Nature et Structure de la Justification en Mathématiques / Categories et Dynamiques

0:00 Thank you. We have an omega, we have two puissance omega, and we say that this is the letter A. This is a hypothesis. That very first part of this This tape was recorded on the evening of Monday, the 24th of March 2003. It is the opening few seconds of the part of Gerhard Heinzmann's talk on the nature and structure of justification in mathematics, the example of the transfinite, which was recorded in the Sal Ceylon of the École Normale Superior in Paris on the evening of Monday, the 24th of March 2003, and unfortunately I missed the first 20 to 25 minutes of the talk through making the mistake of taking the bus from my hotel in the Garden or to the Luxembourg in heavy traffic when it would have been much quicker to have jumped on the RER as I usually do and walked from there. The next part of this tape is the recording of Giuseppe Longo's talk of Wednesday the 26th of March

2:30 2003 in the Salcavalles in the Salcavalles In Charles Aldoni's seminar on pensée des sciences, the series on category theory, the title of his talk being Categories and Dynamics of Thought, the Importance of an Open Conceptual Framework, and Several Specific Applications, Categorie et Dynamique de la Français, L'Importance d'une Carrière Conceptuelle Ouvert, et Quelques Applications Spécifiques. before this seminar or after it I must tackle him on the issue of whether Pierre Cartier would be willing to come to the November 2003 meeting in Florence in honour of L'Orvia whether Charles O'Leary did record any of the seminars that took place in February that I missed whether he would be willing to record the one on Wednesday the 2nd of April Heisenberg, Lizine Galli-Teter Heisenberg, Lawrence, the Laurent Notal seminar, and whether he has any record of previous talks that he should every longer has given in the series of talks in his list of publications that he, that is on the web, of which I have copies in my section of the archive on Longo's writings and talks. Thank you very much.

5:00 Thank you. you found it where you found it then you found it Thank you very much. about the inequalities of the world. So it will not be until the end of the day. I think it's the quartier. It will not be possible. So today, it's Josep Boulang, who is here. I'm very happy that Josep Boulang termine this series because he has been working for years since he was a mathematician here at the university, and he is part of these mathematicians and logicians who have never separated their work the science theory, the scientific questions, the philosophies, the philosophies. He was quite close to Gene Chattelet. He also worked on the aspects, the enjeu of the biologie,

7:30 Thank you very much. Thank you. On a visage page, vous, par exemple, qui est petit au montage de végé en général en matériel. Bon, c'est un peu plus, non pas d'un catégoricien, je ne suis pas si on parle, et je suis un mathématicien impliqué par l'économie qui a utilisé certains outils de la pluie de la force et du fruit des catégories qui se tiennent un peu très bien, puisque ça c'était le charme de la poche, in a field of informatics. For me, the informatician consider this part of the mathematics and the theory of the theory, the theory of the programming, the theory of the language, etc. However, after some time, after some time, with a solution to the method, and we have always tried to illuminate with a question, with a question, with a question, with what is happening in this theory. So, I would like to try to make what is possible, with a perspective, and why we also have a question. this is a category, but in the case of the problem of mathematics. For the moment of mathematics, in fact, there is not only an interest in terms of philosophy, but there is an impact enormous, not only on the previous knowledge and on the other, on the informatics, since the science of the capitalization of the modern science is done entirely by the case of the knowledge.

10:00 And I continue to think that if you want to think about the next machine, It is to do it through the reflection of the philosophy of Adam Matthew, as it was done in the past. Even the horrors are made in a cadre of the mechanism of the 50th century. Well, let's start with the Noir. It means that for a few centuries, we have had a formidable cadre foundation of mathematics, of physics, which could be in this context. At the heart of this vision, there is an absolute space and time, The states-et-temp who are there is a void in which there are some subjects. The state is at the heart of a conclusion scientific, it is about to look at the place of the imagination of the physical science, which, the paradoxes, continue, well beyond the 16th century, at the interior of a philosophy philosophy, which is the subject of mathematics. C'est-à-dire qu'on reprend ce paradigme avec sa part, auquel il y a un absolu, un espace absolu, où tout va se situer. Mais l'absolu dans ce cas-là, c'est le théorique des ensembles, et la logique, les lois logiques, les lois de la France, signifiant. which is very necessary. The 4th, which is prolonged, which is a good motivation, is certainly a part of a shift between the physical and the physical and the mental health. The 4th has a responsibility, but it is not the first which is the debate that is made? Well, it is already said that the situation of space, the rapport of the physical, space and climatic, it is in the book. It is in the book of the book in 1884.

12:30 So, the development must be done on the book with its logic logic, which is an induction, which is a logic logic, which is a logic logic. And the reduction of the algorithm will be the only path that we can do. We have lost the certitude of Grinland, but we have not lost it. In doing that, we will be able to make, and this will be a sequel, not in this sense, entièrement cette révolution qui arrive avec Riemann, la référence de Christ et Riemann qui s'enfant à Herber. Riemann était comme Dieu, il est mort, il s'est dit comme Dieu sur terre, à Göttingen, le docteur, il est fait à l'époque. Et il y aura cette filière qui continuera à penser à la philosophie style Descartes, with the results, and it's often, the results of the technique, in this case, absolutely important, which will be approached by the ancients, the results complete, of course, the results of which the most important are the results of the technique, as far as far as the theory of the technique and the theory of the prependance, and with the more important, the more important, which is the essence of the pathology, which is a theory of absolute, The base of data is exact. It is very clear that the information is for that. The base of data is exact and perfect. It has nothing to do with the problem that explodes with the dynamic The problem is the approximation of the internal which allows the fluctuation, the perturbation, which is a rapport with the dynamic. This is a dynamic, which is very different from the physical activity. And finally, the re-utilization of the construction, which is correct to the relativism. When it comes to the development of the algorithm, there is this vision, which comes from the geometry, which is based on the relativization of the construction, in relation to what? The choice of the invariants, the invariants structurels, the choice of the transformation that they preserve. That is very important. When it comes to the development of the algorithm, because it's just in this point of view-là, fixer the environment, the structure, the measure, fixer the transformations, expliciter the transformations that we preserve, that this attitude of geomètres,

15:00 which is the Riemann, which is the Gleimann, which is the Sassasson-Kamenheim, they arrive in the field, and that is Mecklenheim, which is the wheel, the wheel, the roten, etc. And finally, the situation is recomposed. in the sense that we have a tentative convention, a tentative of a cadre constitucional, which will recover, heureusement, this sense of reconstruction relative, which I will explicitly cite in my presentation. For to say that the importance of the curriculum is interesting, of course, because there are no numbers, for example, that's very important, because, I don't know, we work habitually in the capillories assemblis, and it's a bit like someone who says, but listen, the system planet the most profound, it's the solar system. That's it, I'm not sure, I'm not sure. We work with the solar system. But if we say that it's the only system that we have in the universe, that, for example, isn't it? There is an idea in the concept of the universe if we present a duality that is concrete, and normally we work in the universe as we work in the universe. The point of view is introduced by the catégories, but just in the parallel with this analogies that I have done with the physics, and that is what I did earlier. because there is like a plurality of universes who are not construed at the same time, they become a friction with the real problems, sometimes even in the pure fantasy matrix, and we recuperate an unit in explicit terms of the interpretation of the one and the other. It is to say that an unit dynamic that is constituted by this plurality of universes. So, each one is motivated by what counts, by the vibration of the structure. The invariant of the structure, which is, at the heart of the math, because it's the grand contrast, it's the problem of the structure. The duality of the structure is, by the expression of the points, and they look at the fact that the theory of the ensemble is a theory of the points,

17:30 which makes the mathematics, first of all, destructuring, and then destructuring after. And I think that the mathematics mathematics is exactly the same. The mathematics is normative in the world, it is to say that we have a certain regularity, we transform them into an invariant conceptually, and they structure, they become normative from the physics of the universe. And so it's this reimbursement that is operated by the idea of the technique and the component in parallel with the sensibility that is constructed in physics and in geometry, because of the visualization of the 20th century, which is the motivation of the pond. It's the concept of the concept. I'll talk about it, of course, of the possible reduction, of the possible production. It's something that we can see in a technique. What is the concept of this concept? It's the perspective of the philosophy and the mathematics that they have. The crisis, in fact, in theory of the ensemble, begins, I cite all the two episodes, which is very important, by English, José Quintor, at Derby, in 1877. In fact, Quintor wrote at Derby Millet des Esquelles. Ecoutez, j'ai démontré quelque chose de terrible. J'ai démontré que si vous prenez les plans de notre construction du continu, les plans à la quinte de l'aise, disons, les points que vous venez de définir la loi 13, construction absolument fantastique, marqué l'histoire, je peux vous donner une correspondance objective entre le plan et le droit. Pour la réponse, qu'est-ce que j'ai fait ? J'ai décrit la notion cartésienne de dimension. and with a single parameter, I can represent no part of which point is in an space of dimension. He is really fracassé by this question of what I have done. Caldor was a genius and psychologically, so he was trained on the personal side of the life. Well, I think he thinks and he responds, 15 years after, it's a very remarkable result. But you know, you don't worry about it, this transformation is everywhere discontinued. So, what it doesn't count is not preserved. He doesn't know what it is, but from there, it begins an analysis which is called the analysis topology of the dimension.

20:00 It is to say, we discover it, we discover it after, It will show you, after, and obviously inspired by these things, that the dimension of the earth is an orthopedic. But what does that mean by the result of these things? Well, that really, it's the sign of what is the theory of the category. It's the dispersion of points, the structure of the mathématiques, like this, of points, where those who count the isomotivism of the category, I ask the colleague here if he knows the results of the material, which is the carbon, which is very complex. So the results you use, etc. There are no results, because the math is not that. What is the difference in an space, the three dimensions, is what is the dimension voisine? Yes, yes, on the right, who is the fact of the structure, the topology, the matrix, the theory, the space apart. and the fact that there are points, it's a construction that is made, interesting for the remarque of the cardinal, but it has no interest in the problem of the cardinal. This is the first one that was negative, and it continues. And there were reasons, because we still seek this universe, this universe absolute, to the Afrikaans, against this catastrophe. The disappearance of the certitude, and the case we don't have to. So, there is already a crisis, and how can we continue? For example, in a category, we can't even construct that. It doesn't have any sense, because the only thing that we give, It's not necessarily a structure in the sense of what I said before. So we're going to take attention to the recul, like this structure, the structure of the cabinet. But now, we're going to go very far. And we're going to motivate these structures by things that are not as simple. In fact, disons, I've been structuré, par exemple, justement, d'un produit et les objets mathématiques. D'une façon, par exemple, il faut comprendre la conception logique, la conception logique. Disons, l'habitude naïve, en simpliste, d'accéder à l'ir, ça sera à l'intersection.

22:30 Immédiatement, une perte d'information radicale, puisque si vous interessez là comme l'intersection, When you have an intersection, we give you an intersection, and you don't know A and B. What does it mean to know A and B? It is to preserve what counts in the construction that we do with objects. What do we do with a category? It is to understand that the conjunction is in the public. What is it? If we give you an object A and B in the category, and it exists, it comes with two projections. And for all OGC, and for all pairs of morphisms F and G, there is an unique morphism F and G. You understand? It is very different. We don't look at it as a structure, and we look at the construction of a new object that gives us what counts. What counts? What counts here is to be able to recover the information that we have and to have this universality of the construction which is given by the existence and the authenticity of this organism. When you think about a specific case, you see that you really have what you need. If, for example, your spaces ALB are topological spaces, you apply automatically this definition and you will see that on the product, you have exactly the topology of Weinstein that is a project that is continue. It's very important. It's like a category, etc. So, attitude. You see, it's rather an attitude, an approach conceptually. Evidently, there are categories of ensemble. But all that is less important. in the case of IA, the universality of the constructions, which is interesting, but IA, which is the structure signifiant to transform. Our motivation, is that a very important notion in mathematics finds its naturality in its category, which is to say the notion of duality. What is the construction dual? Well, it's simply to reverse the flèches. It's easy to reverse the flèches. So if we are going to invest in this construction, we take an ETA and we build an object dual, which we can noter like this, which is called Corpovid.

25:00 What does it mean to invest in the fletch? It exists an inverse fletch. As for all the morphisms, there is a unique morphism that rend the diagram to the object. I say that when I give a notion, a category, it's the same conclusion, the same notion. It's not like a game, but is it something? And here, miracle! Very important for the rest of the history. If you take the disjonction intuition, it fits perfectly. It is to say that this thing is very popular, that the intuitionists told us that to have demonstrated a disjonction, which means to have a demonstration of the A or the B, and to know which, that to the point of view of the classic, you know that it is not necessary because you believe that the A or the M or the B, always, without causing the problem which is the problem of having a friend of the one or the other, and then this construction motivée, in a way of philosophy, let's start by Crowder. All of a sudden, it's the dual of a construction of the biochemist, the product, the conjunction, and the conjunction, and the logic, which has its interpretation all of a sudden, like dual, and the product, it's-is the co-product, in the category which is adapted to interpret these occurrences. So, you see, this study, without regard to it, but using it in the structure of what it is, it gives immediately a regard unitary to something which, we have the immobiliations, which is what we do. So, this is the duality, and I'll give you an example. that we can define a morphism, the catégories are in fact, the morphism, which is well understood, is an epi if, for all F and G, you have this equality, an epi is equal to G. What is the catégories of the ensemble? It's a interjection. But it's obviously more general, and it depends on the structure that you have.

27:30 a category of space continuous with a notion of a sub-ensemble dense, a epi, it's not a... a epi, it's not a sub-section of a sub-ensemble dense. It determines the functions of continuous. So it depends on the version of the topology of the algorithm. It is necessary to have this notion to be universal. But then, if we are inversed, action inversée, c'est-à-dire si on a un contenant h tel que on puis f égale g, c'est une notion de h, mais ça c'est un moment, non ? L'équipe, il faut le fixer correctement, il faut changer l'équipe, et ça marche. Donc, automatiquement, vous avez une autre notion pour y voir un moment sur la vérité. which we understand as an algebraic. The points... Again, it's very important to have the points, but it becomes a derivative notion. It's not at all. First, since the Ensemble is one of the categories, like the solar system, it's important, it's not important, it's one of the categories. But the points we have ailleurs. First of all, you need to know what is an object terminal. An object terminal is an object that for Punta, it exists and is unique. It exists as unique morphism, an object A, for the terminal. Once we have an object terminal, we say that if you take two functions of morphism FPG, if for all h they are equal for all h you have this diagram commun it is to say f h is equal then it indicates f is equal it is to say that you can parkour your object b by the morphism h of the terminal When Sénat caractérise du morphisme en tant qu'ego, ça veut dire que ces flèches nous donnent assez de points dans la pièce.

30:00 Donc voilà, on voit qu'il perd une notion certainement importante. Pourquoi très géométrique ? Pourquoi ? C'est un peu la façon de Clyde. Clyde donne la définition du point, sans doute. Il dit que c'est quelque chose qui n'arrête pas. But after, they derive the existence of the point through the segments of the one. The first term of the first chapter is created in segments, built in plain and in the middle of the earth. The formalist, in 1908, he is an historian, he is a version of the Kennedy, but that is not shown because he has not shown the existence of the point. No, the point is that the intersection of the line continues to terminate. For Euclid, there is a definition of segment, and then, of course, he remarked that the extrins of the segment are the points. It is the consequence of the fact that the segment is terminated. For him, it is a continuum, certainly not a continuum. It is not a continuum ensemble, it is not a flux. And so, in a certain sense, in this situation, we construct the point as a notion of the idea of the construction. Quelle est la notion dual, d'abord ? C'est intéressant, je ne sais pas, je ne suis pas réfléchi. Non, je ne sais pas. C'est un peu générateur. Ah, mais c'est vrai que c'est un générateur. C'est un générateur. It's more intuitive. For example, the object 2, 1 plus 1, it's a co-generator with an opportunity to do it. Yes, I didn't think of it as a generator. I think it's only categorical. It's like everyone. He has a categorical sense. there are some examples that don't have any sense in these terms of configuration, but you have to think that these things have not started like that, in this category. They have started in a row, in a notion of transformation naturel. These are things that maybe you have seen, I don't know. But first, there is an ancient fronteur between two categories. The fronteur, it is something that has the property to transport d'une façon cohérente F et RG d'une catégorie à l'autre.

32:30 Donc vous avez que les types, pour ainsi dire, sont préservés. and here you have f of g composed with f which is equal to f of g composed with f of g. But in addition, you have the notion of transformation naturelle and the point of the beginning of the equation was just like that it was the place where the preservation of the invariants by transformation this applies to the level, let's say, the most rich. The transformation naturel is given between two contours, which are between two categories, and you can tell that Tau is a transformation naturel of F to G. If you have F1, FB, is equal to F. Here you have the Tau of A which is A with F So in fact, it's a family of transformations which commutes par rapport à this diagram. In other words, the accent is made on this transformation and preservation. Now, let's go a little closer to all of this, in this case, which has been made with some success, with first of all the effectiveness of the transformation. Why all of this is related? Well, you have already noticed that the effect that if we obtain a good interpretation for the disjonction intuition, that is already an effectivity, so that we can do it in a way of nature. In fact, what is happening is that what has been construed is in a diagram of relations mathématiques very interesting, very useful, because of which I have earned my salary for a long time. C'est-à-dire qu'il y a une correspondance en théorie de la preuve constructive de la première notion, la tric-catégorie, et de l'âme de calcul,

35:00 qui est une théorie que la calculité n'est pas, mais disons celle que, du point de vue mathématique, est la plus riche. C'est-à-dire que c'est la théorie à laquelle il y a des théorèmes dépendants de l'âme de calcul, comme en aucune autre. The machine is very good, the machine is formidable. There is a distinction of the material which is the core of the informatics. But let's say, there is no internal results in the machine. When we use it for a very complex, at the end we say, but that is independent of the machine, it does not work for a big formalism for the federal government. So, there is a number of results, and there, a lot of people said, but also a lot of people said, but also a lot of people said, and what is the most important thing about it, is that it is the the preuves and the preuves and what is happening is that, unfortunately, the language function which has been very long dominated the panorama, the theoretical, and the computer also and they are directly affected by them. The connection is a little more detail, it is that the flèche, which indicates the morphism in my diagram, is also an application logique, it is also an equipage, an equipage in terms of transformation, so the three things have lieu in the same structure. Now, if we look at what we're doing now, we can see what's the base of this relation triangulaire. In fact, a beautiful way of introducing the logical implications of the logic is through a passage of the meta-torials. It is called the deduction of nature, this rule is called the fletch of the interruption. What does it say? It says that if you can write in your meta-torials B and A,

37:30 you can put in your language the formula A and B. What is nice is that it fits perfectly with a rule that gives the title to the term of the length of calculus. You could say that if I suppose that x variable is a proof of A or a proof of B, and of that I can derive that n is a proof of B or a proof of B, Alors j'appelle tandapeak.m la preuve de cette application, ou si vous voulez, des programmes qui chaque fois que je donnerai une donnée de pipa va me donner une donnée de pipa. Donc j'ai mélangé les trois choses, c'est très sympa. La notion de programme, la notion de titre, la notion de proposition, ce sont tous pareils. In addition, there is something very practical, the modus pons of my ancestors and the Romans. I don't know how to say it, but maybe it's not an explanation. And the modus pons, what is he saying? Well, if you have A and B, you can say B. But this is also very nice to do the calculation because you say that if I have a term of type N and I have a term of type N, N applied to N, A is the type B. Or if you want, from the point of view, if I have a preuve N of A and I have a preuve N of A and B, what is the preuve applied N ? If we apply to the preuve N, well, it's a preuve B. So you have at the same time the notion of PIC, of program, of preface, but it works also in capitol. If you read this as amorphism, and you have A, which is the same here and there, you will see that this application has a sense categorical. L'écriture n'est pas plus bonne, je dis plein de choses, c'est très rapide, mais les sens, elles suivent évidemment même. Mais on a besoin toutefois, pour mieux comprendre qu'est-ce qui se passe avec une structurelle dans une catégorie, quand on veut comprendre ces mécanismes politiques, de faire un passage qui est là, absolument crucial. On est passé de l'extérieur d'un système informel à son intérieur.

40:00 C'est-à-dire qu'on a dit qu'une déduction métathorique qui n'est pas dans le langage, qui est faite en dehors du langage, donne lieu à l'intérieur du langage, incommule la flèche d'indication. Eh bien, ça se voit très bien pour les catégories. Quand on a un ensemble de morphismes qui va être adapté par la catégorie dans laquelle on travaille, This passage corresponds to internalizing the space of morphism with a representation interne called l'exponentielle. And this gives, with certain properties of morphism, naturel, etc., the category cartesian term. This is an interesting adventure, because these categories, with certain properties of l'exponentielle, the properties that we can see, and all the exponentials, they were not very studied in mathematics. For the reasons, which is the most useful category, topological, topological, topological, etc., they don't have this little theory. The theory that we have to verify is that every time that we have C x A to B, the space, this is the natural and natural, the way that we have that in the case, for example, of the space topology, it implies that a function of multiple variables, when it is continuous, compared to each one of the variables, it is globally continuous. And that is not true. It is not a topology. It is in many other categories, for example. So, here, we have two variables. In fact, it corresponds to a logical operation of a normal calculus, because when we have several variables, we write all that in a normal calculus, L'abstration, une barrière de la fois, et ça donne globalement une abstration. Si vous voulez employer la calculabilité, par définition, on pense un calculable de plusieurs variables si et seulement si elle est calculable d'une barrière de la fois. Ceci correspond très bien justement à cette opération interne cathédagogique qui s'appelle la fameuse cathédagogique. Donc à partir de ce moment-là, ces cathédagogiques existaient déjà, explosion from the point of view of the success of the public, of the kind of the work that is done, in the view of this connection with the technology and the informatics of the other. But we are going a little further, in the analogies of the notions,

42:30 typages, in which information and categories, on this parallel, if all this is a communication, a bit of notation to be rapid. So, the formulae of the system of the objective is to have a category, but vice versa. We can see a category as a category in this case. The derivation, the fact that F is a program or a term of proof of A and B, it's a morphism. The axiom of the identity, it's an identity. The U-pure, the formula B, you see, this U-pure plus B, it's the composition. On peut voir que ces propriétés, qu'elles sont dérivares. Un connecteur logique, on peut le voir comme un fronteur. Par exemple, si c'est un diffonteur, c'est un connecteur binaire. Puis les opérations sont ce qu'il faut pour donner la fronteur réalité. Ça, j'ai utilisé deux arguments puisque c'est plus intéressant avec une technologie, normalement on va avoir au moins deux. So the punctuality, it is the operation at the interior of a system, for example, the conjunction in this case. The fact that the identity is preserved, it corresponds to the preservation of the axiom of the identity by the compteur. And then, it demands a little more detail, in detail, la cupure sur une formule, ça correspond à la poncturalité par rapport à la composition. C'est-à-dire, encore une fois, il y a ce relance d'un côté ou de l'autre des structures typiquement catégoriques et de la logique. L'aspect constructive était dans la forme de la discussion que je vous ai donnée, conjonction et objention. Je ne réponds un instant ici pour souligner la constructivité même de la conjonction dans un cadre cathéorique. Vous voyez, si on introduit, on élimine la conjonction logique d'une façon constructive, c'est-à-dire en trainant des proof terms, des termes qui sont des prêts, ou des programmes, si on est dans ce cadre informatique, et ça, c'est l'écriture formelle de cette propriété-là que j'écris tout à l'heure. It's simply an écriture intuitive, an approach constructive to the notion of a product,

45:00 which corresponds exactly to what is the notion of a product cartesian in a category. We can even continue the game and give a signification logique to the notion of transformation naturelle, or vice versa, to understand as transformation naturelle, the passages, all of our words, from a point of view, logique. Regardez ces deux deductions, et cette deduction-là est une deduction qui me dit que la conjonction de ces deux formules va en connecteur f, je déduis en connecteur g avec ces deux mêmes formules. Si j'appelle phi à b ce passage éduitif, si je suppose que f est une deduction qui va de a dans un bim, and I can consider G as the transformation of the connective to the morphism, to the F, which will pass from these two hypotheses to the connective G with A' to the plate R. You see, it is A who leads to A' to F, but transformed by G. That's a theory. But we can use these two hypotheses and use them in another order. of F, we place A by a prime, so we have F over F1, and then we make this passage which is in the top. To say that these deductions start from the same hypothesis and arrive at the same conclusion are equivalent, it means that we have defined a transformation of the correct. So, we have to have a good correspondence between 1 and 2. So, in general, on a lot of results that show a lot of results, that are not going to be a characterisation in terms of the preuble preuble preuble in certain categories as interesting. Since we can work on the truth theory, that gives us the result of the preuble preuble preuble preuble preuble preuble preuble preuble. And that's what it is to say. This is to say, it's just a level abstract of the transformation in preserving certain variants of structure, which give a look at the point of view on the technique and the applications. This is something that has been done

47:30 and I think The most important thing in the informatics were the suggestions of language, and extensions of language. And we could add some constructions of language, since there was a termite or a termite category which gave sense to these extensions, sometimes even inapprendues. An informatician would not have thought about it. Allons maintenant un peu plus loin sur ces applications. Pourquoi, certainement, on peut faire du codage ? J'en parlerai bientôt en samedi. Qu'est-ce qu'on y perd et qu'est-ce qu'on y gagne ? C'est ça ce que je vais bientôt discuter, mais regardons en l'instant encore quelques applications. Sur le fond de la truie de la peur, il y a eu des choses très importantes quand c'était fait sur les ordres. So, when we do the math, we do not know the variables in the first order, it is very interesting to use the variables in the second order, which will vary on the structures in the second order, which will vary on the sub-ensemble. In particular, we need to do, for example, from the logic point of view, the analysis. Because in the logic, with this monomania which dominates the element of the arithmetic, we need to do an analysis and take care of the sous-ensemble of the entire. Because in reality, it's not a sous-ensemble of the entire. So, the analysis, we need to have like a form of arithmetic. the second part. This is a motivation for the work and the conductors, and a motivation for a system extremely powerful, which is the systemet of Girard. And this is a part of it. It's an incognito. It's quite incognito. But what happens is that there are types of types that are not canonics, not details,

50:00 but in plus, when we have a type 1, Ah, on peut s'y construire des dits suivants, en prenant la quantification sur la collection de tous les dits, et dire ça c'est quantique. Vous voyez l'implicabilité qu'un doit s'y formalise, puisque ici vous êtes en train de l'expliquer aussi, si vous voulez un élément de la classe des dits, en utilisant la quantification sur la collection de tous les dits. then the formalist will be done with this. And it's very interesting. If we are at the level of language formel, and they have right, at the level of language formel, we are trying to define formellement a class of signes which will be informed by the script. We do the sense of the script, we are talking about the atom, we are talking about the flesh, we are talking about the product, and then all of a sudden we are talking about the classification on the class is on the end of the film, it's not that important. But here comes the theory of catégories that we help. And, in fact... You can explain what's in your description, what's in your description? Well, that's the collection of the types at the end of the film. There's an intérieur of the base, the entier, etc. I put in the intérieur, of course, I can put, if I want, and then I will do it by a quantification of all the types. This is all x, which is a type. This is a quantification of all the types. This is a quantification of all the types. I would like to be like this, if I would be specific. I would like to say that x is equal to all the types of b. if I wanted to say that, I did the same operation, but not based on the other type, but based on the production of the politics that I was just trying to define. So you can use the construction formals, I don't know if it's formalistic, or if it's not formalistic, or if it's not formalistic, or if it's not formalistic, but in math, in fact, there are plenty of secularities of this genre, because, I don't know, In topology, when we define an intersection, we know that it is possible that the intersection is not part of the family and is not part of the intersection. It is not part of the problem. It is like that we understand the theory that we do.

52:30 In fact, there is a lot of work on the sémantique of this language which is extremely important in the chronology and which has been remarkable when we do the information. Why? Well, because it's very useful to have the types of parameters. Because if you want to write the function identity, for example, if you want to write it one single time, then you say, well, you have the parameters for your type. And your type will be like that. It's the identity. You can write it everywhere. You can write it in a program. It's called the volume of view. In this program, you can write it in a program. That will be parameter by the type. So if you want to write it in a list, you want to give it first as an argument. So you apply this to A, lambda x, lambda x, lambda x, lambda x, x, you apply A, and you know that this is the specific identity of A, which is going to be A to A. This is uniform, this is a type of programming uniform, and this, it is very good in a case of That's very good. But it's perfect. There's a lot of logic, even in the syntax, it's very hard, very difficult, very important, which is called the administration, that guarantees that, formellement, we don't have any contradictions. But, let's say, the circularity, at the level of syntactic, is not entirely accessible. It's not quite difficult to explain in detail. But what I can say quickly, First of all, the quantification is a very good interpretation of the geometry. The quantification, if you want, is a generalization of the product. of the product, because to say for all, it is to say that in the same time, it is a conjunction generalised, in the same time, all the predicate instantiés are true. It is an infinite product. And we can see it as an adjunction of the diagonal, but it is a bit detailed. But if we generalize this idea of the problem, we can define the problem,

55:00 we can interpret it as an agent of the diagonal, and it exists, which is very beautiful, as an agent of the gauche of this same compteur. So it gives a symphony that exists in the structure of mathematics. It's an idea of the categorization of the categorization of the math. Well, in this case, what a certain number of people have shown is that in the sense categorical, it's important that it's important that it makes the fermeture of certain categories. We can construct certain categories and we have proposed a category that we call the first university, which is an ordinary institution, but I think there was someone who was one of my students who was the first to have been asked. And so, the first model of this kind of structure is an example that doesn't exist in a category which is called Ethics Complete, which has the quality that they have of the sub-categories, which are the same category interne. The category interne is what? We can represent a category as a category interne as an other if there is a couple of objects, such as one represents the objects, the other the morphisms. That is, the structure of the entire category is due to the structure of the morphisms which is only one of the objects. And what is important, the theory of the fermeture is not evident, as we have demonstrated, is that the quantification, the quantification on the collection of all the objects, since this is a sémantique, when we do a quantification which is introduced on all the elements of this category inferno, we rest in the category itself. This category is fermed by a product of an infinite. Like we said, the rationales are fermées by division and modification. And in this case-là, the category is fermée by a infinite product which remains in the end of the category. So, we understand that the value of the formal is an ode of the confusion of impossibilities,

57:30 So this is an example. Can we see it in a cabin in an ensemble? Yes, yes, after we can see it, and I can see it in the moment, but we don't see it. After we can see it, but if we can see it, If we don't have an actuality, a notion of adjunction, which is really at the core of the comprehensive category, we don't have to do it. I would like to see the immersion of the guillemets. We have to do a codage. But we have to think about what it does and what it could be for a codage. Ah, well, I've been avancée now, at the same time, before this kind of work. There is another application important, which is one of the categories in construction. You see, I've talked about these deductions which correspond to a language type, in saying that if I have x and y in A, then I can say that x, y is in B, but we can play, like Church in 1932, in a structure of the century. It is not only to allow X to apply for this hypothesis, but also to allow X to apply to X. In 1932, Church proposed the language of the century as a structure of the century. It is not that you have a problem, but it is not that you have the application function. You can see what is a function? It's something that when you apply A to A, you have f of A. This is explicit for the dependence function. It means that when you apply A, your number of x eats A and puts it in place of x. This is the operation of A. The operation of A, and R is very efficient. If you apply this and that, you understand that it is defined. and it's done. You can take two parts of the second part, it's like a diverse part. It's not a problem, but it's essential to the calculability.

1:00:00 In continuing to play with that, you see, Church is completely like all those who read about the promise or a little bit of a synonym. Because, you see, he lived in his language in a negation. Now, I'm going to take a look which is not really respect to the story, but it's not just that. And what's he did? Eh bien, alors, Bami, disons, Cary a dit, bon, alors, écoute, moi, je prends la négation de ton xx, je prends la négation de ton xx, je le mets là-dedans. Qu'est-ce que ça donne ? Eh bien, ça sera égal à la négation de lui-même, ce qui n'est pas très joli si cette négation veut dire complémentaire, quoi. It's not good to have a system where something is equal to its own communication. That's why it's called paradoxical. Look at what it looks like. When we look at it, we look at it in this error. This is what we call it. It's exactly the same thing as the paradoxical race. because that is not... If you have changed now the auto-application by the auto-appartenance, you ask if he belongs to himself, if he belongs to himself, he belongs to himself. And so, it's changed the application by the pertinence and the function function by the science individual. So, it's a function function that is part of the antinomy of Russell on the territorial movement. Well, but instead of blocking the situation, Sassoujian is very jolie at Harry, in 33-34, he has replaced it by a y, so he has replaced the x by a y. So he has replaced the x by a y. Par an effect, he has given a name to this operator,

1:02:30 which is very powerful. This operator is very powerful. This is not the most contradictor, but if we have an indication, he has to say that for the term F, we have that he doesn't, he doesn't, he doesn't, he doesn't, he doesn't, he doesn't, he doesn't, he doesn't, he doesn't, he doesn't, he doesn't, in a way of mechanical, you have, I think it's very important, math, you have an important point. And that, and that, it's the same thing. We're going to do the equations. We're going to do and we're going to do the equations that we use the equations that we use. And so, this is just because it's a calculus of 100 piques. So, we don't have to construct the term as preuves in this sense-là. Very good. that is the capitalization of the mechanism. We are in the end of the end. In the end of the end of the end of the show, we guarantee that the point of the form formel is that it is, that we do not show the legalities, but it is a form of cohesion formel. We say that it is a system that satisfies the hypothesis that we have in terms of cohesion for the form of the humanist, because we limit the regulation. in an angle of the regulation that the program is in this system. It's a very high-paying technology, but the system, in fact, is a system of lack of power. Problems of signification. What does that mean by the point of view of the stock market? This is the case that we have seen in some cases. The risk of the netras in the language of the formation of the 1960s and the crass of the calculus. which is very good in mathematics. But what is there behind the constructions? You have to have an universe A, which is the endomorphism, because here, the terms are functions, they are applied to them. I made F applied to F without problems, like X applied to X. There is an universe, so that the functions of them, the endomorphism, with the elements of the universe human. It's to say, you need to have an isomorphism like that. It's not easy to prove that it is not trivial. If you don't, of course. But you don't have to do that. You don't have to do that. for at least for all the sentiers, for all the sentiers there.

1:05:00 So there is an universe that contains all the sentiers, and who is this property? Well, for the reasons why there is an entity, the category of the ensemble will not be the same. the telescope, it is very catégorique, limit inverse. It is a limit, but it is organized in a way inverse, as we say, in bringing an object fini, in fact, with a series of inversions inverses. It is, let's say, it is a construction dual compared to a a notion of the limit is quite ordinary. And what we obtain, at the end, is an infinite object. These operations are, in fact, which are the retractions even more than... It's an immersion with a surjection. At the end, we obtain an infinite object which is an infinite object, which is typically categorical, which is a power in the text, which is, precisely, who has this property of the misomorphism, which allows to see the misomorphism and the other people as inhabitants of the same universe. This is 42 years after the invention of the 19th century, in 1970. And what is that? Ah, pardon, it's a notion of a typical reflection, It's an immersion with a surjection of J, and J composed with J, it's an identity. J, what is it? It's a morphism in a category where we get a unique inverse. But D plus D, it's only D plus D? Yes, D plus D. So I'm not sure. It's an object... D plus D plus D? It's not the only one, it's not the only one, it's not the only one, it's not the only one, it's not the only one. And it needs to have this immersion. In fact, we talk about something very simple. We talk about it. Yes, we talk about it specifically. And at the limit, we show you what it is. And so the regard to the construction of the limit is categorical. We'll do it. Let's do it. We'll do it. Because you can do it in the category, there's no morphism.

1:07:30 Because, obviously, this is not valid. In fact, all the constructions that I know, there's no problem with that, are the types of... It's to say, the recession is made in a structure of the topological structure topological, which we can take in the category of the space topological, but we can do it by all, because we need to look at the cartesian. The cartesian, that's the idea of the primaire, for some categories which are interesting, because it's rich, we can't consider all the types of ecologists, which are formed by the exponentielles, which are cartesian and fermiers, and which don't give us a lot of morphism. To have this isomorphism, we don't need to have a lot of morphism, because we don't have a lot of morphism, because we don't have a lot of morphism. And the functions continue, so that's how we move, which works perfectly well. The QG Star, there is no other way possible. We work like that. Okay. and I have to go to the conclusion. And so, as it is in this case, we chose the structure that we have, we analyze the materials that we need, the transformations that we preserve, the invariants and the constructions that are made, and this is in a way of realisation. So, for example, it is the base that we have not an universe absolu where we have all the mathematics the past, present, future, and the future of the children of the children and the children of the children, there is a plurality of the divers on a construed into a new world, which is important, which reconquête an unit through the hunters who move from the category to the other, the transformations notary that they tell us that these hunters are wrong. It's an universe of expansion, always motivated by the sense of the structure that they come from. Therefore, the catégories of 100, certainly, it's the solar system the most important, and it's the end of the system. Or, you know, there's a very low terrain, which is like that, which tells us that when we give a catégories C, there's an immersion between and fiddles in the catégories of the concrete on the ancient, which is in fact, in fact, a catégorie simpliste, are associated with objects, so to control, indexes, etc.

1:10:00 We can do it with a variant or a variant. Yes, it depends on what it is. Pardon? Ah, yes, of course. And so it's an immersion in the art, in the art of science. It's a very good result. But it seems to be a lot, and it's not to say that, that it is a priority anthology. You see, it's like when you give an Euclide, Descartes, Newton, and you say, well, There is a variety of Riemannian, I can't say that in the interior, but there is a sort of Riemannian, or a geometry hyperbolic, which is known as the Kufia de Beltran. The Bonnet de Beltran. It's an interpretation of the negative. It's amazing. These are the results of relative coherence. For example, used in the book of 1989, it's called the relative coherence of the different options of the chemistry. But you see what happens. When we do this, is full and fidèle, we are full and fidèle, but there is a lot more to do the possibility of this immersion that there is a certain period of time, or logical, it's like if we were to say, yes, in fact, the universe is the 4 minutes, then there is a certain curve, but we are going to immerger, it's not that the attitude of the physical, it's not that, there is a degree, a concept, then there is an interpretation of the energy, and it doesn't give any priority immunological or morphological to the structure of the newton of the space. In addition, we go to the same way. This is the same way, the catégories of the senses, as one of the catégories, in the same way, as one of the particulars. In fact, with the advantages and the défauts. The particulars, in the sense that it is a singularity, It's a situation of singularity.

1:12:30 It's not a corbure constant. It's not a corbure constant. Egal 0, it's a singularity. And it is at all the two sides of the absolute. It's to say, when there is a corbure, for example, constant, suite to the correlation between the corbure and the tensor metric, it gives you a measure metric. And that's already a lot. We have a reference. So, we don't have a parent or a parent, but we have a measure. When you give a structure of categories, it's what happens. You give a information pertinent to a format that you are trying to do. We don't have the list unstructured, the list of the creation of the ensemble, which is exactly the same as the absolute unstructured in the space. There is a structure, there is automatically a matrix. There is already a structure. So, these kinds of things are very informative, as well as the work of Klein, who gave this kind of interpretation. But first, we come back with a confirmation of the generality of the human being. Which is the fact that we have 4 categories, 4 and 5. And when we come back with the category of the ensemble, On le comprend comme catégorie particulière qu'elle est similaire. Pourquoi ? Puisque c'est un topos classique et puisqu'il y a assez de points. La analogie n'est pas parfaite. La singularité d'Eden est particulière, mais, disons, c'est tout à fait comparable comme temps de généralité de la construction. Well, I think that the value of the category, plus the technique, plus each time, is to emphasize the importance of this reference that we chose each time, and in general, this capacity is still open to the construction of the world. because the culture that is behind the absolute ensemble is really the culture of the New York Times, which is formidable, the report, which has disappeared in the history of the sciences. And there is also the culture of the cottage, that is another grave story that is appropriate to the culture of the New York Times. L'exemple du cantor est typique, mais lui il s'en rend compte, et lui il s'en voit pas, l'histoire de pouvoir coder,

1:15:00 c'est quoi la culture du codage ? C'est ce qui est passé après, parce que vous voyez la nourriture des nombres, la nourriture des nombres, la nourriture des nombres, c'est vraiment quelque chose à intérieur, comme on dit de hiper la nourriture du encendant. Ce qui compte, ce sont des points dispersifs, discrets, discrets. C'est cette culture du discret qui parle de la typologie T discrète, but the topology of the key is that it's really interesting. Sur the entier, there is the topology discreta. So that's what we call, we call it all. And it's what we call, and I call it, which are these things rhythmically, which are changing the world, which is a terrible, and that's what we call it all. We call it all, or a big band of machines, you can't do that. No, not at all. On can't do that. We can't do that. We can't do that. That's what we're learning about. That's what we're learning about. It's very important. When I'm a teacher in my education, I need to know what's going on, what's going on, what's going on, what's going on, what's going on. You know, it's a way to make the knowledge of the brain. When a student says that I can use the knowledge of the brain in the machine of the Turing, it's just to have an attitude cataclysmic, and to say that it's a good test, because the structure of the brain, the stars, the volume, and to make the machine of the Turing, as well as the power of the Turing, and to make the machine of the Turing, The physics is a catastrophe, because there are some things that say that if you create a different dimension, you have the same spécific. It does not exist in physics. The dimension of space changes the physics. The physics of the physics is an enjeu of the passage between the dimensions of the dimensions of the core. So, this is a theory of codage, which is, I don't know what it's called, but it's like a theory of knowledge, image of the world, which is really something catastrophic. So, you have to look at the point of view, corrective sense, and at the same time, use it for the best.

1:17:30 and I think that's enough, I think, to look at it and to look at it and to look at it and to look at it and I think that the point in common with the physics of the XXe siècle, which is the geometry that is behind. I believe that the invariants of the transformation Klein and the common bias which is behind this whole structure which has marked the philosophy of the physics and the path of the physics of the 10th century and at which it has the same geometry and the theory of the capital. These four concepts are the prime of the importance of the solar system Solaire qui est pour nous plus actuelle et l'académie de Sarsan. Merci. Vous avez compris ce qui est important avec cette synthèse. J'avais une question sur le point de vue Bête, naïve, et comment les informaticians ont appris à la théorie des catégories adaptées aux problèmes de l'informatique ? So, let's say, there is obviously no use of the language. We see people who really don't use the theorem for the computer. In the computer, they just use the definition, and say, well, it's not true, it's not true, it's normal. Three points, but it's not the theorem. D'abord, the importance of having the theorem cartesian is very liée to this story of the capitalization which is defined by a component. which is already in my clays, but it doesn't matter. It was an interesting example. So, it was a development of a theme that was interesting there. And the informaticians learned that it had to do a bit of a problem to understand the language and to give it an extension. I think one of the impacts was in 1975, when Jean-Louis Nune which was made for 15 years,

1:20:00 a paradigm for information functionally, which was vendu by the whole world, he was dessiné as a language of information, and in particular, he had put certain operators on their language, and some other companies, because he had a type of mascot that he had a sémantique that justified his extensions. So, he had a language of information, and he had to do extensions to the subject, and the problem is that there was a structure that had this property. So, the formalized in a raga, it was an extension. And so these people had to know, they had to know, about the mill, the mill, the mill, the mill, the mill, the mill, the mill, the mill, the mill, the mill, the mill, and the mill, the mill, the mill, the mill, the mill, and then they had to develop a whole field, which was very complicated, But then he developed a whole field, which is very complicated, as we say, categorical, which is very special, which is where these properties of the fermeture are, for example, that is a test of mathematics. For example, his results on the small complex are made by the motivation. It's not a system of logic, but his success is intervented by the informatics. The problem of the informatics was really remarkable. It came from the logic. to develop an informatic, the signification is categorical. It's not simple. It's the result of the literature, the construction of the category, and the demonstration of the literature that it's important that it's categorical. Of course, it was a group, the review that I dirige since 1990 at the University Press was born around this group which had some of the structures in my head. Of course, with a certain economy. Let's say that now, the challenge is not, because everything that I have raconted has a limit that is in a sense of a sequence. The calculated of 1930 to 1990 calculated sequence, even if there was already a problem. But since 1990, and this is because of the network, there is an explosion of problems. The space and the time, finally, are arriving at the moment, because the machine is part of it, and the time is a caricature of the time, there is a roll-off, the machine is coming, all of a sudden, the space and the time are coming, because the engineers are not, we are not mathematicians, but the engineers are coming to the machine.

1:22:30 They are starting to make the market in concurrence, information for the processus. Eh bien, on n'a pas la théorie de la probabilité pour ça. Donc, ils sont les grands problèmes. C'est une théorie générale des processus concurrents avec le testeur mathématique. Comme c'était la calculabilité et sa sématicalité. Et il y a là différentes propositions formelles, nombreuses formelles, quelques-unes catégories, pas simples, pas simples. Je travaille remarquable sur les catégories and that in the 1930s, we have given a plurality of systems of calculus, and in 36, we have shown that they were all equivalents. They were in six years. They were born in four or five or six systems in six years. In the same time they were construed, they were very equivalents, but now they are in four or five or six systems very interesting, It's a situation that I don't care about it. With, I'll tell you, an effort very remarkable, but it's difficult to tell you, because I've been talking about it very quickly in this problem-like, and I don't have a preference for it, and I find that none of the authors are avisés in particular, and they are not active yet. They are not active yet. I would like to come back to the... I have an ensemble, I know that it was very... very Mexican, very action, very little. In the system solar... I'm a little bit tricky. I know there's a book of MacLeod in the upper world, where we can't do it. We can't do it. I don't know. But in the example that you have given, there's an ensemble. It's an ensemble. So it's not even an ensemble. Because the morphism of one another, it's an ensemble. And I'm not sure

1:25:00 The metaphore, the plongement is varied in a different way, but there is no one, there is no one, but there is no one. It is really canon, so we plonge all the categories together, but it is not a plongement, like we plonge the sphère where it goes, no matter what. So I think that the metaphore is not... Yes, that's the two of us. There are two points. There are two points. There are also two points of view. It's on the duality. It's a little bit the seminar that follows, because the relation of the incertitude, it's nothing else than the duality. It's to say that in an space, they are dual. If we work at the same time in an space and dual, It can't be the case, it can't be the case. But for me, the duality, it's not the opposite. It's not at all the same thing. First, the duality, when you have an espace and a dual, the dual, when you have an A and you have an dual, it's not A, it's dual, it's on its feet. It's true in dimension infinity, but it's not true. And for me, the duality, it's the category... It's like the functions and distribution. the functions are the functions. And for me, the dual has a category. It's the one that I'm marked here, it's the category of the fonctions or the fonctions of the orient, and we're all together. And we're not on our feet. And the duality, it's to say that we have two things, and we're all together, and we have an observer, and we're all together, and we have a nom. So we have a category and a fonctions, and we're all together, and we're all together. to do the duality, we have two things and together, we have a woman, and we have an infant. We have to produce something else, the duality, an experiment, and an unknown. So, for me, the opposite, the symmetry of the space, but it's not at all the duality. I'm not sure that the duality is specific to the structure. In fact, even the junction is a form of duality, and it's already different from the duality. The categorical dual, it's a specificity of the duality relative to the categorical duality. This is also a duality,

1:27:30 like the symmetry. Translations, there is also a translation which is a specificity and a symmetry. The categorical dual is a specificity, which is perhaps a poor, which is the example of the mission, but this is an epinome. I agree that the canonics of this dimension is not because it's an inventive, it's not because it's an inventive, it's not because it's an inventive, but from the point of view of the dimension, pleine and fidèle, it's an analogie which is bizarre. Non mais la dualité, on peut aussi dire un autre phénomène qui est la covariance et la contravariance. Dans l'Homme on le voit bien, tout à l'heure tu as fait une petite hésitation à droite à gauche, mais là il y a une dualité, et ces dualités correspondent bien à une dualité physique, qui est la covariance et la contravariance. And the fact that it is not perfectly symmetrically, it is that the richness of the phenomenon. It is because we are waiting for it to be symmetrically, we are not to talk about the same category of the ensemble, but not to be dual. But it is the difference, it is the difference, it is the duality, and the asymmetry, which is all the richness of the mathematics. But it is still a duality. It's not because it doesn't matter. If we take the word duality in the sense of the term, yes, of course. Duality means the same thing as the same vertical. But the symmetry is the same. Yes, it's a certain symmetry. It's like the symmetry of the translation. It's true that the dualities that we find in mathematics which I would like to be categorized, they are never a symphony perfect, but they part of the idea of a symphony, of a desire of a symphony, always contrary, and it's very different in the research of what is going to happen. But again, I think that there are a lot of space and space that we don't have on this side. The covariance is the covariance. It's the space, it's the space. It's because it's not perfect.

1:30:00 I mean, for me, the reality is that it's an engineering, an accouplement. There are two things on the north end and it gives you something an antithesis. But here, there is a duality between, not like you said, but between the vessels, the pre-faisseaux, plus tôt, c'est-à-dire les objets de la 7-C-O et de la catégorie 7-C. Alors là, il y a une dualité réelle, c'est-à-dire qu'il y a un produit entre eux, un incouplement, comme tu dirais, qui produit un ensemble. Si tu as un souci... La même objection, c'est que la catégorie comme suffisance, tout ça, ce sont des dualités, pas. C'est un souci. Et qu'on se joue... The catégories dual are not all at all symétriques, it's a default symétriques, which is... And in this effort, there is another thing I would say that you have mentioned, but perhaps not as evident. What I have said is that in terms of the category, there is only one operation, which is the unit projective, which is the dual, which is the one that is the one that is the one that is the one that is the one that is the one. It's the only thing. It's the only thing. It's always that. It's a very important thing. It's a very important thing. It's in this symmetry. That's how it works. To take an example, you talked about an object terminal in the generator. Because in reality, it's not just like that. It's an object for which there is only one morphism that goes to him, each time. There is duality, the initial object. But if you take it as an object terminal, duality, you find something else, which is an object omega, like the people who do it all. In the case of the ensemble, it's omega equal 2. The ensemble has two elements. But what I want to say is that there are two ways. The duality is not an absolute thing for all the time. If it's a point of view of something, at a moment, there is no absolute duality.

1:32:30 Yes, that's what I said. It's specific in the structure. I was very happy on the initial terminus, I didn't want to give you the actuality, because I didn't want to pass it on top because I don't want to see what I'm doing. I don't want to see the whole thing. I don't want to see the whole thing. I don't want to see the whole thing, that's what I'm doing in my case of top, or something like that. The difficulty is really analogous to the symmetry. I don't think we can give, once for all, a formalization of the symmetry. The symmetry is specific in the context. It's very varied. It's almost like the message of the leading word symmetry of Herman Biles. It tells us that there is this plurality and the experience active of the symmetry in the world. and there is a multiplication of mathematics but it is always a construction that is a derivative of a certain structure which is enormous. It is a phenomenal idea and it is almost like

1:35:00 Thank you.