Lawvere Axiomatic Cohesion / discussion

0:00 Thank you. It's easier if there exist the number of the buildings, right? Yes, but it's not the number of the buildings. I don't know. Aucun dans le site. Aucun nom de l'ébattement. C'est un test. C'est une bonne testation de votre sens direction, de votre sens géographique. On a toujours le projet d'essayer de monter un psychologue ou une école dans les catégories. Oui, bah oui. Dans quelques temps. on arrive à faire une synthèse de tout ce qui se passe en catégorie inclusive on a abordé pas mal de projets, très différents en fait mais c'est toujours un peu, c'est toujours des commencements il faudrait pousser oui, mais moi dans tout ce qu'on a vu, ce qui m'a plus entusiasmé And it's actually lower, even if it's not what I have the most understood, but it's very deep, so if you want to deep into something, it's all right. Perfect. So... I can't believe it. There's no doubt in which other Catholicism is interested in the physics. I'm sorry, I'm sorry. John Baez, I'm sorry. Pardon? Who was that?

2:30 John Baez, in the last year, he was in the last year. John Baez, in the last year. He's interested in coming to me here, maybe? I don't know. In July. Ah, okay. Ah, actuellement. On avait regardé les trucs de fait et on avait conclu que c'était un peu superficiel, que ça n'est pas très loin. Non, pas compliqué, finalement. Non, ça n'est pas très loin au point de ce qu'il a besoin d'avis. Mais actuellement, l'Ukraine donne un très bon et bel exposé à cette... L'Ukraine donne un très bon et bel exposé à cette colloque en Londres cette semaine dernière. And tell me, he will be here in Paris several times this year because his wife is French. So maybe it's possible to arrange a presentation here for him. Thank you very much. Lovir the last time, which I didn't mean, but I didn't want to make an explanation complete, detailed, definitive of this exposé, which was very dense with a lot of things. So, the but, if you want, well, at the lecture, I was aware that there was something quite nice in the whole tentative of Lovir, which could be formulated, finally, as a problem mathématiques à propos de la physique qui serait le problème suivant comment formuler mathématiquement la notion physique de cohésion de l'espace alors c'est sur ce point là qu'il

5:00 nous a dit des choses la dernière fois mais qu'il a dit des choses aussi dans des textes anciens donc j'ai été voir un peu d'autres textes où il parle déjà de cette question donc qu'est ce que c'est the approach mathématiques of the cohesion. So I'm going to read. So I'm going to read the definition minimally of the cohesion in physics. So that's why I would like to... My moyens are very limited. I looked at Google what the physician could say in Wikipedia or ailleurs on the cohesion. I found a certain number of things on the cohesion of the material, and things like that. But I'm not a physicist. The word cohesion is never used in physics. fundamental. It is used in physics, in chimics, etc. But it's not a term that is used for the experience. So I think that, we will see after you can hear what the physicians can hear there. In particular, the question of how can we hear what is going to hear would be the following. Is that, from the point of view of physicists and practitioners, they can, without in order to have a notion formelle, to engage, let's say, the idea that there are several states of cohesion possible of the matter or of the space, or of the space, of the space. I don't know if it's the matter or of the space, or of the space, or of the matter, but is it that the idea that the cohesion is relative, is something that the physicians have a problem and that they have something to say, and that they have something to say. Because it's that the way, it's this thing that, at the point mathématiques, all of Yes, it seems to be the moment to place the debate, because I don't call it a cohesion, but I think it's this problem that it is. It's the eternal debate, which is at least from Newton-Lenitz, between the existence of an space, or an space-temps today, absolu versus une conception purement relationnelle de l'espace, c'est-à-dire selon laquelle ce qui existe, ça n'est pas un espace absolu indépendant de la matière, mais ce qu'on appelle l'espace, c'est la conceptualisation de l'ensemble des relations entre tous les objets, ou pour l'espace-temps, entre tous les phénomènes. Donc ça c'est un débat qui est très vieux. Jusqu'ici la physique fonctionne sans y répondre, c'est-à-dire quand on est newtonien, si on well Newton is in part for an absolute space, but at the limit we can reformulate a little bit the physics newtonian in terms of relative. The relativity, she is construed in supposing an absolute space with certain properties,

7:30 but we can also have a vision of relativity relationnelle, and Einstein already had this vision relationnelle. And where the question is posed today, it is when we want to go further, which transcender both the physical quantic and the relativity, it's to be a physical quantic or something. The question is to ask if we should consider that the space is something that is substantial and that, like all that is substantial, should be quantified and that gives a piste, or if, at the same time, the space should be purely relational, that is, finally, the properties of the material. And in this case, we don't have to quantify that material with these properties. And this debate on this question, which in English we call the background independence, in this moment, it makes a fureur. In general, we could almost say that there are two camps. There is the camp of those who are not interested in the background independence, and that is the camp of the string and the super string. And in other words, the camp of the other, it is those who defend the gravity quantic, the networks of spin, with just Barrett-Krein and the approach categorical, all the stuff around that, HTCAR, Rovelli, Smallin, etc. So, this debate is often posed in these terms. Well, obviously, it is not as caricatural as that. So, the big question is, is there a space as an object physical, independent of the matter with an interaction with the matter, or is it what we call space? It's just the properties of the material. And just to finish, quant to the general relativity, we can adopt these two conceptions. We can say that there is an space-temps which is a different variety with its properties, munis of a matrix pseudorimaniac. So what we call gravitation and what is also a measure of distance or the relationship geométrica between the objects that are related to the space-temps and we have interaction between the space-temps and the material. But we can also have an approach relational which is difficult to conceive with what D.G. Einstein had in the head which is to say that what we call the metric, the tenseur metric, the tenseur of corbure, in fact, are not attached to a variety of all sorts which would be the space-temps,

10:00 of all the material, which would not live in the space-temps, but would only live on the one compared to the other. This is the concept of the relativity already, which would be purely relationally and purely machian. In this way, there is the principle of Mach also. I do not pronounce the word cohesion, but I think that it is this question that there is behind it. Well, it's clear. Well, for what you just said, I remember reading the book of Smalling, on the last one, on the contrary, which explains, which is a little more in detail than what you said, but it explains that there is a great alternative to do a physics with a fond, an space that serves the fond, which is what you want to say, In which, in which, in which, in which, there would be a phenomenon, even if we can manage things, even if we say that there is not a single space, but that there is an space for observance, and that these spaces are raccorded. But there is still the idea that there is a font. And then, how do we have a physical where there is an entity of the start? An entity of the start would be the champs, which would not be the champs in an space, which would be the champs as an entity primitive. And then, we would build the space, after, as a product of the function of these champs. So, it's an alternative. It's exactly what you... Exactly. And how he did it? Well, for this alternative, it's actually the proposition, the question of the cohesion in terms of the theory, finally, we don't have to trance to know if we're in this perspective or this perspective of the physics, because he speaks of cohesion, certainly, I said earlier, in terms of cohesion of space, but it's also cohesion of calculus, if you want. It's to say, the cohesion that he tries to speak, I don't think I understand, There are quotes and parentheses to all of that. It's the one that will allow us to take a certain type of calculus. The calculus différentiel, especially. And the calculus variation, as you mentioned. Like conditions of possibility. So, what is the cohesion? So, of space, if we start with a space space and a phenomenon in it, so directly from the system of the champs that we put in place, Quelle est la cohésion nécessaire pour qu'ensuite s'enclenche un travail mathématique de calcul, de différenciation, de calcul différentiel, de mise en équation, etc. Donc ça commence à rendre la question un peu plus précise.

12:30 Mais je pense que, disons que par rapport à l'alternative fond ou pas fond, on peut dans un premier temps ne pas s'en avoir à trancher. Je précise un peu techniquement la chose, c'est-à-dire que, justement, par le biais des topos, on peut aussi bien penser à un topos comme un espace fonctionnel, généralisé, c'est-à-dire déjà un système de champ extrait, que comme un espace. Et justement, le langage des topos nous permet justement de ne pas trancher. Maintenant, ce sera peut-être intéressant, justement, de mieux voir dans les topos le côté espace. Alors, est-ce que le côté espace qu'on évoque là est lié à la notion de ce qu'on appelle topos spatial ? I think it's not necessarily automatic, but I think it's between the space and the space function. And the topos, in the end, I think it's one of the techniques of his possibilities of thinking, which is not between I am a topos space or I am a function on an other space. So, it's the first point. Yes, just to make the basculement, here, the choice, the relationship, the substance, the space of the fonds, or relational. And here, with the Raman de René, who introduced the topos to say that we don't need to trancher, independent of the tool, what it means is not to trancher, it's not to statuer on the object, but it's to place the question on the lien of the connaissance. So, this, independent of the topos, in a more general, in order to say is that I have to have a statuette on the object or a statuette on the space to construct the knowledge, I can be neutered and not have as first question, this one, and then I can question it on the line of the knowledge. So this is what I call the topos, but also maybe other tools. So I mean, it's not the topos that allows us to move through everything, but it's to move through the practice of the knowledge, and to get on the line of the knowledge. because here in the interventions, exist-t-il? Well, no matter what exists, when I say that, in fact, exist-t-il? It's a question of where begins the mise in scene. Yes, it's a question of imagination rationally. I'm sure that it will have impact on what we're going to do, but if there are some imagination rationally on the question of the media, we can leave these questions in response and work. It's just a moment, I think, for everyone, because it's true that when we do the relativity, we don't have a concept absolutist or rationalist.

15:00 But if, for example, I want to quantify the relativity, what does it mean to quantify? Well, it's a vast debate, but in general, the idea is that if we quantify a theory, we quantify the real thing. of theory. So, we can't say I can't find a theory without identifying the objects that we consider as real in our theory. So, is it in the relativity, we consider that the variety space-temps is a real, which is a reality independent of the matter, and in this case, we must find it, or not, and in this case, we must find it. So, I'm sure in the practice of relativity, we can't fiche, and that's the force of the physics, of course, to be able to function without having to answer complicated but if we want to progress through a new theory then we have a choice and the two options are really different for what we are going to do in practice. They are very different by the way that has created two communities that are not supposed to be in reality. The result is what you say when you say reality, there is a fact that we have to identify things that are not necessarily observable and that we can not represent it as a sign of a jauge and you are aware of it. There is the same problem. When you have these degrees of freedom which are supplementary, which in fact exist not, and that we have an invariance in terms of these degrees of freedom, which is what we call the jaunes, well, there you have the choice. So, first, you take off and you quantify what remains, which is real and all the same. So, if you don't know how to do that, you pass by an etat that is, but you say explicitly, you say, I'm going to quantify things which are not real, so these and then you have to remember that you have quantified things that are not real, so you modify your theory. But I would say it's a surprise. You can see that you can see the symmetries after thinking about it. Yes, and that's why, by the way, we don't know. Smolin, in his last seminar, he said that the notion of symmetries, we don't want to. Because if there is a symmetries, it means that there are things in trouble. And if, for example, there is a symmetries joke, and that's what I always thought about, I was very happy that someone says that there is a symmetry compared to the variance of phase, that is to have a function, that is a phase that represents the same physical object. Well, just simply, what I say is that the phase, it doesn't exist. We were too bad for finding an expression, to make sense of this thing without having to intervene the phase, but it's something that, in other words,

17:30 for all the coordinates in space-temps, it's exactly the same thing. What we want to try is the physical object that exists really according to the relativity, it is, for example, the tenseur metric. But the tenseur metric, we express it in a system of coordonnées by its components of G-minus. Or, a G-minus very different can express the same tenseur metric simply because we express it in a system of coordonnées different. It's a difference. But it's terrible. But I don't think so. I don't think so. That's why, if we take the example of the phases, if we use the phases of things that are inobservable and things that are inobservable, then perhaps not. Because there is a practical problem with the linearity. I don't know. But it's not because we have something to explain. There is a problem of rationalization of these objects. it goes through the inverse of these inobservables, or even in the face. Just a example of that, which is very simple. I just dessine this. Have you seen this? Now I just dessine this. What difference is between the two? Well, it's not observable at the end, because you have a point and a right. Sauf that there I did this, then I put the point on it, and then I put it on the point and then I put it on the right. This means that if what I observe is this, the history of its construction can depend on something that I can't observe after, which is the order in which it is done. And this difference there is no matter in mathematics, because it means, for example, when you build a tangent to a curve, or you take the direction and you touch the curve, which is the point of view of the touch, or you have the curve, you fix the point, and you take the position limit. So you have fixed the point, then the right, and then the right, and then the point. And the change between the two, it is called the transformation of legend. So there is something which gert a difference inobservable, and all the idea of the transformation legendriery and the transformation of contact is already there, and at the beginning you have something that is the same thing twice. Well, I think, I imagine, because I'm not competent, but I imagine that these stories of choice, for example, have the same role of describing a sort of story of construction which will allow, just, the ambiguity, and if there is an ambiguity, in the sense of... I don't know how I've done things, I don't know if I've done this first and the other first.

20:00 Well, this is a sort of theory of Galois, if you want, there is a kind of ambiguity here. It's like a group of ambiguity. It's important to note that there is an ambiguity mathematical, it's the same, but not physical. Because physically, we can see... Well, physically, it depends where we place the physical. If the physical, it's the observation of the trace finale, of course. Well, now it's true that if we look at the thickness of the tach, we can see... If you look at it, you can see it on the other side, because there is a thickness. But this would be another parameter, which would be this thickness, which would be the role of the phase, in a sense, which would be a thickness inobservable in the ideal. But in a certain level, in a different dimension, in the case, because you could create a thickness here, which I put in the first, but in a different dimension, this parameter exists well and well. And it allows to observe the history. Just to remember the... No, it was for the story of... And then, we can be obligated. The point mathématiques technique, it's that these things are not a fioriture, so these stories of symmetry, which go with all that, are not a fioriture, because we can be obligated to be obligated to pass by this direct caractere indirect of the gestion of the ambiguity. But there, the question is, is it what we are obligated? As a physician, I can't... I don't have an idea here, I can't talk about the point of view mathématiques. No, but, just the debate... that is, is that we are obligated to make an appeal to the notions that disparaissent in every application of physics and then give them a reality, at least necessary mathémically or not? I don't know about the physics, but for the mathématiques, I would say that in mathématiques, in the mathématiques, in fact, we are constantly obligated to make an appeal to things, to objects des nominations des notations qui disparaissent au cours de la pratique oui toute la question c'est est-ce que ce qu'on est obligé d'introduire est arbitraire ou pas si on est obligé d'introduire un arbitraire mais oui mais c'est ça le problème de la symétrie oui oui tout à fait c'est qu'on introduit un truc que ce soit c'est bon mais il faut faire quelque chose alors pour les deux autrement quand il y a une soi-disant symétrie en physique évidemment il y a deux éléments qui seraient hallucinables on dit dans ce cas là c'est très bien il y en a qu'un mais le problème qui est assez analysable aussi en termes de topos précédent between the two. It's a natural way to choose. That's an information in plus. And if this information in plus for choosing between the two doesn't exist physically, we don't have the right

22:30 physically to say that there is no one. That's all the debate. It's true. It's like the observation of the system of the particles indiscernable between them. The statistics are not the same if we numerate it or if we numerate it. And in fact, the experience is strange. In favor of the discernibility. So there is a symmetry that is there, in this example. Well, from the point of view of the topos, from the point of view of the mathematics of the topos, these questions of the discernibility and of choice, canonics or not, are quite well in terms of treatment of the axiom of choice inside a topos given. We have a topos and then we ask if the axiom of choice is true or not in this topos. For this kind of thing, for example, we can easily describe a little bit of a topos in which the ambiguity of the class will appear. Wait, but if it's true, the action of the choice, it's only the topos of the ensemble? Comment? It's only the topos of the ensemble. Ah, no, it's not the ensemble. No, no, no, it depends if we make a theory of the ensemble with the action of the choice. But let's even start with a theory of the ensemble and the action of the choice, we will have very quickly a little bit of a topos of the gentil, like, for example, the topos of Z on Z ensemble. If you take Z on 2Z, the group has two elements, you take the topos of the actions of the group has two elements on the ensemble. Well, that's a topos in which there is no action of choice. It is Boolean, there is no action of choice. Why? Because what is an element there-dedans? It's an ensemble with a kind of function of mirror. Each element has an envers. So, an object of this, you will have to define it like this, if you want. You will have elements like this, and you will have elements like this, and you will have elements that will go by double angles. Because the action of 1 on an ensemble will let fix certain elements, and other elements will exchange them by 2. So, the typical image of an object of this, it is this. So if you take this image here, which is an element there-dedans, I'm going to call it 2, if you want. But D, it's not the same thing as that. You can call it 2. So there, there are two elements here, but it's not the same. And the two elements that you have there-dedans, you're not able to choose, in the sense of the action of the choice. It is to say that if you have the element 1 like this, you can create a morphism that comes there-dessus or there-dessus, but you can't create a morphism here.

25:00 Because the element should come here, but because of the symmetry of the goal, it should come here. So you can't create the choice of an element there-dedans. Because normally, the negation of the choice is always the existence of an abstraction Yes, exactly. Yes, exactly. Exactly. Exactly. Exactly. Exactly. Exactly. So here you see this question of an element indiscipline. there is no element that allows me to choose naturally, it has a translation or a presence in the theory of the topos, and the logic of the topos is quite well manipulated today. But it's not bad. When you write the category 7, the category of the ensemble, it suppose the action of the choice, or it's independent? Well, it's independent, because it suppose the action of the ensemble in which I put or not put the action of the choice. Non, dans la catégorie des ensembles, je me sers d'une collection d'ensembles qui sont réputés former un certain modèle de ZF. ZF, je peux y mettre ou non l'axiome du choix. Dans ZF, il y a l'axiome du choix, mais on sait très bien qu'on ne peut pas le mettre et ça nous fait... Yes, but also because when you see the encadrement of Tropos, you understand that the axiom of choice is liable, very formal axiom of extensionality. It's equivalent to the separability relative uniform. Is it in the category of the ensemble, the axiom of choice has a link with the existence of an element terminal or initial? Existence de section globale, oui, mais section globale. an ensemble section ensemblistement c'est ça ça c'est faire ça c'est utiliser l'action du choix bon

27:30 le faire un traitement dans une catégorie c'est au lieu de partir de ça partir d'une flèche absolument un hippie quelconque et de dire qu'il a été la même section mais encore une fois c'est pour ça que je vous ai donné ce petit modèle il n'y a pas besoin d'avoir des modèles sophistiqués pour voir que l'action du choix n'a pas lieu normal en théorie des ensembles donc l'action du choix the choice would not suppose that if the d like that, like you have done, there would be an infinité. But then, do you suppose that, potentially, the d like that, there would be an infinité, because there is potentially an infinité of particles, or then you have still a physical variant of the choice, which even in the case of a non-finity of d, it needs a choice? Well, it's true. For the ensemble, there is not this option... Yes, I know that in the axiom of the choice, there are several formulas. There is one that says, for example, that there is an infinite product of non-vide and non-vide. But there is another that says that every flèche epimorphe or subjective has a section. So, to contradict the axiom of the choice, it is necessary to find a epimorphe, even if it is, between guillemets, that there is no section. So, it is true that I have always a little hesitation on the axiom of the choice choice, because I have even seen in some books, a book called Kemke, K-E-M-K-E. Well, the theme of the choice, it's in an ensemble, I can choose an animal, point. Ah, no, that's it. That's what he says. It's like that he says. Well, it's not the same thing. I mean, it's not the same thing. I'm going to say, well. Now, what is the question? The question remains, because we can imagine that there are like that, there are there basically it's supposed to be Well, it's lié to the action of the choice, as we express it in the topos, in terms of epimorphism, but it's true that we have to finish at a point. It's better to respond to your question, it's sure. I'm going to talk to you about your question. Yes, just to move on to the cohesion. I'm going to talk to you about the cohesion now. You mentioned that you're different. Can you understand that as a sort of generalization? because also I'm talking about this idempotent noyau, how do you say it? Yes, it's possible for me to ask a question. It's my impression, it's my impression very strong, when I heard the exposé de Lovire,

30:00 that they understand this concept of the axiom of the cohesion is very liable to the negation of the axiom of the choice. It's a mark of the absence total of the cohesion of the axiom of the choice. Yes, it's true. It's true. It's a mark of the absence of the cohesion. Exactly. Oui, c'est comme ça que j'interprète aussi, comme toi, ce qu'a pu dire Levire, dans le sens où effectivement, quand on a l'action du choix, on peut justement fabriquer la section tranquillement, puisqu'il n'y a pas de cohésion entre les différents choix. Exactement. Il n'y a absolument aucune cohésion, donc le choix est possible. C'est ça, c'est que dans ce sens-là que... Et aucun point c'est possible de devenir un autre point avec aucune attention de le mode de paramétrisation de la motion, comme ça. Yes, this is the relationship with... That's interesting, because it allows us to relate the question of the symmetry, etc. with the question of the cohesion. In fact, it's all quite judicious. So, we understand a little, for different reasons, why, already, it's with the topos that we express ourselves. Already, there is this question, I don't know, that we were talking about earlier, but before, there was this alternative between the font or the function. I have another analogy to make it easier to understand this distinction between the function and the function, because it is implicit, I believe, in an article of Le Vire that I will talk about a little bit, which allows us to understand a little bit what he says about the cohésion. Well, it's about the way that we practice, I say well, that we practice to enter the probabilities. Well, of my side, I wrote a book called The pulsation mathematics, which explains the math technique on a kind of movement between a tenue, or a very, in the same way, a sort of ouverture. and then there is a probability called Laurent Matziak, who wrote a little article, very simple, and very just, I believe, on the pulsation probabilistic. I think it is well in jeu here, this pulsation probabilistic. l'idée c'est que lui il prenait d'abord sous un angle de l'enseignement des probabilités mais je pense qu'on peut l'apprendre tout à fait sur le plan épistémologique comme quelque chose

32:30 d'un noyau irréductible alors il ya deux choses dans la probabilité telle qu'on veut les enseigner ou les comprendre d'une part aujourd'hui il ya le grand oméga le simple space l'espace des dessus ça c'est le modèle de kolmogorov en fait c'est noir sur blanc c'est écrit chez boule les historiens les probabilités ne savent pas c'est dans les voies de la pensée donc mais enfin il y a cette idée qui allait au grand oméga les événements élémentaires la probabilité dessus après quoi on construit les fonctions les random variables aléatoires comme des fonctions sur This is the first position, the first way of doing omega. The second way of doing it is that it is a point of view where there is a sort of space that plays the role of space, not the omega. And then after, we construct the functions there-dessus. This is a point of view, probability with fond, if you want. And then the other point of view, it is that we start all at the way of Paul Levy, for example. There are variables alayatoires that are coming from the ciel. X, Y, Z are variables alayatoires. And the only thing that we can do for these variables, it's right to build their laws, the laws of probability of a variable. And it's not said at all that the variable is a function on omega. It's not said explicitly at all. Well, and what Mastriak points, it's that in fact, at the point of the teaching of the probabilities, it's all about to maintain a tension and a va-et-vient between these two ways of seeing. because it is obviously, for something to be found, to see the variables on the omega. But in general, this omega is practically inaccessible, inobservable, etc., etc., as a kind of space of universe for physics. But in practice, what we are interested in is the law. And then, to calculate the law, we will be able to change the omega for the same variable. The variable in fact is not attached to the omega. The omega can vary according to the calculations that we have to do. Calcul de loi calcul d'espérance etc donc la situation allait bien dans une espèce entre deux entre un espace qui sert de fond et des fonctions qui sont plus des fonctions d'ailleurs Alors moi j'ai envie de mettre ça sur la table pour vous au point de l'alternatif qu'on a dit tout à l'heure en physique Et en plus parce que je crois que ça touche à des points qu'évoque l'ovir dans l'article que je vais vous présenter maintenant There is also the geometry non-commutative. Because if you go to an space, you have your algebra of functions, which is defined as well.

35:00 You can show that if the algebra of functions... If you have, by ailleurs, a C-star algebra commutative, you can always construct it. And then, if you take a C-star algebra non-commutative, it's not an algebra of functions, but it's an algebra of something on something. And that's what we call the geometry non-commutative. So you already have this duality, and it is really rich, because it is sacred, the generalization. Yes, all right. You have the duality algebra and geometry that you try to push the best possible. Well, in this kind of move, like you just said, on the jump commutative, or like we said on the probe here, or all a little bit on the question of the symphony, well, there is a sort of meta-thesis implicit which should, I think, animate William Lovire, that the theory of the topos allows us to choose this subject because a topos can be thought as well as this as this. Well, the problem of the alternative will remain in the topos. For example, if we talk about a topos, we will try to construct the points for eventually reconstruct the topos as a topos of function sure. But also, in other circumstances, we will try to see the topos as being an espace. You see, when we search the points, there are two things in the points. The points, it is to see the topos as an espace, but for then to see the same topos as a new space on a concrete space and so to see it as a function function, so the points of the topos are particularly ambiguous since they also serve as well to étayer the fact that it is an space, the space of these points abstracts that the fact that it is in fact a jeu of functions, the functions of the faisceau on all the points d'accord et on est dans cette alternative toujours que tu évoquais avec la géométrie commutative aussi de représentation bon alors je pense que ça c'est une remarque générale qui nous a pris pas mal de temps mais qui est peut-être assez intéressante pour lire l'ensemble de ce que fait l'ovir d'abord pour se pénétrer dire qu'on peut peut-être quand même regarder sérieusement son point de vue de travailler avec les topos la souplesse à ce niveau là que permettent les topos So, on the cohesion, there is an article that was the most clear article called Volterra's Functionals and Covariant Cohesion of Space.

37:30 So, you can find all these articles because there is a site, there is a homepage in which you can find these articles. So it's the function of Volterra and the cohesion covariance of space. Now he talks about the cohesion. It's a very interesting article from an historical point of view, because it explains, for example, that... So it's an article of 98, 97, a version of 98, so it's been 10 years already. But we are obligated to turn 10 years back to understand, because today when he talks about cohesion, he doesn't come back to that, obviously. So for him it's like established. And I think that it's there that we will understand the physical sense. It seemed to several authors, and I have even doubts also when he gave us the conference, that he didn't have a lot to talk about physics. So for understanding that really he had a lot to talk about physics in fact, I think there is to see how in the word cohésion he already met a lot of physics which correspond to the texts more ancient than he. So, he talks about cohésion and I'm going to try to explain a little bit what it is, this idea of cohesion that is in this text. Then, the second point that I would like to explain, it's the recours that he has made to the function of the adjunct, adjunct, adjunct, adjunct, you see, it's what we call the strings of adjuncts. If you look at Google Adjunct Strings, you will be able to see a lot of articles about cordes et vous ne trouverez rien concernant les suites enfin les chaînes d'adjoints bon alors il faut faire un peu autrement il n'y a pas beaucoup d'articles là-dessus je vous indiquerai alors l'idée d'exprimer des contraintes et des propriétés par le fait qu'il ya un certain foncteur qui a un adjoint qui a lui-même un adjoint etc je pense que là c'est pas très clair savoir qui a fait ça le premier je pense que en un certain sens c'est quand même l'ovir also, but in a way systemically it was done, and I will explain how I will explain, by Rosberg and Wood, who, at the end of this simple notion, have given a very, very jolie characterisation of the category of the ancestors. Wood is a dysfonctionary, or ? No, it's a not, yes, no, there is a wood, there are several woods in mathematics, no, it's a cellulite, he's called Richard Wood, he was a professor at Halifax. Wood and and Robert Rosberg are all the two students of Lovinas.

40:00 They followed the course of Lovinas in the 70s, in the 69-70s. So this idea of chain of chain, because it's a concept relatively abstract of the way of contraint, it's to say that it plays an important role because it is actually, in a way, it's expressed uniquement in terms of, in terms of art. I would like to say that there is a function of a function of a function of an adjoined. And when we are a physicist, we need to get a kind of intuition very naive behind that to say that there is just that an space is connected, or just that there are things palpables in the sense of the way that is ordinary. So, there is this idea that we need to give some examples of this. Well, Lovir had introduced to this subject, I will show you all the right, a notion called UIAO. It is a suite of three functions. A function which has an adjoint X, which has an adjoint W, with the function Y, I don't know if it's Y, yes, Y is fully fixed. When he has a situation of this kind, he calls it a U-I-A-O, which means something that is attached to the philosophy hegelian revisited by Lovir, which means, you excuse me Unity and identity of adjunct opposites. C'est comprensible, mon anglais, oui. L'unité et les entités des adjoints opposés, en opposition. Alors ça, c'est une notion qui, dans le fin de compte, après, va être généralisée. On peut voir ça comme ça. On peut voir que l'idée générale d'une chaîne de propres plusieurs, comme ça, c'est cette idée de chaîne d'adjoint. Il n'y a pas seulement un contact à un adjoint, mais lui-même à un adjoint. Puis on pourrait continuer. what it allows to express the more and more. So I will talk mainly about these two things with examples. So, one part, the idea of cohesion, and other part, this way of expressing through these chain of chain, and it's all that I would say today about the exposé of the last time. But we will still be able to understand a certain number of elements l'exposé grâce à ça j'espère alors première partie donc basé donc sur cet article sur volterra

42:30 alors le vire dans cet article dit qu'il doit l'impulsion initiale pour cet article un autre article d'un certain gaetano fichera qui a écrit un article historique sur volterra qui a promis lovire sur l'examen du travail de volterra alors le travail de volterra en analyse fonctionnelle pour la physique c'est un travail qui a été tout de suite porté au pinacle et considéré comme extraordinaire par adamard alors chaque adamard plus tard dieudonné jean dieudonné a expliqué je comprends pas pourquoi adamard trouve ça c'est extraordinaire c'est pas terrible le truc de and of course the same terrain, it does not work because it is true that there are the principles of variation in general who do not work. So, what we call Fichera, and detailed here in the paper, is that it does not work, but because you, Mr. Diodonné and some others, you are part of a garage. However, if we take things better, in the case, from the point of view of the topos that we have developed, it will not work very well. That's what they demonstrate, what they demonstrate in this paper. And the heart of the question, it's precisely the question of the cohesion. Because, so, at Girol Terra, there are spaces, we will say. The idea is to have spaces, spaces, functions. function. So the emblem for the whole of this paper is, what does that mean? Three objects in the same category, the exponentiel and the function. The function abstract in general is like this. But how can we build it, calculate it, calculate it and calculate it? And in In which context? That's the problem. I'm going to say that because it's not for an instant. So Volterra introduced the idea of function, and he introduced it as the site, I'm not going to show you the details, but as the point of being able to hear, he did very expressly to treat some problems of physics, and especially the famous problem of Dirichlet, trouver connaissance la fonction qui donne la température au bord d'un domaine calculer la température à l'intérieur et pour traiter de ça il faut le faire avec ses fonctionnels sur

45:00 l'espace de fonction la fonctionnelle qui est à des fonctions sur le bord ici pour moi ce ça dans les années 1880 82 c'est vers ça qui va permettre effectivement ensuite à ilbert vraiment of the problem of Dirichlet. I don't like it. But without the abstract formulation of Volterra, I don't think that the movement of things would have been the same, at least for Hilbert. Well, the point is that there is obviously a function of continuous function. That's the point. It's not to say a function, well, I have an exponent, I don't know what it is. Well, you can build an space of functions from A to B, as an space, in mathematics. Just a space, and then the piece and point have a certain cohesion. The conflict is like an espace, and then the function has a certain character. So the words that will come from habitually, it's space, function, and function, continue. That's going to become, in the practice of mathematicians, since the beginning of the century, with the development of this analysis function. I don't know who introduced the analysis function. It's Frechet, no? It's very true, in all cases, it's Frechet who does that, yes. Well, the word analysis function, it's about this question-là that it comes in place. And so, the use in place is automatically with two implicit. The implicit is that what we have to think, it's the space 2, and the continuity. Also, when we say, ah, well, the notion of space in general, it's the notion of topology. the topology and then the continuity is the corresponding notion of continuality on the topology we will not come back to it and we will consider that of course it is the good one which is what is what is what is critical of it precisely because what we want is a notion of cohesion between all the elements of the space of the functions which is not necessarily expressed by a topology we don't know And then we have a notion of continuity, but not necessarily this continuity to which we are accustomed to. I think if we were to call Lacan, we would be very critiqued. Because we would say, ah yes, but we would turn the sense of the mathematics of continuity, etc. Like when Lacan talks about compacité, the word compacité exists since I don't know how many times in the vocabulary of philosophers in a precise sense.

47:30 Then people, knowing that Lacan uses the word compacité in a way consequent, but who is not in the sense of math, he doesn't understand the math. I think that it's a good problem. On this point précis of Lacan, on the compacité, I think that the critique is wrong. And there, we could make a bad critique in saying he has not understood what is an space topology or what is a continuality. Precisely, he is in the way to pose a new problem. The problem is that with the topology and the continuality in the sense of ordinary, we have a good choice for the analysis in the space function. And his answer is no. the continuité parcelles de cohésion ou c'est plus subtil ? Alors c'est plus subtil. Ce à quoi il arrive tout de suite dans le texte, c'est que, en fait, la continuité est une des possibilités de cohésion. Mais il en cite d'autres. Excuse-moi, pour ce qui est des topologies, il n'utilise pas tout ce qu'on fait maintenant comme des topologies faibles, des topologies... They cite, this article is very dense, they cite a lot of things, they cite a lot of things, they cite a lot of the topology, and also on the, how do they call it, on the topology? The topology. Well, the topology. They cite a lot of these, and they talk about Grotendick, because Grotendick, precisely, is one of the first, with his work in his thesis, to say, that on these spaces of function, there is not a topology, there are plenty, they are all as legitimate as the other. So at the moment where there are many of the legitimes, it's that the notion of topology, perhaps, can be made in cause. Well, the notion of topology can be made in cause, for what it is to describe the cohesion and the continuality, here. Continuality cohesive. The notion of topology can be made in cause, but even within a choice, which would be the choice of topology, we will also have several alternatives possible to topology, according to what we want to do, the kind of continuality that we want to do. it is not to forget that we want to talk about continuity which corresponds to what's happening physically. The continuity that we want to put here is the continuity of physical like in the problem of Eric Leclerc. We want to have a definition here which corresponds really to the fact that the temperature inside of the body varies, between the end of the body, in fonction of the temperature at the body. The temperature at the body is described as a function. So it is necessary to have on the space of these functions at the body and on the space of the body these cohesions which allow us to say that in fact it is continuous. You know that the problem in such a way is not solved. Because if the board is something topologically very irregular, with the epines of Lebeg, the pointes, the paratoneur and all that,

50:00 well, the fonctions of the board do not need to describe what happens next to the ear. If you have a very irregular board with an epine in 30, very pointy, in exponential, like that, it is called an epine of Lebeg. You have an infinite contact of two exponents here. There-dedans, the problem of Dirichlet is causing problems, because of these epines. If you imagine that you don't have one, but that you have an ensemble dense and rationnel, you can see the caca in which we could be. The epines in the question, it's flat, it's not? Yes, it's that. It's not plus or minus x, d'accord? Yes, d'accord. But it's too many of them. Comment? It's a rencontre... No, it's a rencontre at the origin, like that. and the contact is infinity. The plus and minus x. Moins x. This is an example that trouve Lebec. I cite it because it is an example of these years. At the time of the problem of Dirichlet, even though it is advanced in the place for the terrain of the problem, there are all of the types of objections which are posed, including those that propose Lebec to say that there is not a determination. This continuality, if the board is like this, the functions on the board and their rapport to the interior is physically extremely, mathematically, not really resolved and physically even more. What does it mean? Well, we can say that we make a choice, we put a topology there, there, and what we do is continue. In general, if we do it like that bêtement, this is not the case. So, it continues all the time. So, there is a reflection more profound to do on the choice. I think that's what it means in this article, the choice that we are going to put in a scene and in this genre of particular element, like there was a choice in a genre, which would be topology or a certain topology, but in fact it is more vast. We have to choose a genre of cohesion, so in which category these things live? So that's the question. Now, to respond to the question of the topology, is it cohesion or not? The answer they give is the following. They say, there are several genres of cohesion. There is, for example, the one that is given by the category of topology, stop. There is also the space that is given by the space, I insist on this example, the space pornology. The pornology. A pornology, it is an space in which there is an abstract family of bornes. It is never used in the vector. It is infinite, in fact. It is infinite in this case, when we do the pornology. What is it?

52:30 It is infinite, especially in the case of the infinite dimensions. It is interesting, especially in the case of the infinite dimensions, in the space vectorial topological. But the bornology, in such a way, is to have a family of parties in an ensemble who are reputed to be the parties bornées. So, a reunion of parties bornées is born, a point is born, I think it's about all, a reunion. Reunion finite. Reunion finite. I said the two. I said the two, the finite. Yes, yes. And it's not a big thing, so it's very few things, the bornées. An example of a bornée, if I take a vector vector, I can define a bornée like what is contained in a bowl, it would be a bornology. also on the vectorial topology local and convex and then in this case it's especially when it's in infinity that it's interesting but in any way, the goal of the game is to have a function of a function which will be in infinity then the vectorial topology it's another kind of cohesion for the view a kind of cohesion possible it's the topology the vectorial topology it's another it was a lot studied in the 70's 60's even They give the reference, but especially after the inventors that I'm obligated to cite, a little bit later, there are the French who did the bornology in Bordeaux, by example. There's Christian Ozel. Who did he do that? Who did he do that? He did a course at the UNS on the bornology. The inventor was Walbrook. Ah, it's that the name he did, Walbrook. But other than there are people who did the test of the Etat on the bornology in Bordeaux. I don't remember his name. One guy, one guy, one guy, an African who had a name in two pieces. Well, so that's it, but for the view, you see, it's a riponase, also, the logiciel. Three example, we can give an ensemble with an ouvert, we can give an ensemble with an ensemble with an ensemble, we can give an ensemble with a notion abstract of convergence of the suites. We have the suites convergentes which are declared converges, So we call these spaces at limits, but at limits, very limitless, if I could say, limitless, limitless, limitless, limitless, like that. Because there is also another notion more abstract, which is the notion of limitsraum, space at limits general, where we give a limit for all the filters. then there it is more general and we will also admit that in the discussions there were a few years ago on the analysis functional between Ehresmann and Lovir because Ehresmann was very interested in the Mimus-Rome, there were discussions

55:00 on this subject in the 70s, I believe, where Ehresmann talked about the Mimus-Rome it was to enter into his survey and do something but the Mimus-Rome it is much It was introduced by Kowalski, I believe, in this room, but he doesn't talk about this article, but it would be, in the eyes of the article, it would be a different possibility. So, here it is also very interesting, this example of the spaces to suite, the limit of suite to court, of suite to innumerable. Because if you look at the article on Universalis, you will see that there is an article, who is it who has done that? CAC? It's an article in which the author chose to place expressly in this context for all the theory of distribution. It's very light and very invariable. Well, well, the point that you want to hear is that this, this, this, and this, and others, that is to say the category of ensemble, but there it is already a cohesion already, and as it has been remarked earlier, if we put the action of the choice, it's even less cohesive. But so, his idea, I think in this article, is that these different cadres, which are given at each time by the specification of a category, which are for the analysis function, but which are for the cohésion more or less strong. For example, they consider that when we have an space vectorial topological, an space vectorial topological, we can go to the vectorial-bornological space. There is a function of an oublier. In a certain way, this cohésion is a sort of presentation, like a presentation by generator and relations, you see, the collision of E.V.T. is too precise than that. It is to say, when you present a group and say, well, I take the group, and then you give the generator with relations. You give the trop, because it's still a problem of writing in trop, because the group, it's what is engendered by that, but it's also what is engendered by that, but it's also, it's, finally, you forget the particular way that you get it. when you engage in this particular way you give an information in plus which is not as isolated from other possibilities of particular which is not special to the group in question well there it is a bit that the idea is that this would be a presentation of this a way to engage this so it's like a sort of quotient

57:30 reality with relations also so but it's two kinds of different different. The goal is to do an analysis function, but in the same time, we have to do it, it's to do it, but also to do the calculation. It's to do the variations, the qualifier. So, from that, to that, to do different categories of cohesion, we say, there is something in addition to surveying what we can in the elaboration to do something that will give us a calculation of exponentiel, It's to say that the category that we can obtain is, as we say, cartesian fermé, with the exponential of the object. So the categories that we envision as a good category of cohesion, well, these are categories where there is an exponential which exists. On this point, there is a remark that is pertinent, which is much easier for having something cartesian fermé. It's much easier for the morphism to be defined in a covariant than contravariant. If the morphism is defined by the contraintes covariates, for example, in terms of limits, the image directe of a limit is a limit. It's good. In terms of WR, it's less than WR than WR. It's contravariant, it's good. Or, it's because of the title of the article, and it's special about the cohesion covariate, the cohesion which is representative by these objects, but rather those who are in the covariance the beginning. which allows a different view of different fields. There is a lot more accessible, I believe, about the text of Lovir on the geometry differentials synthetics that you have already seen that this text on Volterra. So, I pass. And then, we get to the idea a cohesive space would be a category. These are not the two categories, these examples. Well, you can see that. Well. Then, of course, we're going to have more than these categories. These are things that are better, more than, more than, more than, more than, more than a point of view of math. They're going to calculate the exponential, and why not, but not only that Cartesian, with exponential, but that even the topos. So, but that, they don't talk about this paper. In this paper, the essential thing I wanted to tell you is what I just explained is this kind of relativity of the cohésion. And so, I'll end up here.

1:00:00 In this context, in a function like this, we see that this is a sort of presentation of this, as I said, but we also see that there is either a perte or a gain of cohesion between the two. This is more or less cohesive that this one. For example, when you have the topology and the function double to the ensemble, this is more cohesive than that. and if you forget the ensemble with action of the choice it would be certainly even more cohesive like it was said earlier but now we are on this idea which has been used in the exposé we start to arrive at exposé certain things are said in the exposé we are on this idea that of the cohesion something which represents a cohesive space and as we said at the beginning so viewed in the space, so viewed in the function of the angle. Because the intérêt of that is that if you say that it represents an space, EVT is the space of the physics, the space of the physics would be EVT, by the way. He says that it is not that, because there is also an aspect constituted linéaire here, in the EVT. There are applications linéaires, so it is not sufficient. But let's say that EVT would be the space of the physics. Well, it is a certain type of collision which is made in scene, but as well as of this space that of the way for the fabrication of the spaces functionnels since the fabrication of the spaces functionnels A, B, C, A, and B are des EVT it's the fabrication of the exponentiels in this category it's why the idea of presenting the cohesion as an space so it's not an space functionnel it's something that I would call an space it's a presentation of space functionnel with a slash it's also the one that the other this presentation is a category it allows us to not have to choose between the one and the other and then also to be able to compare this is the point that I did not at all in the exposé that he made to compare between the cohésions if it is a cohésion it is another and a component like that it is a comparison so we can imagine that a component who goes to one category there were often factors of E, R, S. I think that it is to think about it in the last one. It is to think about it as a comparison of cohésion. The last one was with these examples, but with E, S, and Topos.

1:02:30 There is no need to force the trait to say That's not the topos, so we decide from there to work with the topos. If we are a little savage, we can do it. There is no way to do it. The pre-fessos on AVT, it's a format topos too. It's a bit too big because it's a bit too big. But Lovir is also on the fact that it's interesting to have the little topos, not the big. In other words, there is a way to pass the category of topos by plungement of neonatal. Non-flask. on the passage of the prefaceo on the category of investing. When you have a category C, which is quelconque, the category C-chapeau, which is 11% CA, is a topo. You know that, it's the prefaceo on C. So, when you have, for example, like this, it doesn't make a chapeau here. And of course, it's purely formal because we don't know what we do. it's to say that, the fact of having a topo, it's not necessary to impose an avance. It would be better to have a kind of generator of the situation. On a put it in what we wanted. And then, on prend a topo engender by that. In prenant of the tissue, etc. Or, it's what happens in geometry-differentiel-synthetic. D'ailleurs, all the models of geometry-differentiel-synthetic that we know, mainly those of Eduardo Dubuque, and it's him who has made the most, are made like this. On prend a concrete category which has for object of certain variations, or certain spaces of different variables, of which you want, applications c-infini between them, for example. And then we say, this category serves as a site. I mean, I'm going to take a vessel there-dessus, for a certain topology, and it's in this topos that I'm going to do my different geometry. So, in general, at least at the level imaginative, or at the level of technical, there is certainly a subject between these considerations-là at a moment of a year and then after the fact modulo this kind of transfer that I just told you, the fact that it is possible to pass at the two posts. So, we are now at the point where you can see if I haven't forgotten something. You mean to say that this construction that you call the comparison of... The comparison, that's it, that's it, that's it, that's it. We need to think about the analogies of the ferrant-fusseau. From this puncture, we will be able to make a puncture from EVT-CHAPEAU

1:05:00 to EVT-CHAPEAU and then... I'm going to have a need for the second part, so I'm going to explain it. For the second part, I'm going to explain it. It's a passage of the pre-fessor and the same, which is also... Yes, that's another thing, yes. I'm happy to be able to fabricate a topos, so that's the pre-fessor for the instant. It was to be sure that everything we say can be seen in terms of topos. But of course, the topos of the pre-fessor is not necessarily precise. So if you have a category like this, there is a factor which compare it in a sort. You can take ans puissance CO, ans puissance BO. So here you have a function of u. So here you have a function of u, which is the composition with u. If you have an object here, which is a function of u, which is an object called a, you can precede u. and you get here the A on U and it's a A that you have to use A on U so you have a function of C in D and the composition with this function gives me a function in the other direction here the composition that I have described and now this function has a joint to the right and to the left the joint to the left as we have noted that there is U It's one way to read it, and the other way to read it. It's called U. So you can say that when you go from the U to the C to the D, that finally you get a function, that here there is U, which goes from C to the A to the A to the A to the A. If there is U, we will also note U to the A. So you put the A to the A to the A, which you can see from the A to the A to the A to the A. which I said a little bit earlier, but since the question is posed, I would like to précise it. So, there, there is this idea, so this problem of the cohesion is very interesting,

1:07:30 because it is now to try to understand, not what is the cohesion, but what is the variation of cohesion, rather. Sachant that each one of these categories that we mentioned earlier, or and another. Now we have a note of S, because it's tradition. We do like if it was CEP, like if it were the ensemble. And here you have a perte of cohésion, we pass from E to S. And what is interesting is this perte of cohésion. It's not the fact that we have decided, once for all, that it's the one or the one that we serve as a model of space physical cohesive. So that's the point that I want to introduce in the debate for the physics, it's not a model of the cohesion of the physical space. It's a model of how to study the variation of the cohesion, of every model of the physical space that we would have. Which is not the same. There is not a model determined of the physical space. If you take it as a model, or if you take it as a model, between the two, there is a difference of cohesion, it is necessary. And here, I think, the history of the parents of Cantor Canthor has something like this, like a distillation of this ensemble? Yes, yes, there is something about it. The passage of Canthor says that Canthor does a kind of distillation of cohesion because it's not very cohesive, but it's even less cohesive than that, it's the cardinals all courts. If you replace the ensemble by their simple cardinal, you have a thing that has completely lost, absolutely, and finally you are aware that there was still a little bit before. And why there was a lot of cohesion between the ensemble? Because between the ensemble, there was an amorphism. Between the cardinaux, there was no more isomorphism. So, finally, it's the category, it's the richness of the morphism and the comparison that gives you the cohesion. And when you lose, the way you lose, it's finally to reverse it. They cite it also, by the work of Gabriel Elmer on the category of fractions. You have a category, and you reverse formellement some flicks there. then you lose the cohésion. To return to the symmetry, if you have a group of symmetry, if you decide that all this group... Pardon, if you have an endomorphism, and if you replace it by... If you reverse it, you will pass to a group of symmetry, so. But if you go a little more and you replace it, that I don't do it, but you replace it by identities, then you will completely disappear. And you have all the equalities between the cardinal, point. So it's this game

1:10:00 one category to the other, of the cohesion that it interests. It's not to say, I found the model cohesive space. It's this relative. It's very different. He will quantify the cohesion by the number of flashes In general, yes. After, from there, with his model UEO, he will be able to propose a sort of definition of the cohesion through the fact that the function in question un adjoint à droite et à gauche et la flèche canonique entre les deux est un iso ça ça veut dire effectivement on va le voir sur des exemples que ça tient qu'il ya des relations entre qu'il ya suffisamment flèches enfin on va dire comme les idées important tout à l'heure il ya suffisamment de flèches dans la catégorie mais ça c'est une proposition que j'ai envie de détacher de détacher de ce de tout ce mouvement de pensée là qui est dans l'article sur volterra opposition They are always Yael. This paper is coming to a dizaine of years. On the comment with other things. I had a pleasure in this paper, to read the name of Arzella, which is one of my mathématiques, for a very simple reason. I was very interested, as a young student, by all the theory of Ascoli, the convergence of uniform, etc. And one day I discovered the theory of Arzella, in fact, when is the limit of function continues and continues on a compact? Arzela has given this response, and I think it's bad that, I'm happy that he is here, he is quite aware of the pointage of the nom, so the point is that he has a reason, point this nom, because it's a moment in the history of the analysis function crucial that we have this theorem general d'Arzela, even if it's a little bit complicated to announce, It's strange in our ensignments that it didn't disappear. Norges, he did it in his course. Well, he does the report with Gontendik, I've already said, all the topologies. He explains that there is a difference between, for example, the theory of the topology of the topology of the topology and the theory of the topology of the topology of the topology of the topology. These are two theories that have a look like this, but who, from the point of view of the question I don't understand why, I haven't had the time to really think about it, but I will indicate it because it has to be taken care of it. The question of covariance, I've talked a little bit about it. Then there is a question of the quantity, the quality that he has talked about last time, but I don't know if we will be able to go very far.

1:12:30 For example, I'll give you a last situation that he examined a lot, which he also examined in other contexts and in articles anciens. It's the idea, well, there is a lot of people who have very well done the theory of I think he said it, it's Prolisher, the abstract of the space, in the sense of the following. Imagine that you have, in a certain category of space, for instance. Moralement, if you voulez, it's a category where the objects are, for example, different varieties, like this point of view. You have an object V, a variety, a second, and you want to say what is the morphism. You want to define the morphism. The morphism is a variety of a certain type and a variety of a different type. For example, the applications C-infini, the V in W, if V and W are the real varieties, C-infini. And then, for defining the morphism, the applications C-infini, how do we do? that this fletch can be well on the chemin, for example, here, you have an element particular which is the interval 0,1, so you can take the chemin c infinity here, and you demand that this fletch A, if you take a chemin c infinity, you demand that the component fc is also c infinity. This is a definition, this is a definition, which is ancient, but and that the efficacy has been demonstrated by Froelicher in the 70's article, also at the end of the 70's article. It is to say that these objects are already defined, but after the definition of mortis is by the fact that they transform in good objects, in good objects, in good objects. So, to define this, it is necessary to define this, but only those who want to deliver. naturellement, you could also do something of contra-variant from the other side with, for example, the functions, and for the different variations, we can do one or the other, the two are equivalents. The chemin differentiable or the functions differentiable, the one or the other can test the differentiability of the functions. When we are in the different dimensions, it works, it's a theorem, but it's... Now, when we are in the dimensions infinites, for the different dimensions infinites, Well, that's what Froelicher has taken as a definition of a variety of dimensions infinites, of a differentiality, I would say.

1:15:00 The morphism. In exigeant the two... In exigeant the one or the other, and then in the case where he was placed, he demonstrated that the one was equivalent to the other. So that's also a point that interests me very much, because that's also... The categories that the morphisms are construed in this way are also, for him, examples of modernization of the cohésion. which would be good because you record what we did here yes it would be good to pass the thing to hear for you to get all the bêtises yes of course if you want yes with your dossier or even soon good interaction well then on this article there in fact I have the impression that it is still an important entry if you want in the thing because otherwise it is so abstract Well, I told you that the role of the space nucleus was not clarifying. Then, there are quantities of extensives, certain types of categories. Well, that touch to what we have heard. We talk about the double duality, which is a little bit, let's say. I will pass now to the second point, which is the point of the adjunct, adjunct, adjunct, adjunct. I hope you have done a little what we can imagine being the problem, we will say it like that, the problem of the cohesion in this way. But in doing that, he talks really about physics. This issue, the issue of the cohesion that he has, not only does it remove a certain number of shapes, I would say, well, there is a topology general and all that before doing math and physics. It's better to see this very qualitative thing and see what we represent the cohesion. Now, why... The second part is to understand why So, with this kind of material, we can actually represent what we would like to say by cohesion, by cohesion, by quality, etc. So, I told you earlier that there was a good theory, I will start by you announcing that. So, it's the theory of Roosbrug and Wood.

1:17:30 I have a version that is on their site, which is in a preprint, I don't know where it is after. I have the version that is after, but on their site you will find the article that I cite. But it is obviously been paru since 1997 or something like that. You take a category B. So, you can consider the category 7. I wrote Hans earlier, but since we are with their text, I'm going to take it in notation. It's the category of the little ensemble. And you consider the plungement of Dionéda, that you have a Y, of B, in 7, to B. So, this plungement, I'll remind you how it's done. b, you associate a b, which is so a forcteur sur bO to a value of 7, and the one who b' vaut, well, the ohm in b, I'll put this in the middle, the ohm in b, the b' v. You represent a b of b, like a forcteur sur bO to a value of 7, which is this forcteur I have outlined that. I have outlined that. I have outlined that. And then there is a B. Maybe with a B. Like that, it will be better. A B of blackboard. Blackboard. How do we say it in text? Well, you are going to see this one. This one is fully fed. Fidely? Well, this one could have an adjunct at the left, and this one could also have an adjunct at the left, and this one could also have an adjunct at the left. So if we stop here, and if these adjunct exist, we are in the situation that I mentioned earlier of UIAO. You have a function at the middle, which is an adjunct to the left and to the left, which is completely fidèle. So this is the U-A-O to the value. U-I-A-O to the value of all of this time. The theorem of Roosbrough,

1:20:00 this function is called x, this function is called w. I suppose now that the w-t exists, that the function y has an adjunct to the left, un adjoint à gauche W, que W a un adjoint à gauche donc qui remonte, et pour terminer que celui-là a encore un adjoint à gauche, l'appelait U. Ça paraît un peu invraisemblable. Alors le théorème de Rosemont-Geroux dit, ça, ça arrive dans certains cas, mais ça arrive quand la catégorie B est équivalente à celle-là. D'avoir tout ça est équivalent à dire que the category is equivalent to 7. So it's a characteristic of the category of the ensemble. You have an arrangement, an arrangement, an arrangement. There, five times. There are five people. U, V, W, X, Y. You have this. Well, their article is not completely invisible. It's because of 7. C'est à cause de Seth, à cause du fait que ce soit des ensembles qu'on a cette situation. Ah bah oui, oui, oui. Tu veux dire, si tu avais autre chose que Seth ici ? Oui, certainement c'est pas très bien. Mais t'es connu depuis longtemps avant que c'était vrai pour Seth ? Non, non, il le démontre aussi pour Seth. C'est facile à voir pour Seth, je vais vous le montrer pour Seth, pour que c'est vrai. Mais je pense pas que c'était complètement connu qu'on allait jusqu'à lui pour Seth. Ah voilà, voilà comment ça se passe, ce théorème justement pour répondre à la question. Voilà ce qu'ils en disent pour montrer un peu, parce que j'ai besoin de vous montrer un petit peu ces choses-là un peu à la main, How Lovir talks about cohesion in terms of... He's talking about it. I'm going to try to construct this schéma. In the case... I'm going to say it like this. I've made my drawing in the other direction. It's good. Ah, oui. I don't know if you will be able to rectify. I prefer to rectify and give things like I said. I'm going to turn my tableau. If you take B, if you take a theory, I give you this equivalence and I don't want to demonstrate this theory. Now I'm just going to look at what is this suite of things

1:22:30 there for certain values of b. So I start with b equals 0, b equals the catégorie b, and on I made a lot of random notes. I will explain what happens here, and then I will do a little bit with... If you have to examine this, we will see that there, it has to do with the schema that we will see. You have the category of ensemble and the category of a single object. The category is reduced to an object and only an identity on this object. So you have a terminal terminal here that I note like this. This is obviously an adjoint to the right and an adjoint to the left. on note 1 des 0 parce que vous voyez la catégorie 1 c'est en fait l'ensemble on va noter comme ça l'élément de 20 que notait l'étoile c'est la catégorie qui a un seul élément étoile donc l'étoile je l'envoie sur l'ensemble à un élément ici, l'art 1 et par 0 je l'envoie l'étoile sur l'ensemble vide et ce sont bien des adjoints on va le vérifier pour les gens qui ne sont pas ce que je veux dire adjoint ce sont bien des adjoints droit et à gauche It means that if I take an ensemble E here, if I descend, it comes to E. Now the junction here means that E. remonte to the whole of the whole. And the fact that the junction that we want to show, is to say that there is more than more than more than more than more than more than there. Well, here in the middle there is an A and then there is also an A. So there is a good function, the property of the junction has a good place here, between what is E on E and what is E on E on E. That is the junction of this side. And then the junction of the other side, because this time it is a new one with E and E, and this time you see this one on E, 1 is the ensemble of elements, and so an amortism of E to E to E, is not enough that the amortism of E to E. So, just simply, what we are saying is that in the category of the ensemble, The object is the initial object and the object is the terminal object.

1:25:00 This means that we have a junction like that. The sense of the flèches is not the same, it is not the same? The quelle flèches? The one? The one? The one? The one? The one? The one? The one? The one? It's the same thing with the morphism of the radar. No, but it's the other side. That's why, here, the joint right and the joint right, the joint right and the joint right.