Cohesion in Physics

0:00 The 1, which is here, it's the Yeneda, which is the Y, the Y of 1, what, Y of 1, what, right. Yes, d'accord. Well, just the situation in general, I'll show you a little bit what's happening. But there are some more after here, which are not very interesting, it will be interesting if it's more complicated here. In the case where B is equal to 7, what do we have? You have finished looking here? If I descend, you can't see it. I'm going to write it down. You can climb and climb and climb. What can I do? I'm going to climb and climb. I'm going to write it down. You can write it down. It's hard for me. Oh, my poor boy. I give you the sequence, we call it the sequence of Rose-Bogood, the sequence that we interest. Now, you start, so like here you start, instead of putting 1 here, you put 7 here. So you will have 7, and here you have the Yeneda of 7, which goes into 7 to the power of 7. That's to say that it's the puncture contravariant on 7. The puncture of 7 in 7, but which are contravariant.

2:30 You have this puncture here. Then you have here, I introduced my notation earlier, you have here the 11th power. The ionized A7, I have introduced the implantation of 1 here. The 1 is here. I can take the power of that. I can take 1 to 1. I'll show you so that we can see. And that makes a function of there to there. And that's a joint, effectively. Well, after... Why do you create 1, 6, 7? 1, it's the function that I've seen here. 7 and 1, it's the same thing. 7, it's the same thing that 7 is here, it's the same thing that 7 has the power 1. Yes, it's 1. Ah, j'ai mis 1, c'est 7. C'est 11, donc 7. Oui, c'est 12, 7. Oui, dans mon cahier, j'ai écrit. Dans mon cahier, j'ai mis 11, c'est 7 de temps en temps, ça change. Alors après, vous en avez un autre. Alors celui-ci descend. Vous avez un autre adjoint encore qui va être par là. Et qui va être 11 puissance, enfin 7 puissance, l'exclamation qui est ici. Donc à partir de cette adjonction-là, en fait, vous en fabriquez un bout ici. Ensuite, vous avez ans à la puissance de 7 à la puissance 0. Il y a des hop toujours, hein, qui descend. Et pour terminer, vous avez ici, puisque ça c'est un puissant de quelque chose, il y a un adjoint du genre « il existe » à gauche, hein, et qui s'appelle « il existe 0 » dans les notations de logis. Bon, je ne vais pas réagir plus les choses, mais voilà ce qui se passe, en fait. If you part of the category B equals 7, you have the plungement of Yoneta. This plungement has an adjoint to the left, which has an adjoint to the left, which has an adjoint to the left, which has an adjoint to the left. That's right. And then, the job of the thing is that, the reciprocal, which is a little more, is to demonstrate that every category

5:00 which would be like that, from a Yoneda, a succession of 4, which is in fact equivalent to 7. And that is to show you that the existence of an adjunct and adjunct and adjunct, it becomes more and more contraignant, of course, but to be the point of contraining the catégories to be the same. If you are part of a catégories and you have all this, it is necessary to understand how, So how, in fact, you are obliging the category to be simply a category of an object where there is no structure. There is a presence of structure in your category that should disappear, it should be opened, ecrasated, by the effect of its successors. And is that we can demonstrate that... You can destroy the coherence. Is that the UVWX, of a certain way, looks like this? ah bah oui a posteriori oui mais c'est ce qu'il faut démontrer c'est ça oui a priori tu pars de là tu as le rayon et d'un trait il n'y a pas plusieurs suites comme ça je veux dire ah non maintenant l'adjoint il est unique donc tu pars du rayon et d'A ici celui-là il n'est pas astray c'est le rayon et d'A de B là-dedans maintenant si tu as un adjoint ici c'est forcément ça c'est forcément a posteriori ça non c'est pas d'avance mais il faut le montrer ensuite etc oui la suite c'est la même la même c'est en remplaçant Yes, it's that. There you go. So if you have a suite like this, the theorem says that in fact B is 7, and of course, the yoneta B is yoneta 7, the x is this. With the isomorphism 3, it can be that. Of course. Of course. All the joints are unique, there's no problem. So that's the point by which I wanted to start this series of successions, it contraint, even if you haven't seen the demonstration. Now, so, an example much more simple. I'll come back to this question. I'm going to go back to this simple little picture. I'm going to put it like this, 7. Here I put the point of exclamation, 1. Here I put 0. Here I put 0. Here I put 0. Here I put 0. Here I put 1.

7:30 Here I put 1. So on this simple picture, Lawyer had made a very interesting variation at the Coloque on the categories at Tours that I had organized at the time with Pierre Danfousse and to show how, from the same data abstract, so of an UIO of this type, we could, in the case of 5, we could replace 7 by 7, and in this case there, there was a configuration of this giant, from which it was very jolily, and it was also directed to the physics, but how we could do a different calculation from the fact that, part of this data and these two functions adjoints here, if we impose that there is an isomorphism here, what does that mean? What does that mean? What does that mean? Obviously, it's not here. 0 is not equal to 1. And now, take, for example, the category of the group Abelian. An isomorphism between what and what? Well, you see what I've done here, here, for example, when I take an ensemble, an object here, an object is equal. Like I have an identity here, it gives me something between 0 and 1. And then, like I have an identity here, by the junction from the other side, it gives me something from there. So I have a fletch here, from 0 to 1, which is composed of two. And what I'm interested to hear is, what is the nature of this fletch? Is it injective, surjective, epimorphine? These are the contraintes on this fletch that will express more or less cohesion. Here, it's not cohesion at all, because 0 is not equal to 1. The fact that 0 and 1 are distinct, does not make a lot of cohesion. If, in revanche, you take the category of the Amelian group, AB, and you could say the same thing, the category 1 and the element, you would also have 0 and 1, but as you know, in this case, 0 and 1, the terminal is the initial object, it's the same, it's the group, the group nul, it's the one who has an element. So, the difference between this and this, is that here, the flèche of 0 to 1 is not...

10:00 I don't know what they ask about in their conference, that they are epimorphes or something like that? If someone has notes of this exposé, they ask something about... I have an article. And in the article, you might say it? I will see it in my notes. If I find something in my notes... The point important, it's that. It's the fact that... Even inégalities, for example, to talk about the types of qualities. Well, for example... epic. That's what they ask, I think. That it is epic. In a first time. In a certain moment in the conference, they ask that it is epic. Well, that means, in all cases, that it would mean zero equal one. Well, then... Now, the abstract that they envisage, that would be the schema that looks a bit like this. And not so cool. That's a lot. That was an example that I made for you to show the interest of this idea of having a joint. In the tradition, we don't do that. In the tradition, we observe what happens. For example, when you have the category of the vessels and the pre-vessos, and you have the inclusion of the pre-vessos, the vessels in China, in the space X. That's a category that is included in the category of the pre-vessos. If it's an adjoint to the left, it's not an adjoint to the left, but it has still properties of exactitude. For example, if it commutes to the product, it's an adjoint to the left. So in this case, there is an adjunction that exists, but there is not the second, but there are properties of exactitude of the adjoint to the left. So this situation between the vessel and the vessel is not at all exemplary of what is there, and even more than what we will say later on the UIAO. It's quite a good one. Well, here's an example of UIAO. Very good. So, Levir will, in the exposé, talk about the extensives qualities. I will remind you what it was, and try to understand it from the point of view

12:30 of what we are talking about. So I'll take my notes from the conference of Levir. At a moment, they show us two things. They show us the quality type. We have an E0, a functer... Ah, yes, he makes it monter, attention. The E0 to E, the functer, in the middle, is a functer that makes it monter. there I have to descend, but it doesn't make sense, there will be an adjoint to the left and to the left and to the left. And they ask that the canonical flash, which goes from there to there, is a ISO. C'est-à-dire, il demande 0 égale 1, si vous voulez. D'accord ? Donc, je l'avais écrit comme ça. Alors, ça, c'est ce qu'il appelle un quality type. Donc, ce quality type, ou type de qualité, il le réutilise ensuite pour parler des qualités extensives. C'est surtout là que ça devient intéressant. Parce qu'il va avoir, par exemple, c'est la situation que je reprends de lui, qui est comme ça. Vous avez S, E. Alors, il faut bien penser. all these things are wrong. E, what is it? E, it is a topos, all this. It is a topos abstract. But, just a moment, I think that it is necessary to understand as a kind of space of cohesion, as I mentioned before. It is a cohesion and a other cohesion, and it is a comparison between the cohesion that is in the jeu. It is a cohesion that is deconstructed or that is reconstructed. There is a cohesion here, there is another here. And the rapport with the attainment. So I do the... There, there is a puncture, a joint here, a joint here, and then, what I'm interested is to do that, from this S and this E, and then we'll talk about something. We're going to talk about quality extensible, extensive quality. Here, at this level, the button goes from left to right, for each object, it is not necessarily an identity, an ISO. It can be a button that goes from zero to one.

15:00 And the function of the part, is it how you choose? Well, it's all the situation that is like that. In general, he thinks that it represents the points and the components of the connection. I have an example that I can give you, if you want, of chain, because I have not put it in the example, I'll give you the example. Because maybe it's a point here in connection with the original question of Mark. It's a relationship with all this, with the question of the cohesion of the space. If I understand, all the agents in this construction, the agents on the left, are the agents of a component of the space, of a category of space? or not? In this example, no? It's not explicitly... I'm going to give you this other example, between the category 7 and the category 4, for example. Here you have a function which shows which is the function object. In a category, I associate the whole of these objects. This function has an adjunct to the right. It's the function which shows a whole of the category indiscreet, It's a category gross. When I have an ensemble, I can always see it as a category, but a category where the objects are the elements of the ensemble and the flèches are all couples. The flèches of x and y, it's the couple xy. And then, here, I have the component, the discret. from the other side. Discret, grossier. And then, if I continue, I have pi zero. The components of connects. The components of connects in a category, it's the joint to the discret that is the same joint to the right to the object. There is this sequence, which I would like to tell you earlier, excuse me, because it's an usage mental, at least imaginaire, completely current. When we think about a chain like that, if you

17:30 Un discret, objet, discret, pi zéro. Pi zéro, c'est ? C'est les composantes connexes. D'accord. Donc, tu as une catégorie, tu fabriques l'ensemble de ces... Deux objets sont connectés s'il y a une suite de flèches tête beige qui vont de l'un à l'autre. Voilà. Tu prends l'ensemble des composantes connexes. Alors, c'est ce que fait Loire dans l'exposé, parce qu'il... Là, il ne met pas objet, il met ici, il ne met pas objet, il met point. You can see this, which is for 7 and 4. Well, you put it in a little abstract here, on E and S, all the opposite. The right thing thinks like point, it thinks, finally, and then pi zero. He thinks like the object. Point for object. In the middle, it's the discrete, here. And at the left, it's the pi zero. So, this is for the imagination. in the case of 47, there is a sort of choice. It's all a different way, because here 4 is not a topos, 4 is not a topos, but it's for the imagination of things. But there is a choice, let's say, naturel, yes, sort of object, and what is the analogy in general? Well, for example, here, you see, when you are in this situation-là, here, uniquement On peut se demander quand est-ce que la flèche qui va du discret au milieu... Donc ici, quand vous avez une catégorie, vous avez d'une part son P0, de l'autre part ses objets. Puis il y a une flèche qui va du P0 vers les objets. Si vous demandez que cette flèche soit un ISO, qu'est-ce que vous demandez ? Vous avez une catégorie. You take these components connect, one part, and the other part, you take these objects. You want to have enough components connect that objects, so there is no fletch. It's completely decomposed by the condition. Because it's always on this condition of the fletch that goes from left to right, from 0 to 1, that it works. And here, it's the same, it works on that. But here, we are at a level abstract. So, in fact, for defining a type of quality, it starts with two topos, EES, and then the material formel to express it with a certain collision or not. But it is not real, it is just in place.

20:00 And so, all this, if I try to interpret it, it's a model for a variation of collision. That. What is the variation of collision? The changement of collision. the central, it can be given to the start? Yes, what we give to the start is this function, and then we have an adjoint to the left and to the left, yes. This function of the start, like earlier we had the function of EVT to EVB. I would say something like that, with an adjoint. The fact that there is an adjoint to the left and to the left will allow us to have the transverse which goes from the left to the left, which is not invertible at this moment, but which will become, We are trying to define the extensives. so now we are going to have a Q here, we are looking to have a Q for D, and this time that this is compatible with the information, we want that Q is compatible with Pi0, compatible with Pi0, So, from this side of this, we will be able to reconstruct something analog to this, with something like this and like that, these adjonctions here. But in addition, we will now demand that at this level, here, there is an inégalité. The flèche that goes from there to there, the famous flèche of 0 to 1, which is the double adjonction, we demand that this flèche becomes an inégalité. So, we demand that this thing, that this thing is transformed, and we will try to do it in a universal way, because it is not entered in the details, but we are talking about this, which is just a type of quality, and now a quality, really, it's what will be built from a type of quality, so that it becomes, I would say, connected. It acquires a certain way. But it's another topo. Well, yes, I think it was a little bit, so I think that all this is a topo. So, his major definition in the show was this moment, but it was relatively abstract.

22:30 He gave me 10 examples, because he said that it was a dispute, I don't know what, I'm not very good in English, I have to understand all this he can say to the letter, I mean, I have an interpretation sometimes a little global, but... Well, there are examples of all of this, but the key point is that we are not able to understand the exposé, but perhaps to understand the issue of the exposé, which is the treatment mathématiques of the variation of cohésion, through the description of what we would call an extensive quality. after he talked about the intensives, that's another aspect. He said that there was an adjunction between the two. This difference in relation with the physical sense of intensity and intensity? I suppose, yes, I suppose. So, between the forces of force which are intensives and then the vectors which describe the displacement, which are between the vectors and the forms. I think that's it. Or, between, we talked about it at the end, between homology and homology. You have to demonstrate a different message with duality. Yes, it brings us to this story of data entry and data entry on an space. When we have an space, we can make figures in it. So, it's extensive. We extend something in space, like a vector of déplacement, for example. And then, on the contrary, we can have a champ. a function on the space available in R, or in other things, and this is for example the homology and the homology will be like this, we will see the homology as a treatment of figures and the homology as a treatment of forms, forms differentials, but we can so, more or less, it would be a question precisely, we can maybe envoy this idea of intensive and extensive to these two registers, but that is still another aspect, There were several aspects. There was first the treatment of the cohesion and the variation, and then there was a double of the thing between the intensives and the extensives. Sur what I would say nothing more for today. Well, that gives a bit of an idea of what there is to understand. I have a problem. I have a question about the distinction between the quantity intensive and the quantity extensive

25:00 is liée with the functionality contravariant and covariant of the construction. We are in this case. The case covariance and contravariance on intensive and extensive, it's what I mentioned with cohomology and homology. Yes, exactly. But we can also see in... It's also something that we can find... The case covariance and contravariance, intensive and extensive, I'm going to say a word if you have a little bit of time. here is how it can happen in a scenario, we have time there? Yes, yes, yes, yes, you can't do it, you can't do it, you can't do it, no, no, no, no, the next time. Ok, ok. I can't answer this question. No, no, it's fine. Ah, merci. Merci à vous. Prenez-moi la main. Je pensais à la dualité recteur-forme que tu as évoqué. Et en fait, on peut considérer que quand on a une métrique sur une variété, c'est justement quelque chose qui donne un isobanthisme entre les deux. Donc, tu donnes une espèce d'adjonction, oui. Oui, tout à fait. Oui, parce que Lovire a évoqué de manière très vague en disant, bon, il a dit entre mes deux définitions, là, there is certainly a way to put in place a pair of function adjoints which allows to pass from another, so an adjunction. He said that, but it's true that it's true that it's true. But it's true that we can take this idea of force, of form and of vectors, it's to say, the idea of Herman Weil in the space and in the matter, it's to say, he has put in place the duality between the core and the in physics. I suppose that the intensive and the extensive have to do with that. Now there is another possibility, it's the link to the question of the treatment, of the implementation of a scenario of cohesion. It seems that, as he said earlier in the exposé on Voltera, the question of the intensive and the extensive, or rather the question of the co- and the contravariance, which is perhaps a little different, because it's different, I'm not completely clear, but the question of co-variance and contra-variance we have a good or a mauvaise mise in place of the cohesion.

27:30 As I said earlier, when things are described in a co-variant, it is more easy to have a structure, it is immediate, even if it is, it is more easy to have a structure cartazine fermée, while when they are not, it is more complicated. I will give you an example of my crew to this subject. It is an example of a situation which I worked with Pierre Danfous for almost 10 years now which consists of studying non plus the category of the space topology but a more abstract the space topology, but there is no axioms there is no axioms, but there is no axioms so we call it the space of quality it is to say the data, it is the category corresponding, we have three of the moins and we have another three of the plus qualmoins a for object an ensemble x and a family of parties x sans axioms it is essential that it is without axioms what you call it without axioms we don't ask that it is by union we don't have any axioms it is a family of parties d'accord ça c'est ça l'objet d'accord alors les éléments de x rondes on appelle ça des qualités et si un élément petit x appartient à un élément de x ronde si un élément petit x de la si petit x appartient à grand x qui est un élément de x ronde je dirais que petit x a la qualité grand x e c'est e maintenant non mais ça c'est juste pour un x qui est dans e ah un x qui est dans e et maintenant si jamais x est dans x alors je dis que petit x est la qualité grande x alors on avait introduit ça pour des questions de traitement mathématique de la théorie de la traduction par exemple on veut traduire des verbes en verbes ou des séquences grammaticales en des séquences grammaticales analogues ça c'est les qualités qui peuvent se chevaucher qui peuvent être multivocs ambigus So, when you want to translate, when you want to translate all the words on the alphabet or all the words on the words, to translate the phrases, there are phrases that will be composed in different types. These are the qualities. But now, when we do the translation, we ask if we do the translation direct or indirect. Is it what we want? Or is it what we want to do between the beginning and the beginning, or the contrary?

30:00 The theme or the version. So, there is a notion of morphism, which corresponds to the topological notion of application continuous. Application continuous, the morphism is here. If I have an e, x and f, y. f is an application of F which has the property y in Y, is in X. That is the analog of the image inverse of Y. The image inverse of Y is in Y. The image inverse of Y is in Y. c'est celle-ci, tandis que celle-ci et bien ce sera pour l'image directe donc ici l'amorphisme ce sera autre chose ce sera la condition, ce sera toujours des morphismes de gré dans f donc g mais telle cette fois-ci quel que soit x dans x l'image directe g de x appartient à l'application, ça c'est l'analogue Connaître ? Comment ? On peut mettre la fin beaucoup de choses, parce qu'au départ, si on pense que ça généralise stop, c'est continue, ouvert. Mais si maintenant on pense qu'on met dans les qualités, par exemple, les parties connexes, l'image d'un connex et connex, c'est-à-dire toutes les applications continues transforment un connex en connex, mais il y en a peut-être d'autres que les continues, donc ce n'est pas la même notion de morphisme. Mais ce sur quoi je ne vais pas vous faire une exposition sur tout ça, This is just a response to the question on the co- and contravariant. Because here, we have something that touches to the idea of Ovir that I have mentioned, which is the idea of it exists and it is what it is. We saw that Ovir, he knows that. He knows that if we take an ensemble of E, and that we take an ensemble of E, And then, if we have an application of E in F, I will have here at the end of E and the end of F, if I have an application of E in F, I have an application of E in F,

32:30 I have an application here with an adjacent to the left and an adjacent to the right. I call it f. Here I call it inexistence. Here I call it inexistence f. I'm going to put it like this. Here I call it f. Or f-1. Then there I call it inexistence f. That happens in the article of Levere of 1969 in the act of the Congress of the International Council of Nice, he explains that the quantification is expressed in terms of adjoints. He explains that. And if you want, that is a thing local. E and F are fixed. So there is a rapport between the image direct and the image inverse which is explained here. And you see that his idea on the I.U.O. is already exemplarized here. Now, the question that we have treated, Pierre Danfouss and I, here, is to prolong this thing, which is local, in a global question. Because if you want, in general, it is the general calculation of the inverse images. And that is the general calculation, but with an ensemble EF which varies all over all. On n'est plus à E et F fixés dans des sous-objets. Là, E et F varient, et ce qu'on fait avec F, c'est du calcul d'image inverse, et ici, du calcul d'image directe. Donc la question qui était envisagée, c'est quel est le rapport général, catégorique, entre le calcul d'image directe et le calcul d'image inverse ? Well, just as it said, in the article on Volterra, in a certain way, this is not bad, because it is clear that, as it is defined in a way covariate, this category is easy to see, we have seen it in the article, this category is cartesian fermé, it is exponentielle, it is cartesian fermé. Now, the real rapport between the two, it's another theorem that we have established, it's that the catégories what the plus and what the moins have a rapport of duality. It's to say that the catégories what the moins is supposed to add a function of nature towards what the plus, which determines this like a catégories of algebra but in the same time, we have another situation with the plus and the moins is changed. so I can put it like this,

35:00 minus over plus, and plus over minus. The theorem that we have is the following. Qual-op is algebraic, in the sense of the algebraicity of the monads, a little technical theory, which is defined by the operations algebraic. Qual-op is algebraic on Qual-plus. and equal plus is algebraic sur equal moins d'accord qu'est-ce qui établit cette double adjunction qu'est-ce que c'est ce foncteur ici c'est le foncteur qui justement si je pars de x enfin de e avec x c'est le foncteur qui me donne les parties mais de manière contravariante ce qu'on a si vous voulez deux puissances e et puis un truc qu'il faut construire which I call the X-Cup. The important thing is that, for those who know, just for those who know, there are people who know here the duality of Stone between the algebra of Boole and the stone space. And in particular, from a point of view catégorique, the beginning of the category of that, it's enonce. The category Hans-Hop is algebraic on Hans. It's the beginning of Stone. and then after you have to know concrètement what they are. Well, here it is the same thing, it's this term of stone which is enriching in a certain way, in saying that, if you want, the term of stone, it would consist to be interested in the family vides and the family vides, all the time. We are not only the families vides here. We will find it in science, and the term of stone will be inside of this one. But here, it's a sort of term of stone generalised which gives an algebraic of the one on the other. So, the response, excuse me du côté un peu technique un peu long mais la réponse c'est que le le côté co et contravariant sont adjubriquement dual l'un que l'autre dans ce contexte là mais dans cette dualité il ya quelque chose qui se brise c'est le côté cartésien fermé en effet parce que celle ci est cartésienne fermée et celle là ne l'est pas bon donc ça peut-être que si on veut les reprendre les indications de l'ovir sur sur la construction de cohésion cartésienne fermée évidemment il from a kind of genre, and then interrogate that, which is not cartesian fermé. But the duality between the co- and the contravariant is latent. And if we just take all of them, it's not the prefaceo.

37:30 It's not the prefaceo. And if the base is cartesian fermé, then it's cartesian fermé also. Well, the prefaceo, of course. Yes, the prefaceo are always the topos, so cartesians fermé. But why don't they say that it's... because these categories are duals l'une with the other in a certain sense but there is one who is cartesian fermé and not the other but there is another property she is co-cartesian fermé so it's the dual justly it reminds me John Baez distribuía the categories between cartesian fermé and the other based on the analysis of the catégories of the internet, we will talk about that the next time, and it's the mien, perhaps, it's a delir, even though I think Lawyer is not aware of the vision of the model. Yes, I start to find a part of the difference between Bayes and Lawyer, it's that in fact, if the first elements that I put on the table are solid, It's that in fact, Louvier, he does really do the physics in s'intéressant to the modernization of the cohésion, to his studies, and he does not do an external model, in a certain way. There is something exterior, in the proposition of the Bayes, there are N categories, things like that. It's something external, but we don't really do the physics. There is something very different. I think Louvier is a physicist in the mind. in his intention. I'm not saying that it's a physicist, a pure juice, but I'm saying that his intention is really the physics. And he wants to study the physics of the cohésion with the mathematical tools which he seems to be useful for that. You know, at the end of the exterior, John Bess is not a physicist, even if he works in a physics laboratory, but in his business with the category, he is not a physicist because his business first is to import the physical tools in physics. He spends his time to say that he is a mathematician and not a physicist. Yes, that's what I think, to read outside. And so the paradox is that the physicist is the one who read. Well, the paradox... And, historically, the paradox begins in the beginning of the book, as a physician, it was 40 years ago. In your presentation, it started to start with that. It started to start with the grand tour of Norley Truswell, the continuum of physicality.

40:00 And of course, we see that the difficulties are not for... It's a major part of the motivation. I think that the physicality is relatively clear, this idea of the cohesion. What is it for the physicians? I mean, it's something that is palpable, this issue of the cohesion. But on the other hand, the mathematical, the mathematical, is more delicate because... I don't know. We have the impression that it corresponds the notion of information, of structuration, of richness, of content, the richness of the content informatics, you see? I will translate it like that. Well, in the last exposé, in the last exposé, because it was the one that I wanted to comment, in the last exposé, in my sense, it's the question of this question of the variation, of the cohésion, as I said earlier. So, there, we are more in how the information, the information, the information between objects se détruisent se reconstruire tiens comment ça le mot français c'est tout enfin en anglais c'est holding la malade tenue des choses il s'intéresse à la variation de la tenue dans ce texte là mais dans les textes précédents et il explique bien que ces différents modèles c'est au post il va faire dedans du calcul différentiel c'est une idée de l'ancienne synthétique et que ça sert et que ça sert pour la mise en équation c'est un truc relativement ordinaire de la physique alors j'ai I don't want to insist on it, but in the paper on Volterra, I said before he did it without insisting, but I revient. Once he made his retours by the explanation of what you need to hear by cohesion, he presents at the end of the paper, he says, in the topos, the difference of the differential synthesis works, and then we have an exponential, an exponential, which is built by exponential, and so the principle of Volterra, we can write it quietly and it works. So he also ends up giving a real model of this calculation to respond to the critique of Diodonné. But to respond to the critique of Diodonné, he also does, and I will cite it again, because I did the problem of Dirichlet earlier, but he also cite another problem of which Volterra especially interested. So it's true that it's a way to use the category for physics, which is obviously different from the base. That's clear. We understand that they don't have the same attitude.

42:30 I don't know what you think. But it's a very good exposure. Thank you very much. and co-advocate to a construction. I explained it with your exposé this morning, but it's also a question of the foundation of the topology and the refonding of the topology on the notion of the figure adequate for the generation of space. It's another subject, but it's perhaps Maybe it's in a sort of continuation of this aspect too. But there are several, several reasons. I understand what is envisage between the theory of the category and the physics. This aspect of the cohesion, the cohesion, the reaction, it's very important and very central, but it's only one of the reasons he envisage. It's absolutely incredible, their vision of a very rich person. Thank you. Very good. We need to work. Well, of course. Now that we've got a little bit, you can't come and explain it. And then we'll try to figure it out. Thank you. Merci encore. Thank you.