Itineraire d'un mathematicien / Some things I learnt from Ehresmann (1st 20 mins only)

0:00 The red light is flickering, which is as it records. Mr. President, Mr. De Maille, who has problems with transportation, he can't speak French, so he can't be here today, so I open the day in his place. So, first of all, welcome. Welcome to everybody. If you want, I can speak a little bit of English with you. Thank you, Mr. President, for being a friend. I'll make an effort to speak some French. So, thank you very much for coming all this way to the University of Paris-Le Verne. We have three days. We are very happy to have everyone here. I hope you will enjoy the day in front of you. Now, Mr. Paul Carson, who is a professor at the Université d'États Unis, will present the speech of the president. Mr. Carson is the director of the Interculturelle and former president of the university. Then, he will present the speech, which I think was given late, in the first place, and then we will move on to the interventions, which are for today, in general. The president, unfortunately, is caught up in traffic, not reference, train congestion, so he went to New York, so he'll start loading after today in his place. Mr. Paul Tasson is... An ex-president of the university will be presenting a few purposes of his speech, and then he himself, as director of the faculty, will also be making a small speech, and then we'll get on to the start of the second seminar of Charles Ezra today. Thank you very much, and have a good meeting!

2:30 5,000 people since the creation of the university have experienced that their return remains in the daily life of their fellow students. Through the education of the students, either in our theater, or in our place, in the establishment, with the hope that we will remember during these 3 days the importance of science, as well as in the international community, and in the life of students and in our university, is one of these. We are on the ground and we are on the space. It is a teaching and research to respond to their questions in which we find ourselves. It is necessary, therefore, to pay tribute to those who have left such important traces in the mechanical research and in the work we are doing. Also, we have been influenced by these many works. In topology, I have described these works in several ways, such as the cyberspace, the tributary canon and the collection. Varieties and qualities that are important. Based on his theory of effects. In the material, these works have read and are in the course of our applications in physics and cosmology. It should also be noted the importance in theory of categories on local structures species, internal categories and the theory of species. These works by Charles Westphal have combined applications in mathematics, in systems and more recently in geology. In your initiative, madam. Délite Artan, who directs the hotel, Charles Alassman, occupies a privileged position in the French mathematical and scientific community. He was also, in the 1960s, the president of the National Committee of French Mathematicians.

5:00 The recognition of his work was lost thanks to the work of his disciples and many mathematicians. All that has to do with mathematical research is to harmonize the scientific community thanks to the revolution that began in 1958. The key terms may include topology, geometry, algebra, analysis, Penrose, Atiyah, Witten, Connes, Hawking. The laboratory at Miemois, the fundamental mathematical and applied laboratory, is the private place in our building, the work of the students and successors of Mr. Erasmus. You will make me happy, madam, to serve you wholeheartedly and to be able to thank you for the quality, quantity and activity that you have deployed and that you continue to deploy to serve the memory of your country and in this way. I would like to thank the researchers who are in charge of this educational and work. Thank you very much for your attention. I wish you particularly good days. I would like to thank Mr. Édouard Lisbourg, the president. Mr. Charles Espagne remains, for all the researchers of our university, one of the great scientific experts who have made this dream come true. As a professor in the world of science, in the development of science, in its development and in its realism, I remember We have organized a number of seminars with his wife, André, as well as scientific journals in Saint-Pékin, where Charles S. Spahn was our university.

7:30 This weekend, we organized Charles S. Spahn's ordinate valaisessence. But also, this is our university. Please allow me to link these two seminars and these two destinations, which contribute to the resumption of our university. Here we are, a few weeks after my disappearance. You can see on this image the passage to Denoine. In July, Charles S. Mann joined the initiative of his debut. He honored this ideal, a great number of his extraordinary travels and, as I mentioned earlier in my speech, in this film, with his most famous works, the canvases of your attention, so that this connoisseur Charles S. Mann understands, and thus continues this dream without end, As we have already said, these are the director's orders. Thank you very much. Thank you, Mr. Paul Persson, for everything he said. It touched me a lot. And I would like to thank all those who helped us in the preparation of this lecture. We can't mention everyone, but there were materialists, financial experts, experts of all kinds.

10:00 And we would also like to thank you all for coming to commemorate the 150th anniversary of the birth of the cat. So I'm going to talk about both his life and his work by following the diaporama that Jean-Paul Montmange prepared, constructed and realized, and so we're going to see the different periods of his life with the works associated with it. I speak in French, but for those who have some problems with English, a lot of everything I will say is indicated in English on the different slides that you will have. Charles was born in Strasbourg in 1905. At that time, Alras was under German domination. They were French, but Charles was raised in an environment where they spoke, the Alsatian and the German, so he spoke French only a lot later, he was not really a maternal. He was in a fairly modest family, his father was... His father was a gardener in a hospital held by Protestants and so he lived in a Protestant environment. He was quite influenced by them. He also had anthroposophic friends who influenced him a lot. He liked to go on vacation with his grandparents who were peasants. He took me for a long walk in the Vauds in particular. Then, of course, he wanted to study and he went to Ecole Normale Supérieure in 1924. So there, in the period of economics, he was in a promotion where there were quite famous people, literary people, and he had a lot of good friends with the literary people, since it was the same promotion, there was in particular Raymond Aron, with whom he had a lot of experience, and he was quite friends with Raymond Aron, there was also Jean-Paul Sartre, in the same promotion, but he had close relations, Jean-Paul Sartre and Paul Lisant, who were rather a bit different groups.

12:30 And then, after a normal exam, he went to the military service, and he went to Morocco to do his military service for a year. He wanted to travel, and he was very proud because he had been an officer in his military service, and these soldiers had made him a magnificent mall for his departure to Morocco. And I still have this mall in my home. Then, I asked him to show it to me, and he agreed to do some research on geometry in the direction of Hitchcock. So here we have a view of the different members of that time. So here are the essentials. I tried to draw a general picture on the essentials of his work. So you see, I tried to put... These are two branches of his work which, starting from a further point, diverge a bit. One that will bring him, as he did in his thesis on heterogeneous spaces, to the geometrics of the past, with, for example, the filmed spaces and all the applications, of course, that have been done in the past. We will come back to that later. And another part that is more focused on local structures, localization, quantum localization, Then, structured categories, internal categories, sketches, and therefore all the works of the second part, the last part of his life. And the three columns indicate three different parts, one part where there is essentially topology, geometry, the part in which he tries to take what he has done before, he has to meditate in the case of the end of the categories, and where, in my opinion, there are the three main articles that will be at the base of the whole... In this part, they considered what they were doing to be part of the human race. We are in between 1930 and 1934, he prepares his thesis, this thesis goes to Paris, but he goes to Göttingen to meet Hermann Weill, to be quite disappointed because he had really worked with Hermann Weill but there was not a good contact, and then he left for Princeton where he spent two years, from 1932 to 1934, he really prepared his thesis, and besides he had the opportunity to meet Einstein briefly at Princeton at that time.

15:00 So, Sartre is on homogeneous spaces. For those who do not know, a homogeneous space can be defined as a group, as a space with a group of transitive transformations. The definition is in Sartre's book. If you consider Huygien's geography, you can consider it as being the study of a plan space, for example, with the unit of the group of Huygien's displacements, rotations or plantations. If you look at the Atiyah-Witten, it's the same plane, but it's a flat plane, but it's connected to the Atiyah-Witten transformation group. And Klein, in his Erlangen program, had proposed, in a general way, to call geometry the study of a space connected to the action of a group. It's not sensitive, that is to say that between two points of space, there is always a transformation. Between two points of the plane, you can always find a transaction that goes from one to the other. So homogenous spaces are spaces made up of a group of transitive transformations, and what he does in his thesis is to study the topology, i.e. the groups of homology and homotopy, of different homogenous spaces, and in particular of the varieties of races. So he spent his time in 1934, he stayed in Paris. He is the ancestor of the CNRS, you know, it's not really called like that. And I think he will have a rather intense life with philosophers because he had met Caballès and Lothman at the Normal-Superiori school. They were not in the same class but they were in a similar class. And so he stayed with them in contact with them in particular. He discussed a lot with them and Pierre-Alonso will explain to us a little this afternoon the links he had with philosophers. And he got married at that time. And he lived in Paris. He was named only in 1939 professor at the University of Strasbourg just two years ago. So these are different people with whom he studied at that time, among whom there was, for example, a very well-known mathematician, so there are different people who are more or less well-known. So during this period, he continued, he did not continue on homogeneous spaces, but he extended the works he had done on homogeneous spaces to locally homogeneous spaces.

17:30 The homogenous space is a space where at each point there is a surrounding that is isolated from the homogenous space and this will lead us to think about the notion of localization, which will really be the root of all his work on Earth. So, what is the notion of localization? It will define the notion of a pseudo-group of transformation that Glenn M. Whitehead had given a definition but which is not so precise. So, he will make this definition more precise and then he will define spaces locally that are associated with a pseudo-group of transformation. If we have a space E, it will be associated with this group of transformations if it can be covered by Ui spaces, which are covered either by a bi-universe correspondence, by a Celi map, with one of the Ui spaces on which we have this group of transformations. The space D of things, on which there is... The group of pseudo-groups of transformation is on this side, and this is on the other side. We have a unit here, and it is in relation to another unit. So this unit will transport the structure that we have on this opening of the pseudo-group here. And if we have another uj, it is the same thing. We have a card, cj, and the change of the card has the right to contain a group of pseudo-groups of transformation. It is transformed from TI to CI to MIX, and from TI to CI to MIX, which is an indication that there is a certain CI to MIX, and that must be in the transformation code. This transformation code is defined by the data on a topological space, the data of the domeromorphisms between topological spaces, which are stable by composition and by collocation. So, in particular, for example, if in this group of transformations, we consider the group of transformations between homeomorphisms of RNAs, let's say, we can define in this way an ecological variety. Let's take for example an example here, if we consider the sphere, we can, as well as the ER2, well, the sphere is locally isomorphous to L2.

20:00 This would be the stereographic projection from the North Pole, for example, on S2 here, and then the same would be on the other side of the globe. So now we come to 1939. Of course, 1939 is the war that will be mobilized. At the time, the radars were not yet used and he found a method that was approved by the commandment to better identify the stations in the sky. This is why he was named a little before the armistice as a professor at the military school in Paris. So after the war, after the armistice, normally the Germans would have tried to pressure him to go back to Strasbourg, but they didn't, and he went to Léon Perrin, where he was expelled from the University of Strasbourg. So he fought the war at the University of Strasbourg, except a little bit towards the end, where there was a raid of all the Tabassians at the University, and so he went to which he escaped by miracle, because in the morning he had phoned that you should not go to do your job, and so he escaped by miracle in Strasbourg, and at that moment he hid. By the way, before all this, he had always remained in contact with Cavaillès, and so Cavaillès had asked him, who was a resistance, he had been defeated by the Germans, he had asked him to wash a reneutropic machine in his house as part of a calendar. Let's go back to the time when we used to make machines to make tracks. So, during this period, he did a lot of mathematical work, and it was at the time, in 1939, that he defined the notion of a locally trivial space, which he introduced at the beginning of the war, at a time when communications were cut off with the United States.

22:30 And so, he did not know at all the works of Steenrod, who made a general definition of fiber space to be analogous to that time. It is not to say that he also had a son, his only son, Jean-Marc. So, to define locally trivial fiber spaces, he defined them by using the structures associated with this group of transformations, such as fiber spaces that are isobars to a product. And there, what he really introduced was to introduce an ecological structural group on the infrastructures. And this is what is still used a lot today in physics. So, an infrastructure, we have to think about it, is to give a base B, U, and S, and we have the cases we talked about. We already talked about this earlier, but there are other cases. This is a product of a green zone with two sides, and if we have two cases, the 130 cases must be such that one of the A elements here is applied by one of the cases on the Z, and by the other case on the same Z-prime, and we change the Z to the Z-prime, so the application of gamma belongs to a certain level of transformation. And the fact is that when there is a structural group of people, the idea is that this pattern of cells and batteries is done by a non-action, that is to say, the batteries are concentrated in this group of cells by a certain geological mechanism. And one of the benefits of taking topological structural groups is that a fiber space is associated, the main fiber space is associated, which allows us to better understand the group itself, operating on itself, by itself. So the notion of a fiber space in a topological group, the main fiber space of many of them. In two examples, for example, a cylinder is a trivial space, i.e. a bus of the product of an index by a segment.

25:00 The Tau, yes, the Tau of the plus L is a hybrid space which is obtained in the form of two elements. So it is obtained, you see, locally it is a product, but when we turn, there is a change of case which returns to convert two points. So he developed the theory of fiber spaces and, in particular, the theory of relativity and homotopy, which are the first he did in collaboration with his first student, Falco, who also died during the war. And he developed the theory of fiber spaces, and one of the problems he introduced was the notion of restricting the structural group, which in particular leads to the study of quantum spaces. So then, after the war, After the war, he went to Strasbourg, where he became a professor at the University of Strasbourg. He had an intense activity there. On the one hand, he organized a seminar, which at the time was relatively rare, especially in the province universities. He invited a lot of people to this seminar. For example, René Thorn talked a lot about the influence of this seminar. He also has different students, and in particular Webb, with whom he develops the theory of variety of qualities, so we call it the Webb thesis, which he will then define in more general terms a little later. Then he can also pursue his studies in fiber spaces by studying fiber spaces. We study the connections on a space-time. The connections have been introduced by Yves Cartan, but now he will be able to generalize them in the case of specific mathematical spaces,

27:30 to give a definition that is very manageable and therefore can easily define the derivative and the curvature of a connection. And there is therefore also space-time. So these are the works that have probably had the greatest physical application, since the notion of the main fiber space is used in theories such as the Winston-Salem theory, or in all the legal theories of Gauss, in these theories. The theory of Jaws comes from a study of a principal space on which a connection was given. So the potential of Jaws being the connection and its field being given by the connection curve. So, for example, the standard theory, the art on this bar, had a symmetry break, and the symmetry break in terms of principal Hilbert spaces comes exactly from the data of the restriction of the structural group, so the problem that he had introduced in terms of Hilbert spaces. So, there, really, the physical applications are very important and they were, by the way, I had taught them to physicists in Paris. The physicists asked him to give lectures on fibrous spaces so that he could understand the notion. Then there is the theory of connections, which means that without it, there is a difficulty to talk about the notion of derivative, the differential of derivatives. These notions that involve coordinates do not multiply and without them, we would need something more dynamic, which will lead to the notion of derivatives. So this is also a career where they work a lot, and we are also starting to find that Bobati demands a lot of time, so they decide to get rid of Bobati, and that's what made Bobati restore an art of art very recently, not too much against the fact that there had been defections, in fact less than Manavichar, and... So he travels a lot throughout this period, since he will spend about 6 months of the year abroad, he will be an expert at UNESCO in Brazil,

30:00 he will then go to the United States, to several places, to Yale, to Harvard, he has been invited to different places, and then he will also go to India, to Iran and to India. Many of them were very influenced by Buddhism, especially vegetarianism, which he did not do immediately, but he did for a few years, in recent years. So, the definition of g, by thinking about the notion of a derivative, manages to come up with the notion of g. Today, the notion of g is infinitesimal, it is quite folklore, it is commonly used in differential geometry, but at the time it was something very new. So, the definition of a g has two applications, let's say, of Rn in Rn, with the same g in the case at a point x, if they have the same derivatives up to their case in this case. And so the G is the balance sheet for this relationship. So this can be defined in Rn, but in Rn it is not so much of an interest because you can learn a fixed coordinate system anyway, but it also allows you to do it in a variety and there this definition, while in principle if you want to define derivatives you have to go to coordinates, there this definition of the G allows you to avoid going to coordinates. So it makes a calculation without coordinates of the differential and that's what will allow us to develop... So, I have developed the theory of exponential value based on the idea of g, in particular to define the g-variance in the case of a g-variance in the case of a g-variance in another g-variance. And so, in particular, the variety of 1 to g, these are the tangent vectors, so if we consider, for example, the sphere, we have tangent vectors, tangent vectors in a fiber space form a fiber space at the base of the sphere, the fiber being above point B corresponding to the tangent space B, and it is a vectorial fiber space. And more generally, the different spaces of G will be divided spaces, and this will allow us to define G-spaces as a general composition of G.

32:30 This composition has properties that later will be referred to as a category. It is the first place where there is really a general category, but at the time, the definition of a composition was in the category area. We didn't know the category theory very well yet. And then, the entire series of extensions of the ancient edicts, which he considers to be the study of the hybrids of G and eventually of the groupoids of G. With the notion of G, he defines the notion of the groupoid of G as the notion of the groupoid of G, the notion of the groupoid of G, as a generalization of the notion of the groupoid of G. So, he developed a differential geometry that has probably been applied today, let's say, by everyone, and which has also influenced the development of synthetic differential geometry, and we will talk about it in a moment. Then we come to the point that I think is the turning point in his work. This is the period between 1957 and 1968, the period when there are the three articles I mentioned in the study. And so in this period, he is in Paris, so he goes to a seminar in Paris, where he will start to have many graduates in this period. And so I want to talk to you about it. This is an article published in 1956 and which he published at the conference of mathematicians of the Latin Exhibition in September 1956, and this is where our couple was formed. In this article, he will take up previous works where he had spoken about local structures. Jean Benabou and Nédita will talk about them during the day. So his notion of pseudobouffes and social spaces of transformation, he had at the beginning that in this case, in the pseudobouffes,

35:00 we did not use points at all, and only open ones. So he wanted to define a local structure notion of topology without points. But they are not formalized in a complete framework, so this concept will be formalized in the framework of the categories. In fact, he had reminded me that the notion of category, he had seen it when McLean and Ellenberg, Ellenberg and McLean had introduced it, but without really paying attention to it, and that it was when he was also on a course in Brazil. One of these students, who will then lead you to Paris, Constantino de Moros, mentioned to you that his composition of the Gs seemed to give a category. So, that's the part that really got him interested in category theory, and so he tried to see, and so in this article, he will contrast the pseudo-group of transformation by a local egoism, and more generally a category of cases, and the associated spaces will be given by a kind of local structure that is complete in a certain sense, which is a good recollection. So, let's take a look at how we associate them. First, we have a category. For those who don't know what a category is, I'll give you the way in which Charles saw it, at least at that time, and which was important for the development of multidisciplinary work. He saw a category, we gave it a name, a name, and there may have been several names. Speakers are generally amorphous. And on the Earth, there is a composition that is associated with a sequence of successive So it's under this format that you see the categories and so, in fact, the idea of the category is represented here in a schematic way, it is the data of C, the set of morphisms, of the set of limits of C0, of the two applications associated with the arrow C2, which is called alpha and beta, so there, the biota associates an object, the arrow and the entity, and then the composition of C2, the place where we can do that. Thank you.

37:30 So, here I speak in terms of the whole, there were some questions on which he thought, but from a certain point on, he said that we could not take the theories of the universe, so he avoided these questions of the foundation. Thank you. There is a set of structures above the category C, i.e. the category C, a set of E, a set of S joined by a projection of P towards the objects of the category, so each of the objects here is associated with an object of the category, and we give ourselves a composition which has an arrow F and an arrow S which are projected by P on the source of this arrow, associated with an element F and S which are projected. This composition means that if we have F, here Fs, and then we put G, Fs, we get the same thing as if we start with S and we take the composition with the composition of F and 2. And now, if we have a local category or a set of local structures, well, the ensembles are replaced by ensembles equipped with a local structure. A local structure is a gold structure in which all parts of a major city have a separate branch. So today, we are talking about locales, which has been studied a lot. So, I would like to point out that in the notion of structure theory above C, the P is not necessarily an indication of all the objects of C. And precisely, the theory that appears in your article is that we can extend, when we have this one, like this one, we can extend it by putting a structure theory on this one. In fact, it is a construction of the extension of Kahn, a little particular, at the same time, so it was done a year before the article of Kahn appeared. When now we take a kind of local structure, so we consider that on C and on D we have a parallel structure, therefore local,

40:00 The main theorem of the article 57 is the theorem of complete enlargement, which associates a kind of local structure above C, a kind of local structure on C, but which is in addition complete, that is to say a property of collocation, which is a bit similar to the theorem of associated vessels. And these notions will be generalized in a very long series of papers on orderly categories with orders that are more or less strong and also orderly structures and extending each time the order of the centuries. The next important article is the topological category, the differential category. This is the first of these articles that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my life, since this is the first article that I have written in my When we first looked at groupoids, it was to study the groupoids of fiber-on-fiber isomorphisms in a fiber space. And we saw that the groupoids of isomorphisms had particular properties. Well, these are the properties that will be completely characterized in the notion of local groupoids. So, a global polygamous group is characterized by a property of local sections and shows that there is an equivalence between the theory of global polygamous groups and the theory of algebraic attributes of principles. And re-associated spaces to a main space are simply types of topological structures, differentiable if we want to have re-differentiable spaces, associated. A type of topological structure is defined by the fact that we have topology and that the way of composition of the application P and the way of composition we saw between the structures and the base is also topological.

42:30 So, this completely brings the theory of fiber spaces to a theory of a kind of structure, of a particular cohomology. And so, it will be one of the things that will be listed here. The third article is still an article. It was written while we were in Vienna, where we spent three months. In this article, the idea in Gauvaki had defined a notion of structure. It is approximately in these years, a little before, that the result of Gauvaki on the theory of ensembles was made. And this notion intervened in the scales of the whole. Schwarzschild said that it was not very clear, so he wanted to try to give it a more strict and therefore clearer support. And so that's why he introduced what he called photocopies. And what is important in this article is that this has led to the notion of double category. So, the double category is the data of two laws of composition on a set of two different laws of composition, which gives us a category, where we have a horizontal composition and a vertical composition, and the thing that makes us have a double category is the question of permeability, that is to say, if you first compose horizontally these two there, and these two there, then vertically the compositions obtained, you will obtain the same thing as if you first composed vertically these two there. The two categories of double categories are the two categories in which the units of the objects for this law are also the objects for the law. So, they are built by the square on this material transformation category, which he called a quintessence, a quintessence is therefore for him the data of the four factors, you see, here it is a factor, a mass, which is here, and a natural transformation, which is the composition of these two factors towards the composition of these two.

45:00 And since he called it a quintessence, and it forms a double category, well, it is essentially, he defines the mathematical structure by the study of subcategories of this category. From this point on, the papers he published are essentially works on the category theory. In fact, under Jasson, they are all more or less motivated by ideas for geometry. By the way, Charles considered the works on the category theory to be part of geometry. But, indeed, at first glance they are very abstract and often they are realized in an abstract way, so that we can not always see the intentions that are in place, unfortunately with annotations that will often pose a lot of problems because they were not correct. So, during this period here, it is a period where he will receive a degree in the field of science, which was given to him by me. He also studied at the University of Moulins in 1967. And it is at this time that he begins to develop a research team with a lot of researchers, in particular Bernardineau, Ferrer, who was also a professor who died many years ago. During this period of time, we had a very long trip, from six months to Kansas. Oh, Kansas. Why Kansas? The state of the University of Lawrence, but in fact, we arrived in Kansas City. That is to say that we have realized that, in fact, one of the three questions is clearly not easy to answer. What I'm going to present here, it seems obvious because I'm going to insist a little on this aspect, but at the time we did not realize that also the notions of a local category, or generally ordered categories, the notion of topological categories, of differential categories, and even of double categories, can be considered as particular cases of a general notion which is called a P-structure category.

47:30 We give ourselves a category, which is often called a concrete category, that is to say an H category, divided in two parts by the category of the subjects. And a P-sculptured category will consist of a category and a structure S on the axis of the axis so that the ideas of the category, what I call the idea of the category, i.e. the alpha, beta, delta, theta, are naturally related from S in the category A. So, what does it mean to relate in a natural way? It has to be, or at some point, to get to the point of a definition that is relatively light. So, the idea is that, starting from S, there is a P sub-structure of S, S0, which is projected on C0, and such that alpha and beta relate to each other. The relation is unique because we take a concrete vector, so we assume phi of S. And at the same time, we take the hybrid product of A and B in the category H, and we obtain an element S2, and the composition law will be raised in an arrow K of S1 and S2. So a structured category B only comes to raise the idea of the category in the category H, and in a calm way, let's say. So, the definition here is a little more complicated. It forced us to define what we heard from a sub-structure analysis, from an article on sub-structure analysis, and at that time, we were looking for general terms. So, if we want to find the local categories, for example, we have in category H the category of applications for mathematical algebra as local. So, we seem to have become local. For topological categories, we will define the category of continuous applications in topological space, in the differential case, the category of differential applications. For double categories, we will define the category of categories, i.e. the objects are the categories and the objects are the factors. And at this point, if we reiterate the operation, that is to say, we obtain double categories, but these double categories, the German German is a category, if we take this new category, we will have triple categories, and then by reiterating, we obtain the middle category.

50:00 All this is defined in his article 63, in a long article of 63 in the journal of Jean-Paul. And so, there will be, in a certain number of years, essentially, facts that will develop this, that will try to find general theorems. So, the problem to find general theorems on the category of structured P is that, obviously, it depends on P. If you consider, for example, the applications of a financial table. In this case, the B factor is bad in a certain sense, since there are not many fiber properties and there are bad linear conditions. So, if you want to find properties, good properties, you will have to see what are the properties that can be put on the table, defining different general notions, general cases of factors, factors under states, for example. So these are the subjects in which we can extend a certain number of theorems that we had either on topological categories, or on differential categories, or on total categories, in terms of structure aspects, since the same thing is done for structure aspects. All this can be extended in certain cases, so... A lot of articles with a lot of details. We are obliged to see small properties. For example, often theory of existence, free theory, or adjoints, which are more precise than visual theories of adjoint existence, for example. So, a whole study, and Charles thought that the theory of categories was ... Little by little, we are moving towards the study of counter charts and of all the properties that we can have for such a counter chart, for such a counter chart, for such a categorist. It has also been developed a bit in the work of the German school of categories, topological categories, well, topological categories in the absence of certain categories where we can remove some of them. And so we see that there is always the same idea, since the founder of the differential application, Léonard Trébaud, since that's always what he had in mind, was to find theorems that could be applied within the framework of differential applications, because for him, he always considered that everything he did could later have applications in differential geometry.

52:30 The main part of these works was a totally different subject. And so our idea, since it is not good, is to complete it. Then there were a lot of people who did that, who did not focus on the differentials categories. I will give you two links to the categories, differentials, math, etc. But of course, so the theories of consciousness. So there is a large number of artists, in particular a lot of ours, who have studied science. In different theorems of completions, categories, enlargements, especially the completions of factors, there are such and such properties, enlargements of factors, and in particular generalizing theorems of completions, as we saw earlier for the aspects of local structures, to all kinds of figures. But in 1864, while we were in Pintas, we thought a lot about the notion of structure. So, there was the Benabou thesis, which in 1863 defined the notion of algebraic structure, in the case of category theory. In 1866, when we were in Pintas, we studied the Benabou thesis. who also studied, who generalized the rules of Planck's theory and so the idea was, the problem that arose is, can we find ourselves the notion of structure that can allow, therefore, of algebraic structure but that would encompass the category theory. In algebraic structures, usually, we always had non-complexions everywhere. But in the categories, the word composition is partial, so it cannot be explained in exactly the same way. So, in his article by Kinshasa, he defines a general notion of species. which will then be made more particular, and in particular, which will be categorized by simply taking a category made up of cones, either inductive or projective, that is to say that it comes to the category of marked categories, which is done at the same time by Chevalier, or in the table where there are only projective structures, equating to the projective topology of Benin.

55:00 Thank you for your attention. Now we are going to try to construct an art on this idea. We will then go on to a category. The idea was rather what he called a multiplicative part, that is to say a category where the composition law is not defined necessarily on all the elements in the state. So that's a little bit... So the idea is to add the arrows so that the only data of these arrows allows to have the actions. So the two actions of the categories are associated with each other. So what are we going to do? In order to be able to answer these questions, we will ask the students to give us the course. Here we have a square, and it is going to be the result of this. When we go to the realization of this graph in the education category, we want it to give a category. And finally, the A, B, B1, and B2 must be applied to a co-digit. So here we have the mu-1, mu-2, which is the co-digit of a co-digit of B, which is given here. And we have the modern rules, but it is enough to have an arrow that corresponds, that factorizes. All of these terms are related to the identity of the two groups, the alpha and the identity of the other two groups. For the associativity, it's a bit complicated. We are going to ask for the 1 and the 2 that we have here in the table. We are going to take the hybrid product here. And we are going to ask that we have the hybrid product. We are going to take two applications, so two classes, kappa 1 and kappa 2. These are the main categories of subjects that apply in the category of applications, and that apply in the category of subjects that apply in the category of applications. So here we are entirely in the notion of a category.

57:30 If we apply this in a category of ensembles, we obtain an ordinary category, but if we apply it in a category H, we will obtain a category that is more general. In the case where H is a concrete category with an interface that could be a structured P category, but which is actually useful in any category. So the idea of ... The idea is to define a generalized categorized structure, which is the name given to him at that time. So, it simply speaks as a realization, a model of this synthesis of mathematics. A model of this category in a categorization will be a generalized categorization. Today, it is called an inter-category. Note that Jean Bellavaux gave another totally different definition of an inter-category, but using the notion of monoid as a monoid in a certain category, which requires that the category in which you take the categories is productive. So, this is the generalization of the phenomena. By the way, the category statistic can be seen in a more complex way, but we have not seen it at all at the time. This is what we have seen as being the category or the category opposite of the official category. This is not the case in recent years. And so, since we have talked about structured P-categories, we will try to generalize them in the case of these generalized structured categories. In particular, in an article we wrote in 1979, we defined monoidal structures based on structured categories, which will be generalized in the context of inter-categories. Of course, internal categories are categorized as closed monoidal categories, where the given conditions show that the internal categories are closed monoidal categories. And at the same time, there is another thing that we have done in the past years, a very long article that makes an entire volume of books that we published in 1972, we write in the category of sketch or something like that, or the first article that is written with usual annotations, rather than the more complex, more complicated, more elaborate annotations.

1:00:00 In this article, the first part is built with associations in each section. There is also a mix of technical and objective categories, called a mixed sketch, a prototype and a type. A prototype and a type, allowing for a more intrinsic representation of the categories in the structure we maintain in this way, the internal category. For example, a representation of the internal categories, the type of the category. The theory of the type corresponds to the notion of theory. From 1969, I worked mainly on categories. These categories, I call them now, as we normally call them, internal categories, and on theoretical ones, so of compression. And in this period, we developed the research team of category theory and application, Paris-Amiens, which was its official title at the time, where we had a lot of 3rd grade teachers, a lot of people who were teachers, who were teachers in Nain, who spent at least during these years, who spent their doctorate... Their third degree, or even their state doctorate in this team, and this team has had a lot of... Activities around the world, in particular the organization of two large international lectures on algebra and categories in 1973 and 1975. Another one in Chantilly, which was founded at the time, where there was a Jesuit school, which also depended, which had links with the university. We also had one there. And then we organize, practically every month, mathematical days, so TAC days. There were one or two foreigners and several of you were able to attend either the State Zones or the conferences that took place at that time.

1:02:30 Charles was 70 years old. At the time, 70 years old was the age of retirement. In higher education, we could not continue. There were professors at the University of Paris, then the University of Paris 7 in 1968, because there was a gap between the universities of Paris-Sorbonne and other universities. So, in 1955, Charles was forced to retire. But they didn't have time to retire, they had a lot of teaching, they wanted to continue an activity, and so here, the Mathematics faculty of Damien was very welcome to come and be charged with training here. But in theory, of course, he could only be charged with teaching because he was retired, but in the end, he was completely enclosed in the life of the faculty, since he was even the deputy director of... The Faculty of Mathematics as a period. He was also representative of the Faculty of Scientific Council of the University, Director of the Scientific Council, so really, we are completely ... our whole life has been based around ... So, from the research team in 1969, and then, at that time, in 1975, all the faculty came here, and even at that time, we transferred the topological and differential geometry files here, which had been created in 1957. Created, by the way, in such a curious way, because initially, in 1957, there was a seminar, so there was a seminar. At that time, Bell Vaudet was the great master of seminars in Paris. In this seminar, in the year 55-58, I personally gave lectures on polyethics, on which I worked at that time. And there was a problem because there was the subject of the three qualities. And here is the third test that we asked to submit. Something like 60 copies. And Bergdoller, who had a little bit of a fixation on what we could do, did not want to give the additional 60 copies to be able to submit the third test. So Charles has failed, of course, and at that moment... Of course, Helioterre did not give up, but it was hard to find a way to be at home, to do human studies.

1:05:00 And he met, in the halls of the university, at the top of a tower, an old machine of arithmetic. I don't know if you have the slightest idea what a typewriter is. It's a kind of printer, a diabolical machine, absolutely diabolical. I used to use it for years, and you had to type your text on one side, and to have the justification, you had to type the same text on the other side. Of course, if you have the slightest idea, pay attention, if you wanted to put Italy or whatever, you had to change the little bar in front. So it was a work of agreement. However, the first topology papers, which were not exactly like this at the beginning, were made on this machine, the 261, to have a secretary who typed the topology papers. It did not take so long to get a good part. It was like that when we started with the topology papers. And so, in 1975, when we wanted to come here, after his retirement, we decided to come and settle in Amiens. And to settle in Amiens, of course, we had to bring the Ritz-Piper. So, and this particular machine did not belong to anyone. Theoretically, it belonged to Beth Goddard because she had left it on Thuyen, Thuyen, at the time, she had more than 15 years of existence, so she had seen it normally in the inventory. And so, one day, with a clear question, we opened the machine by Cooper and we left. With this machine by Cooper, we brought it here. A few days later, Charles received a letter from the director of the Mathematical Institute in New York. I am very happy to be here today to talk to you about the research that we have been working on for the last 10 years. We have been working on a lot of problems. Our research team had become a bit porous, but it was not always the case for everyone. So there were some problems that certainly affected the fact that we were able to come and install you in a place like this. I am very happy to be here today to talk to you about these things. So, in this last part, when we arrived in Amiens, we continued to work on the categories of the articles.

1:07:30 So, the last series of articles we did was a series on unique categories. Unique categories, as we have seen, are defined from double categories as an iteration. And so, there are four articles here. I would like to say that, essentially, we apply the different results that we had seen on the monoidal structures, for example, and the monoidal structures that we have on the categories of the SCC, well, we apply it in the context of the multi-categories. And then, a while ago, a person made a link between Charles and Jules Verne, and in fact, I looked a lot for links with Jules Verne, because this year we could have all the credits we wanted, if we could talk about Jules Verne, if we could have all the credits and give it to Jules Verne. A few days ago, I realized that there was a relationship because Jules Verne, who has lived in Amiens for the last few years, is buried in Amiens, buried in the Amatinsch cemetery. And the Amatinsch cemetery is a very particular cemetery. It is a large park, a magnificent park, where we also go for a walk. And so Charles liked it a lot. We walk a lot. During the period when we were in Amiens, we walked a lot. Charles always liked to walk. And then Amiens was remarkable for that. You walk for five minutes and you find yourself in the middle of the campaign, so since when we walked like that, we would sometimes go for a walk at the Amatel Cemetery. And in particular, I remember that when we were talking about the last article on multiple categories, we had a problem. We couldn't find exactly how to see this monolid structure on multiple categories, we had to do something, a kind of cult of cults, in fact it was quite complicated. We have no other idea. And one day, in the afternoon, we found ourselves going to sit on a bench right in front of Juppert's classroom, in the middle of the morning. And that's when we had the idea of how to do the hecatechumensis. Unfortunately, we didn't know at the time that it was going to happen in July 1970, we didn't know that three months later Charles Combré would be sick and that a year later he would be buried a few meters from the tomb of Jules.

1:10:00 So, he fell ill at the end of 1978, and essentially he had problems with his prostate, which degenerated into an insufficient kidney, and so he died in September 1979, the 24th of September 1979. But his illness continued for a long time, because there was even a time when he was a little better. In February 1979, we came to Manici for a meeting of the faculty's scientific council, who was the director, and then there was the thesis of Juanita, which was passed. So at that time he was already very sick, we could not go out, and I had been authorized to pass the thesis at you. So we had, and there was a thesis, so everyone came, about 40 people, I think, and so the thesis was supported by a lot of people. Recently, I have decided to forget these complex works. We have often talked with Charles about the publication that would like to resume these different articles that have been more or less answered, as we see in the journals. Some of them are well-known, others are less known, and he had also made a small collection that probably a certain number of you have seen, of folklore, of different geometries, I think it's called something like that, and so he had collected a certain number of these articles of different cultures, but he had obviously never seen anything like it. So I decided to study these different works. And as I realized that there were problems, since there had been denotations throughout the theory of categories, there were problems in the sense that these articles used denotations that had never been known, so they were really difficult to read. In addition, there were a few wonders, in particular the way in which he did not use from the beginning the notion of joint structure, so he found again a notion of structure linked to the work that was done initially, essentially with repetitive theory, so the joints, when there were joints, it was done in a really quite heavy way, and so I said to myself that to make them more understandable, also to make the link, because Charles, what counted was not the past, it was the future.

1:12:30 So, to make the link with that, I decided to read the comments. In the first part, which is about topology and differential geometry, something I didn't know much about, it wasn't something I had really lived with him, so I asked specialists, who are here today, to make articles on this issue. In all the others, in the three other parts, Since these were results that were either done in front of me or that you had explained to me in the first days of our meeting and you started to tell me everything that you had done recently as a work and so I didn't know all that very well. I made the comments myself by trying to first correct a few little things, there are always a few little things in mathematics, and then also by translating them into a language. All of these have been published as a supplement to the topology of differential geometry from the 80s to the 80s. And then, well, what still has been seen, it was the happiness that the University of Picardy decided to give its name to L'amphi, where we are today, and that was in the 90s, and there was an open solennel location of L'amphi, so at the end of the 80s. Shall I pre-analyze the theories of the Mubakki? Did Charles write the theories of the Mubakki? No, he has not written this. He has written some preliminary versions of the Mubakki on algebra.

1:15:00 But he was present in the discussion on structures in Bovaki and certainly he had been influenced and he had also been influenced under Bovaki by Henri Carton and Dubonnet in particular, but he was deceived by the way Bovaki had handled this problem of structure. It seems that it is only the volume of the results which appear on the theory of the self after a short time, and it is because there were very, very atymoeous discussions in Bobaki, some who wanted it to be given, others thought that it was not impressive, and so on, and I think that one of the reasons for which Charles Monbaki was exactly this that he has said to me. It is always the man who speaks the most time that takes the decision. And he was a very peaceful man and he did not like to speak at a high speech. So that is one other example. One question. What do you think of Monbaki? It's just to complete the answer here about the Bourbaki theory des ensembles. The Bourbaki theory des ensembles is, I claim, the worst... Book on set theory ever possible to write, so I'm sure that Ernst Mann would not have it. I will just say one thing about this book. There is a beautiful piece in this book about universal problems. That is nice. And that is due to Samuel. But as a book of set theory, it's impossible to dream of such a world, I say.

1:17:30 And in fact, you mentioned the theorem of Samuel, and you have yourself mentioned this theorem of Charles, and this theorem you have mentioned of Charles, which are the basis of the existence theory for three structures he has developed in this room. So, you see, there is a direct link from Samuel to Charles for you. What is the difference between Charles and Brodendieck? Because I think Brodendieck loves category very much. Or I have no connection. No connection. I think it's a different generation. Brodendieck was much younger and Charles was not in Paris. When he came to Paris, Brodendieck was not yet in Paris. These are real occasions to meet, so that we knew more or less some of the works of Grotendieck, but there were not physical connections between them. But it's more, I think, the fact that it was not in the same university in Paris, so that there were no real connections. I think it is probably, I regret that there has not been more connections because especially after having read the books of Roth and Link, I think there are many ideas in common, not only on the mathematical side. Thank you for your attention. This is a very good example to associate with the field of mathematics and which is due to Charles Westphal who is a very good example of the combinatorial theory. Yes, it's true, I haven't talked about it. And it's true that I went through the first few works very quickly.

1:20:00 So we can say that precisely these two works, more or less, on the combinatorial theory, or even on the calculations, the way in which he did the calculations of homology and homotopy in his head. By the way, there is a subdivision of a space with cells of a very particular type, which is the kind of thing that was then used in the theory of complex CWBs. Of course, it required to enter into the discovery, and we can recognize it without a doubt. I know less, so you can see that we do not have too many ideas. Precisely, these cells are now called Schubert's cells because they are associated with the Schubert cycle. In fact, I think that it is Erasmus who introduced the cell of Schumann. Yes, that's right. But there are a number of things that he introduced that then carried different names. It happened quite often, as there are things that carried the name of Erasmus. In particular, we often talked about Erasmus's grouperies. Now it's a subject of study. In the past, when we talked a lot about Erasmus's grouperies, we often introduced grouperies. Grouperies are big, it's just that grouperies had an interest in what they were doing. I noticed a very difference between the category theory as sort of exposed by Erisman and the American school of category theory. Do you think this comes from the background that Erisman and Charles Erisman had in analysis? Whereas the background in category theory from America comes from out and around their anthropology. The idea of category was first a group and, in fact, a generalization of groups, so that it's one small category, while in America, the idea of category has been to, as a universal thing, more of a large category and to look at problems of a completely different type. So, I think this is one of the reasons. In fact, for Charles, essentially, categories generalize groups, and he always thinks of categories as groups and then molloids with several units. So, I think that this way of looking at categories is very important in the sense that it is a direct link to the notion of structured categories and internal categories.

1:22:30 Which came in the states much later. While in the states, the first things which have been, and also for Potemkin, the first things which came were enriched categories. That is, categories on which there is not a structure on the set of all morphisms, but a structure on the whole set between two objects. And so I think it's really, but it's also possible that this is an hypothesis, pure hypothesis. That is the difference between the French way of looking at the world in general and the American way. The French are more individualistic and want to keep an individuality and so only one category rather than many structures and different functions. The connection with analysis, because it's interesting that Grotendieck came initially from functional analysis, and in analysis the idea of local and global was at the very standard, and Grotendieck took the local ideas in analysis and then transferred them to accurate numbers theory, where local meant at the prime. I was just asking, what do you feel about the influence of Nilekata and the initial training in analysis on Charles Ellison? I think that it's true that all the works on category as I've tried have been also influenced by Annalise, certainly. Not only because he himself was a geologist of my differential theory, but also because myself was initially an analyst. To apply some of the notions that Archer developed, for instance, in the species of local structures, in analysis. But generally, this notion was not exactly the good one that I wanted. So we tried to find something a little more general. For instance, Charles had introduced, after speeches of structure, the notion of a system of structure, where there are only partial compositions. And this was came because I needed this notion in the works on distribution theory, on distribution theory. I needed this notion. So that many of the works, there were many interrelations between the works.

1:25:00 Some kinds of work in analysis and the works on show for category theory and so on. I have given some ideas in this way in some of the comments. But can you give me a good explanation? Why American Pathology is more popular than Chalice? Well, I can give another explanation. Thank you. In the first explanation, Charles was considered as the very well-known differential geometer, so that when he began working on category theory, he had many links with geometers, but no links with the categories, with categorization, and in particular with American categorization. He did not know that term at all. So that is the words, and the words were written in French. At this time, the internet did not exist, and there were much difficulties to have printed papers, so that, for instance, we waited for a very long time before obtaining the scissors of Louis, for instance. Because it was given to those who knew that it was not known in this circle. There were longer mathematicians generally and more who came generally from algebraic topology in which they had not worked for a long time. So this is one reason. The second reason is the fact that he used notation and in part because of rigour, in particular the category in his first work was written C with small dots for the composition law. And also the composition, he wrote the composition aqua law in the other way. So that these notations were in French, plus notations which were not known, were difficult to read by people who did not know him and who had not real reasons. So that the first thing which was really well distributed was the book Categories et Structures, which appeared only in 1965.

1:27:30 And this book was not very well received because of the notations, and more and more working... So, inside a research team where many research students were just new category theory only by what Charles did, there were some problems for me. So, I think that is another reason for which I have added English comments in there. I would like to state that I think this distinction between American category theory and French This narrows the scope of what actually is in the subject because, for instance, the whole school of chromody does not fit into this distinction. Chromody works with large categories as well as with small, ample dialectics between them. So this alternative category theory in the style of Ehlersmann and American category theory does not exhaust what is in our subject in the early days. No, probably not. But I think the distinction is between the work done by Charles and the people around him and the American... The original definition that is in the text of McLean, the first definition, is exactly the same as that of Heisman. Yes, it's true. It's exactly the same. It's only after that. But after... The writing with the screen, etc. But the first definition is the same as that of Heisman. Yes, exactly. So it's even historically inexact, the definition of Heisman, that it is the original definition. It's true, it's true. But for the fact of insisting that there is a law, the definition of the origin was the data of a set with a law of construction, there was no law but it was not indicated, that is to say, I think that the geometric way of seeing categories as laws with an additional property, I still think that this way of seeing things is wrong. There is one question about it, why this was more popular than the way the language looks like, American or European, that is a very different language.

1:30:00 The expressive power of categories with objects and powers is easily separated. It is easier to use as language, it is more expressive, and thus it became popular because it was, for many people, it was easier for them to see that there were categories around, just as language, while every man was a category as a concept, as a category that serves an object. And these have not expressed the power of language, but the others we can talk about. Yes, of course. I would like to point out a difference which is not in language, but in the view of what can be done with categories. The difference is the following. And home sets. You are immediately ready to define and reach categories, which I did, but that's up to you to do, okay, that's a science question. So if you have this viewpoint, you go immediately to enrich category theory, you place home set five groups of points. It's also interesting when you don't have them, and thinking in terms of generalization, because it's the other one which permits to define internal categories in a categorical feedback. It's the other aspect. You may have objects, etc., but you don't have objects, whatever they are. So, while on one side you have enriched category theory, objects and objects, you replace the objects with what you want. On the other hand, you have internal character theory. We don't have any kind of generalization concept, but it's the global practice. So both are meaningful, distinct, and important.

1:32:30 Can I just make a small rhetorical point about groupoids and categories? Because I remember Whitehead saying he was very impressed in the Islander's claim data by the axioms of a category. And Cockroft told me, who was in Chicago in the 1950s, that Adam Mervyn MacLean was strongly influenced by the notion of a croupoid. Well, this is what Cockroft said. And in fact, the notion of a croupoid was well known to the Chicago school. For example, it occurs in Howard's book on the structure of categories, structure of algorithms. Because it's used in the notion of orders, which is very fast to do with astro-structural algorithms. Now, the curious point that I can say is that, A, the notion of group-void does not occur in Allenburg's McLean paper. Surely McLean must have known the notion of group-void because he was in the algebra school in Chicago. Were they influenced by the notion of groupoids? And he categorically denied it, because he said, if we hadn't thought of it, we wouldn't put it in as an example. Probably, I learned certainly well the notion of the roboid since you have participated to some of Bobati's reunions and the notion of the roboid is an exercise at the end of one of the chapters of Algerois of Bobati. I think that probably Charles had the first heard of Bokoits through Bourbaki when this exercise was put in the book Algebra of Bourbaki. The question is that they did not associate the notion of Bokoits as an example of a category. It's very curious. Ah yes, very curious. Of course, the people know that the notion of groupoid is part of the, should be seen, as from its definition by Brandt, as part of the legacy of Gauss, because it arose for Brandt the describing composition of quaternary quadratic forms, as against binary quadratic forms, which were K-Arbedian groups in Gauss's Discursiones.

1:35:00 I have a question about this theory of species. What are the geometrical examples leading to introducing the theory of species in structure? Essentially, the idea of the form of Hilbert space first, which corresponds to the topological species of structure, and also the notion of structure, to have a good notion of structure, has to be defined by the particle. The word species of structures came from the fact that it is a formalization in a language which are thought better of what tried to do Bobati when he spoke of mathematical structures. But they have a legitimate need in order to generalize hydroponics. Yes, to generalize hydroponics. I think that it is one of the reasons. The two reasons are the notion of local structure which came to give reality. Categorical framework to the notion of structure of mobility and the other reason for the theory of hydrogen. I think we can move on to the next speaker, Jean-Bénard Jaune. Jean-Bénard Jaune. Jean-Bénard Jaune. Jean-Bénard Jaune.

1:37:30 Both to French speaking, but also to English speaking. I think I'm totally capable to give a talk in English in mathematics. I'm going to talk about mathematics, but not mathematics. And that's much more difficult. I speak in another language. Fortunately, the beautiful talk that André Erzmann gave us made things easier for me. So, I have the sad privilege of being one of the former students of Erzmann, and I think the oldest from his parents' field. In Strasbourg, he had Reb and others. But in Paris, the first, I think, was me, working with Erasmus. As I mentioned in my abstract, nothing about mathematics. I was trying to find if I would eventually become a mathematician, but the only thing, very few things I had read, a few books, of Bourbatou. If that had been mathematics, I would not have become a mathematician. Fortunately, Erasmus had just been appointed as a professor in Paris, and I was wandering, going from one course to another, trying to find... ...convinced, rigorous, this kind of...

1:40:00 Everything was deep, clear, working. Essentially, in differential geometry, and for other things I want to say, or seminars, or the very few short notes he wrote, an incredible amount of mathematical little, I thought I knew nothing, and asked me to make a review of differential geometry, I won't quote the name, and I told him, come on, I'm a beginner. They were able to report on such an important book. And they told me, come on, everything that is in this book is contained in the four pages of the Compendium Note, which you have studied, I must say. It was true. So that is how it was.

1:42:30 So clear, so I started working in these domains and then for me, Haynes, what really was the term. He began talking about local structures. He invented what has now locals. Everything I was doing before, theory, etc. which came later. It was, for me, and I had never believed, and I still do not believe, that space, eternal time, and I still don't believe that, creation of points, the real line, and here came, for me, the possibility to evacuate and yet have some notions of continuity, etc., etc. More for me personally than the mathematical aspect, the reason which made me forget there was also a couple of the same attacks for locales at three o'clock in the morning saying, you know what we talked about? Counter-example. And then I called back after at 3.30 and say, your counter-example is not true, etc. That tells you how much I'm excited. It turns out that we've been through it.

1:45:00 There is also a theorem which says that even if one is interested in more general topos and topologies, Locke asked this sufficiently to give information, but this came much later. This was the beginning of my work with Kevin Mann. Although he was such an important mathematician and he had a young student called Jean Benabou, he never asserted his superiority or anything like that, he was always willing to listen to me and give me opportunity to talk, to explain why there was such, later, much later, interest to explain.

1:47:30 I will explain maybe some of the things we discussed here, because I lived this period, came from, came to category theory, differential geometry, fiber bundles, foliations, etc., etc., and in that he was years in advance compared to other people who worked in the domain. He created, so he would use, he created the notion, he would name them. He created many new things in categories that continued adopting the same attitude. He never what other people in category theory were doing. This had been justified in differential geometry, in fiber bundles, etc., etc. It was not completely justified in category theory. Even if he brought important ideas, other people had been working in this domain much earlier than he did. Therefore, to the notion of adjoint factors, which existed a long time ago. He invented not the general joint functions, but their specialties, namely what we call reflections or coreflections, invented these things, not knowing, in some cases, something which had been done by Kant earlier.

1:50:00 And so there was this discussion about American category theory and French category theory, In category theory, people who work in category theory accept Mann and the people who work with Erisman, and this could not continue without conflict, and the conflict I am scandalized by the attitude towards not only his direct students in category theory, but something which happened to me, Jean Benabou, not to Erisman, and much later. That was early 70s, a professor in Lille. And there was a job, there was a new university, which was created near Paris, and there were jobs in mathematics. And I applied for one of these jobs. Since it was a new university, there was no faculty members to decide, so they gathered mathematicians from all the universities in Paris to decide who will be appointed. Application was Benabou, who works in category theory.

1:52:30 I want to talk about me, I'm talking about the atmosphere in France, and maybe some of the students of Erisman and Madame Erisman will understand what I'm talking and why I say these things. Six months later, at the same university, with the same board of persons choosing application, there was another job, and I applied. During the six months, it took me a few weeks, three pages, with just Google, it took me a few weeks, it was just for me a very simple application of Beck's criterion of triperability and things like that. I had been working in category theory for 15 years and I had done, I think, substantial work, and I had been rejected the first time. Because in this note we generalize the theorem on descent or scrutiny, the same person's report was Benavu works in category theory, but his categories seem to be useful, and I got the job. This is, you have to be aware of the situation. You have to be aware of... And really, not talking about Erasmus, but about the progeniture of Erasmus, about his students. I was one of them, but even the younger one. There was, for a long, long time, this kind of hostility towards, but more so, had been influenced more or less by Erasmus' work. I mentioned the locales, but the sketches, but many other things. It's good that his former students have been able to learn both languages and to express themselves in both languages,

1:55:00 because otherwise they would be dead now, dead as mathematicians. I think some of them will agree. Was an important mathematician and also a very proud person. He knew he was an important man. He could not accept the students, but certainly internally he could not accept trying to study both types of category theory, which I think there is one. I was convinced that I had betrayed his school to become a prophet. I had never met him. I didn't care about him. To Charles Erisman, a great mathematician, I must say, but also, as I said, to his progenitor, and I'm glad that this meeting takes place and that with the help of André Erisman, the work he did to explain, publish his papers, but also give translations, which is most usually adopted by him.

1:57:30 People who work in category theory, whether they are French or American or English or Australian, that doesn't matter, that is what is adapted. She did this work and I think that in differential geometry, on fiber, bundles, on foliations, etc., etc., acknowledged essential. His work on category theory was just not known. Thanks to André Erisman, people will realize that even in category theory, he brought very important ideas. And I would like to make just a few observations. First, Gluckendieck used category theory a lot in his work. And he himself had a thousand in that. Among the academic directors, there are different standards. And some of them did not like the work of Rotenberg because he was using category theory and abstractions too much. Another remark that I would like to make is that the Russian school of mathematics, we all know that the Russian school of mathematics is very strong. And they have no objection to using category theory. They are completely free when they think that a notion of category theory or theorem is useful. They just use it and they produce marvelous mathematics. My own opinion is that on the long term, category theory will be completely accepted because it will have more and more applications and it will be more and more useful and it will become completely accepted by the mathematical community. I agree with this. There is even a danger, which I am very much aware of. Again, I am talking about France, not others.

2:00:00 Because category theory in France was for so long considered as very superficial, as a proof, it is that there are in French universities, very few universities where courses of category theory are given. Maybe you can maybe mention two, I don't know. I have been a professor at the university for 35 years. I've never had the opportunity to teach category theory in whichever university I was. So this is what the situation was. And I mentioned the danger, and the danger is what? Now people are beginning to realize that category theory can be a wonderful tool. For many things, and in this meeting you will see all these things, in some circles, is that people who have never worked in category theory rush, they want to write, they know nothing, they write very superficially, and the danger is that these people will, in a sense, make the law in category theory. I hope they won't. Okay, so I'm glad that you confirm that this situation I'm depicting in France is not only a French situation. It is a pity to see some people who for ages have despised category theory, who know nothing or very little, There are many papers about category theory, or even the history of category theory, which they know nothing about, and it's to us, categorists, the real ones, genuine ones, to try to stop this...

2:02:30 Concerning category theory, Borelli wrote about his years and he sought to say that Rottenberg had presented the first plan to develop a category theory and then Burbanking rejected this Rottenberg proposal and then Rottenberg left Burbanking's more or less. So, in a sense, Burbanking had decided at one point, after a lot of reflection, for instance, that he had spent a few years thinking about, decided that Catherine's theory was not going to be developed by then.