Causal Sites as Quantum Geometry — Seminar re. Crane & Christensen

0:00 The American Pronunciation Guide Presents Âu Léguerre Leclerc Leclerc Leclerc Leclerc Leclerc Leclerc Leclerc Leclerc Leclerc Leclerc Leclerc Leclerc On va essayer de mettre au clair le contexte de cet article, j'ai passé un certain temps à extraire les définitions qu'il donnait, il y en a beaucoup, et j'ai passé aussi à peu près autant de temps à essayer de les relier aux définitions classiques. What do you think of today? It takes time, but I'm going to show you what I've understood. In fact, for the moment, there is a first exercise of translation of their language. So, a language a bit more classique, or a bit more diagrammatic, I would say. So I'll try to make a drawing, so that both the physicist who is in us and the mathematician who is in us can find his part. Well, in general, the idea that I have understood in this article, which I think I have understood, is that if we consider only flèches of inclusion between regions of space-temps, we describe this topology. It is to say that we have a classic site with the open and the inclusion. The idea of the authors is apparently to add the flèches of causality, to say that a region preceded another region in the space-temps, in a certain sense that we can define it, and, in addition to these flags or flags of inclusion, we will try to configure a category on which we would have a topology Grotendik interesting. So we will try to see if it actually satisfies the actions of a topology Grotendik. You know what I've searched, I haven't found it. So, yes. De toute façon, la structure causale décrit déjà une topologie qui est relativement triviale, puisqu'elle définit une pseudo-distance, donc sur l'espace-temps, il y a ces deux topologies. D'accord. Apparemment, ils veulent les marier. Apparemment, ils veulent les marier et voir si on peut donner des actions un peu plus abstraites, un peu plus généraux qui pourraient étendre un peu le cadre à la fois de la topologie sans structure causale et de la topologie causale. Donc, c'est ça qu'on va essayer de clarifier.

2:30 I think in the text of Quinn, there is always a sort of confusion, almost terminologically, because they are not using the technology that they don't use. Exactly. There are only cases in particular of inclusion, exactly, of their potential. But maybe I'm not sure that it's, well, we can say that it's the classic, but still it's more large than the topology assemblies classically. It corresponds to what we call the local STO, etc. D'accord. There is the idea to marry the structure causal and the structure of inclusion, to see if we can enter something, and see if we can pose an axiom enough clear to be able to build models a little larger. And the fundamental problem to which I'm sure, is to try to understand the definition of the Groton-Lie, The best that we can do is to verify if we have the axioms of a base for a topology, or to try to engendrer a topology with the regions and the flèches causales, and the flèches of inclusion, and even that, it does not work. If there are three axioms to verify, that the authors keep well to verify, they propose, they speak, but it does not exist. It is really the arlesian. So, I must admit that I'm not, I'm not aware of the whole article. I have well understood how they define their flesh, how they use it, but they don't make any diagram of catégories, nor diagram of space-temps enough clear to make a translation. So what I'll try to do is to translate the text in French and in French language, which is comprehensible by the maximum of people, so both the diagram of space-temps and the diagram of catégories. So, in a first time, I will remind you some definitions that he gives us on the classical structures, and we will do a little drawing in the space of Mikovsky 8-dimensional space. Ensuite, je donnerai la définition des sites causaux que eux donnent, en faisant pas mal de diagrammes pour voir à quoi correspondent leurs flèches. Donc, définition des sites causaux.

5:00 Et dans un troisième temps, on verra le lien éventuel avec les topologies proton-diques. Donc, je rappellerai au moins les axioms d'une base pour une topologie proton-dique, c'est-à-dire vraiment les familles qui vont engendrer ensuite la topologie. There are a lot of things in the future, but I think that as it is not assured, the rest does not hold it. It is to say that we can build a sort of bicatégories, try to put the pre-fessos there-dessus, the pre-fessos of categories, what they call the fibres, etc. But if we don't really have a topology Groton-Dick or something like that, I don't see any interest. So there are a lot of things in the future, and it's only if we really do the link with the topology Groton-Dick, we can go further. What do they say about the structures causal, things that the relativist knows well? So if I say these things, Mark me correct. In the first time, concerning the variety Laurentian, they don't know if it's dimension 4 or whatever. In all cases, the diagram is in dimension 2. And they remind the definition of what we call the diamants. I have a question, the question of dimension is very important, because the idea of that is that the dimension of the space-temps should be released as a result of the theory. So, we fabricate that, for a moment, in a way abstract, that is to say that the variety of space-temps is not going to use. We will just use something that we will modellize on the structure causal. And the idea is that then, in doing the calculations of the site causal, we will find some results. And, I have read but I haven't read, but they pretend to find the fact that the structure of the space-time, the dimension of the space-time is going to be the case. Okay. It is one of the results that they pretend to have. D'accord, but they do it in this article? Yes, there are a lot of articles, two and others. It is perhaps not exactly the same approach, but what we need to know is that the approach of the site causal also is very répanded. It is to say that there are dozens and dozens of articles on the site causal, but without the approach catégorical approach.

7:30 I think there are a lot of articles on the ensemble COSO, on the terrorist Sorkin. Yes, on the structure Causale Ensemblis, we agree. On the site COSO, I don't think that. Yes, I agree. We don't have any confusion between... Because I said, they have written so much. I think that what was written by Sorkin and his school, it was on the ensemble COSO. they have so many examples of the reasons Rafael Sorkin, S-O-R-K-I-N which has been a long time it's been a 20th century so it's him who has defined the ensemble Kozo the ensemble Kozo it works, it's a discreet of the variety Laurentian in general then they can have some results on the dimensions but there is no There is no more calculation possible, and we will see that, at least to prevent the fish from se noyer, they are going to be at all. Franchement, I don't know how to go. You see, I'm not sure, just that this sequence, this approach, which is central for him, because for him, as an efficient, and I repeat each other in the article, he could improve the effect of the theory of sorting, the theory of the ensemble coso, and using the cipher coso and not the ensemble. So for him, the history of Grotonnik, I think, is marginally compared to the principle principle. Yes, but in particular, he says something that he doesn't enter into the definition. He says, at a moment, our category has no pullback, so it's not enough to try to find the definition. Or, we can define the topology Grotonnik without pullback. So, I think, we have some definitions, we have some tools which works. Now, there is not to do the fine bouche with things that work. I think it is at least to try to satisfy an embryo of action of the topology grodeny. So that's why I want to see if we do it well. Or maybe after we arrive at other things, but I see it in particular that it doesn't satisfy even the action of transitivity. And there, it's not easy to do. I can't do it. I can't do it. So I'm going to make some pictures on these things, I'm going to remind you of some of these. So in a variety Lorentzian, we make an hypothesis that we need to have an orientation temporal global,

10:00 otherwise it's a problem, and that it's not confirmed of the type of time. So there's no eternal return. So in this variety Lorentzian, we define the diamants between two points. a diamond between two points P and R, which is all the events of the event, which is the point Q, that P is related to Q and Q is related to R, It is the one that exists in the future, which is the PAQ and the other that is the QH. So the whole point Q is called the PR. So the structure is the fundamental. Here we are in Minkowski. I make the drawing in Minkowski R11. Not in a variety Lorentzian, but at each time I will make it to two dimensions. to give an idea of this structure. And he defines the notion of region bornée, a region is bornée if it is contained in a finite number of diamonds. Effectivement, if we have a region bornée, in the sense of the term, we want to enter a finite number of diamonds, because the diamonds are born in the habitable, but he defines the notion of bornée independent of the compacité. and it is true that if the variety is adjacent is global hyperbolic it is to say if we can see it by the variety of dimensions 1 in fact it is equivalent that a region is relatively compact or that it is born in the sense where it is contained in a number of diamonds it is equivalent so there I have not found the demonstration exact of this theorem on the structures global hyperbolic but we have confidence that the notion The region bornet corresponds to the intuition habituelle, whether it be defined in terms of diamonds or in terms of compacité-relation. Next, it's about everything that we use, only if it is contained in a finite diamond, there is a unique diamond that we use? Yes, all right. No, no, no, no, no. No, no, no, no, no, no.

12:30 C'est des losanges, on finit par faire rentrer ça, c'est un grand losange. Je pense que c'est même vrai dans toutes les trois composantes. Voilà. Bon, enfin, ce que je veux dire, c'est qu'on va essayer de se rattacher à l'intuition qu'on peut avoir d'un espace-temps bidimensionnel, Minkowski 1 plus 1, pour extraire les définitions générales des psychosos. After, having defined the notion of Diamant, he will define the notion of Liens Causal. I think he does directly the Liens Causal between two regions. I'm going to change a little bit the notation for the article. Because in the article, the idea is to start from the regions of the space-temps and to say that there are two types of relations that we can have. We can have the sub-regions, B' plus B, and then we can also have a causal relation, which is defined in the way the following. We say that B precede A, which is like this, B precede A. c'est seulement si pour tout point petit p élément de b et pour tout point petit q élément de a il existera un chemin causal il faut prendre causal de type temps il faut vraiment que le vecteur tangent soit pas de norme nulle il faut vraiment que ce soit une particule massive qui puisse joindre tout point de b à tout point de a si on ne prend pas le cas limite So if all points of B can be joined by a chemin like this to all points of A, then we say that B precede A. In general, what does it mean in terms of the distance of space-temps? It means that if I take the intersection, I always do this in Mikowski, if I take the intersection of all points Q in A, in fact, in B, in A, then the region B is inside the intersection of all points of light. So it means that we have this intersection of the light, it's like this, and the region C is there. And at this point, we will see B precedes A. So, it's really on retrait. The intersection of the light, it's like a other light.

15:00 In general, it would be much more complex. But it's a certain code. And B should be contenu there. In this case, we have B preceded A. I'm going to change a little bit the notation. In lieu of B' included in B, I'll put B' with an inclusion in B. And I'll put B, like this, like this, a signal that propagates to A. This allows me to write a diagram catégories a little more simple. It's not necessarily a signal of a photon, it can be a massive particle, but it allows me to write a diagram with flèches ondulées or flèches accrochées. So, this is my notation. I prefer to use this notation. We can write a diagram with this. So, he has two directions of order on the entire region of space-temps. and he will define abstractly the six causes in terms of these two relations of order. And it's there that, for me, there is a confusion. In any case, I haven't managed to clarify this point. When he uses the fiches which would be a six causal, we don't know very well if he uses the fiches causals or the fiches of inclusion. Sometimes there is a confusion. And when we try to describe the axioms that we know that they work, then we don't know which fiches they use. Maybe we'll see that it's either that it's inclusion or that it's causal, but there's a problem at a moment. So, he uses structures causales, which define the relationship precedes between regions. Ensuite, on va essayer de la décortiquer en petits morceaux parce que là c'est une dizaine d'axiomes qui sont à peu près naturels, qui définissent les relations d'ordre et des compatibilités entre ces relations d'ordre. So, in a site causal, in fact, it's going to give a class of objects, entre guillemets, the regions. Now, I don't know if you consider the regions or the regions born. If you want to abstraire the notion of region, the notion of region born, you know.

17:30 We'll see, there are some places where, at least the drawing I've made, it's always with the regions born. No, it's not a percent of the regions between guillemets. — Tout à fait, régions entre guillemets. Mais est-ce qu'il a en tête d'abstraire la notion de région générale ou bien uniquement celle de région bornée ? Bien sûr, on fera uniquement une petite image pour faire des diagrammes avec des régions bornées, mais à chaque fois que j'essaie de représenter ces actions, il n'existe que des régions bornées. Donc apparemment, peut-être qu'il suffirait d'utiliser des régions bornées. On va voir. So a site causal is given by a class of objects, we think about the regions, and two regions. So, the regions, we have the class of regions. We have two regions of order, so a type inclusion and a type causal. If we only take the inclusion, then we have a partial order, R unis de flèches d'inclusion, c'est un ordre partiel, avec la transitivité, etc. On suppose qu'il y a un objet initial, la région vide. Donc on peut inclure dans n'importe quelle région. Not for inclusion, but for the other relationship. For the other relationship, it would be an object nul. At some point, it seems to have an object nul, but... It's an object nul. It's at both initial and final. Yes, yes, yes. It's the fact that it is initial, for an instant. But for the notion of a new element... For inclusion. No, no, for the notion of a precedent that you have given. It seems vides. Yes, all right. But what happens is that it excludes the order, it does not define a part of the region, but only the regions non-vides. So in fact, it does not serve to be an initial object. But it is not... This object 0, it is known as the Allemands, and it is not clear. All right. But it excludes the R, and it considers the regions non-vides. The front is minimal. The front is minimal. The front is minimal. The front is minimal.

20:00 So there is no one in it. So it is only initial. But it is not the object of R. R is the problem. When he takes R and the causal relationship, he does not put the object in it. So in fact, the object is a little bit. The proposition 2.6 is exactly the same. So there is an object initial. There are some reunions. So if I take the diagram... So R, the inclusion means reunions. If I have two regions A and B which are included in C, then there exists an object in a union B, so that a union B is included in C. So that's it. The union, how do you define it? Well, he defined it as being... It exists, there is. There is an object that corresponds to... Is it... I suppose that it exists, each pair of objects AB can be associated with the reunion. This is an image of the minimum that we can ask. This is for all the notions of inclusion. that doesn't pose a problem. Then R, minus the region of the region, muni of the causal relationship, which is a part of a strict order. And what is the product or the product here? Here? It should be this in terms of the category. Is it really a product? Yes, yes. It's a product. I don't know. I'm always talking about product and co-product. So it's a co-produced in this case, but it's not a reunion disjoint, because habituellement the co-produced assembly is the reunion disjoint, so that he takes a reunion, he notes like

22:30 He uses apparently a reunion, or is it possible that... No, but it's not a reunion. It's not a reunion. Yes, it's not a reunion. You can't forget the space-temps. Yes, but if it's a co-produced, well, it's a category co-produced. That's why these are the regions in the middle, and not the middle. Yes, of course. So, R, the entire region, the object R minus the region void made of the causal must be a partial strict. In particular, the region will be present for the same. The initial object cannot be part of the R. It is present for the R. There are relations of compatibility between these two types of flèches. That's to say that the relationship of compatibility is if, for example, I have a sub-region A of B which is the same preceding the region C, then I have A preceding the region C. if I have a region A' which precede a region B and if I have a region A which is included in the region B, then A' precede also the region A. Because in terms of the cone of light, if I reflect on the cone... I'm going to make a little picture for you to be sure about the fact that it's elementaire. In terms of light, if I have A, which is a low region of B, the intersection of these light curves is greater than the intersection of light curves of B. So if after it sits in the intersection of light curves of B, it will be in particular in the intersection of light curves of A. And then, if I have A, which prescède C, and B, then... the reunion which precedes C. We're going to make a little bit of a picture

25:00 so we can reassure the significance of this thing. So, B precede C, so that means that I have C here, I have an intersection of these corners of the line, I have B which is inside, and I have a sub-region A, like this. Well, A preceded C, yes, because of the inclusion, at the moment when we translate it in terms of intersection of the cone of the lumière, we see naturally that it has to happen in the space of Minkowski. Then, the second, A' preceded B, A preceded B, A preceded B, and then A' precede A because the intersection of the light of A is bigger than the intersection of the light of B. And then the last region, which is the same region. A' precede C, the intersection of the light of light, A' at the interior, B' at the interior. So it's very rassurant to see that. I have sometimes confusion in the diagram between the two types of FLECH, when I try to verify the axioms of the topology protonics. But you could do it to you-même complicate the tache in oubliant l'inclusion and the precede. Yes, at the moment, there is to do the part. Tant qu'on rest with the notation inclusion precede, we think in terms of ensemble. And if we really want to think in terms of category, we are obligated to use FLECH, between these two poles and we can also come back a little bit but there is no opposition how do you do it? there is no opposition because there is no opposition but there is no opposition but there is no opposition but here, normally we always have a different nature between these two types of flashes There are two laws of composition different. There are the composition of the flèches Causa and the flèches of inclusion.

27:30 So that's why we have problems with the transitivity. And when we try to put them together, it's a bit like the oil and the oil. We'll see. On dit que deux régions sont disjointes. Il y a un axiome qui est important. C'est le fait que si je prends une région C qui est incluse dans B et qui précède A, en fait je peux définir ce qu'ils appellent la coupure de B par A. Alors je vais faire le... Je vais me dessiner différemment. Ici, il y a un axiome qui se traduit en termes de limite en fait. On peut l'exprimer comme une limite d'un type de diagramme. Si j'ai des régions B et une sous-région C qui est elle-même noiselle une région A, donc ce que j'ai B, j'ai C, et puis C présente une région A. B is not necessarily A, but C is in the middle of the intersection of these points of the line. In fact, what he says is that there is a large part of B that precedes A. The most large part of B is A, which is this one. This is what he calls the coupure of B by A. This part is also called a flèche. Yes, exactly. It is always in such an object. Ah, yes. The flèche of A in B. That's it. Every time we try to translate that into a category, there exists a flèche. of all the diagrams of type. They tell us that the diagram of this type put a limit in a category. But there, again, because it's a bit hybrid, there are always two types causal types. In general, you need a category with two types of flashes. After, you have a big category, but it's on the space with the path. So it's a bit more complicated.

30:00 Is there really a structure of bicatégories? I haven't looked at it enough to really verify the actions of the bicatégories. But it's probably something that it should not be satisfied. Well, it's not finished. There's still another one. I don't know about it, but on the notes it's numerated. So there's a notion of coupure, there's a notion of joint. Two regions are jointed. if each time we have a common region, it's a bit more than A, it's a bit more than B, if the object is null, it's an ensemble vide. Noted ensemble vide, but the region is vide. So it's the notion of nature of disjonction. So, he puts a collection of axiomes, you'll notice that he calls site causal something, but he does not verify that it's a site. But we'll see if it's at least one of his first axiomes. The notion is important, after it is the notion of raffinement, between families and regions. So, there we have given the definition of their disposons, now the notion of raffinement. The cid is what? It is R with the two relations? Yes, all right. R with the two relations of order, which is called the principal. Now, what is what? We are going to consider two families S. S is a family of regions, a family of objects of R. the Tj, which is another family of objects of R, and he will say that T is a raffinement of R? Well, he takes 2 A2 disjoints. the same for the tj so the si are two to two disjoints entre elles the si are two to two disjoints entre elles and the tj are two to two disjoints entre elles and he dira que t is a raffinement of s If...

32:30 If... I'll make a lesson. If... If all regions admit a sub-object of the type TJ. How do you say this? So in this case, it exists, so in this case, it exists FI in J, t, I said FI, which is a sub-objet of S. So what does it mean? It simply means that if the regions S are the wrongs, and T, it is the square. Here I have SI. I have always a sub-object of type T. I have an exterior one in plus. So we can pass from S to T in increasing the region SI and eventually in STJ. So this is the Tj, the J prime, etc. So this is SI. So, this is the notion of raffinement. So, every time I have a point in a SI, I can take it in a TJ. If we think in terms of points, of course, we will try to forget the point. So, this is important for the notion of raffinement. And the other notion, it's the notion of the causal chemin. Well, the chemin causous, it's simply a family of flèches causables, successive, composable, donc le chemin causal, c'est une famille finie, A0 etc, AL, avec des relations causales et on dira que c'est de longueur L, il y a L flèche composable So, just simply, I take a region, so I can put it in the right direction, AL, I take the intersection of the lumière, I take another region, AL-1, and then I take the intersection, and then I take ATL.

35:00 We can use them to make an integral of the chemin. The importance is to integrate it on the structure. We are far from these structures. We can remove the half of the words in the title. There is a causal, there is a geometry between the two. So, these are the important ones that we have seen. So, the causal notion, the coupure, which is not really useful, so the raffinement, causal chemin, and then the notion of completion, which, also, is also heritage of the cause. So this is a complexion for a causal line. So a causal line, I'm going to note how I press A, C, A, B, C. On prend une flèche causale A flèche C et on dira qu'on a une complétion de A flèche C si on s'est donné en fait une décomposition de cette flèche en A flèche B flèche C, deux liens causaux successifs tels que, en fait, si tout chemin causal qui mène de A à C peut être décomposé, peut être raffiné, We have a causal chemin that passes by a sub-region of B. I will describe it properly. We say that this is a complexion. This is the two fletched composable. This is the two fletched composable. This is a complexion. A fletched C. If all the causal, not necessarily for a long time,

37:30 which begins in A and ends in C, with n'importe what between the two, We have a causal path which passes from here. Causal path from A to B' which is a source of B. to see what signifie the fiction. Well, in fact... It's not a propriety universe. Well, is it possible to translate it into a universe? Yes, when there is everything in the definition of a universe. But what type of diagram is it? Well, I'll do a diagram of space-temps to see what it corresponds. So I'll have C, which is in the top. I will have A in the middle of the intersection of the light future, and in fact B should be a complexion, it is to say that every time I take a type of time, it will pass to the interior of B, So in fact, we need to be in this intersection, both in the intersection of the light of the light future of A and in the intersection of the light of the light of C. So it should be inside this parallelogram. So here we have a complexion possible. We need to take all the parallelogram. So that would be B. But if we try to draw a path, we are obligated to pass by a sub-region B'2B.

40:00 It has to be enough large to intercept all the signals from A who arrive in C. Is it a complexion? There is no complexion. There is no complexion. We take all the possible paths of time, which go from A to C, but for the definition of a complexion, there is no complexion. We choose a complexion like this. maximale, par exemple. Alors, c'est important. Dans ce cas-là, parce que je fais ce dessin-là, mais dans le cas général, dans le cas abstrait, où il suppose qu'il y a une catégorie de région avec deux ordres partiels, etc., il ne suppose pas, dans le cas général, qu'il existe une complexion maximale. Et là, je prends un espace-temps ultra-saint, dans ce cas-là, effectivement, il y a bien une complexion maximale. It's what? The diamond? The diamond, yes, all that. It's not all that. But maybe even in the Lorentzian varieties, we could have something a little exotic where the maximum complexion doesn't exist. Have you seen things like that? Or even with an ensemble coso, just simply. With an ensemble coso, we could have a little exotic in which we could not have a maximum complexion. That's why there's no hypothesis of existence on the maximum complex. With this definition, the diamond is open. It's not open, it's not open. Yes, yes, of course. Yes, sir. Yes, sir. Yes, sir. causal de ce type, ABC, il existe une complexion de AC. Donc le dernier axiome, c'est l'existence

42:30 the complexion. So for all the causal A, F, B, F, C, there is a complexion of A, A, C. It doesn't mean that B is a complexion, because from the moment where there exists a region intermédiaire that precede C and that A precede, there must exist a complexion, which is a complexion, which is a complexion, which will serve as complexion of the causal A-C. So, we have... In fact, it's the raffinement of families of objects correspondents. When you have a family of different regions, you look at the objects indépendamment of the flèches. Yes, I haven't been precise here, but we have defined the raffinement of families, but in particular, if I have a family of objects, I look at the families of objects and I say that a causal is a raffinement of another. if the family of objects in the center is a raffinement of the family of objects. But to say that a raffinement exists, it's not just a raffinement? No, but it doesn't include the existence of the raffinement. It includes just the existence of a raffinement, a very particular type of raffinement, which is the complexion. So for the liens cause of the length of 1, only. But it doesn't mean that there is a raffinement. It's always his own raffinement. It's only a very particular raffinement that he needs, it's the complexion. To be able to intercept the signals that go from one region to another. Otherwise, maybe there are some sauvageries that he would avoid. Well, that's all the axioms. I have 7, I have number 7. in A7, in any case, if there are other notions that would seem to be more pertinentes or

45:00 what. Well, after he defines the notion of site local. So what he says is that if we look at just the regions that are just a region given, then he defines it as a source site. Okay, so... Well. So he calls it the site local 2. Well, you can't believe it, but you can imagine that in all the intersections with the light of a region, all the relations are preserved. The notion of object and the notion of causal will be preserved. So, without checking, I think it's plausible, we'll say. Il donne quelques exemples de sites causaux qui sont fabriqués avec des objets classiques du type soit variétés minkowskiennes, soit ensembles causales, variétés minkowskiennes discrètes, et il dit voilà de quoi il s'est inspiré en fait. Il s'est inspiré des ensembles causaux et des variétés minkowskiennes pour extraire cette définition générale des sites causaux. Ensuite, il affine un petit peu sa relation, la relation précède. Il définit une relation précède faiblement entre des régions, parce que là on a vu que si A précède B, il faut que les régions soient suffisamment éloignées temporellement pour que les signaux puissent avoir le temps de parcourir toute la région, entre guillemets, pour arriver à n'importe quel autre point de l'autre région. But, more generally, he defines the relation precede-fablement, so a precede-fablement. And, this is what? So, this is a little f, like that. in fact, in this relation, in all cases, in the article in the article, but if it exists a' and b', a' under the object of a, b' under the object of b, tell que A' precede B'. Non, non, ce n'est pas A' precede B' seulement.

47:30 Il manque plus que ça. Il faut que A' precede B'. Non, non, non. J'ai une bêtise. Oui, A' precede B', A' equal B'. He also has this possibility of having A' equal B'. This means that some signals from A could be like this. So here we have a region of B. when I put it in the intersection, but not necessarily, I could get out of it. There is a region of B' and A'. At this point, we have A' which precedes B'. At the interior, there is enough time. At the same level, it seems to be understood. It's a little bit evident. In Minkowski, it works. But, we have to see later on the pertinence of this property. To see if we can do something. Apparemment, it doesn't work by the end. The relation precedes it. No, not even. He's not even in the definition of the complexion force. There, we have taken the relationship of complexion standard, in the future. We can't suppose that A precede A? No, just not. We don't suppose that A precede A. It's to say that R, the order of causality, is a strict order. So A precede A. I don't think that A doesn't precede A. That would mean that A doesn't precede A because it would mean that it would respond to BQP. So for having a signal that it would be in the same region spatial. Because after this B, it means that every event of A can emit a signal, or a signal that propage, for every event of B. That's why you can't do it with the same region.

50:00 So it's true that it's not an off-partiel. An off-partiel free can eliminate... Is it true ? No, no, no, no, no, no. Si, si. Inclusion A, c'est bon. Or, c'est un vrai ordre partiel pour la relation d'inclusion. Non, mais en général, si A précède tous éléments B, A et B sont visuels. Donc, ma question, c'est tous éléments A. Voilà, parce qu'il a en tête que c'est vraiment tous les éléments de A qui peut émettre un signal qui a tous les éléments de B, et pas seulement une petite partie comme ça. That's why he has a little bit of the notion, the notion of the precept, but he doesn't do much after. Next question... The format is a problem because it's more catégorical. This one? No, but it's an order if you exclude... Yes, if you exclude... Ah, yes, of course, it's not catégorical if you enlève it. It's why it's going to have problems, because in fact, I have the impression, and this could be an interesting conclusion, that the notion of causal relationship is quite bad at the categorization, finally. That's perhaps one of the reasons. In any case, in this way, the relationship of causal effect is quite bad at a categorization. catégorique de cette manière il faudrait un peu plus malin contourner ou jeter certaines choses tout à fait enfin moi je le présente comme ça peut-être qu'on pourrait encore s'éloigner j'espère mais pour l'instant tel que j'essaie de comprendre ces axiomes j'ai effectivement représenté diagrammatiquement et peut-être que c'est encore trop près de minkowski ou de la variété But it's not enough far enough to verify the topology. What remains there again as a definition? So, the complexion-forte, the complexion-forte, which is not more, but it's a simple thing to remember. So this time we will consider the intersection of the light of each event of each region,

52:30 the light of light past and the light future. We will see the reunion of the light. And it is defined as the completion of force for a relation causal A-B. The completion of force. It's a region... You can see what it is... It's a region... It's a region... It's a region... So we have a sub-region A' of A and a sub-region B' of B. So, every causal path of A' of B' It can be raffiné on a causal chemin which passes by a sub-region C'C. A' C', and B' with C' in the region C. And the diagram of space-temps is inspired to give this definition. We have A and B. It's something like that. So it's at the interior, it intercepts all the signals, but it's even more. In fact, if we look at the 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0. the intersection of the future of A' is more large than the intersection of the future of A,

55:00 because A' is more small. So, in the inverse, in fact, the flèches, in order to take the intersection of the future of all the events of A, it will take the reunion, and at this moment, it will have a future more large for A, in which it will be considered the future of A', as a reunion of the future of A'. And so, in this way, if we take a path which goes from A' to D', then C is enough large. C should be enough large for that all path causal should be refined by a region. Because if I try to contour it like that, I'm obligated to get more than the light, and then it's not possible. To return to the density, there's no problem. In fact, there's no problem. in the parallel to the structure of the Goraï, there is an amorphism in the history of the Goraï, if A is causal or A is equal to B. In fact, these identities are there. Yes, they are there, but they are there. Yes, they are there, but they are there. D'accord. Oui, de toute façon, on peut toujours mettre... Après, dans les problèmes d'inversion... Le mariage par le maillement, ça fonctionne. D'accord. Oui, oui. On peut compléter un petit peu ces choses. Donc, voilà pour la notion de complétion forte. Après, je ne vais pas aller tellement dans cette direction, mais il définit la notion de site causal euthérien, avec une condition de finitude sur les chaînes décroissantes de régions. In fact, we suppose that every chain of the region is finished. If every chain of the region is finished, attention, chain of the croissance is not for the link causal, it's for the link of inclusion. That's why they change the flag, and so it's a bit a direction orthogonal to the general article, because, according to him, it would be the idea that there would be a length of the plan, There is an idea that if we could satisfy it in certain sites, it would correspond to a notion of minimum length. And that seems so fumeux that I can't write the definition. In particular, the aspect quantic or minimum length, I think there is a terrible confusion, and there is no need to add an entropy mental already generalised.

57:30 So that means that we can always, in a region, we can never define more than a certain number of regions, is that ? Yes, yes, all right. There can be a lot of... The number is always finished, is that it? Yes, because even if there is a number ornée... On a toujours cette intuition assemblée, qui nous dit que c'est un ensemble fini, point, c'est tout, mais on a l'impression d'avoir... Non, parce que c'est un ensemble fini. Ah, un ensemble fini, oui. Alors, il donne un exemple d'ensemble avec N, et l'ordre d'inclusion, c'est effectivement qu'il n'est pas neutérien. Il fait un ensemble discret, mais non neutérien. Enfin, psychosal discret, mais non neutérien. So, there is a lot of attention to this notion. But, for the moment, my objective, but maybe we will change if we are in agreement here, but for the moment, my objective is to try and justify the title. Is it really possible to approach the notion of topology bretonique? She works. In fact, in my intuition, in a case like that, it's not a finitude, but it's a complicity. For the moment, how does that mean? But if we talk about an epistemology, it's a question of authenticity. It's not the definition of notarianity. Yes, but the definition of notarianity which he defines, he is really drawn from the ensembles causals. Well, the ensembles causals are locally finis, but they come from the ensembles causals. That's why he tries to generalize the notion of an ensemble causal. In fact, the question is, what will be this change in the opposite of the logic in general? It just means that if we have two elements, there is not a single flesh, like I said, but there are several. And after the question, is it the physical sense of it? Well, we'll see, because I think that what I'm worried about, is that at the end, they're going towards the belles constructions with the pre-champs, or the champs, or the infibrils. No one has never verified that it's already pre-champs in other categories, but so let's say, what I call pre-champs are pre-champs in other categories,

1:00:00 but if we want to at least construct that, it's a site. So it invalidates the second part of the article if we don't have the minimum of having a base for a topology. We have a question about if the event A is cause of the event B. D'accord. Now if there is an event A and cause of the event B, in different ways, is it reasonable to say the point of view of physical? It is to say that we have several reports of causality. That's what we ask for the logic of physicality. I can tell you, excuse me. There are several reports of causality. Yes, the other thing is depending on the category. Yes, it's a category where you have one flèche. All right. Two flèches. Plus qu'un flèche. Now, if we want to do the technique, it should be said that there are several flèches. There are two flèches. No, but that's the flèches of two types. Yes, the inclusion. Yes, it's more than that. Here, there is a notion of... There are several ways of preceding. There are several ways of preceding the region. There are several ways of preceding the region. in a certain way, the flèches of inclusion might be multiplied by the relations of causality of A to B. Because what is fundamental in this case is that there are correlations. And that's why even if there is no relation causal between A and A, or between two regions spatially barred, we can still have There is a flèche causal, but there isn't one. But in using the idea that there is flèches causales, which is the relations causales between the two regions, it enriches the simple fact that there is one flèche causale from A to B. I would consider all the flèches causales from A' to A' to B' and A' to B' and A' to B' to A' to B. The composition of the flèches exists. So if you have, for example, But between the cross-a, and the cross-a, and the cross-a, and the cross-a, and the cross-a. Yes, if you have, for example, A, C, B, and the cross-a, and the cross-a, A, D, and the cross-b, the two only means that A, B, but it's the same cross-a. In terms of A precede B, it doesn't depend on what there is inside.

1:02:30 Yes, but if you have A precede C and C precede B, it means that there is a morphism of composition that will be A precede B. If at the place of C, you put D, you have another composition that gives the same morphism at the end. So it's always the same. Yes, it's the sense of the order, it's the notion of the fact that there is A, which precedes, and which precedes with the intermediate things, between the two. But why would you... Yes, but why would you... The logic of the index, it would have to have, I don't know, maybe we don't have to have the logic of it. Yes, for him, he thinks that he has a vision of a physician. He formulated things in the end of the category, just to the point where he can do it. But it's true that the phrase topology of Mothendik appears once in the text. Yes, but if you want to build a pretexto, if you want to build a pretexto, if you want to build a site, you don't want to build a site, but if you want to build an inclusion, at this moment, Well, there's a category of hybrids on the space-temps, okay. But what he wants to do is something better. For example, in the motion of the movement, you have a motion of point 1, you can move to three, and you can move to three, and you can move to several weeks. Is it possible to do that? Well, it's exactly, you can multiply the causal flèches in using the sub-objects, eventuously points or small regions. So, you can use the category of the sub-objects? because we have the order partiel, which are small things, it's poor. It's poor. We have a structure of category with the order of inclusion, and a structure... Anyway, with the inclusion of inclusion, it makes a lot of sense, but with the causality, it's really the millenorme. I don't think I could have a question, but... There are two things here that... No, no, the notion... I'm not going to let you know, but if you want to start, if we imagine, if we imagine, if we imagine that we have a cone for each point, when we have a cone for each point, future, past? Well, my question is, in this case-là, what do you do with the specific properties of this abstract operator?

1:05:00 Or is it something that is something other than a binary relationship? Is it something that includes an ensemble of three points? Well, he does a lot of reunions, in fact. And so, what is he doing? He does a lot of reunions and intersections. Yes, but it doesn't matter, because... No, but the operation, just like that, for the moment, what I've drawn, is that every time I've drawn it, that each region will have some sub-regions, even some small sub-regions will have some points, some events, and... I tell you that once you get into the state of the state where you have presented the show, in the intersection of Cone, etc. all that you say works well for all these actions that you say that's my question. The operation Cone is perfectly abstract. There is no axiom for the operation Cone for what you say to model what you say. That's my question. But Cone is an abstract. It's a relation binaire quelconque. Non, non, non. Il y a tant de propriétés... Non, non, mais dans les propriétés... Non, non, mais c'est ma question. L'opération qui, à chaque point Q, associe l'ensemble cône de flux, l'ensemble des... Mais cône, c'est à moi, entre guillemets, il n'y a pas de... Mais c'est le concept central, en fait. Je suis dans l'idée que c'est une relation binaire sur des états, mais pas d'états, là-bas. It's a model with the Etats. At this point-là, is it a binary relationship with the Etats? It would be a model of all the Etats? No, it would be a model of all the Etats. Well, that's all the properties. Well, that's all the properties. That's all the properties that he said, result of the logic interpretation with the intersection of problems, etc. And that's exactly what happened. It's really the démarche of everything. which is gravity. It's to say that the only thing that we concern in the space-temps is that we want to forget the space-temps and we want to find it at the end. So we want to depart from something. And what we say is that the only thing that is really fundamental in the space-temps and that we will keep in the beginning of the theory is precisely the causal structure, but reduced to its plus pure squelette, which is precisely a relationship between partials. So the idea of all these projects is that with simply the data of a relationship between partials,

1:07:30 We will find all the possibilities of the space-temps. Now it's ambitious. Yes, it's ambitious, but there are things that have been done in this field. In this field, for example, we give an ensemble, which is simply an ensemble with a relationship of the space. In this case, we can already reveal that it implies a certain type of space. because, for example, we can define a kind of surface that we have in the space, we can define a kind of form of a plane, so we can examine, for example, what are the conditions that we have in the science of the marine, which is even more exact, so that we can define something that we have in the space. And we hope that we can do it with that. It's a bit different. Yes, it is. If the ensemble coso works well, we can see in which way this construction of cone is a pivot role. But, in part from this notion, I believe we have to forget the cone and say that it's a part of the relation, that's all. But it's called cone. It's not necessary, but we can find another maximization from rather than the notion The question is, is there a need to, at a moment of a moment of a question to Mark? The question is, is there a need to put a concept of physics explicit inside, a sémantique or a physics precise, a cone, for that all this stuff works, or is it on the contrary? No, no, but I don't know. I don't know. I don't know. I don't know. But the reality is that, I don't know. Well, after, I would like to make a more precise because it's a big deal for the ensemble pose, which I don't know, but the idea is to consider two structures d'ordres, and not really d'ordres. That's a unique idea that was developed by Erasmann in 1957-1958, with an ensemble with two ordres, and, notably, for a few deflectages. We know the interest of the spectacles that we have to do. It was an example of the spectacles of the spectacles, which is a spectacles that we have to do for the occasion.

1:10:00 But then, we have studied the spectacles in terms of bi-ordres. There is a topology, and then there is an order on the ouvert of the topology. I call it a paratopology, it's a terminology that's been published at this time, and there are not many people who have a test on it, that's how it works. Excuse-moi, I don't understand. When there is a topology, there is already an order with inclusion. Yes, so you add a 2nd order. 2nd order, d'accord, that's it. Yes, it's a simple thing. No, but I don't know. I don't know. No, I don't know. and then when you make it happen, it's anonymous, it's written. You place, according to the paper, an ouvert, which will be included in the paper, or by déplacement. So it's the physical law. So it's a good thing to do with the quotidian. I think it's a memory from the point of view of the physics. But it's not at all the point of view. So, this is the following, I'm going to write it in a way more abstract, in terms of paradoxicality, and then it's precisely what it's even more abstract, but it comes from there for him. These are the categories, not the bicatégories or the two categories, but the categories double. And then it seems that we have just one category double. Okay. I'll tell you. I'll tell you. Yes, yes, yes. Because in fact, what they use, what they use, what they use, what they use, what they use, what they use, but when there are two types of flesh and some incompatibles between the two, this is what they use. It's this animal-like that will constitute the base of the category double of the situation. There's a composition longitudinale and vertical. And the actions that you've announced, they're going to write well with it. They're going to write well with it. But they're going to write an ensemble bicentlicial after. And the goal is to represent two categories, because it's going to represent the conditions on this ensemble bicentlicial, so that's the nerve of the two categories. I'm not very well following the conditions of Segal, but apparently... Effectivement, c'est cette cellule-là qui va être fondamentale dans le reste de sa construction. Mais avant d'en arriver là, je tiens à vérifier et à voir si j'ai bien compris

1:12:30 s'il y avait encore une topologie crotonique avant d'avoir un ensemble de synthétiques. Parce que ça, ça permet de mieux écrire tout ce qui est écrit avant, dès le début. Par exemple, quand on écrit les axiomes de compatibilité... Je vais réécrire... I'm going to put it on my cahier. You can put it there? Go ahead, go ahead, go ahead. On a dit, it suffirait de remplacer la notion de deux catégories par celle de la catégorie double. Oui, par celle de la catégorie double. Voilà, c'est intéressant. Directement pas compatible. Dans les catégories doubles, il y a des notions de limites, de deux limites, de limites verticales, de limites horizontales. Oui, je pense que la plupart des constructions sont liées certainement diverses. Alors par exemple, quand tu faisais l'axiome, A, B, C, A, C, on va le décrire plutôt pour le temps. Après, non, c'est pas ça. A, B, B. Alors ça, par exemple, like this, the second is written down, it is written down in the form, if I have a B, I have a piece of paper like this, and then a vertical like this, then I can complete with this identity, and then here, something. So the axiome is written down in the form, every time I have a board, if I have this board which is written down, but it's actually part of a complete cellule. And all the rest is created in the avenant. I have an SI, which is one of the stages of the SI. The TJ is there, and the raffinement of a chemin, because it's why I asked you a question earlier, because you have a raffinement of an ensemble by another.

1:15:00 But in reality, I think that on the chemin, normally it would be a little more precise, but the raffinement of a chemin by another, it's really the fact that I have a other chemin here. And so that each thing here is included. Yes, it's exactly that, yes, I put it on the family, and that's exactly what it is. Well, that's really an idea of homotopy, but two-dimensional, that's what makes homotopy vertical and horizontal. And at that moment, the B, the conclusion, it's a particular thing, like that, and which is the property for all the way, for all the way, for all the way, for all the way, for all the way, there is a special way to find out, because for all the way, there is a raffinement here, with one of those, which is in B. And so, we see that what we are trying to build, is just a homotopy between all the way and the way of particular to the length of 2. All right, so all this is written relatively well. It's a very semblance to call two categories of inclusion, in the paper a little bit further. So, is it a double category, is it a two categories? It would be written precisely the actions to make sure what it is really? Well, if we had a vertical category, or if we had a vertical category, and we had to take simply the square committal with vertical and horizontal the same thing. What we do here is really the terrible homotopy of the category in question. And what we do here is the suspension of the category, of the omega. But here, what did not seem to be clear, it is that there are two types of flashes, so it is rather a category double. Or, he says that his ensemble bis-saint-picio satisfy the conditions of Segal for that it is a bicatégory. Yes, but the bicatégory that he talks about is not a bicatégory with these two flèches. It is a bicatégory with the flèches between the chemin and the relations on the chemin. So, a path is one path, and then the path between the path is two paths.

1:17:30 But the question is, it's the link with the glissamptitious ensemble, because he has a very bizarre phrase. He says, since the regions of the site have two ors partials, But there is a correspondence, there is a description, there is a description naturelle of the subcoso in tant qu'ensemble dissentitieux. And that is not very obvious. But it's simply... In fact, yes, can we take the two nerfs with two relations of order and not with two categories? you can use it in both ways. Because in fact, if you have a category, you have to take the nerf, you have to take the ensemble symbiotic, the ensemble symbiotic that satisfies the conditions of the gap, and you can come back with the cocoïde, but if you... Can we do this construction with a double category? Because at the beginning, there is no category, there is a double category with these two types of flesh. And it's for that I have noted exactly that. But just now, his chemin is different. He doesn't try to build a big category or two categories with these two flashes. He uses these two flashes to say, I have, in my theory, the ensembles bisimpliciaux. Then on the ensembles bisimpliciaux, he builds, he builds, and then he builds another flèche sur les chemins causaux. Et c'est la flèche causale plus la flèche sur les chemins causaux qui entrent dans la bicatégorie. C'est cette bicatégorie qu'il appelle inclusion. Oui, oui. Et c'est cette structure qui est très avancée. Justement, je peux écrire la définition, mais ça me semblait tellement lointain que c'est effectivement un niveau plus avancé qui va commencer à pouvoir construire des préféceaux de catégorie sur la catégorie d'inclusion all the cones of the past for all the regions and all the cones of the future they define the past, the future but before arriving there there are many things to clarify before we can build this bicatégorie Is it a bicatégorie, it's the same thing as two categories? There is a difference in fact there is one that is faible and the other that is not faible that the composition is only a little bit different than the bi-catégories.

1:20:00 It's not the bi-catégories, it's not the bi-catégories. In fact, today, when we talk about bi-catégories, it's already low. There's a bit of a glissement, James. There's a bit of a glissement. In the current parlance, if we say two categories, it's rather low, but initially, it's a bi-catégorie and a two-catégories. It's a-t-a-dire... I think the right question is what he said in section 5, page 10. Because until then, he explains all the history with the flèche of inclusion and the flèche causal. And then he goes to the structure simplicial where he says there is a natural correspondence between the two. And then, from the assemblies, it's there that they built the categories of inclusion. But the fundamental correspondence that allows them to pass, it's that. And you have to look at if there is a reason or not to talk about the correspondence, the description of nature, as we said in page 10. To which point? In page 10, it begins by and it has two partial orders on them, they have a natural description as the vertices of the bicyclical set. And that's not very obvious, I think. Because if it's true, if it works, after he uses a theorem of I don't know who, to put the assembles of assumptions in correspondence with a figure, and it's like that. So what happens is that with these two orders partiels, he has rather a double structure, and not a bicatégorie. And, as he said, this phrase is on which one can gliss, it's that this correspondance... For creating an ensemble bisimplicial, is there a plus to gliss it? Is there a way to construct an ensemble bisimplicial based on a double category? It's the same thing that the same thing that the same thing that the same thing that the same thing is. D'accord, so there's no need to have a specific catégorie of the same thing. When you talk about the same thing, I think it's the same thing that is associated with it. But I'm not sure you can go into that text. D'accord, so it would be more interesting to see the same thing that the same thing is associated with it. Yes, because it's this section 5, because after he uses just a theory of I don't know who,

1:22:30 just to say that once he has the ensembles symbiotic, he can go to the bicategories. So it's automatic. He has to impose when-même the conditions of Segal. Yes, the conditions of Segal. So it's a clarifying. That's what would be the signification of Segal. Because these conditions of Segal have a signification of Segal very strong. Yes, yes. They play the key to be able to come back to know that we have an ensemble symbiotic which is a good thing to do. which is something cool. So there is this construction that I don't know. How to do with a category double, with a category? Construire the donor, and then... And then there's the theorem... But then there's the theorem of Dapsamani. That's what's behind it. That's what I'm talking about. It's very clear, because there's a category, which is, you have to say, it's a frontier for the ancients. It's not 2 estis, it's a category. Sigma 4. Sigma 4, it's a category simplicial with limits. A category simplicial. You can use certain limits, delta-hob. And delta-hob. When you say that you have a unique limit, you add something? Yes, you add something. When you have an ensemble simplification, you have E0, E1, E2, E3, you simply say that the E3, the two flèches which are here, they form the product human, and the two flèches which are there. That's exactly what you want to say in saying that if you have triple flèches composables, these are the flèches which follow. So that means that you add a condition that you have a certain point of view. So in fact, it means that one category is a function of delta-of-normances which has certain properties to transform certain diagrams in limites. So the question is that we have what we call it, and we call it the category. So in lieu de dire qu'on a un compteur comme ça, on va dire qu'on a un morphisme d'esquise. Le sigma 4 vers Hans. Evidemment, d'une fois, c'est complètement trivial qu'une catégorie, c'est un ensemble officiel. Alors, l'interaction au point de vue, c'est un truc qui se réitère.

1:25:00 Après, vous pouvez faire ça pour les catégories d'O. Rien ne t'empêche de regarder maintenant sigma 4 dans 4. That's the category. But it's not even to say that I can make a sort of product sensoriel if I make a product of the same, it's a product sensoriel in my case, and it's delta-op, not delta-op. So, to give something like that, it's the same as to give something like that, but to give something like that, now it's delta-op, delta-op, So you have the bisimplicial, which corresponds to that, which corresponds to that, and then you have the bisimplicial. So the categories double, the categories are placed at this level, at the level of simplicial, but the categories double are placed at this level, at the level of bisimplicial. What does that mean? It's not that the piscine is here, but it's what you see. It's like the cells of the type of piscine, and it's here. So I think that's what it's about. Yes, yes. In which language we're talking about. The point of view catégories is that when you are sure that you look at the catégories, it is in effect to be associated with these catégories. So at some other catégories, or at some other catégories, or at some other catégories. Because if we want to build, That's to say, his but apparently at the end, it's to build a catégory, with the pre-champ. And for that, he needs to represent a bi-catégorie. And is he able to do that? He can define his catégories on a double category, not necessarily in representing a bi-catégorie. If his objective is actually this, for the category double, perhaps. Just for the beginning, because of course it's abstract. The same, the categories can always, more or less, because of the phenomenon of Gioneta, they can always see the category for the FSO, that they can only do the same. When you talk about the category C, you can always do the same, as you say, So, for the category 2, what can we do?

1:27:30 We can put 4 here. I don't know how to explain it. What I want to explain is that this thing, we have a concrete example, it's the following. I take a prime category. I take a fromcter here. I take the figure here. Here we have 4 categories, 4 fromcter. And between these two constructors and these two, I put here a transformation naturel. I put it like this, between this composite and this one. With these blocks, I can create a double category. I'm going to create like this. This category double is also considered by R.S. Manat. It's called a category double. Because there are five things. And in fact, we can demonstrate, that is a thing that I have done in the same time, that every category double is plunged into the category of the catfish. So that means that it is still a concrete and universal for that. So we can always think, we can always imagine that the sommets are the categories, and the flèches here are the concrete. D'accord. So it's a way to associate it. Because after, at each region, it will associate a category, and of course it would be a region which would correspond to the future and the past. He has a two concrete future, two concrete past. I have a question of terminology. what he calls the two-categories weak. That's what you call a big category? Yes, yes. That's strict. But in the future... Because in the theory that he uses, of Tam Samani, it's about two-categories weak. That's what you call a big category. Yes, yes, yes. Today, just two-categories, that's what is already weak. D'accord. bingosibouriste et benabo qui proposent ce terme après on a un faible et aujourd'hui c'est le genre qui est often je voudrais quand même il me reste à peu près une Kyocles je vais quand même rappeler juste le minimum des actions pour une base d'une topologie crotonique et vous montrez le problème

1:30:00 que moi j'ai entre the authors are present, and there are difficulties to make the link with the actions of the topology of the autonomes. So, we will not use the scribble, in fact. For the base of the topology of the autonomes, there is no need to be scribble. After, we can look at the scribble engendring, but we just look at the notion of family couvrant. How do we define it? We define the notion that a region A couvre a region B. On revient à notre structure causale pour voir comment, avec le gain de causalité, on peut définir une relation de recouvrement, une notion d'objet couvrant, de famille couvrante. Just after the completion of force, the definition of A becomes robust. And in fact, A precede B, and all the cause of B, in the other direction. He uses it in the other direction. At a moment, I invert it in my notation on my paper, and I realize that after we will have to compose, so I will use the sense. And then after we have A, B, C, B, so we have B. B, three powers. So, B precede A, and it needs to be, for the definition, that all the cause of which ends in A will be able to be raffiné in a causal which passes by a sub-region B' of B. the causal which is about in A will be raffiné. He doesn't necessarily pass to the last step, but to a certain step, he will pass to the sub-region of B. So here I go, blah, blah, blah. At a moment, I pass by B and B, and B, and B.

1:32:30 Then I continue the other chemin, and at the end, I arrive in A. That's why and we say that a family is more important if the reunion is in this sense. The diagram of space is still in this case. The diagram is possible in all cases. So B is already there, and B is already there. So B is at the inside. It's not that it's going to pass. So I'm going to take something like this. and that would be more important for A. If a family is more important, it could be like this, for example. This is B.I. or B.I. in the reunion of A. if we find a causal which ends in flat. This is quite evident, but, in fact, these regions such as this, like this, like this, we can always raffine the path to something smaller than to pass by the sub-region of the prime 2D. So I have the impression, in fact, intuitively, the notion of family couvrante, which is central if we want to have the base of the topology, it is to have a region the largest possible which is contained inside the cone past, large in the sense spatial. It is not possible, it is not possible, it is not a signal of any region preceding or preceding A, it is not possible. It is rather the surface of Cauchy,

1:35:00 The surface of Cauchy should be feuilleted, it should be used to use. A surface of Cauchy is not... But for A, if you restrain it's A. A surface of Cauchy relative to A. Yes, so maybe we could have a definition. It's to say that the data of everything that happens on this surface is enough to determine what happens in A. For the propagation of the Klinger-Done, for example, the Klinger-Done. So it's with this diagram that there is a problem with the actions of the base of the Klinger-Done. What I say is wrong. In terms of the surface of Cauchy, it depends on how we define the surface of Cauchy relative to Cauchy. Look, the real cone of the past A is much larger. What you call the cone of the past A is the reunion of the events. Yes, so the cone of the past, which would allow everything that happens in A, is that something that covers, like that. Yes, that's right. That's right. That's right. Yes, yes, yes. But at the level of propagation of the signals, we need something more large. So this surface B, I don't see very well... Physically, it doesn't correspond to a surface of the technology? Well, but in any way, maybe in using... This is perhaps an important point, but if we come back to the fact that causally we have to need, rather, the surface of Cauchy, maybe it's with these cones larger that we could satisfy the axioms of the base. But for the moment, I'll just remind the axioms of the base of the topology. I can't remember the definition of A. A would not be true to B. So, B recouvre A si... So, B recouvre A si B precede A. And si, tout chemin causal qui finit en A, qui finit en A et qui passe par une sous-région des primes de B. You can influence all the data? If it was a point, yes, but...

1:37:30 Yes, it's a little bit more large because you can influence these zones-là by the way you can see the outside of the universe, the intersection. There's always this intersection and reunion of the economies of the past. In fact, if one thinks of creating a site, then it should be different. And not to introduce this relationship on the recouvre in the events, but rather it should be looking for a definition of family. All right. In fact, he defines the notion of family couvrantes from the notion of object couvrant. All right. He's collated to the reunions. He's collated to the reunions. A priori, it's a family couvrantes. There's an espace on-dessus. There's an espace on-dessus. But it's not that. It's still a key to the ensemble. But this encouvernment is not a encouvernment. It is built from a chain causal. For the notion of family-couvrantes, the family-beis-couvrantes is the reunion of the beis-couvrantes. So, like Andrew, we are collated to the ensemble. But maybe it would like to enlarge the notion of family-couvrantes, indépendamment de la notion d'objet courant, ce qui est subordonné à cette notion d'objet courant qui a priori n'existe pas non plus. Et même si ça n'a pas fait de bon sens physique non plus. Et même au niveau physique, il faut quelque chose de plus large. Donc juste, je vais rappeler quand même les actions de la base. So, for example, we give a category C, and for each object U of C, we give a family So we have U, for example, we have a property of an antique, in general, which is what we have to do. We can absolutely say something, but in this case, it will suffice, because we have an ensemble. And then, K, maybe that U is an object C. On a KU qui est un ensemble de familles dite ou dite ou dite ou dite ou dite ou dite ou dite ou dite ou dite ou dite ou dite.

1:40:00 f of U' which is an isomorphism. If I have an isomorphism of an object to another, then the single tongue contains an isomorphism, it is already an isomorphism. If f of U' is an isomorphism, then the single tongue f is an isomorphism. The single tongue is an isomorphism. The second axiome, if I take the FI of U of U, and if G and any flèches of C and V of U, Alors, il existe H.K. So here we have a family to move on to D, so an element of KD. This is the case of D, an element of KD. Each flèche G, R, H, K, so G, R, H, K, it goes from V, K to U. Factorize. Factorize. Factorize. Factorize. So in the diagram, if I take the output of the unit, the output of the unit, and the output of the unit, if there is a subject in U, in fact there exists.

1:42:30 Like this. and then there is a factorization for each case, an I2K, by which we can pass. And this diagram, I don't see how to satisfy. And then the third axiom is the transitivity. And here I see how to satisfy. The third axiom is that if I have family F ou. c'est une famille qui couvre U, une famille qui couvre UF, Si J alors l'oct hogy Z worked H, des G e G, you crois ? No, you ... the family... but... Raffinement. Oui, tout à fait. Raffinement ou transition. Oui, mais ça marche avec le raffinement. Avec les flèches causales ? Non, mais eux, ils disent que ça ne marche pas. Ils disent que ça a l'air d'être authentique, mais... Oui, mais la phrase qu'ils utilisent, c'est qu'ils disent dans notre catégorie, il n'y a pas de pull-back. Non, non, non. They say that even if there is a pullback, it is not good.

1:45:00 They say that there is no pullback. But if we look at it, there is no pullback. So, let's go ahead and see if there is at least an action for the base, even if there is no pullback. They say that even if the pullback exists in this category, like we saw in the example. Quelle page ? Le pullback d'un recouvrement n'est pas nécessairement un recouvrement. À la fin, page 5, à la fin, ça a l'air d'être une typologie d'or authentique, mais si le pullback existe, le pullback d'un recouvrement n'est pas un recouvrement. Yes, but it's not a pullback what we use, it's simply the existence of a composition. That's why they say that it's an alert to be a prologue. That's why I wanted to see the pullback. It's a square, you have to write properly. You have to be careful of the fact that you have the same image. by the way. By the way. The general definition of the book... And it's the only place in the text that he talks about the topology of the European Union. After he built the vessel and everything. Sauf that he uses the word site. And site is synonymous with the topology of the European Union. He uses the terminology but... But after he uses... What I want to say is that we try to have the heart net. Is there something, is there a link or not? There is a link, eventually, in a little bit, in a little bit, in a little bit, or in something like that. And so, in particular, the action of the neutrality is not satisfied. And we can do a diagram of space-temps which shows that it does not be satisfied. It is that, with the simple families, we will take simply an object U' which covers U, So, the price is not covered. So, we'll take the flow, like this. Well, here's an object. The price is not covered. Now, if I look for an object of a second, or a family covered... Well, it's not this action-like,

1:47:30 This diagram, we don't know how to interpret the flèches. The flèches G, we don't know if it's a flèche causal or a flèche. If it's a flèche of inclusion, it's trivial. It's just intersecting in the good past, and we restrain the family of the good past. There, there's no problem. If it's a flèche of inclusion, it works. Well, for the flèches, it's not interesting. But if it's a flèche causal, it doesn't work. And that's where the difference is. At this moment, what I want to say is that if we want to build a prefaceo in categories, we need something that works for the prefaceo. But what you want to say is that when they say that it looks like an anthropologist, it's not true at all. It's not true at all. It's not true at all. It's not true at all. It's not true at all. It's not true at all. But... Yes, it's still funny. If it was published, the text was written, the letters, the revues, etc. Yes, I know. In general, people publish their own revues. I don't know. In general, it was an opportunity for me to say, is there really something that looks like? And really, if we take the axiom the most simple, Johnstone said even, it's just to verify this axiom. And already, if we have it, we can build an axiom. But the central one is the square. It is not satisfied. This flèche is not satisfied. The flèche is ondulé. Here, yes. Here, we are going to put the flèche ondulé to define a notion of family couvrante. And here, if you take a flèche d'inclusion, it works because it is necessary to intercept the family couvrante by the cone of the lumière and the sub-object V. And if you take, in revanche, So, V is going to be the prime, but you don't have to be the root of V, but it's not the root of V, but it's not the root of V. If you try to find a family of the U, you take what you want in there. All the families of the U have more or less than the root of U, but with this, if I compose, I don't see how we can factorize by a prime.

1:50:00 No, I don't have a flage causal. That's it. The flage G, it means... If we only take one flèche of inclusion, that works. That's enough. Yes, but is it really what we want? Tels that it's formulated, all these flèches are the flèches of the category of which we can use the site. So we have to take the same flèches, vertical and horizontal. Yes, it's not a classic, but in the context of what we want to do, it's something like that. Well, at this point, it's probably why they have introduced it. to deform the action, not the usual actions. I see. Because in the category double, there is an intuition geometric that is behind, compared with the first one, and the second one, it's the idea that, if you have a flèche ondulée like that, you have an intuition, it's the idea that it represents, it's the ensemble of the square, It represents the idea that the flexion du lait in the top is a sort of restriction of the flexion du lait in the bottom. That's it. Like for example, when we have an application that we limit to the sub-ensemble of the part and the arrival, and that we have a function of A to B, or we limit to the sub-ensemble of A, that goes into some sub-ensemble, and then B, it's a restriction of a function. That's it. That's it. That's the idea that we have here. And it really means that it's like an image inverse. So maybe there will be an inclusion. Because that would be a good idea in the context of the image inverse. So here, all the horizontal horizontal, it would also be an inclusion. Yes, yes, yes. So we always have the idea that the element element is that And then, yes, it's in this context, I think it would be reformulated. There is nothing that would be universal. If you put the left-hand side on the right, an inclusion in bar, and you ask if there is an inclusion in bar,

1:52:30 and you ask if there is an inclusion in the left, that it is universal. That it is universal, that it is to be an inclusion in the right. Well, it's a factorization, which is a little more delicate, because it's a very endogonale. So, there is a leçon of doubt that exists. D'accord, that's true. D'accord, that's true. The thing to verify, it's the factorization. But at least we can see if, in space-temps, it corresponds to the intuition that we have. In gros, it sounds like it sounds like it sounds like it's fabricable. D'accord. There's a distribution of the site, but it's not a site of the site, because there's two types of flèches. D'accord, so that's it. So, I think the group has no idea of the flèches because it's another thing. It's perhaps an object parallel, which has an interest. which would be the topos, because the effect of the spectacles are the spectacles. D'accord. They would be the topos, which would be the topos. But there, there are two flèches, but we can forget that they are of two sorts. And there are some flèches... Because they are not. Ah, they are not. They are not. They are not. They are not. They are not. They are not. If you want to be the topos, you can do it. but once you've used the term of changement, I've used it earlier, because when everyone is in the quintet, I'm like, okay, now, you have to use the flèches on the left and the verticals on the left. But, it's a bad thing. D'accord, but you have to add the two flèches. It's a demand for the two flèches. It's the 5th element. But, it's not on page 2 where it says exactly that the definitions are not. It's on page 2, entre parenthèses. We warned the reader. C'est le troisième. Page 2. Page 2. Enfin, je ne sais pas. 8ème ligne. Oui, nous avertissons le lecteur, vous voyez. Entre parenthèses. Ah oui. Ça marche pas tout à fait. Que notre définition du site n'est pas tout à fait celle de Rottenberg. The question is to know for which reasons it doesn't work and how it works. I don't say that at any point they don't have a token.

1:55:00 They don't have a token. The problem is that it's still hard to use the word site all the time, while there's no site. It's a thing that makes people laugh. If we show that with the two things there's a token, I think it's really the fact that we have a double category and we have to take into account, it's a sort of categorization of the notion of flowyting. There's a parallelism between the feuilletage and the space topology that we have to do. It's a bit more subtle, a bit more fin than a topology grotonique. To come back to what you said, a categorization of the notion of the feuilletage. It's interesting, but I'll come back to what you said before, that the high-flash was always a sort of restriction of the low-flash. Yes, it was a very different restriction. It means that you are going to define a weightage and when you are going to go through the line of the lines of the lines, you are more or less restring. If I mêle both things. And so there is an evolution. No, no, no, no. It's not that the line of the lines of the lines of the lines? It's not that the line of the lines of the lines of the lines of the lines. Because the restriction is the transverse. Well, so. In any case, it defines a kind of flow, which is the one that says that you are going to have a description of the plus or plus restreint. Or this flow, of course, when we go back to the space-temps, it will be, in general, a flow temporel. So it means that there is really a direction of flow temporel. It's not at all neutral. All in one sense or in the other, it seems to not be the same thing. It's not at all. It's not at all. There's a restriction on the group, and obviously after, there's a double-stress. What if it's a true inclusion? It's not at all. It's not at all. Yes, that means that from a purely causal structure, you arrive like that to build something

1:57:30 which is a sort of plus temporal, which has a non-trivial property. If there is something that makes when we move with the plus temporal, On va vers une description de plus en plus, je ne sais pas comment dire, restreinte, c'est intéressant. Bon, ça me paraît une ouverture. A priori, there's no relation between the two types of fish, because they don't have an issue that they don't have a problem at risk, but there's no reason to do it like that. In this case, there's no reason to do it like that. So it's not because we don't have a problem with food that automatically we have found a restriction in a kind of food, because the compas are the compas. I don't understand what I mean. Well, it's very simple. We had to do that for a reason. I don't understand that. I don't understand that. Because, I don't understand what I mean. I have a question for you. I don't understand what I mean. The notion of causality that we have just here in the end of the day is strictly liable to the propagation of the sign. It's totally inspired, at all. Are we considered other reasons, other notions of causality? At least the ensemble causal? Not that I know. In the concrete sense of the physicality, we have other notions of causality? non non non on physique il faut savoir aussi on a une relativité disons en théorie de relativité et la notion de causalité si on a traduit des relativités quantiques par exemple mais pour le moment on ne sait pas justement la notion de causalité la notion de causalité en relativité quantique ce sont les ensembles causaux et ce qu'il faut savoir c'est que si vous avez une variété disons que thodo-rymanienne, loranthienne la notion de causalité c'est la même chose que la structure conforme It is completely determined by the conformity, and it determines the conformity. The conformity conformity is the same thing. But what you would ask is that in physics we use the conformity? And the question, when we want to do the gravity quantic,

2:00:00 if we want to, at least in the geometry geometry, we want to, in general, that the geometry is quantic, that it is to fluctuate. The question is, is that everything fluctuates, or whether the topology is fixed, and it is simply the geometry that fluctuates, or even the topology and the structure conformes, that it is to be fixed, and there is only the part non-conforming of the geometry that fluctuates. and there are different approaches. We could try to formalize the fact that the causality exists in the micro level, and that the causality in the habitable sense is an effect resultant... Ah, well, that's already formalized. Because if you want... The causality in the habitable sense is formalized. Yes, but already. You could have a point of view. No, no, but the definition of this is the same, and then, globalement, there can be several tests, because the rayons lumineuses are the same. What do we do next time? We're going to continue that. I don't know if we don't have a very serious basis on what we can do. There are many definitions that we could adapt, but on what we are, on what topology we are.

2:02:30 I think that Tamsamani is equal to this kind of thing, but maybe after. But this is the category homo-topic, Thamsemanie, Seigal, it's also a work that is a physical, but you don't have to do it. Yes, but maybe we can stay in it. Yes, yes, but maybe that's it. But maybe René can continue with what you want to do. I can do a little presentation of the causality in relativity, but it will take a lot of sessions, I think. But maybe you and René. On the date, on passe à jeudi. Comment les noms ? Voilà, comment tu l'as fait ? Ça, c'est un premier point. Après, il y a des petits programmes de l'adaptation. Alors évidemment, suivant ça, le bord, je ne sais plus ce que c'est. Parce que si au lieu de l'appeler V1, V2, V3, tu vois, si tu prends V1, V2, V3, tu le tires là, et les 24, V5, V1, V1. Et donc, tout le bas est un peu là-dessus. Et donc, déjà là, après. it is Well, that's Christian Senior. Thank you very much.