1/4 A Crash course in topos theory : the big picture
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Joyal, André
About half of the topos theory of SGA4 is devoted to categorical generalities. They are now subsumed by the modern theory of (locally) presentable categories. I will sketch this theory, stressing the results that are important for topos theory. The category of complete lattices and sup-preserving maps is a toy example of this theory.
 I would like to thank the organizer for asking me to give a course in topos theory that courses I share with Olivia. We had some discussions with Olivia and me on what to do. I 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 Grosjean Dzik and his students. Maybe it's time to understand what's going on. I will be separating my course in three parts. There are four hours. It's a bit of a problem because it's difficult to divide four by three. I will do my best. I will start with the theory of locales. I will start with the theory of locales. I will begin with topos theory for the second part. I will do again some topos theory tomorrow. I will finish with the topos theory. It's a bit presumptuous of me to believe that I can describe these three theories in four hours. I will make some kind of sketch. Sometimes I will give proof when the proof is really easy. If the proof is a bit difficult, I will discuss the idea of the proof. The theory of locales was invented in the 1950s. Many of the things that happened in topos theory can be already seen in the theory of locales. The theory of locales is a good start for topos theory. It's very elementary. It's quite elementary. It's much simpler than topos theory. Many of the phenomena that happens in topos theory can be seen there. If you know the theory of locales, sometimes when you learn topos theory, you say, aha, that's something I know with the theory of locales. It's the same thing. When learning a subject, very much on the idea that the bottom approach, not the top down. Unfortunately, books in topos theory now start straight with the topos theory. They give a chapter, maybe chapter 9, chapter 10, the theory of locales. I very much recommend if you read these books to go straight to the theory of locales, maybe chapter 10 or chapter 9, and then start to read the beginning. Maybe a little bit of philosophy to start to begin about geometry and algebra. Well, algebraic geometry is precisely the subject about the interactions 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. Let me just look at this a little bit to start. 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 a sub-abroad variety by describing sub-object by equations. So this is a big discovery. I guess this is due to the cat 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 bad action between, let's say, if you have a ring A, then you have the spect of A. And there is a bad action between the map from x with respect of A and the ring maps from x into map xr. I hope I'm getting it right. Okay. Now, this duality between the rings and space does not work perfectly in the sense that very often it's not 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 notion of space, a notion zero of spaces, and a notion zero of rings. And you have a counter, 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 notion 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 sides 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 locales, 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 Sirpinski space. So the Sirpinski 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 Sirpinski space is that if u is an open subset of a space x, maybe I should write it like this, u is an open subset. Then there is a unique map, a continuous map from x to the Sirpinski 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 u, zero, otherwise. And so there is a projection between a map of x into the Sirpinski space and the open set of x. And so mapping the Sirpinski space into the Sirpinski space gives you all the information about the open subset of a fix. 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 so so what are the operations on the Sirpinski space? If you follow the general ring, I mean, we are interested in operations, the basic operations for the general ring are addition and product and multiplication. So what are the operations that exist on the Sirpinski space? Well, general operations, well, I can give you a couple of them. Maybe I should write just s. 0 1 is a pull set. So you can look at Suprema and Enfima on this pull set. So that gives you, for example, the Suprema, Supremum of two things will 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, the distributivity law. So of course, you have much more relations, many more relations than just distributivity law. But also you can consider infinite operations because you can take infinite powers of the Sirpinski space. It's also the political space, just the product apology. 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, to be, actually, it's called a frame because we are, 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 distributivity law distributivity law holds. In this equation, it's important to observe that the, the infimum, which is called the meat, this is the operation here is called meat of X and Y is derivative over arbitrary supremac. And this is very important for the notions of a frame. And this notion was introduced by Erasmus, I think. Excuse me. So the first example of a frame is of course the lattice of open subset of a particular space. And maybe I should see what a lattice is. A poset P v l is a lattice. If it is a complete lattice, if it admits a supremac, arbitrary supremac and infima. So that's the notions of a lattice. And it's important to observe that it's enough to have supremac or infima. Because the infimum, the infimum of a subset, so if you have a subset S of the lattice l, you can express the infimum 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 typical space is closed under suprema, because the suprema is the union, the supremum of the 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 infimum too, and the infimum 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 infimum 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 two political space give you a map in the other direction, which we could call f up or 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 frame A and B. So Fee and A and B. It is a map which preserve the operations arbitrary Suprema and finite and FEMA 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 frame is x itself is the unit is the unit element for the intersection. So this is how the notions of morphisms of 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 r m for the category of frames and morphisms. And you get a functor that goes in this direction from top and let's call it O. It's a contra variant functor. Maybe a word about the terminology here because some people say that a contra variant functor is a contra variant functor. I'm sorry, this is a co variant functor. Co variant functor is the same as a contra variant functor from C to D. So when you say that a functor is contra variant, you don't need to write up. 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 contra variant 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 contra variant map. There are two possibilities. Okay, so here you have a contra variant functor. It turns out that this functor has a contra variant 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 to apologize its home 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 the of the of a point. And you define the notions of a point of the frame by looking at morphism from the frame to one. And you get an adjunction between two between two contra variant functor. Maybe I should write explicitly the form of the adjunction. So, so if you have x at the topological space and a frame, then you can look at arm of of A into O x. So this is in the category of frames. And this will be in projection with the continuous map from the 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 is 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 sometime pointless topology, because the points in a local are not that important in the theory of locales. They are important, but some locales don't have points. And they are important for in the theory. So I will maybe I don't have any time to give to discuss examples, but I just tell you that some locales don't have points. Okay. Okay, so what is how does it work, the theory of locales? Well, a look a frame. Oh, I forgot to tell you what a local is in this picture. The local 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 local is a fiction. The reality is that you 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 frame. It's a little bit like in Azerbaijan geometry. Sometime in the theory of rings, it's very good to think of a ring as as an affine scheme. And so we but in this case, there is an actual notions of a general notions of a scheme, which is partly independent from, well, not exactly from the theory of commutative rings. So, so this factor here, which was a contra variant, once you use this duality, you get actually a pair of covariant functions. If you change the covariance, if you replace frame by local, what you get here is two covariant functions. Yes. Well, in principle, they are defined here. We could call that a character, you see, of the local. The local is like a ring. And this is like the base ring. And you look, 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 local is a character. Maybe I could say a little bit about it, because yes, right. But maybe I can say, I can say a little bit about the notions of a point of a local 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. Okay. 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. The 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. Okay. So what is so special about this open set is that they are meet irreducible. So if I call u the k of phi, the maximum element in the kernel, u is meet irreducible. And this mean what? This mean that if you have, 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. Okay. 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 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 subobjects in the locale. This is just the opposite lattice. Okay. Is joined irreducible. So it's a, it's a joined irreducible. So joined irreducible means that if f is f1 union f2, then f 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 subobject 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 vertical space is that the morphisms of locale, they produce closed subobjects that are irreducible. And I don't know if this answered the question. Okay. Yeah. In, in a topological space, right, this is true. Yeah. Yeah. Yeah. That's an example of of closely reducible, but there are in a topological space sometime closely reducible that won't have that don't have a generic point. This is called or a generator. Okay. Yeah. Yeah. Okay. Now. Okay, 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 community rings. So, but it's somehow it is simple. It is simple. It's a kind of simple versions of the theory of community rings. So the category of frames is an algebraic category in a sense it is, 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 a morphisms of frame, you can decompose it into three morphisms. So write five. Like in the theory of rings or whatever. Okay. First, you take the image of the morphism, you get here a subframe. That's the notion of subframe. There's an equivalence relation on a defined like the usual, the congruence. There is a notion of congruence in the theory of frames. Okay. 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 sub, the sub-object of the product, but it should be a subframe of the product. That's all. Okay. 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, injective morphism. Okay. 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 local. Okay. Let's call it z, a quotient. So what is the notion of quotient local? The notions of quotient of cow 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 morphisms of frames has a decomposition as a quotient followed by an embedding. Okay. Okay. 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 locales. It's just an order preserving map or a covariant map between two poset in the category of poset. And this is just this follow from the fact that five preserve Suprema. Any map preserving Suprema between two complete lattice has a right adjoint. It's a, I'm sorry, which I will write our star from B to A. Oh, I'm sorry. I confused myself of frames. Yeah. Okay. Okay. And the existence of this right adjoint makes things simple about the composition 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. Okay. And I'm supposed that F is left as 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 equivalent 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 common adjoint operator. It's the dual of being a closer because a closer operator can be called also a monadic operator. So a common adjoint operator is just a dual. It's an order preserving operator that satisfy. And you have the compositions. If I look at the set of sigma flows things in P, and Q row as the row open elements if you like, or that's the fixed point of row. Then you have the composition of F has a map first to P sigma in order to Q row and inclusion here. And here now the morphism. That's you have the compositions like this. And there is a similar to composition the other way for G. So for G you have an inclusion P sigma and as a morphism and Q row for the fixed point. I mean in sometimes in abstract Galois theory, such a thing could be called a Galois connection. And if you have a Galois connection, you get a bad action between the closed things on the two sides and that's really no more than that. Now that's exactly 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 another morphism in general. So that makes things more complicated. It happens that if to make it short in the case of a morphism of phi between two locales, so I will write phi like this and you have the right at 1 phi our star. Then if you compose phi over star in phi this way, you call it sigma. It's an operator from A to A. But in addition to the conditions that I wrote for the closed operator, it preserves meats. Simply because phi preserves, it's in a morphism of friends, so it preserves meats by definitions of the morphism. And phi over star being the right adjoint always preserves intersections, so we put closer in here. So an operator, a monadic operator which preserves meats is called the nucleus in the two-year frames. And there is also a co-monadic operator which preserves meats for the same reason. I'm going to call it co-monocleus, 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-monocleus. It seems to be a reasonable name. And then if you have a nucleus, you could look at the fixed points. I'm going to call it asigma. And the fixed point of a nucleus, a formula of cal. It's a post-sat. 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 morphism of local. Is it a frame or a question? Thank you for it. So it's a quotient frame. Thank you. Okay. Maybe I should just stress 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 the family of closed elements, union 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 the... 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 closed map is a morphism of the frame. A sigma is a subframe of A. So in the context of topological spaces, is it... So when you have a sub-space of a topological space, is it a situation of what you consider that is 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 a closure of the usual closure on the subset of the topological space because the usual closure operator does not preserve intersection. The interior preserve intersection, but not the closure, okay? So, okay, so that's... This is really what people now call a group of the topology. I mean, this is what the group of the topology is. A group of the topology is exactly exhibiting an operator like that on the frame which preserve meets. And this is like the sheaves with respect to the group of the topology. Okay, so on the other side, you have B.R.O., the fixed point of the cone of the cone of the tail. And now, the inclusion is... This is a frame, and the inclusion is a map of frame. This is a subframe. The inclusion preserve the union, and there's an atom of it in between the two, just induced by the map bar. So, there is a 2RM here which is that there is a projection between the quotient frame of a frame A. There's a projection between the quotient frame and the nucleus. And there is a dual theorem saying that there's a projection between the subframe 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 three 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, Burbeck notation, I guess, for the free commutative monoid on the set n. So, these are commutative monomials 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 function. 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 function and the free monoid function. And if you combine them, you get a distributivity law, or maybe a 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 functor as a functor from set to set, just because I want to see it as an end of functor. So, you take the free aberrant group functor, and you forget it's an aberrant group, you get a set. Okay? This is an example of a monad. So, it has a multiplication. It's called a monad. There's a unit. And there is also the free commutative monoid functor. Let's say, CM. Again, I forget it's a monoid. I see it as an end of functor 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 the right adjoint. You get this monad structure. It's like the monadic and the common adjunct operator before. Now, it happens that the composite of CM, if you compose CM with ad, that's a monad. Okay? Because this is the free commutative ring functor. So, it must have a monad structure. So, how is it that a composite of two monads is a monad? This was observed by John Beck. One is it true that a composite of two monads is a monad? Yes, it is true. It was observed 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 composite with B is a monad? So, what do you need to get a product on A composite with B? You would need to say, well, I'm going to do that. I want to exhibit a map like this from the simplifications of A and the simplifications of B. But compositions of functor is not commutative. So, you cannot switch A and B. You would like to switch A and B and replace that by, by, and then you will be in business because you will be able to use the simplifications on A and the simplifications 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 this switching operator, it's a little bit like a Young-Baxter operator, if you like, but it doesn't need to be full. 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 take the three-a-billion group followed by the three monoid. 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. I'm sorry. And distributivity law is a precise notion. It's not only an operator like this, but it must satisfy some compatibility relations with the simplifications of A and B. Okay? So I don't have the time to write it down. Okay? Now, in the case of commutative rings, see there is a distributivity law. And in the case of frames also, there is a distributive law. So in other words, you can construct the three frame in two steps on a post-set. 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 lattice. Lower semi lattice are just an enphemum operations, the finite enphemum. You don't have Suprema or general enphemum and you have a unit one. So this is what defines the lower semi lattice. So you can forget that the frame has an addition and then keep forgetting and goes into the category of post-sets. Now, this counter has a left edge wind, which consists into adding finite meat to a post-set. The construction is very explicit. And this out of functor also has a left edge wind. And this left edge wind is very simple because it coincide with the left edge wind from Sup lattices. All 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 post-set, P, is the set of lower sections of P. S is a downward, let's call also 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. Okay. Now, this post-set has Suprema. I mean, it has Suprema and it's the left edge wind to the forgetful functor from the category of complete lattices and Suprema to post-set. There is a left edge wind. And here it is. You see, this is the same D. So there is, I should finish with that, with the analogy. So we have sets, a billion groups, and commutative rings. Okay. So that's a sequence. And we know that if you want to study commutative rings, you need to study a billion groups first, or modules or something like that. But now we are going to extend that. We take post-sets 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 has tensor product, home, 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 infinite operations that makes this strongly additive. And it's, 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 Dooling 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 local is really like the theory of schemes. Yes, of course. Yes, that's true. Frame a particular object. That's true. That's true. The general rings here with Sup lattices are called quanta. That's, they don't need to be commutative. And the commutative one are called commutative quanta. That's exactly true. But there is a nice connection between the two. The inclusion of frame into commutative quanta has both a left and a right adjuvant. And it makes life very nice. If you, it's very easy to transport theorems about commutative quanta to theorems about frame using this adjunction. I guess I should stop here. Thank you very much.
 I would like to thank the organizer for asking me to give a course in topos theory that courses  I share with Olivia. We had some discussions with Olivia and me on what to do. I 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 Grosjean Dzik and his students. Maybe it's time to understand  what's going on. I will be separating my course in three parts. There are four hours. It's  a bit of a problem because it's difficult to divide four by three. I will do my best.  I will start with the theory of locales. I will start with the theory of locales. I will begin  with topos theory for the second part. I will do again some topos theory tomorrow. I will finish  with the topos theory. It's a bit presumptuous of me to believe that I can describe these three  theories in four hours. I will make some kind of sketch. Sometimes I will give proof when the proof  is really easy. If the proof is a bit difficult, I will discuss the idea of the proof. The theory  of locales was invented in the 1950s. Many of the things that happened in topos theory can be  already seen in the theory of locales. The theory of locales is a good start for topos theory. It's  very elementary. It's quite elementary. It's much simpler than topos theory. Many of the  phenomena that happens in topos theory can be seen there. If you know the theory of locales,  sometimes when you learn topos theory, you say, aha, that's something I know with the theory of  locales. It's the same thing. When learning a subject, very much on the idea that the bottom  approach, not the top down. Unfortunately, books in topos theory now start straight with the topos  theory. They give a chapter, maybe chapter 9, chapter 10, the theory of locales. I very much  recommend if you read these books to go straight to the theory of locales, maybe chapter 10 or  chapter 9, and then start to read the beginning. Maybe a little bit of philosophy to start to begin  about geometry and algebra. Well, algebraic geometry is precisely the subject about the  interactions 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. Let me just look at this a little bit to start.  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 a sub-abroad  variety by describing sub-object by equations. So this is a big discovery. I guess this is due to  the cat 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 bad action between, let's say,  if you have a ring A, then you have the spect of A. And there is a bad action between the  map from x with respect of A and the ring maps from x into map xr. I hope I'm getting it right.  Okay. Now, this duality between the rings and space does not work perfectly in the sense that very  often it's not 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 notion of space,  a notion zero of spaces, and a notion zero of rings. And you have a counter, 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 notion 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 sides 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 locales,  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 Sirpinski space.  So the Sirpinski 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 Sirpinski space is that if u is an open subset of a space x,  maybe I should write it like this, u is an open subset. Then there is a unique map,  a continuous map from x to the Sirpinski 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 u, zero, otherwise.  And so there is a projection between a map of x into the Sirpinski space  and the open set of x. And so mapping the Sirpinski space into the Sirpinski space gives you all the  information about the open subset of a fix. 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  so  so what are the operations on the Sirpinski space? If you follow the general ring,  I mean, we are interested in operations, the basic operations for the general ring  are addition and product and multiplication. So what are the operations that exist on the Sirpinski space?  Well, general operations, well, I can give you a couple of them. Maybe I should write just s.  0 1 is a pull set. So you can look at  Suprema and Enfima on this pull set. So that gives you, for example, the Suprema,  Supremum of two things will 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, the distributivity law.  So  of course, you have much more relations, many more relations than just distributivity law.  But also you can consider infinite operations because you can take  infinite powers of the Sirpinski space. It's also the political space, just  the product apology. 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, to be, actually, it's called a frame because  we are, 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 distributivity law  distributivity law  holds.  In this equation, it's important to observe that the, the infimum, which is called the meat,  this is the operation here is called meat of X and Y is derivative over arbitrary supremac.  And this is very important for the notions of a frame. And this notion was introduced by  Erasmus, I think.  Excuse me. So the first example of a frame is of course the lattice of  open subset of a particular space. And maybe I should see what a lattice is.  A poset P v l is a lattice. If it is a complete lattice,  if it admits  a supremac,  arbitrary supremac and infima.  So that's the notions of a lattice. And it's important to observe that  it's enough to have supremac or infima. Because the infimum, the infimum of a subset,  so if you have a subset S of the lattice l, you can express the infimum 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 typical space is closed under  suprema, because the suprema is the union, the supremum of the 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 infimum too, and the infimum  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 infimum 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 two political space give you a map in the other direction,  which we could call f up or 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 frame A and B. So  Fee and A and B. It is a map which preserve the operations arbitrary Suprema  and finite and FEMA 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 frame is x itself is the unit is the unit element for the intersection.  So this is how the notions of morphisms of 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 r m for the category of frames and morphisms. And you get a functor  that goes in this direction from top and let's call it O. It's a contra variant functor.  Maybe a word about the terminology here because some people say that a contra variant functor  is a contra variant functor.  I'm sorry, this is a co variant functor.  Co variant functor is the same as a contra variant functor from C to D.  So when you say that a functor is contra variant, you don't need to write up.  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 contra variant 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 contra variant map. There are two possibilities. Okay, so here you have  a contra variant functor. It turns out that this functor has a contra variant 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  to apologize its home 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 the of the of a point. And you define the notions of a point  of the frame by looking at morphism from the frame to one. And you get an adjunction between  two between two contra variant functor. Maybe I should write explicitly the form of the adjunction.  So,  so if you have x at the topological space and a frame,  then you can look at arm of of A into O x. So this is in the category of frames.  And this will be in projection with the continuous map from the 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 is 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 sometime pointless topology, because the points in a local are not that important in the  theory of locales. They are important, but some locales don't have points. And they are important  for in the theory. So I will maybe I don't have any time to give to discuss examples, but I just  tell you that some locales don't have points. Okay. Okay, so what is how does it work, the  theory of locales? Well, a look a frame. Oh, I forgot to tell you what a local is in this picture.  The local 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 local is a fiction. The reality is that you 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  frame. It's a little bit like in Azerbaijan geometry. Sometime in the theory of rings,  it's very good to think of a ring as as an affine scheme. And so we but in this case,  there is an actual notions of a general notions of a scheme, which is partly independent from,  well, not exactly from the theory of commutative rings. So, so this factor here, which was a  contra variant, once you use this duality, you get actually a pair of covariant functions. If you  change the covariance, if you replace frame by local, what you get here is two covariant functions.  Yes.  Well, in principle, they are defined here. We could call that a character, you see, of the  local. The local is like a ring. And this is like the base ring. And you look, 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 local is a character. Maybe I could say a little bit about it, because  yes, right. But maybe I can say, I can say a little bit about the notions of a point  of a local 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. Okay. 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. The 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. Okay. So what is so special about this open set is that they are  meet irreducible. So if I call u the k of phi, the maximum element in the kernel, u is meet  irreducible. And this mean what? This mean that if you have, 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. Okay. 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 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 subobjects  in the locale. This is just the opposite lattice. Okay. Is joined irreducible. So it's a,  it's a joined irreducible. So joined irreducible means that if f is f1 union f2, then f  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 subobject 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 vertical space is that  the morphisms of locale, they produce closed subobjects that are irreducible. And  I don't know if this answered the question. Okay.  Yeah.  In, in a topological space, right, this is true. Yeah. Yeah. Yeah. That's an example of  of closely reducible, but there are in a topological space sometime closely reducible  that won't have that don't have a generic point. This is called or a generator. Okay. Yeah. Yeah.  Okay. Now. Okay, 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 community rings.  So, but it's somehow it is simple. It is simple. It's a kind of simple versions of the theory of  community rings. So the category of frames is an algebraic category in a sense it is, 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  a morphisms of frame, you can decompose it into three morphisms.  So write five.  Like in the theory of rings or whatever. Okay. First, you take the image of the morphism,  you get here a subframe. That's the notion of subframe. There's an equivalence relation  on a defined like the usual, the congruence. There is a notion of congruence in the theory of frames.  Okay. 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 sub,  the sub-object of the product, but it should be a subframe of the product. That's all. Okay.  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, injective morphism. Okay.  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 local.  Okay. Let's call it z, a quotient. So what is the notion of quotient local? The notions of  quotient of cow 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 morphisms of frames has a decomposition as a quotient followed by an embedding.  Okay. Okay. 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 locales. It's just an order preserving map or a covariant map between two  poset in the category of poset. And this is just this follow from the fact that five preserve  Suprema. Any map preserving Suprema between two complete lattice has a right adjoint. It's a, I'm  sorry, which I will write our star from B to A. Oh, I'm sorry. I confused myself of frames.  Yeah. Okay. Okay. And the existence of this right adjoint makes things simple about the  composition 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.  Okay. And I'm supposed that F is left as 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  equivalent 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 common adjoint operator.  It's the dual of being a closer because a closer operator can be called also  a monadic operator. So a common adjoint operator is just a dual. It's an order preserving operator  that satisfy. And you have the compositions. If I look at the set of sigma flows things in P,  and Q row as the row open elements if you like, or that's the fixed point of row.  Then you have the composition of F has a map first to P sigma in order to Q row  and inclusion here. And here now the morphism. That's you have the compositions like this.  And there is a similar to composition the other way for G. So for G  you have an inclusion P sigma and as a morphism and Q row for the fixed point.  I mean in sometimes in abstract Galois theory, such a thing could be called a Galois connection.  And if you have a Galois connection, you get a bad action between the closed things on the two  sides and that's really no more than that. Now that's exactly 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 another morphism in general. So that makes things more complicated.  It happens that if to make it short in the case of a morphism of phi between two locales,  so I will write phi like this and you have the right at 1 phi our star. Then if you compose  phi over star in phi this way, you call it sigma. It's an operator from A to A. But in addition to  the conditions that I wrote for the closed operator, it preserves meats. Simply because  phi preserves, it's in a morphism of friends, so it preserves meats by definitions of the morphism.  And phi over star being the right adjoint always preserves intersections, so we put closer in here.  So an operator, a monadic operator which preserves meats  is called the nucleus in the two-year frames.  And there is also a co-monadic operator  which preserves meats for the same reason. I'm going to call it co-monocleus,  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-monocleus. It seems to be a reasonable name.  And then if you have a nucleus, you could look at the fixed points. I'm going to call it asigma.  And the fixed point of a nucleus, a formula of cal. It's a post-sat. 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 morphism of local.  Is it a frame or a question? Thank you for it.  So it's a quotient frame. Thank you.  Okay. Maybe I should just stress 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 the family of closed elements,  union 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 the... 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 closed map is a morphism of the frame. A sigma is a subframe of A.  So in the context of topological spaces, is it... So when you have a sub-space of a topological  space, is it a situation of what you consider that is 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 a closure of the usual closure on the  subset of the topological space because the usual closure operator does not preserve intersection.  The interior preserve intersection, but not the closure, okay? So, okay, so that's... This is  really what people now call a group of the topology. I mean, this is what the group of the topology is.  A group of the topology is exactly exhibiting an operator like that on the frame which preserve  meets. And this is like the sheaves with respect to the group of the topology. Okay, so on the  other side, you have B.R.O., the fixed point of the cone of the cone of the tail. And now,  the inclusion is... This is a frame, and the inclusion is a map of frame. This is a subframe.  The inclusion preserve the union, and there's an atom of it in between the two, just induced by the map bar.  So, there is a 2RM here which is that there is a projection between the  quotient frame of a frame A. There's a projection between the quotient frame and the nucleus. And  there is a dual theorem saying that there's a projection between the subframe 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 three 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, Burbeck notation, I guess, for the  free commutative monoid on the set n. So, these are commutative monomials 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 function. 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 function and the free monoid function.  And if you combine them, you get a distributivity law, or maybe a 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  functor as a functor from set to set, just because I want to see it as an end of functor.  So, you take the free aberrant group functor, and you forget it's an aberrant group, you get a set.  Okay? This is an example of a monad. So, it has a multiplication. It's called a monad.  There's a unit. And there is also the free commutative monoid functor. Let's say, CM.  Again, I forget it's a monoid. I see it as an end of functor 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 the right  adjoint. You get this monad structure. It's like the monadic and the common adjunct operator before.  Now, it happens that the composite of CM, if you compose CM with ad, that's a monad.  Okay? Because this is the free commutative ring functor. So, it must have a monad structure.  So, how is it that a composite of two monads is a monad? This was observed by  John Beck. One is it true that a composite of two monads is a monad?  Yes, it is true. It was observed 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 composite with B is a monad?  So, what do you need to get a product on A composite with B? You would need to say,  well, I'm going to do that. I want to exhibit a map like this from the simplifications of A  and the simplifications of B. But compositions of functor is not commutative. So, you cannot switch  A and B. You would like to switch A and B and replace that by, by, and then you will be in  business because you will be able to use the simplifications on A and the simplifications 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 this switching operator, it's a little bit like a Young-Baxter operator, if you  like, but it doesn't need to be full. 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 take the three-a-billion group followed by the three monoid. 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. I'm sorry. And distributivity law is a precise notion. It's not only an operator  like this, but it must satisfy some compatibility relations with the simplifications of A and B.  Okay? So I don't have the time to write it down. Okay? Now, in the case of commutative rings,  see there is a distributivity law. And in the case of frames also, there is a distributive law.  So in other words, you can construct the three frame in two steps on a post-set. 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 lattice. Lower semi lattice are just an enphemum operations, the finite enphemum.  You don't have Suprema or general enphemum and you have a unit one. So this is what defines  the lower semi lattice. So you can forget that the frame has an addition and then keep forgetting  and goes into the category of post-sets. Now, this counter has a left edge wind,  which consists into adding finite meat to a post-set. The construction is very explicit.  And this out of functor also has a left edge wind. And this left edge wind is very simple  because it coincide with the left edge wind from Sup lattices. All 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 post-set, P, is the set of lower sections  of P. S is a downward, let's call also 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. Okay. Now, this post-set has Suprema. I mean,  it has Suprema and it's the left edge wind to the forgetful functor from the category  of complete lattices and Suprema to post-set. There is a left edge wind. And here it is.  You see, this is the same D. So there is, I should finish with that, with the analogy.  So we have sets, a billion groups, and commutative rings.  Okay. So that's a sequence. And we know that if you want to study commutative rings, you need  to study a billion groups first, or modules or something like that. But now we are going to  extend that. We take post-sets 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 has tensor product, home, 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 infinite operations that makes this strongly  additive. And it's, 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 Dooling 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 local is really like the  theory of schemes. Yes, of course. Yes, that's true. Frame a particular object. That's true.  That's true. The general rings here with Sup lattices are called quanta. That's,  they don't need to be commutative. And the commutative one are called commutative quanta.  That's exactly true. But there is a nice connection between the two. The inclusion of frame into  commutative quanta has both a left and a right adjuvant. And it makes life very nice. If you,  it's very easy to transport theorems about commutative quanta to theorems about frame  using this adjunction. I guess I should stop here. Thank you very much.
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