a scheme. It can be an arbitrary topos, ring topos. And then you can understand the limits in the category of locally-ringed toposes much more conceptually. Namely, they are simply given by the naive limits in the category of ringed toposes, and then given by relocalizing using this construction. Okay. So, and in my last zero minutes, let me turn to Peter Ahn's answer. You can also, like, give a very nice account of the relative spectrum, of the universal property of the relative spectrum, of the relative spectrum as used in algebraic geometry. And then you very clearly see the difference to the spectrum of monikarkeum. Namely, the question in algebraic geometry is to construct the free local ring over A, which is also free over the base over B, over all X in applications. This is a small but important difference. Okay. I hope I've convinced you that using topos magic, topos internal language in algebraic geometry is a very fruitful thing. It allows you to simplify proofs and allows you to gain a better conceptual understanding of the notions and of the statements in algebraic geometry. Thank you very much for your attention. Do you have any questions? Yeah. Yes. Can you go back to the reconstruction of the universal property of the relative spectrum? Yes, this one. How does the relative spectrum compare to the other one? The monikarkeum spectrum. Yeah. Is this a subspace? Yeah, yeah. This is always a subspace, a sublocale of monikarkeum spectrum. Okay, it's precisely the subspace where the morphism from B to A prime is local. Yeah, yeah. And in fact, you know this from Olivia's talk because the theory which the relative spectrum classifies is a quotient theory of the theory which the monikarkeum spectrum classifies because you have one extra axiom. Yeah? It's a subspace and if and only if the base scheme is zero dimensional, then these two notions coincide. So maybe I have a general mark. Yeah. The commutative algebra is very much non-constructive. Yeah, several people here in the audience would disagree. Would disagree vehemently, yeah? So there's a notion that commutative algebra is non-constructive, but in fact it's not. So for instance, these two guys here have a great program where you can just take a non-constructive proof of commutative algebra. For instance, you have an element in order to show that it's null potent and you show it by showing that it's an element of every prime idea. Okay, you take this non-constructive proof, you put it in the machine developed by those guys and several others and out comes a perfectly fine, constructively acceptable proof which explicitly gives you bounds on the nile potent index. It's a rumor that classical commutative algebra is non-constructive. What's the name of that proof? Kokor, Lombardi, and... Oh, program doesn't mean program. No, no, the research program of dynamical methods in algebra. Yeah, yeah. If I Google that, do I find your... Yeah, yeah. Thank you very much.