 I'd like to thanks the organizer for asking me to give a course in Topos Theory, that course that I share with Olivia. For, we had some discussions, Olivia and me on what to do. I sort of insisted to try to give the big picture of Topos Theory because I think Topos Theory has evolved quite a bit since its creation by Glossendick and his students. And maybe it's time to understand what's going on. I will be separating my course in three parts. There are four hours, so it's a bit of a problem because it's difficult to divide four by three. But I will do my best. This, I will start with the theory of locales. So I will spend this first hour and a fourth with the theory of locales. And then I will begin with Topos Theory for the second hour, second part. I will do again some Topos Theory tomorrow and I will finish with how you're Topos Theory. Well, it's a bit presumptuous of me to believe that I can actually describe these three theories in four hours. So I will just make some kind of sketch and sometime I will give proof when the proof is really easy. If the proof is a bit difficult, I will just maybe discuss the idea of the proof. The theory of locales was invented in the 50, actually. Shao Erasmann introduced the notions of locales. And many of the things that happen in Topos Theory can be actually already seen in the theory of locales. So the theory of locales is a good start for Topos Theory because it's very elementary, it's quite elementary, much simpler than Topos Theory. And many of the phenomena that happens in Topos Theory can be seen there. And if you know the theory of locales, sometime when you learn Topos Theory you say, aha, that's something I know with the theory of locales and it's just the same thing. And so I'm very much, when learning a subject, very much on the idea that the bottom up approach, not the top down. So unfortunately, books in Topos Theory now, they start straight with the Topos Theory and they give a chapter, maybe a chapter nine, chapter 10, the theory of locales, okay? So I very much recommend if you read these books to go straight to the theory of locales, maybe chapter 10 or chapter nine, and then start to read the beginning. There'll be a little bit of philosophy to start to begin about geometry and algebra. We, well, algebraic geometry is precisely the subject about the interactions between the interaction between geometry and algebra. We know that somehow a lot of what we call geometry, at least so-called commutative geometry as opposed to non-commutative geometry, is about studying rings, commutative rings. And let me just look at this a little bit to start. So suppose that X is a space and we often sort of understand or study the space by looking at maps from X to some ring R. R itself is a ring object, it's a ring space, if you like, it's a ring object in the category of spaces, whatever category of space you have. And why do we study maps from X to R? Well, that's historically, this is how things happen. You want, for example, to describe subspaces of X by using equations. So you take, so this is a map, F, so you take many of them, F1 to Fn. And you describe the equation so you look at zero, the inclusions, and you pull back and you get the sub-object or a sub-manifold or sub-algebraic variety by describing sub-object by equations. So this is a big discovery, I guess this is due to Decart that somehow a lot of what we think of the geometry is completely understood by, or maybe described by the algebra of functions or something like that. Now, so if you have a space, you look at the maps from X to the ring, and this itself is a ring. So this gives you a funka from, let's say, map X, R from spaces to the category of rings. And very often, you can describe some kind of adjoint, which maybe you could call the spectrum. So you will have then a bijection between, let's say, if you have a ring A, then you have the spect of A, and there's a bijection between the map from X to the spect of A, and the ring maps from X into map X, R. I hope I'm getting it right. Okay, now this duality between the main things and space does not work perfectly in the sense that very often it's not in the category of rings. It's an equivalence. So when it is not an equivalence, you try to improve it either by extending the class of rings you consider or by extending the class of spaces, trying to adjust the two side. So there is a kind of dialectic, you have a notions of space, a notions zero of spaces, and a notions zero of rings, and you have a functor, maybe some kind of adjunction between the two, and you want to transform it into an equivalence, and maybe you extend the notions of rings for that. So you have a notion one, and this leads to a notions of space, which is a bit more general, space one, and again, these things do not fit completely, so you extend one of the two side to make it, and it keeps on like this, and I don't know what is the end result. I mean, it seems that in mathematics we have been doing that six or seven times, and it keeps on. So maybe I will first start with the examples of locales. In the case of locale, the spaces are just the topological spaces, so I will write this top for the category of topological spaces, topological, and the ring in this case could be taken, of course, the complex to be taken to be the complex numbers, the real numbers, but one also has the choice of taking the Sierpinski space. So the Sierpinski space has two points, let's say zero and one, and the open of S is you have the empty, you have let's say one is open and zero one, and the nice things about the Sierpinski space is that if you is an open subset of a space X, maybe I should write it like this, you open subset, then there is a unique map, continuous map from X to the Sierpinski space, such that the following square is a pullback, I take one here, so this map here is called the characteristic map of the open set. If X is in you, zero, otherwise. And so there is a bad action between map of X into the Sierpinski space and the open set of X. And so mapping topological space into the Sierpinski space give you all the informations about the open subset of X. So the idea of the theory of local is to concentrate on the sort of ring that the open set of X is constitute. So what are the operations on the Sierpinski space? If you follow general ring, I mean we are interested in operations, the basic operations for general ring are addition and product, multiplication. So what are the operations that exist on the Sierpinski space? Well, general operations, well, I can give you a couple of them, maybe I should write just S. Zero one is a pull set, so you can look at Suprema and Esfima on this pull set. So that give you, for example, the Suprema of two things, we'll give you an operations and the Enfimum would give you another one. And these two operations behave very much like sums and times. You have, for example, a distributivity law. Of course, you have much more relations, many more, a relation than just distributivity law. But also, you can consider infinite operations because you can take infinite powers of the Sierpinski space. It's also a topological space, just the product topology. And you can consider continuous maps from S to the I to S as basic operations on open sets. And one of them is just Suprema over I. The Suprema over I of a family Xi is just what it is, it's the Supremum of the Xi. So the Supremum over I actually always produce one unless the sequence that you have here is completely zero. And this operation is actually continuous. And this is the basis of the theory of locales. So a locale can be defined to be, actually it's called a frame because when we talk about locales, we are really talking about a ring. So actually, let me draw the picture. You have rings, schemes. So we know that there is a kind of duality between rings and scheme. And here I'm going to put frames. This is a kind of ring-like objects. And here to put locales. This is just like the opposite category. The category of locales will be the opposite category. So what is a frame? So the definition is that a frame is a complete lattice. Maybe I will describe what is a lattice in a moment, which is such that the distribution of state law, distribute t, v, t law, holds. In this equation, it's important to observe that the enphemum, which is called the meat, this is the operation here is called meat of x and y, is distributed over arbitrary supremac. And this is very important for the notions of a frame. And this notion was introduced by Erasmus, I think. R-S-S, ah, oui, pardon. Excusez-moi. So the first example of a frame is of course the lattice of open subset of a topological space. And maybe I should say what a lattice is, a poset p v l is a lattice. If it is a complete lattice, if it admits a suprema, arbitrary suprema and enfima, so that's the notions of a lattice. And it's important to observe that it's enough to have suprema or enfima because the enfimum, the enfimum of a subset, if you have a subset S of the lattice l, you can express the enfimum as the supremum of the set of lower bounds of S. So that's an important observation, which is of course well known. So for example, the lattice of open subset of a topological space is closed under suprema because the suprema is the union, the supremum of family of open set. So O X is a frame. The supremum of a family of open set, of course, is the union, but this lattice is also complete. I mean it has an enfimum too and the enfimum of a family, the enfimum of a family of open set is just the intersection and then you take the interior of the intersection. Okay, so we see here that the enfimum has a kind of weird, apparently weird description because you need to take the interior and this is why the opposite O X up is not a frame in general. That is the distributivity of a union over an intersection which exists in the power set. There's a distributivity also because of the Morgan law. It does not survive when you look at open set. So the opposite of a frame is not a frame. And a map of topological space give you a map in the other direction which we could call F upper star, just the inverse image map. And if you look at the properties of this map is that it is a morphisms of frame. Now, by morphisms of frame, I mean a map, so you have two frames A and B, so phi and A and B. It is a map which preserves the operations arbitrary supremum and finite enfima or meat. And we should not forget that there is a unit element for the meat which I will write one, which is the top element. So one in the case of a frame is X itself, is the unit element for the intersection. So this is how the notions of morphisms of a frame, morphism is defined. And it's straightforward to check that the inverse image map is a morphisms of frame. So you get a functor from the category of frames. I will write F RM for the category of frames in morphism. And you get a functor that goes in this direction, maybe from top. And let's call it O. It's a contravariant functor. Maybe a word about the terminology here, because some people say that a contravariant functor, they say that this is a contravariant functor, right? I'm sorry, this is a co-variant functor, because it's okay. So a co-variant functor from the co-variant, co-variant functor is the same as a contravariant functor from C to D, contravariant. So when you say that a functor is contravariant, you don't need to write OP, okay? So because I prefer to do things like that. And also for posets, I mean, if you have a map of posets, and I will use that, F, map of posets, I mean, it's called preserving the partial order. You have two versions, a contravariant and a co-variant one. And so if I want to say that a map between two posets is a map of posets, I will say it's a co-variant map or a contravariant map. There are two possibilities. Okay, so here you have a contravariant functor. It turns out that this functor has a contravariant functor as an adjoint, which associates to a frame, a topological space, the topological space of its point. So essentially the points of a frame topologize its home in the category of frame, in the category of frames of A into the initial frame. So by bracket one, I mean this frame with two elements, zero. This is a frame. This is bracket one is actually all of a point. And you define the notions of a point of frame by looking at morphism from the frame to one. And you get an adjunction between two contravariant functor. Maybe I should write explicitly the form of the projection. So if you have X at the topological space and A at frame, then you can look at arm of A into OX. So this is in the category of frames. And this will be in projection with the continuous map from X to the space of points of A to apologize. So there is an adjunction between these two categories, but the adjunction is not an equivalence. There are more objects here than there are topological spaces. Actually, there are frames that don't have any point. So this is why the theory of locales has been called sometimes pointless topology because the points in a locale are not that important in the theory of locales. They are important, but some locales don't have points and they are important in the theory. So maybe I don't have any time to discuss examples, but I just tell you that some locales don't have points. So how does it work in the theory of locales? So a frame, oh, I forgot to tell you what a locale is in this picture. A locale is just an object in the opposite category of the category of frames. So locales don't have existence except by being a frame somewhere. So this is kind of a fiction. The notions of locale is a fiction. The reality is that what you have is a frame. But this fiction is very important. When you think about the theory, you want to have intuitions about the space that you're looking at. And so we often look into the opposite of the category of frames. It's a little bit like in algebraic geometry. Sometime in the theory of rings, it's very good to think of a ring as an affine scheme. But in this case, there is an actual general notions of a scheme which is partly independent from, well, not exactly from the theory of commutative rings. So this functor here, which was contravariant, once you use this duality, you get actually a pair of covariant functor. If you change the covariance, if you replace frame by locale, what you get here is two covariant functor. Yes. Well, in principle, they are defined here. We could call that a character, you see, of the locale. The locale is like a ring. And this is like the base ring. And you look at a morphism from A into the base ring. And in the representation theory, this is called a character. So a point of locale is a character. Maybe I could say a little bit about it, because, yes, right. But maybe I can say a little bit about the notions of a point of a locale by looking at the morphism, a character of a frame. So a character needs to send the element one. You see, it must be the characteristic functions of something. Here there are only two values possible. So it must be the characteristic functions of something. It sends the elements one, the top element here to one by notions of a morphism. A morphism frame sends the units to the unit. So the only thing which is interesting in the morphism is the kernel, kernel of phi. Now this kernel is closed under Suprema, because if you, phi is closed under, I mean, respect Suprema, so the kernel of phi is closed under Suprema. So you can take the union of everything that is in the kernel. So let's call that k of phi, the Supremum of all x in the kernel of phi, and x. So this is the maximum element in the kernel. And if you know this maximum element, you know everything about the character. And so you may think of this as an open subset of the space. So this is open. Open in the sense that in the locale, it's an open thing. So what is so special about this open set is that they are meet irreducible. So if I call you the k of phi, the maximum element in the kernel, you is meet irreducible. And this means what? This means that if you express u as the union of two other elements of the frame, then it must be that u equal to u1 or u equal u2. So that's, I'm sorry, meet, yeah. Thank you. And two, you need that u is different from one. OK. This is called a meet irreducible open set. Where does this property comes from? It comes from the fact that this morphisms of locale respect the intersection. If you just play with the axioms, you discover that the element u here is meet irreducible. So the complement of u, which normally should exist, is a closed sublocale. I said nothing about closed sublocale. The complement of u, which is the closed part of a, this is just the a up, actually. There's a lattice of closed sub-objects in the locale. Just the opposite lattice, OK, is jointly irreducible. So it's a jointly irreducible. So jointly irreducible means that if f is f1 union f2, then f equal or f equal f2. So there are two conditions. And the other is that f is not empty. So there is a projection, if you like, between the points of a locale and the irreducible closed sub-object of the locale. So that's one way to understand what is a point. And in the topological space, if x is a topological space, then if you take an element x of the topological space, and then the closure of the singleton is irreducible, is a closed irreducible. So that's the connection between the classical notions of point of the topological space, is that the morphisms of locale produce closed sub-objects that are irreducible. I don't know if this answers the question. OK, in the topological space, right, this is true. That's an example of closed irreducible, but there are, in the topological space, sometimes closed irreducible that don't have a generic point. This is called an originator. OK, so what I want to develop here is the fact that frames are rings. And really, I mean, I'm serious. I mean, frames are rings. In a sense that the theory of frames is very much like the theory of commutative rings. But somehow it is simpler. It is simpler. It's a kind of simple version of the theory of commutative rings. So the category of frames is an algebraic category. In a sense, it's like the category of groups or the category of rings or the category of a billion groups, the category of the algebra. Universal algebra applies. So what does it mean? I mean, for example, if you have morphisms of frame, you can decompose it into three morphisms, like in the theory of rings or whatever. First, you take the image of the morphism, you get here a subframe. That's the notion of subframe. There is an equivalence relation on A defined like the usual, the congruence. There is a notion of congruence in the theory of frames. OK, what is the congruence in the theory of frame? It's an equivalence relation. It's an equivalence relation. An equivalence relation is always a subobject of the product, but it should be a subframe of the product. That's all. OK, an equivalence relation is a congruence if it is a subframe of the product. It tells you everything about the congruence. So given the morphisms of frame, there is a congruence, which I write here as an equivalence. So you can take a quotient to get a frame and here and then here you have the sales of morphisms. So in other words, there is a notion of surjection, the frame, and the compositions of every map as a surjection follow it by an injection, an injective morphism. Well, for the theory of locales, it means something. In the theory of locales, you reverse everything, so you have a continuous map, f from x to y, and the injection here becomes a quotient locale. OK, let's call it z, a quotient. So what is the notion of quotient locale? The notion of quotient locale comes from the notions of subframe. That's just straightforward reversing. And the quotient frame leads to an embedding. Let's say here, this is embedding of locales. So every morphism of frames has a decomposition as a quotient followed by an embedding. Except that in the theory of locales, something simpler is happening because a map of locales, morphisms of locales, let's say five from a to b, has a right adjoint in the category of poset. The right adjoint is not in the category of locales. The right adjoint is not a morphisms of locale. It's just an order preserving map or a co-variant map between two poset in the category of poset. And this is just this pollute from the fact that five preserves suprema. Any map preserving suprema between two complete lattice has a right adjoint. I'm sorry. Which I will write to file our star from b to a. Oh, oh, oh, oh, I'm sorry. I confused myself of frames. Yeah. And the existence of this right adjoint makes things simple about the decomposition because it's a general theorem of category theory that if you have two poset, p and q, let's say a map this way, let's say f, and another map the other way is called a g. And I'm supposed that f is left at one to g. That means that f of x, that we have this equivalence. Then this is well known. You can compose f with g. So you have composed with g and call it sigma. This is a map from sigma. This is a map from p to p. This is a closer operator on p. So closer operator means that it is, of course, a quiverient in order preserving. So you have and that you have sigma square of x equals sigma x. And the composite the other way, fg, let's call it a row, which is just from q to q, is, I call that a co-monadic operator. It's the dual of being a closer because a closer operator can be called also a monadic. So a co-monadic operator is just a dual. It's an order preserving operator that satisfies. And you have the compositions. If I look at the set of sigma close things in p, q row as the row open elements, if you like. That's the fixed point of row. Then you have the composition of f that has a map first to p sigma in order to q row, an inclusion here, and here an isomorphism. You have the compositions like this. And it is not similar to the composition the other way for g. So for g, you have an inclusion, p sigma, an isomorphism, and q row for a fixed point. I mean, in some time in abstract Galois theory, the certain thing could be called a Galois connection. And if you have a Galois connection, you get a bijection between the close things on the two sides. And that's really no more than that. Now, that's why I call that monadic and co-monadic, except that in the situations of having two categories and two functions like this, you have a monad and a co-monad. And this is the composition. There is a decomposition, but the middle terms is not an isomorphism in general. So that makes things more complicated. It happens that if to make it short, in the case of an isomorphism of pi between two locales. So I will write pi like this, and you have the right advantage pi over star. Then if you compose pi over star and pi this way, you call it sigma. So it's an operator from A to A. But in addition to the conditions that I wrote for a closer operator, it preserves meats. Simply because pi is in a monorphism difference, so it preserves meats by definitions in a monorphism. And pi over star being a right adjoint, it preserves intersections. So an operator, a monadic operator which preserves meats is called a nucleus in the two-year frame. And there is also a co-monadic operator which preserves meats for the same reason. I'm going to call it co-nucleus, but I have not been able to check with people developing nuclear friends if this is the standard name for this notion. But I'm going to call it co-nucleus. It seems to be a reasonable name. And then if you have a nucleus, you could look at the fixed points. And I'm going to call it a sigma. And the fixed point of a nucleus from a local. It's a po-set. It's a local again. And there is a quotient map that sends x into sigma x, the closure. So a sigma, the fixed point, is a local. And this quotient map, q, is a morphosome of local. Thank you very much. Thank you for it. Yes, thank you. From the beginning. So it's a quotient frame. Thank you. Okay. Maybe I should just stress the fact that this is not a sublocale of A. This is a quotient locale. I mean, in the sense that the union here is not the usual union because the union of closed things is not closed. So in the union here, the union of a family of closed elements with respect to sigma, okay, that's the closure of the... So this is not a sublocale, but this a sigma is, of course, included in A because by definition this is the set of fixed points. So, but the inclusion is the right edge one to q, okay? But the inclusion does not preserve union. However, the quotient map, I'm sorry, the quotient map is a morphosome of the frame. And A sigma is a subframe of A. So in the context of topological spaces, when you have a sub-space of a topological space, is it a situation of what you consider that the frame of opens in y is obtained like this from the frame of opens in x? No, we should not... I don't know if I understand your comment correctly. But we should not think of sigma as like the closure of... the usual closure on the subset of the topological space because the usual closure operator does not preserve intersection. The anterior preserve intersection, but not the closure, okay? So, okay, so that's... This is really what people now call a group-dictopology. This is what the group-dictopology is. A group-dictopology is exactly exhibiting an operator like that on the frame with which preserve meets. And this is like the sheaves with respect to the group-dictopology. Okay, so, and on the other side, you have B arrow, the fixed point of the cone materials. And now the inclusion is... This is a frame and the inclusion is a map of frame. This is a sub-frame. The inclusion preserves the union and there is an algorithm between the two just induced by the map part. So, there is a theorem here which is that there is a projection the quotient frame of a frame A there is a projection between the quotient frame and the nucleus. And there is a dual theorem saying that there is a projection between the sub-frame and the cone nucleus. Okay, just completely dual. Now, I want to talk about the free frame. If frames are really rings, I mean, there should be free one. So, what are they? Because in commutative algebra, the polynomial ring is playing an important role, so you would like to know what are the free frames. Now, I should say that in commutative algebra, the free algebra, the free commutative algebra on the set of generator is constructed actually in two steps. So, you have the category of commutative rings and you can forget that the ring has an addition and you get into the category of commutative monoid. So, there is a forgetful function from commutative ring to commutative monoid where you forget and you can forget again if you like and you get into the category of sets, so forget. Now, each of these two functions has a left-hand joint. First, you can generate a free commutative monoid from a set and second, you can generate a free commutative ring from commutative monoid. This one takes a set X and it could be defined like I think it's sometimes denoted Xn. Okay, that's a notation. We'll make a notation, I guess, for the free commutative monoid on the set n. So, these are commutative monobiles in Xn. And the second one, you take n something here. Okay. And you take the group, actually the monoid ring of it, Zn. So, maybe I should write it like this. So, the free commutative ring on the set can be obtained by combining these two steps. Now, I want to stress the fact, I mean, the first step is just normal, but the second step has something peculiar about it because Zn is Z tensor n. Actually, it's the free aberrant group generated by the set n. You forget that n has a monoid structure. You take, you linearize n and that's it. This object is an aberrant group by construction. Let's say you could denote it Z to the n if you like. So, you take the free aberrant group and then it happens that this free aberrant group has a product. We all know that. And this product gives the free aberrant group a ring structure and that's the adjoint of the forgetful factor. Now, this is a situation which happens pretty often in mathematics. It is when there is a distributivity law, you see. There is a distributivity law between the free aberrant group and the free monoid function. And if you combine them, you get a distributivity law or maybe a distributivity law, distributivity law or a commutation law. If I had time, I could make it more precise. Let's say, think of the free aberrant group function as a function from set to set, just because I want to see it as an end of function. So, you take the free aberrant group function and you forget it's an aberrant group, you get a set. This is an example of a monad. So, it has a multiplication. It's called a monad and there's a unit. There is also the free commutative monoid function. Let's say Cm. Again, I forget it's a monoid. I see it as an end of function of set to set and you have a monad structure on it. That comes simply because Cm is the composite of a left adjoint with a right adjoint. You get this monad structure. It's like the monadic and the commonadic operator before. Now, it happens that the composite of Cm, if you compose Cm with ad, that's a monad. Because this is the free commutative ring function. So, it must have a monad structure. So, how is it that the composite of two monads is a monad? This was observed by John Beck. It's true that the composite of two monads is a monad. Years ago, so you have a category C and you have, let's say, two monads. And you want to know if, let's say, A composed with B is a monad. So, what do you need to get a product on A composed with B? You would need to say, well, I'm going to do that. I want to exhibit a map like this from the significations of A and the significations of B. But compositions of Fandor Fankler is not commutative. So, you cannot switch A and B. You would like to switch A and B and replace that by, and then you will be in business because you will be able to use the significations on A and the significations on B to get back to A and B. But you need a switching operator. And the switching operator will go from B A to A B. And the switching operator, it's a little bit like a Young-Baxter operator, if you like, but it doesn't need to be fully, oh, it doesn't need to be fully a Young-Baxter operator. But it's a commutation operator which is saying that if you were to do this construction the other way around, in other words, suppose that you're by accident, by distraction, you want to construct this thing and you first state the three-a-billion group followed by the three-monode. You won't get what you want, right? But you will be able to rearrange everything by there will be a commutation operator and this is called distributivity or distributive law. This is so-called distributive law by John Beck. And I'm sorry, distributive law is a precise notion. It's not only an operator like this, but it must satisfy some compatibility relation with the multiplications of A and B. So I don't have the time to write it down. Now, in the case of commutative rings, say there is a distributivity law and in the case of frames also, there is a distributivity law. So in other words, you can construct the free frame in two steps on a poset. So I'm going to construct the category of frames and I forget that there is an addition. I forget Suprema. So we get into lower semi-latest. Lower semi-latest are just an infimum operation, the finite infimum. You don't have a Suprema or a general infimum and you have a unit one. So this is what defines the lower semi-latest. So you can forget that the frame has an addition and then keep forgetting and goes into the category of posets. Now, this functor has a left edge wind which consists into adding finite means to a poset. The construction is very explicit and this other functor also has a left edge wind and this left edge wind is very simple because it coincides with the left edge wind from Sup lattices, right Sup lattices for the category of complete lattices and Supreserving maps and then there is a forgetful functor here and the left edge wind is, I call it D. D of a poset P is the set of lower sections of P. This is a downward section or a Cribble of P. So it means that if X is in S and Y is smaller than X then Y is in S. Now, this poset has Suprema. It's the left edge wind to the forgetful functor from the category of complete lattices with Suprema to poset. There is a left edge wind and here it is. This is the same D. So I should finish with that with the analogy. So we have sets, a billion groups and commutative rings. So that's a sequence. We know that if you want to study commutative rings you need to study a billion groups first, our modules or something like that. But now we are going to expand that. We take posets and instead of a billion group I take Sup lattices and here I put frames and the category of Sup lattices is really like the category of a billion groups. We know that the category of a billion group stands for product, harm, you know, and it's additive. The additivity of the category of a billion groups means that the co-product, finite co-product and the product coincide. It's called the direct sum. The Sup lattices is the same thing. That's additive except that here it's even stronger. Any co-product of Sup lattices is also the product. It's very strongly additive. It's the presence of infinitary operations that makes this strongly additive. And it is symmetric monoidal clothes. And a frame is a commutative ring, some special kind actually of commutative ring in the category of Sup lattices. So there is a general scheme here that I think Duleen has suggested in a paper that maybe algebraic geometry could be generalized to any situation where you start with a symmetric monoidal category which is additive in some sense and you consider a commutative monoidal object in there and deform your ring and you take the opposite and deform your scheme. And these things just follow this pattern completely. In some sense, we could say that the theory of locale is really like the theory of schemes. But it is... The monoidal category, do you want to frame the particular object? Yes, of course. Yes, that's true. Frame our particular object. And in fact, you are focusing on the particular object in that category. That's true. The general rings here with Sup lattices are called quantale. They don't need to be commutative. And the commutative one are called commutative quantale. That's exactly true. But there is a nice connection between the two. The intrusions of frame into commutative quantale has both a left and a right adjoint. It makes life very nice. It's very easy to transport theorems about commutative quantales to theorems about frame using this adjunction. I guess I should stop here. Thank you very much.