 This one item that I was supposed to talk about yesterday was the notions of sight, which is important in the material tokens. So I would like to start today with discussing a bit this notion of sight. And then I will talk about the logical aspect of total theory. So I may use the first half hour for discussing the notions of sight. Well, a sight is, let's say C, J, is a small category. C equipped with a gluten-dictapology, let's say J. So I will describe more precisely what this means. Olivia already said something about it yesterday. But I want to stress the fact that what we do is that we have the category of perceived on C, which is a topos. And we somehow want to take a quotient. I'm thinking of the gluten-dictapology as providing some kind of relation so that you can take a quotient. So you have a quotient map, Q, which is a geometric, well, not exactly a geometric, an algebraic morphism of topos. So Q, preserve, put in it, not add in it. It's an algebraic morphism of topos. You may want, I mean, from a geometric point of view, you may want to see this as a subtropical of that, so you reverse the picture. But I want to take up it as a quotient. And the gluten-dictapology is really what you need to take a quotient of the topos from an algebraic point of view. So I would like to discuss these questions of quotient and topology, first maybe with soup lattices. So suppose that L is a soup lattice. And suppose that you have a set of pairs of elements, and you want to take a quotient of L in the category of soup lattices. So a quotient L is a quotient. So has to identify, you want to take the equalizer of all these pairs. So you want to have the QA equal QB for all AB in a universal way. Well, what you do normally, I mean in algebra, is that you construct the congruence of a relation generated by pi. Then you take the quotient, maybe here. So this is a relation for the congruence of a relation generated by pi. But in the category of soup lattices, something very nice happened. Is that you can construct this quotient right away. So let me write L pi, or let me say the definition. An element in X in L is, let's say, pi-sheaf. For all AB in R, A is smaller than 0. In other words, if you call an element pi-sheaf, the element is not able to see the difference between A and B. To see the difference on this side. So in other words, X is a pi-sheaf. If one this is true, this must be true also, and conversely. Now, if you write L pi for a set of X and L, X is a pi-sheaf. But this subset of L is actually closer under a theme. This is really very simple. Let me very quickly sketch the proof. Suppose that you have a family of elements in L. I want to prove that the infimum, let me write X, the infimum, is again in L pi. Well, suppose that for example that A is smaller than X, then of course all I. And since this is a pi-sheaf, and since this is true for all I, and the reverse implication is true. So this is close under an infimum. And since this is close under an infimum, it implies that the inclusion has a left adjoint. If I call the inclusion U, then the inclusion has a left adjoint to U over 3 from L to L pi. This is true from, I can write a formula for U over 3 of something, U over 3 of NMX will be the infimum of over all Y such that X is greater than Y, and Y is in L pi. That's a formula for it. And this U over 3, being a left adjoint, is a morphism of suplexes. And the solution to the universal problem of identifying A and B. So if you have a map F with the conditions of F of A, B, or all AB, then you will be able to, I think it has a bigger size, so you can expand it. But what about if the lattice is a frame? Suppose that L is a frame. Suppose that L is a frame, therefore, I mean, it means that the infimum is distributed, right, over super... it's a lattice, but with the extra property that we have this distribution to go down. Okay? And when is it true that the quotient there is a frame? You would like to have the quotient to be a frame now, okay? What I just described is a kind of purely linear theory in the sense that the suplexes are like BN groups, they are randomized like addition, and I just told you how to make a quotient of an AB group by something, right? But now suppose that L is a ring, and you would like to know when is the quotient a ring and when is it the solutions of the problem in the ring aspect. So, but you need that five is... so let's say G and L is the same generator. And now what kind of generator? I mean a suplex generator or a frame generator? Let's say there is a difference between the two. I think frame generator. So I'm just now supposed to... maybe I should be careful. I will take suplex generator. So this is a bit... yes. So suplex generator. And I'm going to say that five is... so it's a stable or something. So the condition now, have the condition add the condition that G is contained in five. So what does it mean? It means that G in G and AB in five implies that G, A, G, B. It is stability under put back, or if you like, intersection in five. Then the same construction give you a frame quotient. A frame quotient. And it solved the problem of taking a quotient of the frame with respect to the given system of relation. So this is a little bit like saying, yeah, that's an idea. And I put in the condition here only for a set of generator because very often that's the way you want to check if something... You don't need to check for every element of five. You just need to check for a set of generator. Again, this is not difficult to prove and it's like you will probably enjoy proving it yourself. So I will not take this pleasure from you. Okay. Now, this pattern is just repeating itself in shift theory. This is exactly the same thing. However, I would like here to make a point about the fact that the pre-sheaf on which is D of P. So if you have a pro-set P and you have the unit that embedding Y, D of P is the pro-set of downward sections of P. And we know that this is the completions of P under Suprema as a suplex. This is the solutions of the linear problem. But it happens that D of P is also a frame. Okay? It's a frame. It's given to us as a frame. But as a frame, it's not to be. This is a frame, but it's not the frame freely generated by P. The frame freely generated by P has another construction. We first add finite meat to P. Okay? And then we complete it by adding Suprema. That's the frame freely generated by P. That's this thing. This is how it works. So this one is not free. The generators, of course, are the YX, okay? D of P is generated by the YX with X and P. But there are the relations. So I would like to discuss the relation between the generator of D of P as a frame. The first relation is that the Supremum of all the generator is the top element. You see, this was true also. What I'm going to write will be true in D of P as a Sup lattice. But I cannot write this relation in the theory of Sup lattice because this element here belongs to the notions of frame. In a Sup lattice, there is a top element, but it has no name. Well, I can say it's the top element, but it's not preserved by morphism of Sup lattice. So if I write these relations, I want to write the relation between using formulas in the theory of frame. And I cannot write this relation in the theory of Sup lattice because this element 1 is not a constant or an operations in the theory of Sup lattice. It's not preserved. So that's the first relation. And the second relation is that these relations are very easy to check, right? I mean, the first just means that P, the top element, is the union. You see, if I use, since I am in D P, I can use union, and this is clear, right? Every element is the similarity from this one, okay? You see, why A meets why B is not a generator in general. I mean, if P was a meek lattice, then why A meets why B would be equal to Y of A meet B? Because the unit are embedding preserve intersection when they exist. But when they don't exist, the intersection between the two is not representable. But however, it would be a suprema of things that are representable. And you would just write it down, okay? So these are the two relations. And it's an easy exercise to check that this is a presentation of D P in the theory of Sup lattice in the sense that if you have a frame A, and if you have a map phi that satisfies these two relations where A is a frame, then there would be the linear extensions which is just obtained by using suprema for S downward sections of D. The diagram is not D but P. Ah, thank you. Yes, right. So you have a map, you extend it linearly, and then you prove that phi prime preserved in the section just from 1 and 2. 1 and 2 is exactly what you need in order to prove that, okay? So in particular, the point of a frame, we call it the point of a frame is a morphogen, is a character. So what are the points of D of D of P? Subsets in P. That satisfies such that two conditions. P is 90, F is 90, and 2, A, B in F implies... Well, I should say also that F is upward close. Okay. F is upward, and 2, A, B in F, they exist. C, smaller than A and B, such that C is at F. So this notion here is what we could call the prime filter on P, okay? It's a prime filter on P. And why is this true? It just follows from these conditions. You just... a character will be on D of P, will be... If you restrict the character to P, you will have a map from P to 1. It will be the characteristic functions of something satisfying this condition here. And if you look at this condition, you see that it will be the characteristic functions of the prime filter. So F is the set of X in P, such that P of X is 1. Okay. Sorry, what did you do here again? Okay. I'm looking... Now, first, I'm describing the characters of this frame. Okay? The character will be a frame of morphism. Now, the frame of morphism, if I restrict it to P... Okay, so I think the restriction to P... P, you have the generator. The restriction to P will have to satisfy the relations that I have written here because it's the case where A is the... is bracket 1. And this phi is actually a phi prime. And actually, I'm looking at the restriction here. And it must satisfy this condition. And if you translate this condition, you get these all. Okay. So, in general, when you construct a frame, you may construct it from generators in general. So you have a frame A, and you would like to have a presentation describing the frame as a quotient Q. From some... these are very easy to construct. The D of P. This... so you should know... Of course, this frame is not free. Okay? But you need also a relatively topology, possibly, in addition. So that's the... I mean, you need to take a quotient and then to have the morphism here. So that's more or less the general constructions of a frame. You pick some generator, linear generator, because you want to have this as a subjection. So, some elements having the property that every element is a supremum, linear generator. But these linear generators should satisfy some relations, which are the... okay? And then you pick a topology and... okay, that's the way you construct the... And the topology, very often, is a bit less... not as general. I said D of P. And so having a relation here, you may as well have a relation here. Okay? So it's a relation between the lower sections and the elements of P. And this relation, pi, let's say A, is S cover. Okay? So S, normally, S will be... contain a sub... a subset of our street. And you want to say that S cover A. And that's the... or phi cover. You see that? You introduce a covering relation like this. You want to say that certain lower sections are coverings the generator. And then you need to say that some stability condition that if S covers A, implies that S intersection B, B, B for all B smaller than 8. Okay? That's the stability condition. And then the notions of sheath is defined using that. And that the notions of sheath mean something like this. That T in D of P is a sheath. And for all S covers, by covers, S in T implies A in T. Okay? That's the sheath condition. Okay? So that's the picture for the two of friends. And it happens that this picture just repeats it completely for two volts. It's the same thing. Except that you have to be a bit careful. There are morphisms around, etc. Yes? When you start with a botanic topology and one defines the sheafification. Yes. One is obliged to do it in two steps to repeat twice the sheafification. Okay. What is the analog here? That's a good quote. I was hoping that nobody would ask me this question. You see, I'm a little bit cheating here because I'm making things simple, which is correct. I mean, this is correct. From a conceptual viewpoint, you would like to know how you construct this sheafification. Okay? I said nothing here. Because what I did was to say what is a sheaf? And then I proved formally that there's a best approximation given something which is a pre-sheaf. There's a best approximation as a sheaf, which is I call the sheafification. Okay? So I don't need to worry. The definition is there. But of course, you may want to construct. And then you do as usual. You do as usual. In other words, in order to be able to construct the sheaf, you need to put some extra conditions on the topology. I said nothing about compositions of covering here, which was important in the definitions of what goes in the topology. This is some kind of pre-topology. Okay? So if I want to construct the sheaf, I will need to have an extra condition of the coverage to be close to the compositions, and then I just repeat what we do this way. But what I was asking is, I suppose the same difficulty arises already for locals, that you have to repeat twice. No, because in the case of sheaf, no, I don't think so. Because in the case of sheaf, the first time you construct a separated sheaf. But if the sheaf is separated, you get the first pre-sheaf. If the pre-sheaf is separated, then you complete it. But if it happens that the pre-sheaf is separated, then there's nothing to do. But here everything is separated because all the diagonals are triggered. So in general, to construct the analogue of the composition, the work of covering families in the Rottenbeek topology. Yes. So in this framework, you have to make a process of the composition, but you have to iterate it. Yes, exactly. If you have a cover like this, you may think of this as a family of lower bounds of A that covers A. And then if you want to pull the compositions, you just... So the same thing is happening in topology. So first, you have the solutions of the linear problem. This is really free over a small category C. You take the pre-sheaf, and this is the free co-completion. Now, you may want to take a quotient of that. And how do you do a quotient? You pick a family of arrows in C. In fact, you put it in PSC. You pick a family of maps between pre-sheafs or a set, a set of arrows in pre-sheaf. And then you say that something, five, C, X does not see the difference if the map is invertible, is an isomorphism, for all U from A to B in five. So that's the condition that X does not see the difference, not between A and B, but the object X thinks that U is invertible. And then it happens that if you look at the full sub-category of PSCR, the full sub-category of the pre-sheaf category, those objects are probably the five-sheaf. Then this thing has a left-hand one. This is a complete sub-category, and it has a left-hand one, maybe L5. It's sometimes called localization. And it's the solution of universal problem. So you compose with the Unida polka with L5. If you have a function here that inverts all the maps in five, in the sense that, well, you have to say that F over three is a few and invertible. F over three can mean the continuous extensions of the function of the pre-sheaf category. If this is invertible for every U or U in con, then there would be a function here that is co-continuous and invert, and it's the solutions of the problem of inverting all the arrows in five. And this category is what we call presentable, or in the language of Gabriel and Ömer, it's called locally presentable, but today people have started to call that presentable category. So locally presentable or presentable categories are the same. This is all presentable categories, so-called presentable categories are constructed this way. You take a small category C, you take a small set of arrow, and you do this process and you get, so it's a kind of generalized Topol's theory in the sense that you just get here a category with co-limits, with generators, and that's essentially it, there is no other property than that. The generators are small, so-called, and... Excuse me, I didn't understand just... Can you start from, in this localization, you can start with what you call, instead of fresh sheep of sea, you just need, this is what you call a presentable category, and then you take a set of arrows and you can do this, or do you need to start with a pre-sheep? With heat? To do this process of... Oh, yeah, yeah, yeah, I did it for pre-sheeps, but you could start with a presentable category, start with a set of maps here that you want to invert, and localize, and you have the same theorem. Yeah, you don't need to localize from a set of maps in the category of pre-sheeps, you could localize a set of maps in a presentable category, but the presentable category are somehow defined by localizations of pre-sheep categories. You have actually found some compatibility. No, no, it's completely arbitrary, yeah, that's the good side of it. There's always, you start with... That's it, that's a theorem of the Gabriel energy. So that's the Livia aspect behind Topos theory. These are like avian groups, okay? But then you want to know when is this a Topos, okay? And that's where the so-called growth in the topology happened. So our, all the notions of sight. So this growth in the topology here is the following data. It is a set of monomorphism, S to the representable Y, A. So I'm choosing monomorphism this time, because I want that the localizations be a Topos, and this is called covering seed. A is a representable, A is in seed, and so this S is actually a sub-pre-sheep of Y, A. So you start with the collections of monomorphism like this, and you suppose that the collection is stable under pullback, so condition stability. So that means that if you have something in, if you have a map F and A to be in your category and you pull back the cover, the pullback means to be in fire. This monomorphism means to be in fire again. And now once this condition is satisfied, if you invert universally this class, you get a Topos. And this is called a sight. You don't need that the monomorphism are stable under composition. You need that only if you want to construct using this sheafification. In other words, what happens is that if something is a sheaf with respect to a certain class of monomorphism like this, it will be also a sheaf with respect to the classes that you can obtain by composition. To be a sheaf does not use the fact that the class of monomorphism that you choose is closing the composition. It's only about the constructions of the sheaf where it is useful. It's like in ring theory. You can divide a ring by an arbitrary set, meaning dividing it by the identity of the final set. Yes, right. And it has similar universal property. Yes. It's a universal ring which kills all the elements we are given. Exactly. Exactly. Yeah, yes. It's very, very in the same picture. Yeah. Okay. So, you guys said enough about, let's see, how much time? I still have 25 minutes. For the rest, I'm going to talk about the logic and topos theory. Olivia said something about it, geometric logic in her course yesterday. So, we'll see more today. But I was beginning with higher order logic, which is a different subject in some sense. So, this is a topos. Let me say, okay, a topos, okay, a topos. I'm defining, I'm giving a new, not a new, but yesterday I gave the calculations of what a group with topos is in terms of the zero axiom. Right? I described it. Now, I gave another system of axiom. It's a presentable category, such that, which is locally cappes and flows with sub-object classifier. It means, locally presentable, I'm sorry, locally cappesian. E is cappes and flows, founters, if it is cappesian, so if it has products, founite products, and the founter has a right adjoint. And the right adjoint is denoted as an exponential. So, the right adjoint is denoted as an exponential by A. And the right adjoint is an internal arm? Yes, right. So, you have that A plus B, you see there's a B to C to the A. Right? There's a bijection between these two kinds of lines. And locally cappes and flows mean, locally, it means that, in addition, all the slice circles, if you look at all slice categories, they are cappes and flows too. So, it means, for example, that there are pullbacks, because if this has product, you mean to have pullbacks in E, or further product, okay? Now, the sub-object classifier means that there is a universal, a universal, so this universal monomorphism, the target of it is a term, and the source turns out to be the terminal object. This needs to be proved, but this is, and the point is that for every A and every monomorphism, there exists a unique pullback square, a unique path such that this is a pullback. A nice way to express the universality of the monomorphism, U, is to introduce the category, the category of monomorphism and pullback. So, you look at the category where the objects are monomorphism, and the objects are the monomorphisms in E, and where the morphisms are the pullback square. You look at this category, the category of pullback square, pullback squares of monomorphism, right? And this thing is saying that this category has a terminal object. This is the terminal object of the category of pullback square of monomorphism. That's one way of describing the universal property of the monomorphism. Okay, and... Yes. So if I understand this definition well, it's not the topos of photonyx, it's the topos in matter, doesn't it? No, this is the group of the topos. Presentable. And now, on the students that you could do a lot, just by the axioms, that it is locally presentable and a sub-event classifier, I mean, most of the constructions are many, many could be done, so you drop the presentable. You just start with the category, which is locally presentable, so the sub-event classifier. And that's the notions of elementary. And this is a really, very, of course, a very nice, important discovery, because, let's say, Michel Biennial saw also, afterward, that you could interpret set theory, enthusiastic set theory in any elementary topos. In other words, you can use the usual notation of set theory to work in a topos. You can, for example, the power set, the power set, what the power object is, which is defined to be a definition to be omega to the n, okay? You could say that omega to the n is subset of a, that's the power set. We know that the p to the n is an object in a topos, and we know that it's not the set of subset. However, using this Bire-Boumice language, it shows that you can interpret this thing as meaning that, and as well as all the axioms of higher order set theory, and it's a model of set theory. Because in usual set theory, you can form higher and higher cardinals, so it's like bed, outflow, and so it doesn't exist in, I don't know if that's the right way. Yeah, yeah, so there are some small restrictions of this kind of set theory, for example, you may have a natural number, and then you have p of p of n, and then p of p of n, it's all there, except for, it's all there, but you don't have, you may not have something beyond. Okay? Okay. So, but for most, mathematics is mostly involved with this sort of, okay, you don't need to go much beyond that, so it contains a big chunk of mathematics today, right? Yes? Can low ectopause, which is not a growth and ectopause, see now the growth and ectopause in some weaker model of set theory? Okay, I don't know, I would like to know, yeah, okay, okay, so, now, when I, Benevol, first explained that, I think, in the meeting in Montreal 1974, and I was very excited. I had to work with geometric logic before, but this was not geometric logic, it was the whole higher-order logic, because it was showing that you could do ectopause theory without any category theory, that you could somehow, it's, the point is not that you should not use category theory, but you don't need to use categorical notations all the time, okay? You just use, make reasoning that are just like classical, and this is very good, because sometime in category theory you have a proof and you have a new diagram, arrows are moving all around, and it, and you say, well, what is this, it's difficult to understand you have, or at least I find sometime difficult to understand, but if you succeed to translate what this diagram is saying in this language, sometimes it becomes obvious. So in other words, there is an alternative to understanding. It's not that you should not use category theory, but sometimes the classical notations are more comprehensible, and therefore you should use it when it is the case. What is the main theorem of elementary topos theory from my point of view? It's the fact that if you take e elementary topos, of course the topos doesn't need to be elementary, it could be a topos, okay? But what I'm going to say is somehow through the construction, well, with the elementary topos, then you can do topos theory. You think of e as set theory, now you start again, you do sgfo in e. So for example, you first look at categories inside e, cap e. So what is the category inside e? It is an object of arrows, an object of objects, and a source and target map, and then a composition law that will be like a map like this, you see one. Everything that you can do, usually in mathematics, you can do it in a topos, and therefore you can do cap e, okay? And then there's notions of a site, you can explain what is a site, and you can do topos theory inside a topos. In sgfo what they do, I'm saying that in sgfo what they do is a bit more complicated because they allow to form things like the category of shields of precious, viewed as a site, but it will be a large site, so they use a larger universe, so they use more than what you allow, because in some proofs they need to say we reduce the case of precious, which is the use, which is this site, so for example in topos theory you need the category of set, because you want to have precious, but the category of set is actually a big category, and therefore doesn't they use the universes to make it smaller, and you have always this problem that you have smaller categories and big categories, and usually a pre-sheaf is something in a big category, okay? The same thing happened here, in the sense that you define what a pre-sheaf is, it would be a family of an object over c0, okay, because you think of xa is the union of all the values of the pre-sheaf, a in c0, you think, okay? A pre-sheaf is like if you first a family of set, and then this family of set has an action, right? So you take t of your star of x, and then you will have a map, which is the action to s of your star, so it's, you see, like a descent condition and etc. You can explain what is a pre-sheaf, okay? But this pre-sheaf, you will have a category of pre-sheaf, but this category of pre-sheaf will not be small, it will not be a category of like a key. So you cannot do the game of looking at, like in Nigeria Power, they look at, and Glotendic topology is like on the category of pre-sheaf, so like it, like, they can say that the purpose is also obtained from the category of pre-sheaf, the canonical topology, and if the shapes and things of, you cannot do this. I'm sorry, maybe we can discuss it. Okay. Yeah, so anyway, you have pre-sheaf on a small category c, c is a small category, okay? And this is an elementary topos again, okay? And you can add a topology and you will get a topos. And this topos will have a geometric morphism to e. It will be equipped with a geometric morphism to e. And the theorem, I would say kind of fundamental theorem, or at least one fundamental theorem of elementary topos theory, is that you get all the category of toposes over e, the toposes over e, the topos that are equipped with a geometric morphism. Okay, so there is a category of toposes over e that is equivalent to all construction from internal things here. So I would say that the same thing as toposes in e. It's a bit vague here because I have not really explained what a topos in e is, but it's essentially given by a site. A site is certainly a small object in e, and you can describe everything in terms of internal site, a small category in c and with a topology. So this is kind of amazing. I mean, look, we have a topos which you take as a space. I'm drawing the topos. And then you have another topos, let's say e, and over it, okay? By mean of a geometric morphism. By mean of a geometric morphism. Well, this whole thing over it is completely described by a small thing in e. So should I read that equation as elementary toposes over e are equivalent to internal growth in the toposes in e? They are given by internal site in e. So you can prove things about toposes over e by working in e. So a geometric morphism of toposes in the case of elementary toposes, what is it? Is it a pair of a joint factor? Yes. Let's say e is a geometric topos, so-called, I prefer to call it a topos than a geometric topos, right? And elementary topos, I call it an elementary topos, right? I think that's the right thing to do. So e, let's say, is a topos, that is a geometric topos, f is a number of topos, this is a geometric morphism, and you can, okay. Sorry, so could you explain again the result, but I don't get it. Yeah, yeah, yeah, yeah. This is a bit, a bit vague at the moment. I just want to essentially say, I'm just saying that if you have a geometric topos over e, then you can construct a site internally in e, such that the category of sheaves on that side, on that side, would be, as a more quick way, you have a cumulative triangle type. The first sheet is that quantified part from c to e. Yes. And when you do the notion of a, of a, of a quadratic topology, you have to take a set of c's, and do you take this set of c's in the sense of set, even in the solution e or a set of c's? Yeah, I'm, yeah, I'm sorry, maybe we can discuss that after, but I don't have much time left. Two minutes? Three? Okay, okay. Okay, so in particular, you can do the theory of locales in e. I mean, some toposes are very simple there with locales in e. So if your topos in e is a local, you get a special kind of topos over e, which is called a localic morphology. If your topos over e can be generated, can be described by a local inside e, or if you like a frame inside e, there's a theory of frame. The whole theory of frame that I described to you can be repeated in e. Everything I said is complicated through again in e. And it give rise to the notions of localic morphism. Now, there is a topos which is often denoted like this. It's a classifying object. If the topos freely generated by one object, I mean, I denote it like this because it's like a polynomial ring. It's in the category of algebraic topos. In other words, if you have a topos e and some object u in e, then there would be an algebraic morphism by algebraic that takes biocits to u. And this topos has a very simple description. It's the category of covalent topos from finite set to set. It is a pre-sheaf category. It's pre-sheaf on the category opposite to the category of finite set. And it's kind of interesting to look at locales in this thing. Locales, I'm sorry, a frame in set x is a covalent punter, a team from a finite set to frames, so the category of frames. A frame inside this topos can be described as a covalent punter from the category of finite set to frames. And this covalent punter has various properties. So, if you have a map from end to end, then you have a map from Tn, a map of frames from Tn to end. And this is Tf. And Tf has a left-hand point, which is f from Tn to Tn. And as well as other things. But the point is that T is a geometric theory. This is the kind of theory that Olivia was describing yesterday, but it's the purely algebraic view on it in the sense that Tn is a set of formulas of x, phi, that's the notation. With the variable phi, x contains in the first, you have an infinite set of variables, so you look at, suppose that the variables are contained between the first x1 and phi xn, and you look at these formulas, and you take the logical equivalence, relation given by the heraxioms, and you, okay, and this would be a frame by constructions of the logic, of geometric logic, and it will satisfy all the axioms, and it will be a frame inside these totals. So what this thing seems to say, I will finish with that. Same tactic, a category for some geometric theory that we are to discuss, you say that this category set of x is a syntactic category for some, for geometric theory. This is not the classifying totals. This is classifying totals for one sort. If you have two sorts, you take two objects, that's the only difference, right? But the point that I want to make is that topology maybe can be understood as the study of frames. That's one interpretation of what general topology is. Okay, so now we can develop general topology. There are all kinds of notions, compact spaces, all kinds of theorem, but we can do it inside the totals. We just repeat, but it's not always clear how to do that because the axiom of choice, we can't do that. This is topology without the axiom of choice, and then as a consequence of that, we get theorems about theories, okay, geometric theories. So I think this is maybe a version of the duality idea that is expressed by Olivia in duality between theories or maybe quotient theories, okay? This is a very specific. Yeah, or maybe it's a version of duality or maybe it's an extension of duality. I don't know exactly. But it is in the same picture. I don't know how... I don't know, it's just an idea. Thank you.