 So thank you. So the general idea that, in the same way that toposes are some kind of generalized topological space, c-star algebra are also objects that we want to think of as generalized topological space, so generalized locally compact space. And there is quite a long list of example of geometric objects that can generate either a topos or c-star algebra. So for example, you have foliation, you have dynamical system, graph, certain finite automaton, and so on. And it's quite a natural idea to try to understand if there is somehow a general procedure to go from topos theory to operator algebra or the other direction. And so the first way to answer this question is to notice that both are closely related to topological groupoid. So I won't use that aspect concretely in my talk, but it's somehow told us that the easy direction is to go from a topos to a c-star algebra to try to attach a c-star algebra to a topos, and that's what I'm going to explain today, how we can do that, why the other direction is at least very difficult and more probably completely impossible without additional structure on the algebra. So let just me explain a bit what I'm talking about. So OK, a c-star algebra is just a banar algebra with an until and now involution which satisfies those actions, so the involution will be denoted by star, and a typical example are the algebra of bounded operator over in the space. That's why they're called operator algebra. So in this case, the adjunction would be, if there's a star operation, will just be the adjunction of operator. And of course, you have the example of commutative algebra. So algebra of the form c0 of x for x, locally compact, out of topological space. And c0 of x is for the algebra of continuous function with complex value, which tend to 0 at infinity. So in this case, the star will be the complex conjugation. And it's a very classical result that I guess you all know that all commutative c-star algebra are of this form. And if you define morphism correctly, then you have an equivalence of category between locally compact topological space, out of topological space, and commutative c-star algebra. So let me give a first baby example of non-commutative algebra, which is not very interesting in itself, but which will be helpful to us near the end of the talk. So we will start from topological space. And we will assume that it had made a finite covering by open subset, which are all locally compact and out of. But x might not be out of itself. So for example, the typical thing I have in mind is you start from a topological space x. So for example, the interval one minus one, so it's compact in this case. But with a double point at the origin. So in this case, you have x, which is equal to union one, union new two, where they are both isomorphic to the closed interval. But their intersection is the interval minus the origin. So you have two origin. So in this case, you can associate a c-star algebra to this covering. So basically, you construct some kind of matrix algebra. So it has as many rows as the number of open subsets you're using. And the coefficients are given by function of c0 of the intersection of the corresponding open subset. And if you look closely to that, you will see that matrix multiplication is well defined for this. And this form an algebra. So you can define the involution by taking both complex conjugation and exchanging row and column. And it appears that there is a uniquely defined norm that makes this into a c-star algebra. And you can do something similar. The cover is infinite. But you need to do a bit more analysis as the matrix product of infinite matrix won't be defined in general. So and something you can say that if x was actually outside of itself, then this algebra is morita equivalent to c0 of x. I will define what morita equivalence means for c-star algebra a bit later. It's very close to morita equivalence of rings. So for example, in this case, if you do this construction, you will get a function from the interval to 2 by 2 matrices, such that off-diagonal coefficient vanish at 0. So if you specialize at any point, you will get 2 by 2 matrices, which are morita equivalent to c. So you basically get one point for each non-zero element of this interval. But if you localize at 0, you get c square, which corresponds to the two-point space. You can see into the algebra the double point at 0 region. So this is not a very interesting example from the point of view of non-commutative geometry, but it can show how a non-commutative algebra can somehow end those space that are not themselves Ausdorff. So I will give a more interesting example. If you take basically any discrete group acting on the locally compact Ausdorff topological space, then there is a construction of a cross-product algebra. So it's generated by functions, or by element of c0 of x, and by formal element corresponding to the element of the group, which satisfies a classical algebraic relation. So the generator from the group multiplies as in the group algebra. And you can exchange the size between function and generator from the group using the action from the group of the space. And so you have, again, you can put a norm on this. You can generate a c-star algebra. Well, actually, you don't have a unique norm, but you can choose one and generate a c-star algebra. And what happened is that if the action of G is free and proper, then this algebra is actually more or less equivalent to the algebra of function on the quotient. So it's the same idea that if the quotient by the action of the group went well, then you actually don't get anything more if you see the c-star algebra as a summary of the equivalence. But in most situations, the cross-project will contain a lot more information than just the algebra of the quotient, which in general will be trivial. For example, if x is a point, then you will get the group algebra. The group algebra, but with some kind of L2 or L1 group algebra versions? So if the group, well, you will get the group c-star algebra. The group c-star algebra. What's the norm in this case? So you have two choice. Either you can take a universal norm, which is an infimum of all norms that make the supremum of all norms coming from all representation. Or you can take the norm coming from the less regular representation on L2. That's two different choice in general. And you have sometimes intermediate choice also. So to this situation of a group acting on a topological space, you can also associate a topos, of course. So it's a topos of a quivarion shift that has been introduced in the case of the scaling site this morning. And so it corresponds to the quotient of the topos by the action of the group in the two category of toposes. And it also appears that this topos of a quivarion shift will be equivalent to the topos of shift on the quotient. Basically, also, the action is free and proper. So we have the same situation. If the topos that we construct this way is actually the topos of shift over the quotient, then the algebra is more equivalent to the algebra of shift over the quotient. So this somehow suggests that it should be possible to define the algebra up to more reticouvenance directly from the topos. And so as I said, that's what we are trying to do. So the first point for this construction is that, using internal logic of toposes, we have a very good notion of continuous field of Hilbert space and c-star algebra and banner space over a topos. So we call them continuous field because that's a historical terminology. But morally, there are things that we would like to call field of Hilbert space, shiv of Hilbert space and shiv of c-star algebra and shiv of banner space. And concretely, they are defined just as the Hilbert space, c-star algebra and banner space object of the internal logic. So I will say explicitly what it means in a minute. So as the internal logic of a topos, as André Joïel explained yesterday, is in general interested logic. We need to be a bit more careful on what we call the Hilbert space. Because if you don't have the law of x to the middle, all the definition you know of real number basically give all different objects. So there is a lot of things that need to be made precise. So first thing is that complex number or real number, when they are used as scalar, the ring of scalar, because all these are vector space, real vector space in general. So we use what we call the continuous real number and continuous complex number. So also called dedekind real number. And they correspond to two-sided dedekind cuts. So we define a real number as a two-sided dedekind cuts. So there is a subtlety that norms cannot take value in continuous real number, because they have these little problems that they don't satisfy the supremum property. Like a bounded subset of continuous real number does not always have a supremum. And norms tend to be defined as supremum in general. So if we force our norm to take value in continuous real numbers, then we lose a lot of examples. For example, the algebra of bounded operator of a real-build space won't be a banner algebra with a restrictive definition like this. So we take what we call a upper semi-continuous real number, which are upper dedekind cuts, so one-sided dedekind cuts. And those have very poor algebraic property. But they have all supremum, so we can define norm with value in this. And they contain the continuous real number. So all the axiom of norm can be defined without problem. And the last point we need to make precisely what completeness means, because Hilbert space, banner space, and this algebra need to be complete. So we use the definition by Cauchy filter, because Cauchy sequence doesn't work very well without a countable or dependent axiom of choice. Yeah, and I forgot to mention, it's written that for Hilbert space, the norm are continuous, because they come from a bi-linear form. And to write down bi-linearity, we need to take value in continuous complex real number. Oh, sorry, continuous complex number. So when you define everything like this and you look at what happened for the topos of Shiva or topological space, you get the previously known notions, which I think they are due to fail, but I'm not exactly sure, of a continuous field of Hilbert space and what we call semi-continuous field of this algebra and banner space. So now we can make, for example, those Hilbert space objects into a category. So morphism between them will be globally bandied operator. So what does that mean? So operators are linear map, which admit an adjoint. Because without the law of X to the middle, it's not automatically true that any bandied linear map between Hilbert space have an adjoint. And globally bandied just mean that so they are internally bandied operator, but the norm is bounded by an external constant, so a globally defined constant. So if we have Shiva of Hilbert space over a topological space, this means what we think it is that the norm is globally bounded all the space and not just locally bounded, which would be the meaning of internally bandied. And so the result is that this category of Hilbert space and globally bandied operator between them is what we call a C star category. So it's a several object generalization of a C star algebra. And roughly what it means is that if you take one of those Hilbert space and look at its algebra from the morphism, it is actually a C star algebra. Consider externally. Yeah, consider externally. So this produce the first way to take a topos and associate a whole bunch of C star algebra, which is a good starting point, but it's not exactly what we want. So there is two problems. The first is that all those algebra that we produce are not more reticivalence or not even related to each other. Well, not as precisely as we would want. And the other is that if we look at all the example we had of situation where we actually want to attach a C star algebra to a topos, they don't give a good C star algebra. They give algebra that are a bit too big. Generally, the good C star algebra appears as a certain sub-algebra of those, but there is apparently no nice way to recognize them. So and the reason we get big algebra is actually very natural to understand. So if you look at the algebra of all operators of the ill-built space of infinite dimension, then this is an extremely complicated C star algebra, which is not more reticivalent to the complex number. It works if you are in the finite dimensional ill-built space, you'll get something more reticivalent to complex number. But in the infinite dimensional case, what you want to do is to look at the algebra of compact operator, which are those operators which can be approximated by finite rank operator. And so this is roughly why we get way too much element in our algebra, because we are looking at arbitrary operator on ill-built space that can be infinite dimensional. So I will make a small parenthesis. So today I'm concerned about what I call the topological case, because C star algebra correspond to locally compact topological space. But there is a completely parallel story which could make another story in which I will sum up in about one minute. So there is also the theory of fundamental algebra, which are specific C star algebra which correspond to measurable space. So commutative fundamental algebra is something of the form l infinity of x mu for some measured space and mu and measure, which is the algebra of almost every unbounded function with the subnorm, with the essential subnorm. So yeah, and the point is that b of h is a fundamental algebra, and it's a good fundamental algebra. It's more in the sense of more fundamental algebra, less marita equivalent to C. So in this case, b of h is a good choice. So that's why the story works better in the measurable case. So if we take a Boolean topos, then the C star category we get by looking at ill-built space over the topos is what we call the monotone complete C star category. And so it's almost the same as a fundamental category which are called W star category. And under a really reasonable assumption of existence of local measure over this Boolean topos, then we actually get a W star category. So we get fundamental algebra this way. And now the point is that any topos admit a covering by Boolean toposes. And if we choose a covering by Boolean topos such that the map is injective, we can actually think as those ill-built space over this Boolean cover as measurable field of ill-built space over the topos. And this is a way to attach fundamental algebra to any topos by choosing a Boolean covering, which correspond morally to some sort of generalized measure class. And we get an algebra which can be thought of some kind of generalized algebra of measurable function over the topos. So this approach, this measure theory approach can be applied to basically any topos, but it loses all the information of topological nature. So it's probably interesting for all the topos that come from algebraic geometry, so et al. topos or the scaling sites, and also two topos that are classifying topos for theories that are algebraic. I don't mean in the concrete sense, but that are algebraic in nature. Because those topos are very pro-algebraic topology, and they behave like, for example, the risky spectrum. And we cannot expect to have anything interesting by looking at continuous function on them with value in complex number. So let me come back to the topological space. So in this situation, the idea is that we want to restrict two topos that have nice topological properties that are analogous to locally compact ausdorff topological space. So let's, for a minute, look at a locally compact ausdorff topological space. So let's call it x. And assume we have a Hilbert space over x, which is denoted by h. So we can look at gamma of h, the set of global section of h. So h is a sheave. It's a sheave endowed with a structure of an Hilbert space internally. And so it's obviously come with a structure of a C of x module, where C of x is the algebra of all complex value at function on x. But because h is a Hilbert space object, it also has a scalar product that can externally, when on global section, can be seen as a pairing. If you take two global sections of gamma of h, then you can compute their scalar product. And it's going to be a complex valued continuous function on your space. So you get some sort of scalar product with value on C of x. And then you can consider the space gamma 0 of h of sections such that the norm, so the scalar product with themself, is actually in C0. So it goes to 0 at infinity. And this is what we call a right Hilbert C0 of x module. So I will write explicitly what it means. So because C0 of x is commutative, there is no real difference between right and left module. But I will write the definitions that work for non-commutative algebra too. So that's why I'm presizing that it's a right Hilbert module. So first, it's a right C0 of x module. So it has this scalar product which takes value in C0 of x. You can check that if you have two elements in gamma 0, then because of Cauchy-Schwarz, the scalar product is also C0. So we have those algebraic axioms. So it's linear in the second variable. And I actually mean linear additively too. And if you permute the variable, then you apply the star operation. So it will be somehow empty linear in the first variable. And so when you have the first three axioms, then this actually defines the norms on this space. And we ask it is complete for this norm. That the norm here, you really mean the supremum over all the points, I think. Yeah. Yeah. It's a norm in the algebra, so it's a sub-norm. So it's not a local norm. No, this is external. I was thinking about the norm of the c star algebra. And that's why you choose C0 because in C0 it goes to 0. So it will work if I choose to take all bounded functions, for example. But for what I will state in one second, it's important to have C0. So the point is that we have an equivalence of category between the category of Hilbert space over x and of Hilbert C0 of x module. Well, if we have chosen to take bounded functions and we'll get a continuous field of Hilbert space over the stone-church compactification of the space. So OK, this is actually the kind of the way we want to attach a c star algebra to a theorem. We want a theorem of this form. And so that's the theorem I'm going to talk about today. So I will state it first and then explain what it means. So it's what I call the Topos theoretic Grinjold theorem. So we start with a topos that satisfies a certain list of conditions that I'm going to explain in a few minutes. So it has to be separated. It has to be locally decidable. And its localic reflection has to be locally compact. Then we actually have a c star algebra that I will denote C0 of t, such that the category of Hilbert space over t is equivalent to the category of Hilbert C0 of t module, so those Hilbert space with a scalar product taking value in C0 of t. And so this C0 of t is well-defined up to Morita equivalence, up to unique Morita equivalence. So I still haven't defined what's Morita equivalence for C star algebra, but now I can do it. Two C star algebra are Morita equivalence. It's their category of Hilbert module equivalent. And there is an Adelaugus theory as for theory of ring. We can describe the Morita equivalence as a certain bimodule and so on. And so this last statement is completely tautological because I've described the category of Hilbert module over the algebra, so it's completely useless to write it down actually. So yes, so this is the theorem I want to talk about today. So it's probably just a nice way to get a C star algebra from certain topos that define nice topological condition. So I'm going to explain this condition, but first I want to mention that there is also a version with coefficient. So we can also take C to be a C star algebra over t, so a C star algebra is the internal logic. And then we can construct an ordinary classical C star algebra in set C some indirect with t, such that the category of Hilbert C module over t is equivalent to the category of Hilbert C cross t module. This C cross t is a natural. Yeah, it's a real concrete external C star algebra. And it's also unique up to, unique Morita equivalence for the exact same reason. But I will focus on the version without coefficient because the proof is basically the same, whether you use coefficient or not. So let me explain what those conditions separately locally decidable mean. So we'll start with locally decidable. So an object of topos is said to be decidable if we have internally that for all x, y in x, we have x equal y or x not equal to y, which is a form of flow of x to the middle, which of course does not hold in an arbitrary topos. So externally, it means that the shift or the object x times x can be decomposed into a co-product of two shifts or sub-objects, one which is the diagonal and one which is something else, a complement of the diagonal. So for example, if you have a locally connected topological space, then this decidability condition corresponds to the analytic continuation property. So for example, the shift of holomorphic function on some Rc over some complex manifold is decidable. So decidable objects are important because that's one where you can actually count the element or taking sum index over them. The point is that if you are not even able to tell if given two elements are equal or different, then it's completely impossible to count the element or to take a sum index by the elements. So for example, if you want to define the Hilbert space L2 of x for some set or some object of the topos x, well, you need x to be decidable. So either because you will define it as a set of square sumable sequence and to define sumable, you need the set to be decidable. Or even more clearly, L2 of x is the space which has one generator for each element of x. And those generators satisfy this condition that their scalar product is equal to 1 if they are equal and 0 otherwise. And in order that this makes sense, you need the set to be decidable. That's exactly what it says. So we say that the topos is locally decidable if you have enough decidable objects. So if every object can be covered by a decidable object. So locally topos, so for example, all topos of SHIV over a topological space are automatically decidable. All étendus or toposes of equivalent SHIV over etal groupoid are also locally decidable. And if we want counter examples and pre-shift topos, are most of the time not locally decidable. So they are locally decidable exactly if all the arrow of the corresponding category are epimorphism. So this is actually a condition that we like from the point of view of non-commutative geometry because most of the time when we want to attach the star algebra to a semi-group or to category, we want it to be left-cancelative, which is exactly what this condition of epimorphism mean. And yeah, also I actually don't know any example of a topos which is not locally decidable and which has ill-burnt space that don't come from its locally decidable part. So I actually think that this construction of looking at ill-burnt space over a topos is only interesting for locally decidable topos, but I haven't been able to prove it. I've just observed it on all examples. So this was for the locally decidable part. So it's a really weak condition that is absolutely not the most important in the theorem. The important condition is separated. So it's a condition which is due to Mordech and Fermiland. So geometric morphism between two topos is said to be proper if internally in T. So André Joyal explained yesterday that when you have a geometric morphism, you can think of it as a topos in the internal logic of T, which is morally the idea that you want to work fiber-wise. So you want to work with e fiber by fiber. And so we say that a morphism is proper if this internal topos is compact, where compact mean the localic reflection is compact so it's just the usual finite covering condition. So it's basically we are saying that something is proper if it's fiber-wise compact, but in a locally uniform way and for topological space, it's completely equivalent to the usual definition of a proper map. And so we say that a topos is separated if it's diagonal map is proper, which is analogous to the usual definition that something is separated if the diagonal map is closed with the exception that for topos is the diagonal doesn't have to be an inclusion so proper and closed became different and that's why we're using proper which is a better condition, better behaved. So for topological space, this is almost the same as the ordinary Osdorf condition. It's slightly stronger, the reason why it's different is because the product of toposes is not exactly the product of topological space. But for example, if you restrict to a locally compact space and it's exactly the same and all the T3, T4 and other Osdorf condition imply this toposteritic separated condition. But so for locally, for topological space, it's a very reasonable condition, but for general topos, it's an extremely restrictive condition. So for the topos of G-set, so set and dode with an action of G, the topos is separated if and only if the group is finite and if you consider a prediscreet topological group, then it means the group is compact. So it's very restrictive. And for general group oids, they correspond to the notion of proper group oids. So a group oid with action map is proper. So separated is a very restrictive condition and I will explain at the end of the talk how we can apply the term to non-separated topos in an interesting way. Okay, so let's come back to our theorem. So I've told you what separated and locally decidable mean and the locally-reflection is locally compact. So the locally-reflection is just, so you have the inclusion of the category of locale into the category of topos and it has a left adjoint, let me write it. So you have om tl which is isomorphic to the morphism from the localeic reflection of t to l where t is the topos and l is the locale. And for example, if you are considering a topos attached to a dynamical system, as I mentioned in the beginning, then this localeic reflection is the topological quotient that you quotient in the space by the action of the group as a topological space. So this last condition is also very rigged because for most interesting example, the localeic reflection will be actually simpler and a lot simpler than the space we are starting from. So it's called a Green-Jewel theorem because the Green-Jewel theorem is something that basically says the same thing for topological groupoid instead of toposes. So for proper groupoid, we have an equivalence of category between Hilbert's module over the reduced cistage above the groupoid and a continuous field of Hilbert's space over the groupoid and dode with an action of the for equivalent continuous field of Hilbert's space over the groupoid. So the point here is that because those two theorems exist both for groupoid and for topos, we know that when they both apply, they both give the same result because they are, well, obtumority equivalence, we have the same category of Hilbert module over this algebra. So this is what told us that this algebra, c0 of t is the correct algebra to attach to the topos in all example that we know. So, yeah, there is also a constructive version of the toposteritic version. So it's basically the same, we just need to add some adjective here and there. So separated stays the same. Locally decidable, well, we could keep the same but we can actually weaken the condition and only say that the category is generated by decidable objects, which is equivalent if we have classical logic but it's weaker in constructive logic. It has a site of definition. Yeah. C0 t is an external or an intact? External, external. External. And okay, so the actual thing that we really need to change is we need to add one condition of that the locality reflection is completely regular, which means that there is enough continuous function of it. So in classical mathematics, it's a consequence of separated and locality compact by the reason lemma but constructively it doesn't work very well. And there is another thing that we need to change is that C0 of t cannot exactly be constructive as a c-star algebra. We actually get it as a c-star category. So we can turn it into a c-star algebra in 90% of the time, for example, as soon as the base topos is itself locality decidable but in general it might be a small problem. So the point of having a constructive version of course is to have a relative version of the theorem. So now if I have a geometric morphism like this, which internally satisfies the condition of the theorems and I can relate Hilbert's module over t to Hilbert's c-module over e for some c-star algebra internal to e. So the reason why I was really interested in this constructive version, it's actually how I was planning to prove the theorem first by using some kind of the visage with an intermediate topos and proving it first for this morphism and then for this morphism in a way that I will explain in a minute but actually in the final proof I didn't need to do that so we could prove the classical theorem directly. So I will explain roughly how the proof works because it's actually really helpful to understand what is this algebra c0 of t. So the first part is I have proved an abstract characterization of c-star category which are of the form category of Hilbert module over c-star algebra. So this characterization is analogous to the fact that an Abellion category is a category of module over some ring if it has a projective generator of finite type and if it's co-complete. So basically if you have an Abellion category and an object inside it, you get a functor by taking om from this object to another object from your category to the category of module over the ring or from the morphism of the object and if you assume that this object is a projective generator of finite type then you get an equivalence of category. I think I should have checked that this is true but it sounds like an easy result. So I've proved the c-star categorical version only. So in order to have something similar for c-star category there is two difficult steps. The first is that we need a notion of co-completeness which is adapted to c-star category because I think no c-star category can be co-complete in the ordinary categorical sense. Anyway, the category of Hilbert module over c-star are not but it's actually possible to work something out. It's not really nice convenient notion but it's enough to prove a theorem of this kind. And then we can define an abstract notion of compact operator. So compact operator are supposed to be those which are approximable by a finite rank operator and it appears that the compactness can be characterized with some kind of category or property related to this notion of compactness. And we can define an abstract notion of compact operator in a co-complete c-star category. And then it's possible to formulate an abstract characterization similar to this as, so c-star category will be a category of Hilbert module over something. If it's co-complete, if it has sufficiently many compact operator and if it has a generator. And the proof of the theorem is then roughly the same as for the Abelian category case. So that's the first step. The second step is that we can, it's not really hard. We can prove that for any topos, the category of Hilbert space over chi satisfies this c-star category called co-completeness condition. Yeah, I don't make the difference between co-complete and complete because in a c-star category you have this adjunction that reverses the direction of morphism so I actually have no reason to think that complete is different from co-complete but technically it's more of a co-completeness condition. So we can prove that those category of Hilbert space over chi is our co-complete in this sense. And that, so operator which are internally compact operator and with norm 10 to zero at infinity, so it means on the localic reflection. If you have a continuous family of operator, its norm is actually a function on the localic reflection. And if it stands to zero at infinity, then those operators are compact in the sense of the above characterization. So then there is the difficult step which if I have time I will say more about it so it's a purely topo theoretical thing. We need a constructive proof that if we have a topos which is separated, locally decidable and hyperconnected, hyperconnected means the localic reflection is trivial, reduced to a point. Then the topos actually admit a generating family of object which are finite and decidable. So finite and decidable for the internal logic. So those are exactly object which are internally isomorphic to something of the form one, two, et cetera, two n. So okay, so this is a purely topo theoretical result which is in a paper that I put on archive a few months ago. And actually at this point the end of the proof is relatively easy. So one can use this constructive result about separated, locally decidable and hyperconnected topos. So let's come back to this diagram. So we have t, we have its localic reflection. And so within this topos, this one is separated, hyperconnected and locally decidable. So we can apply the third point internally and we get object which are somehow finite in this topos relatively to this one. And finite object are all we need to construct the finite rank operator which are in particular compact operator. And so it's at this point it's relatively easy to by using some partition function with compact support on this space to actually construct explicitly enough compact operator in the category of Hilbert space and we can also construct a generator. And so this last step is actually relatively easy just showing that everything works together well. So what this proof told us is that this algebra C0 of t is basically just the algebra of compact operator over some Hilbert space over the topos which is big enough, where big enough means it's a generator in the sense that I had to construct in the first point. And by compact operator, I mean in the abstract sense of operator which are internally compact and with norm 10 to 0 at infinity. This is an external. Yeah, C0 of t is an external. So it's the algebra of global section of external on the morphism which internally are compact and with norm 10 to 0 at infinity. Is it true that there is a Hilbert space that this is isomorphic to the edge for some? Yeah, that's the theorem. You mean the global interface? What? So it's a single Hilbert space. There is one Hilbert space over the topos such that C0 of t is the algebra of compact operator of this Hilbert space. But it's not, yeah, it's not a, it's not a classical Hilbert space. Yeah, these are, I'm not an expert of C0 of t but I assume that they are very special, no? The C0 of t? Yeah, yeah, algebra that arise from separated toposes are of a very specific form. They are, what I can say is that they are what we call type one algebra. I don't know if we fall in a more strictly class but then I will explain how to use this for non-separated topos and we can go out of this very specific situation. But also the C star kt bodies are presumably similar to one other or kt bodies? Yes, yes, yes. We also, yeah, they should also have this. It's actually very important if we want to try to reconstruct the topos from this, from this is algebra we need this monoidal structure but I will not talk about it today. So we are now to the, so if I have time I will come back to the third purely toposteological result at the end. So I want to explain how we can generalize to non-separated toposes. So this finite nest result really require separation to work. So there is no way to go around it. In fact, if you take basically any example of topos which is not separated, then you can show that there is basically none, no compact operator of this form in almost every example you can think of. So we really need this separation condition to have any hope to have this Gringo theorem working. So, but there is something we can do if the topos is locally of the form of the theorem. So by locally I mean that we have an object which I would prefer to assume that it is decidable and inhabited meaning that it has local section everywhere. Well, because otherwise you could take the empty object of course, such that this is a slice topos T over X so the etal space of X satisfies the hypothesis of the theorem. So maybe, so yeah by T over X I just mean the category of objects with an arrow to X with Y and T, so this is in T. So it's a topos and geometrically it's the etal space of the sheath X, so if T is a topological space, X is a sheath, then this is the category of sheath over the etal space of the sheath. So it's a space which is locally isomorphic to the base space. So that's why we call this locally of the form of the theorem. And so this situation includes basically all the examples I know of topos where it seems interesting to try to construct a sheath algebra. I cannot prove this of course, but all the examples I know of this form. And in this case, when we can actually construct a sheath algebra attached to a topos which is only locally of this form, basically by using a generalization of the matrix trick I talked about in the second slide. So we had a situation where we had a space which was not itself separated and locally compact, but which was locally separated and locally compact and we were able to construct a sheath algebra forming some sort of matrix and up to marita equivalence it was exactly what we want to do. So that's exactly what we are going to do here. So we have our topos which is, so we have an object X which as I said I prefer to assume it's decidable. We assume that T over X satisfies our theorem. So then we pick a Hilbert space in T over X so we can think of it as a family of Hilbert space HX index by X internally in T because well, Hilbert space over a set is just a family of Hilbert space. And so we take your IH which is a generator of the Hilbert space of T over X which is big enough. So we can actually assume that this family is constant in X so it's actually a Hilbert space in T that we have pulled back to T over X if one wants it's not really important. So if we look at an endomorphism of the direct sum so we can construct a Hilbert space which is the derived direct sum, the orthogonal sum of all this HX so it's a Hilbert space in T. If I have an endomorphism of this thing well it's a map between two direct sums so I can have a matrix decomposition. So let's say we have a H then by composing by the injection and the projection there I have matrix elements which are operator H from HX. And so this is something that live in this topos. The slice topos over X time X it depends on two parameters in X so it naturally live in this topos and I'm considering the algebra of such elements such that all this arrow from HX to HX prime are internally compact operator and such that it's zero outside the compact subspace of these topos. So I'm somehow looking at a matrix with compact support. And so it appears that those such elements with compact support actually form an algebra that I can call C of T. So then I need to construct a norm on this but that the classical theory of algebra of groupoid work exactly the same. I have several choice. Either the maximal norm or the radius norm as we talked about for group earlier and in both case I can take the completion is going to be a c-star algebra. So I'll get either a reduce or a maximal c-star algebra exactly as we have for groupoid. So this produce a way to attach the c-star algebra to any topos that is locally of this form which work perfectly on all example but is not as nice as we could have hoped because for example, at first sight it's not clear at all that this algebra is well defined up to more retire equivalents we need to prove it as a theorem and what I mean is that it will be better if we have a more universal characterization in the locally separated case. So I will say a word about the proof of this point at a separated, locally decidable hyper-connected topos is generated by finite object because it's a really very, it's the most important part of the proof and it's really nice results which I'm sure have other applications and considering new both space and c-star algebra of the topos. So the proof goes into two steps. So let's say we have a T, a topos which is a separated, locally decidable and hyper-connected. So you actually don't need to know what all those conditions mean to understand the proof that's coming. So the first step is to solve the case where the topos has a point. So first, assume T as a point. So in this case, there is a very nice result of Ikemodaik and Femmelen which is basically a constructive version of Grotten-Nigelua theory which say that then T is actually isomorphic to the category of set with, so I remember I need the proof to be constructive. A non-constructive proof of this will be very easy. Of set with an action of a local group and G is actually the group of automorphism of the point. So the automorphism of the point in general, local group. So if we were in classical mathematics, then this would actually, we could actually prove that this topos has finite object as much as we want because the group is compact for local compact group, sorry. And well, if you have a compact group acting on a set, then you can decompose the set into the disjointunior for its orbit and the orbit for the action of a compact group are finite. So we would have all the finite object that we want. There is a big problem with that which is that because this is not a point set topological group but a local group, if the group does not satisfy a technical condition which is called to be open, which is always satisfied in classical mathematics, then the orbit of a point actually might not exist. It's defined as a sub-local of the set on which it acts and it might not be a set. So we cannot use the orbit to construct finite sub-objects. Actually if orbit exists then we can, we are in the case of an atomic topos and it's a complete trivial. But there is still something we can do. So if I have x decidable, so that's why I need the topos to be locally decidable, we have an action of g and if I take an element of the set, then because the group is compact, I still know that the orbit, which is not a set but still makes sense, is contained in some finite set. I just cannot construct actually a set which is stable under g at first sight. And by exact same argument, I can also construct an f prime such that if I look at the action of g on f, this is included into some other finite set. And at this point, there is a really nice combinatorial argument. So it's still true that in constructive mathematics, if you have two compact local k and k prime, such that k and k prime is empty, they are both compact, then k is empty or k prime is empty. That's something that is still true in constructive mathematics and that's what's save us. Because basically every time you have a point in s prime, then if you try to look at the set of pair of elements in g such that g time g prime time x is your element, it will always be empty because it will be g of x, so it will be in f. So there is also a finite set into a decidable set, so they are decidable. We know if we are, when we have an element in f prime, it's in, if it's in f or not enough. So the product of the two compact thing is empty. So we know that one of them is empty. So either our element cannot be written as coming from an element of f. Either it cannot be attained by an element coming from f. So in one case, we can eliminate it from f prime and in the other case, we can actually remove an element in f and in both case, we can either reduce the side of f prime or the side of f up to the point where the two are going to be equal by induction and when the two are equal, we actually find a stable finite set and that's actually work. So this is the case where it has a point. So the general case work by what we call decent theory. So utopos, if it doesn't have a point, always has a point in some covering. All those hypotheses are stable by pulling back. So we actually are going to get finite objects in some covering and then there is a involved decent argument that I'm not going to explain that allow to get finite objects in the base topos from those finite object in the covering. So yes, I think that's all I had to say. So thank you for your attention. Yeah. May I ask, so considering the topos theoretic property that I'm not quite familiar with but the notion of proper, so if you consider the case of coherent topos and coherent morphing, which is usually the case of the right geometry, what does proper mean? For, I hope I'm not going to say something wrong but I think they are always proper because coherence imply compactness property. Okay, this aspect. And then it seems, so this is strange relative to what you said, because if you take then a usual scheme for example, sorry, or spectral space, then you try to look whether it's separated in the sense that you define. So you take the product of x-axis in the sense of topos but this is the same as product of spectral spaces. No. And do you have the diagonal? No? No. No? Scheme are, well, any topos coming from a scheme whether it's Zarisky, Italian, Snevich or whatever you want will, except for case of field, will never be separated in the sense I define. Basically, because the topological parts will always be something like a Zarisky space and which will never be separated itself. The topological parts of the topos will always be a Zarisky like. No, but I ask you before, the first question is for this. I ask you whether a coherent morphing. Yeah, but the diagonal of a coherent the product of two coherent topos as I don't think it's coherent and the diagonal morphing won't be coherent whether something is going to go wrong at this point because coherent topos are not separated. Well, in general, there is some exception, of course, but. I think that the product of coherent topos is coherent. Okay, as far as I know the, because you can write down specific sites of definition and you can see that. Okay, so it's a diagonal which is not a coherent map then. So if the product is coherent then it means the diagonal is not a coherent morphism. Anyway, the product of et al or the Zarisky topos does not relate to the product of scheme. No, I'm not saying product of scheme. The product of just the underlying topological space of the scheme. So you take the spectral space X, make it in, consider the corresponding topos of scheme. Then you have X cross X is the corresponding topostomatic correspond the topostomatic product of this scheme. And I think diagonal map is, and so the diagonal is spectral and so you get real contradiction in convention. So I don't understand what's wrong with what you say, but I can tell you it's wrong because if you take a topos which is separated in the sense I say that it's a really very easy result that it's localic reflection is separated. And those topos is coming from algebraic geometry that a localic reflection is never separated or only on the very specific case like a field. Before the localic reflection it is a house of space? Yeah, it almost the same. It's slightly stronger than house of space. So I don't really, I mean not really an expert on coherent topos so I'm not. Because I looked at, so I need some things there. Okay, so maybe a coherent topos is not proper then. Maybe I was wrong on that, yeah. I don't know. Then if you're wrong, if this, okay. Then it explains my, okay. I'm not really familiar with a current topos. I think that's the case actually. Okay, thank you. Then sorry, I say something wrong. I should have, I should not say it. And so if I want to know about this then it is the paper of the, the image of the moon and moon. What? Which paper, which reference do you have for this? For the misrothoretic case. For the, for this concept of. With the Boolean covering? Then the, for separation. Oh, for separation, yeah. It's a, it's a paper of Maudike and Fermiland and it's called proper map of topos. Yeah, that's it. You have everything in this. And it's also in Johnson's book. In Johnson's sketches of an elephant. Yeah. I just have one, one, which is a relation with the previous talk. Because in the previous talk we saw that for one topos is the notion of locale compact was the same, I mean, from Andre on the result of Julia in the stock. That it was the same thing as being exponential. Yeah. So, but of course this didn't take into account separation. But, so, I mean, this result would suggest that the correct domain for associating a system probably would be exponential like this. No, I don't think so. You don't think so? I should have told you before asking my question, don't answer the question. No, yeah, sure. No, the, the problem is that locale, locale compact already contain things that are not locally separated. You know, but I am, I am saying that it should be part of the condition. Of course there should be separation besides that. Yeah, but it seems very natural that the notion of being exponential should enter into the, the construction. Oh, yeah, sure. Well, I mean the topos that satisfy those hypotheses are exponential. They are exponential. So, I think you should try to push the construction to the largest possible class of exponential topos. Yeah, maybe, but the thing is I really don't know how to go beyond this result. But what I mean is that the fact that there is a concept behind which is exponential, there will be a guide. Maybe, but, maybe, but I, I really don't know. I mean, at some point we need locale compact groupoid in some sense. And locale compact really. Is there a very conceptual notion? No, sure. It tells you that you can, you know, so I mean, somehow it seems very natural. Yeah, what I mean is that you cannot hope to attach a sister algebra to a non-locale compact topological group, I think. And there I, if you don't put this locale separated assumptions then you will get a non-locale compact group at some point. I don't know how to associate a topos to a locale compact group, actually. Well, if it's a pro discreet, then you can look at continuous action inside. Yeah, but then this is not the right thing. I mean, you know, what I have in mind is that locale compact groups don't really fit. Yeah, no, that's the problem of this formalism, of course. So you, you lose a connected isotropy group. Yes, and there is another, there is another thing which is quite important to know, which is that somehow there is a richness in sister algebra which is not in topos. And that comes from the fact that there are sister algebra which are not anti-isomorphic to themselves, or even in the measure theory. And that says that they cannot come from something which is purely combinatorial. Yeah. Well, actually, the two words cannot be the same. Actually, all the sister algebra that we get this way come with a real structure. So they are, they are isomorphic to their dual. In general, you cannot hope for an equivalence of categories. These are simple. Well, I have an objection to this. Using the version with coefficient, we can define the algebra group with algebra twisted by a co-cycle. Yeah, you can twist by a co-cycle. So, but what I am saying is that certainly, you would need to twist by a co-cycle. Yeah. Yeah. Yeah.