Charles Ehresmann & category theory

0:00 I'm not going to talk about the technical aspects, I'm going to talk about the relationship between the two, so I'm not going to develop much on the subject, nor will I be able to talk about the effects either by the animal, or by the intelligence, or by the intelligence of the human being afterwards on the subject. I'm going to focus on the period of 1957-1969, and then I'm not going to go into too much detail on the history of mathematics, on the relations with other mathematicians or other geologists at the same time. No, what I'm interested in for today, and is very special, is what is it that, in Reisman's case, has motivated the return of mathematicians. That's it, that's what I wanted to say. And of course, his past, geometry, which is something else, and then, how did he process, what kind of categories does he have? I'm not going to go into all the categories, if everyone agrees, it's a little different, we'll talk about that later. If we wanted to do a work of history on the emergence, on the continuation of the TV of the categories between 57 and 67, we would have to do something else. What I'm giving here is one piece of the story, but... Of course, to put the same for the continuation of the history of quantum physics, there would be now a classic theory that this one stops and continues.

2:30 To give only one example, we can now introduce with Riemann the stage and then the decision-making, the zoological. We are going to talk about mathematics, but we are not going to talk about the direction in which we are going to talk about it. So, I'm going to talk about mathematics, but we are going to talk about something a little bit... I'm going to try to serve you a little bit in this room, to see what's going on. So, we're going to start the lecture. So, I gave as a title... I would like to say that there will be two levels of mathematical concepts, a fairly general concept of what we do in mathematics, which is very important, because there are more technical, more detailed concepts, on the differential geometry side in particular, and then we will do both in order, as I just said, first in general. So here is the approximate plan. Thank you for your attention. So, more than the subjects, well, then the elements of geography concerning Erasmus and the constellation of the stars, the constellation of the stars, these are very important things. Then, I would like to make a small comment on the expression of the book, which is in the process of being structured. Because in dance, you find interesting things the more you look at it.

5:00 If you want to do it with the categories, it's a project. If you want to do it with the categories, it's a project for the future. This will give us an idea of the point of passage between the differential geometry and the category. This point of passage, we will see it here in the background, but a little further. I will talk about the organization of the work. How does the work of Erasmus disperse? And how does it intersect in the considered area with the first works of André Erasmus? This is very important in my subject today because it is decisive in the passage. Simply, this is where I will necessarily make a small presentation of the categories and sub-categories for those of you who do not know, because then it will be less interesting for the rest. But of course, this is not a story, this is not in the history book. Then I will talk about the last article of Edmund Mann on the subject of the ancient literature, not his last article, but the article on the ancient literature, the article of 1971, which includes the categories of the ancient literature. In three or four extremely simple pages, he says directly why, for the question of the fundamentals of mathematical physics, he was interested in the categories. I'm not going to talk a little bit about that. I don't have a kind of first answer to that. And then, I will try to answer a little bit more on various mathematical details about the future. These five points are not completely independent, and since we have the time, these five points actualize the passage of Erasmus between the two disciplines, in fact, to the left and the categories above, so I will try to explain. I would like to show you, through these pictures, what it means that the passage of the space between the homophobes, the local structures, to the effects of the structures that have been governed, the extensions of the effects of the spaces to which they have been governed, the structures in general, are not governed. These are the passages of the space.

7:30 I would like to conclude by saying that it is possible that in the audience there are people who do not know, who do not know anything about the left. In 1934, Léonard de Saint-Saëns made a theory on topology and algebra, mainly on the topic of topology and algebra of space-time. He calculated the Huber theory, which is still used today, in 1934. After that, we will take a short break. I don't want to go into too much detail, but on the other hand, there is a lot of work going on in the first half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the second half of the Well, that's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, it's not the point, All of these points are very important. So, I'm going to talk about the results. I'm going to start with the results. I'm going to go through the results. I'm going to go through the results. I'm going to go through the results. I'm going to go through the results. I'm going to go through the results. I'm going to go through the results. I'm going to go through the results.

10:00 This is a notion that does not exist in the past, not in the present, not in the past, in the past. The concept of quantum mechanics is also related to the concept of quantum mechanics in the context of the method of Rotterdam. But Resmanach managed to clear the gap between the two concepts in particular and to compare it with a possible example of the difference between the two. The gesture of the quantum mechanics is an intrinsic fiction that is fundamentally radical in its effects. We will say that the two books work a little together, it is his version of the books, it is his version, but more or less, in the history book, it develops at the end of the 40s and the end of the 50s, so we can say in parallel or concurrently with the other books of Lorette. And, of course, there are many others. So, in short, these are the terms used in Coleman. The answer to this question, as I said earlier, is the following. There will be two ways in which we will address the question of structure. In more general terms, all of them. So, after that, there are the category notions. Category domine, category double. Category differential, of the artist of 1973, which we discussed earlier. Thank you for your attention and see you in the next lecture.

12:30 I made some very detailed comments on mathematics, as well as some historical comments. I don't really care about historical comments that relate to the history of mathematics. I mean the comments on the history of mathematics, and then the comments on observation, and so on. Then, there are a number of papers that I will show you in a moment, in the future, in a very concrete way. There is one more thing I would like to mention, which is the notebook, sorry, the notebook of the students, I think, in the schools of mathematics, in the schools of mathematics, but not in the school of mathematics. Yes, it is in the books. Yes, it is in the books. Yes, it is in the books. Yes, it is in the books. Yes, it is in the books. Yes, it is in the books. Yes, it is in the books. So now we're going to try to go back a little bit to our page. So the problem is at the top and to see how it is released, how it is more or less imposed on the students. We're going to do it very quickly because I think I could come back to this if I wanted to. I'm not a scientist, I'm a scientist, I'm a scientist. I'm not a scientist, I'm not a scientist, I'm not a scientist. I'm not a scientist, I'm not a scientist. I'm not a scientist, I'm not a scientist. I'm not a scientist, I'm not a scientist. I'm not a scientist, I'm not a scientist. I'm not a scientist, I'm not a scientist. I'm not a scientist, I'm not a scientist. I'm not a scientist, I'm not a scientist. I'm not a scientist, I'm not a scientist. There are works of the same type in this volume, and note that the categorical works, which are interesting to look at, begin with the second volume, in fact. It is the structure of the calve, but in the structure of the calve, there are some parts that are already categorical. So, from the point of view of quantity, there is a lot of work of nature.

15:00 And, once again, I'm going to give you a very close-up of the two, a very close-up of the two. So, historical references of the two natures. So, in the works of Erasmus, I didn't use only the articles that I had read as I wrote them. There are articles that I found fascinating in the 60s and 70s. In addition, there are two articles on mathematics, one on mathematics and the other on mathematics, which is not in my work, which is a bit the point of what he did in 1955. And then, this article, which I mentioned earlier, which I remember from the conference on mathematics, which took place in 1973-1975. And the extraordinary question that was asked to the lecturers, to the lecturers, and then the articles of mathematics and physics. I wanted to share with you the four works that will be published in English this week, not only the works that will be in the material period, but which will have a major importance, but it is an important one, on this subject. And we found in these works different motivations from those of the different fields. Motivations that, obviously, Erasmus knows, since, with his work, they are among those who work permanently together, always in different works, but in exchange for their works. So the motivations for the papers that we have now are particularly important. There are also historians. These are historical articles. Edmund himself wrote the most important volume, the Comandes, in the works of Madame Azraëlle and François de Saint-Exupéry.

17:30 Then there are other articles by other people. This is where I forgot to give you the key. Otherwise, there are all these articles that have been written more or less at the time of the Collègue des Etats, that is to say around 1980. There are also other articles in the colloques. In the colloques, I only remember the articles on mathematics and mathematics of the colloques. There were two colloques in particular, a colloquium held in Warsaw in 2005 and a media in 2007, with articles of a small number only on the geophysical side of mathematics. In particular, there is a very interesting article on the subject, a very interesting article on the subject, a very interesting article on the subject, a very interesting article on the subject, a very interesting article on the subject, a very interesting article on the subject, a very interesting article on the subject, There is also our colloquium, the colloquium of the 5 Arameans, our colloquium of the 2 Homo sapiens, which is the anniversary of the animal race, which was established by the animal race Arameans. For the material, for the historian, there is already all that to put on the table to examine these two subjects. The elements of geography, very quickly, I will not go through them because they are very small, but there are two parts. So, Gérard Saint-Denis, born in 1865, died in 1864. He was born, normally, in the part of the Atheist, rather after the abrogation of the Atheist, he was a rabbi of the Atheist, he was a rabbi. Thank you for watching this video.

20:00 Here, it was a seminar that had a very important importance, because it was really, really important. It's not only about the physics of mathematics, it's about all the people who went through this seminar, who were more or less educated, who were talking about what was going on in their homes. They started with the physics of mathematics, and then they went on to mathematics, and then they went on to mathematics, and then they started there, and that's it. So it was really a great experience. Well, in 1955, it was the middle of the 19th century, and then he traveled a lot. There are a lot of people here. Here, for example, the little one, in the middle, there are people like that. Next to it, here, there is a group of mathematicians and scientists. The little one in the back is not in full form. The young one is with the big ones, the old one. Here, on the right, you have René de Couperet. And then, this one is with the young ones. So, you can see a little bit of this area. You have René de Cotelle at the bottom, and I think those of you who know him well know that he was there in 1924, so he must have been there for a long time.

22:30 So I'm going to take this picture, there's another one, but I'm not going to show you the other one. And here we are in 1958, so we are in the period preceding the beginning of research, the initiation of research in 1954, when we went to school, and then in 1958 we are going to start now this subject, a little more than this one. I would like to thank Mr. Kellas and Mr. Reisman for inviting me to this lecture. Thank you for your attention. There are several master's degrees in mathematics. This is the definitive coverage. We have, for example, the lecture of Jean-Bézabou, the author of this coverage. At the beginning, it was a seminar and there was a cache. I start this especially because in the world, it is interesting, the detail that is at the bottom right, These are the subjects we are going to talk about very quickly. We are going to talk about the mathematics of the work. Maths, physics, mathematics and mathematics. These are the subjects I would like to talk about. The subjects we are going to talk about are the mathematics of the work. We are going to talk about the mathematics of the work. The subjects we are going to talk about are the mathematics of the work. The subjects we are going to talk about are the mathematics of the work. The subjects we are going to talk about are the mathematics of the work. The subjects we are going to talk about are the mathematics of the work.

25:00 The subjects we are going to talk about are the mathematics of the work. The subjects we are going to talk about are the mathematics of the work. And so on and so forth. So there's a whole bunch of things. For instance, there's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. There's a book called Polytheism and Mathematics. And so on and so forth. In any case, after, of course, there was the generation of the 20th century, especially in the 20th century, the generation of the 20th century. So, I'm not talking about the generation of the 20th century, I'm talking about the generation of the students, not because the tests of the state have been done later. So, after the test of the state, in the sense of the test of the state, there are two categories. The first category is mathematics, and the second category is mathematics. The first category is mathematics, and then there is the third category, and then there is the third category. But there was also the test of practical mathematics, which was at the time of Robbenaert, who had left the subject at that time, and the test of Pascal Thauvin, which was a relatively important test, which is now adopted, which is a very interesting test. You can also measure the importance of this test. There will be an exposition of Pierre Cartier on eucopoids and angiopoids and on the resolution of the third problem of Lie, on the eucopoids and angiopoids. So Cartier is going to do an exposition on that.

27:30 Then, there was the university thesis, and then the 3D thesis. And finally, there is this object. This object is the categorical script. It is the object where we will be in opposition to the theoretical analysis. Earlier, there was a time when the emblem of the theoretical analysis was gold, and there was another time when the emblem of the theoretical analysis was silver. So, in the future category, the book of 1965, we will start with the 3D thesis. I think it's Hubertus, isn't it? It is Hubertus. It is Hubertus. So, the key category of the lecture is the subject of criticism. It is not the subject of a reference to the general and epistemological direction that remains manifest in the lecture. It is a poem with lines that are sometimes a little weak, marked by the end and the beginning. And for the rest, it says something about this idea of structuralism, and precisely this idea that Erasmus brings to us from the research of the structure of our thing, as he says in my preamble. So here we must stop at this question of the research of the structure, because we have an alternative built on the behavior of the work of a mathematical scientist. For example, I think that there was a possibility at some point in this period of time, but that certainly it is on the basis of other intellectual institutions, on the basis of a great difference between the construction of mathematics and there may have been an widening and the fact that there is a maximum of equities between mathematics. Because the alternative to this is either to consider mathematics. It is the resolution of the big problems that are linked to the solution or we consider that it is the discovery of the structure of everything to come.

30:00 We say, strangely, we have a conception that we have to return to the paper, attached to the truth. It is the description of the truth of the paper and the other which is the opening on new problems. There is an article on mathematics, a very interesting one, from Ronald Conybro. The subject of this talk is a quote by Rossa, who has just written the book. Rossa says that we often hear questions like this. What can we demonstrate with differential forms and who asks them? Is there something that we demonstrate with them that we cannot do with them? It seems that no. Everything that we demonstrate with differential forms, we can demonstrate with differential forms. I have heard this question very often, and I can tell you that when you hear this question, you are certainly talking about something important, because it is indeed an indication of the fact that we are not turned in the direction of science, but we are in the opening of a new field. We know how to open a new field for the rest of the past century. So I think that Erzman is completely on the side of opening a new field. And in addition, he could do it with all the honors, because obviously, in his youth period, he had to solve very difficult problems. The German theorems are the ones that you see, they are the ones that you see, they are the works that I did at the beginning for the Parthians. They are not generalities, they are not hollows, they are only problems. But the alternative to solving these problems is to open a new field. At Rezman, I think we hear it a little in this book, because there is still the weight attached, if you will, to history, which is precisely the bad thing of De Kietze, who is still there. He has a sort of integrity, but at the same time, there is the side of Nouvelle-France-Enfant, of the French metamorphoses, of Nouvelle-France-Enfant.

32:30 In fact, in the categories, you can find, we will say in what way, because this categorical management is the same for him, it is certainly not the same for others, but the question of structure in general, of course, it is a little bit in the introduction of the same book, so I would leave you the 1, 2, 3, 4 things that are there, so you see that a program of the book, simply the last alineation, is the program of the following, which is written, but on which there are articles. If you want to develop a homology, you have to do it in a way that is based on that. How can an article be homologated in a way that is based on that? This is for later, but already in the book, we intend to put the ideas, to embody the ideas that we have. They have been there for a long time. You can't say you don't know them. They are simply mathematicians. You already answered them. But the point of view of construction is the structure. How much is there of the structure of construction? How to define a general faction of a structure? This is a problem. You don't answer them at all. You don't answer them in a very classic way. You answer them in a very semantic way by talking about the aspects of the structure. But the classic answer will come later

35:00 with the construction of the system. And then the other point of view is algebraic. We forget how the structures were constructed, we try to construct them, but we use what we have in mind. What we call, by the way, the categoristic point of view. But to deconstruct or not? So then, they make the mark, and so in the common brackets, around the mark of the situation. The only consideration we have in this paper is that it allows us to define the substructures and the propositions when we define a cycle of properties that are not the same, or that are not the same, or that are not the same, or that are not the same, or that are not the same, or that are not the same, or that are not the same, or that are not the same, or that are not the same, or that are not the same, or that are not the same, or that are not the same, After that, it's not just about monos, it's about certain monos, those that have certain properties in relation to a particular type of object. And in fact, it's still very interesting to see things from the perspective of physics. A little more finesse, because when we think about it, we don't necessarily consider the monos. When we consider the monos, for example, for the varieties, we consider them as a mixture of the two. It's about having a good notion of what a conscious mind is in relation to a variety. There is also a series of lectures on quantum mechanics and quantum mechanics of quantum mechanics of quantum mechanics of quantum mechanics of quantum mechanics of quantum mechanics of quantum The book was published in 1965 and ended in 1934. So, at this point, the conversion I was talking about has already been done. It's a kind of analysis of what the categories are and what the effects are. The book was written in 1965, but the book was already published in 1961-1962. In fact, initially, the book was written from the writing of these lectures of the years 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300, 300

37:30 This is a part of the introduction from the beginning to the end. The philosophers, in the beginning, came from... It's the following of the first and the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. It's a part of the introduction from the beginning to the end. On the other hand, there are a lot of people who are very interested in mathematics, even in science, and in the field of humanities. So, I would like to invite André to talk about this topic. I would like to introduce him to a group of people who are very interested in mathematics. And then, I would like to talk about the subject of mathematics. Mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, mathematicians, I'm going to give you a clue. If you had used a colloquium in the colloquium in 2005, now we're going to get to the bottom of it. So, in purple, basically, it's the category of mathematics.

40:00 In blue, it's the period of transition. And then, there are a few statistics. So, what interests me is the distinction between purple and blue. So, these are mathematics books. So, obviously, we're going to see a little bit in detail of the different elements that are at the bottom of the purple area. You can see how they stick to each other due to the small points we talked about in the introduction. But this, as I said, is in the interior, considered from the 50s, but especially from the 32nd century, we can say from the 62nd century, when André-Charles Allemagne talked so much about mathematics in the laboratory that everyone was talking about the work of the other. Well, the two were talking about the work of the other, but they were talking about the work of the other independently, since it is not a subject matter, it is in general. There are a lot of different things, but the interesting thing is that the two sources are going to be a motivation for both of them, for each category, which is going to be the same. So, that's the development with, in particular, which is going to be the main one, the situations, the genes, the varieties expected, and, in this case, the term. In the following slides, I will show you another diagram that I made in the past. This is a diagram where we see the works of the radicals. So, of course, we will not go into details. But the first works are the colliers, the mathematics and the mathematics. Then, the works on the cathedrals, the mathematics and the distribution of mathematics. All of these things are linked to things that are not in the same place, in the same place, in the same place, in the same place, in the same place, in the same place, in the same place, in the same place, in the same place, in the same place, in the same place, But then, what interests me, at the moment, is the so-called epithelial zone, that is to say, the polyhedron in the middle, which is very different from the polyhedron in the middle, and the distributions in the middle.

42:30 So, all of this, the first part is the magnetic field, and the second is the expansion field, with its values. And it's there in 62. In 62, it's the same. But, what you have to see is that, to do this, Mathematical-analytical work, in addition to mathematics, is the engine of a lot of things that come into being. So, a little later, it is the deep motivation for the research of quantum mathematics in general, which, of course, are already at hand in the scientific field, even though it is not the theory of quantum mathematics. It is the theory of quantum physics. It is the theory of the law of law that commands it. But there, really, there was already a need to make a good scientific theory. So that's really... What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? So, yes, as I was saying, my thesis is about the theme of the two, of the two, of the two, of the two, of the two, of the two, of the two, of the two, of the two, of the two, of the two, of the two, of the two, of the two, of the two, Thank you for your attention.

45:00 The notion of internal categories as a category, and then, simply, the category we take as an internal category, there is a counter of oblivion towards that category, because this example was given by the category of mathematics, which is in charge of oblivion towards topology, or oblivion towards children. Each time, there was a counter that was there, and in the phase of questioning all these things, we developed it. Well, as I said earlier, there were also questions about the fabrication of sub-objects, of cosciences, of color, of the scientific symbol, but on the other hand, when it comes to the categories of subjects, it is the internal categories, in the category of bones, and it is not going to do anything. The concept of the subject is not used to define it, but it is used for sub-objects. For example, when we look at the differentiable categories. If we just say that this is an internal category in the category of differential variables, it doesn't work. It doesn't work because we have to specify the two. For example, what we need is that the two applications for CB be transverse. So one way to do it is to say, for example, that one of the two is a submersion. And that's when it's categorical in the category of the two levels. That's what we're trying to do. Or it plays a role, it's a compression, it brings together these categories. Well, the idea of a lecture in a fresh way does not mean to say that it is. That's why there is a contorté who intervenes, but since Jean-Emile, indeed, I do not have to say that, as soon as there is a new term, they are introduced in the scientific field, but there is also an introduction in the space in terms of monolism, monolism in the space, which is a bit boring. In fact, it seems to me. In fact, the concept of structural structure came about precisely because of the categories of the structures. Initially, the first notion that came to mind was the categories that were used in 1962. So, simply, there were structural structures, and then there was, at the time, there was a... There is a structure, but there is no real notion of the product for the science of applications rather than for the case.

47:30 That's it. And to get a good notion of a structure in relation to a conspiracy, the main point of the thesis is that a conspiracy is a topological generation. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. There are two main areas of interest in the lecture, the first one is geometry, and the second one is mathematics, and the third one is quantum mechanics, and the fourth one is quantum mechanics, and the fifth one is quantum mechanics, and the sixth one is quantum mechanics. I'm going to talk about the problems that arise when it comes to establishing the foundations, when it is necessary to establish these foundations, that he will consider to take care of his students. He will do other things for the students, but for his students. So, I'm going to take up the elements in a more uniform way. I'm going to take up the elements in a more uniform way. I'm going to take up the elements in a more uniform way. If it were to be articulated with this category, it would be to articulate it in the form of an algebraic formula and then in the form of an algebraic formula and then in the form of an algebraic formula and then in the form of an algebraic formula and then in the form of an algebraic formula and then in the form of an algebraic formula.

50:00 All of these terms are not categorized in the French language. Passage des feuilletages. Passage du gros ou de la typologie. La conduite. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language. The terms of gauche are not categorized in the French language Well then, as promised, I will make a brief explanation of the categories and the definitions of categories and categories for those who are interested in mathematics and mathematicians, for those who are interested in mathematics and mathematicians, for those who are interested in mathematics and mathematicians, for those who are interested in mathematics and mathematicians, for those who are interested in mathematics and mathematicians, for those who are interested in mathematics and mathematicians, for those who are interested in mathematics and mathematicians. These are the different aspects of the subject. The sociability... I'm going a little faster because I'm on a resume. I don't know if you can hear me. I can't hear you. There you go. So, I'll tell you a little differently now. That is, we start to organize things by layers. There is a first ensemble C0, which is the ensemble of objects. There is a second ensemble C1, which is the ensemble in which the elements are arrows. What I put in green there is an element. And at the second level, the elements... These are the systems of two consecutive arrows, and at the third level, the systems of three consecutive arrows. So, what I mentioned just before, which was the partial composition with the sensations, which was there, that, that organizes itself by following the objects, the arrows, the double arrows,

52:30 which are called two arrows, three arrows, etc. And so, we spread out, suddenly, here, there are certain operations like this, for example, You have a set of propositions. What you have is a set of propositions composed by functions. These are 0, 1, 2, 3, 4, and so on. And you materialize them by functions, like the red one here. The red one is the same as the green one. The red one is the same as the green one. And the other red one is the same as the green one. And here you also have the sources and values. This is a way of distributing the data. These are the key terms that lead us to the idea of category theory in Theatres, or decision of category theory in Theatres. Here is the decision of category theory in Theatres, and here is how it is constructed. What I saw in C0, C1, and C2, to be precise, is that the children in Theatres, as I said in the library, are points in space, and what I get is the arrows between them. And this diagram of four groups with the applications that are here. And, of course, there are relations that translate actions into the best of both worlds. That's what I'm going to explain with the technical facts. There are relations, the diagrammatic translation of the technical and statistical facts, and then there is the description of C2 in relation to C1. Because C2 is divided into two consecutive facts. And that's going to be translated by the fact that there are diagrams in this category that I've put up here. There are diagrams that need to be implemented. I hope that once implemented, it will be possible. So, it's an idea that is quite simple. You have a diagram, like this one, of the arrows. Specific diagrams that need to be implemented, as I said, as limits. But you see here, what we're doing is that we're putting together a list of categories.

55:00 And a model in a category that knows which ones we want. It consists of the realization of an R function, and here is the realization of an L function, in addition to the definition of an R function, that is to say that in fact, technically, it would be a domain of the R. We do not require that we use the properties present in the scheme, but we will have the characteristics of a variable. So, when we have the definition of an R function, it is a little bit like the left and the right, but it is not this possibility that I invented. I don't know, I don't know neither for mathematics nor for mathematics, I don't know how they came up with this formula, it's interesting, but today we can count the appearances of this formula to evaluate it carefully. Well, we have the categories, it's very interesting because that's the category, if the category, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it, what is it. Well, I'm going to be able to make inter-categories, where c is equal to cohomology, for the variables that I put there, in relation to a cohomological category, the categories that I put there. And the point I want to make is the category of the categories. If I make an internal category of the categories, we have to make a double category. So this double category is a point that has a very interesting activity in the interior of the system. To define the two categories, what we need is not arrows, but blocks. What I put at the top right is the basic element of a category or a block. It's a little decorated so that you can understand how it's going to work. So you can see that this block has a source on the right, a horizontal source, I wrote, and a bit of the horizontal.

57:30 And then it has a vertical source and a bit of the vertical. But these sources are themselves seen as degenerate blocks. All of this is a kind of engineering, because it is simply called a core. And then, when the blocks are composed, either horizontally or vertically, there are therefore two ways of composing the blocks, one at the bottom and one at the top. And then there is a termination way, which is exactly the same way as in the case of the two categories, the way of the glasses, if there are four of them, as I said, they are composed first of the two horizontally and then electrically separately. First of all, the verticals and the results, how they were processed by these people. There was a problem in Ukraine when it came to the notation of the verticals and the results. I don't know if it's true or not, but the verticals and the results were written like this, like this. The laws, we know. In fact, it doesn't go very far. We are a little shocked by the... It seems to me that, in our time, instead of a horizontal square, I put an infinite sign, and at the end of the vertical, I put a 8, well, 10 silver, I put a 10 silver, suddenly, I can make calculations with these spaces, so it's hard, at least, at the moment, to have a good sign, to have an infinite sign. Thank you for your attention. So the laws, the axioms are here. So what are the categories? But it can be, as I said before, in terms of internal categories or categories. I can make a sketch of a double category.

1:00:00 I can take a sketch of a category and I multiply it with itself. With, of course, equations. And now this animal here, you can realize it in any category. And that's how you make a double category interlaced. So it's not difficult to see if a category has a sense, if it's a category, in any case. But for nothing more, I don't know, with the conditions I ask you to find out. So now, after this little intermission on the objects in terms of categories, categories and correlations, I present to you the contents of Article 73 of the Act, where it explains why categories can be used for the purpose of adding differences. So the article is very brief, it is two pages long. And these are the four main ideas that we will discuss in the next few examples. In concrete terms, this is the fundamental essence of the method, the mathematical method. The method is to first of all define the category of objects, the category of local objects, and the category of non-local objects. Then, the domain of the world in which we are going to live, and finally, to explain how to build these categories one by one. The second step is to say that if I take two varieties B and B' in the order of B' and B' it means that I take the application of different terms, a germ of varieties. I take a germ of V' and I take a morphism of this germ, there is a germ of V, which is V' equals R, and there we have an enchantment.

1:02:30 A vector now has the property, it's not just a number, it's a function that I've already mentioned twice, it's a function that preserves the submersion and the density of two vectors. Consequently, if I apply this counter to a differentiable category, i.e. a category with a strong sense of direction and with the condition of the dimensions, well, it is still a differentiable category, or a category with a strong sense of direction. But this is not at all a lecture on mathematics or even on cathegories. He will realize, and this is what I have been working on for a long time in Germany, he will realize the cathegories. He calculates the students. It is a calculation. By an indication that I give to a student who is a Christian, who is a Protestant. I tell him that this should not have anything to do with the cathegories. So it seems that so far he has not been interested in the cathegories to some extent, These categories are the big ones, the ones with the structure. And here, it's not so much about categories of tasks, So, even though they are definitely not all the same, they have their own points of reference and their own points of reference. And with this particular case, the category is actually a groupoid, that is to say, the step where the group is a groupoid is the step where the category is a groupoid, I know, it's an arrow with an arrow.

1:05:00 That's the way to interpret it. But, Erasmus, you worked with groupoids before M2, that is to say, because there was even a definition of the definition of a groupoid given by Frantz in 1990. In each case, there is a similar space. Already, before we talk about the categories. For example, when we talk about curves, we are in liaison with space, space, space, and so on, there is no need in these categories of thinking. These are the curves in the way of France. France, B.R., A.M., D.T. Partial operation. Yes, a partial way, it is the generation of groups by replacing the law with a partial way, which is the identity. That is to say, it is very precisely, it is a remark, which is very interesting. The definition of the O'Clewis group is in fact the definition of a historical category. The first text of Edgar Macbeth gives a definition of a partial law. It seems that it is true. Of course, yes, we can work on objects, we can work on parables, but it is a second. The first definition is as if it had been written down on a sheet of paper. So, I say to you, the O'Clewis group is something that comes to him from before the period of Persepolis. It is clear. This is the link between homology, Hilbert space, Penrose, Atiyah, Witten, Connes, Hawking. And on the subject of differential geometry, there is a method test that wanted to categorically and thematically summarize science. And we had to work in this context, with these factors, and by extending, and the work consists in studying the extensions of the characteristics of an aspect. This is very interesting and very useful. We will talk about the program. The first part of the lecture will explain in more detail the five points of passage, which I will approach progressively, with some expectations, a little bit of mathematics, and sometimes the first part. In the first part, as I showed at the beginning, There was a frequency called homogenous space, total homogenous space, space of the particles, and the collections.

1:07:30 So, what I meant in my first five points is that the homogenous space of the particles and the collections, it's not going to take place, but I think it's going to take place. We have to understand how to simplify, uniformity, to a simple concept. There are many examples of mathematical equations, but only a few of them have been used in this lecture. Then, what is drawn and shown is the local and material character, that is to say that you can, at a certain point, as soon as you have a little opening with a small image that is extracted from another object, the violin, which is the I and the J for two positions, and eventually you will have a V in the middle. The first hypothesis is that these films are always the same. The types of films are always the same. The types of films are always the same. The types of films are always the same. The types of films are always the same. The types of films are always the same. You can see the drawing I showed you earlier, except that there is something that is on top of it. The variation is with something that cannot be found in this case. Now, when we examine a corner here, which comes here, in the common area, let's say.

1:10:00 It goes back and forth here. So you see that at the level of the work of these little brackets there, we went from this point to this one. And on the subroutine, I have a binocular in the middle of the map, the change of the map. And on the subroutine, there is only one transformation. It is associated with the field of K and is supposed to belong to an object of transformation called RALC-DNA, or RALC-DNA-RALC-G, which is an object, an object G, which is open to the world. So there is the expectation of the local character, the fiber, the changes. So, the main fiber, what does it mean more than that? Well, it means that the fiber is a C-G, in fact. These are examples of main fibers, I will only give them two. And finally, there is the subject of the subject of the subject of the subject of the subject of the subject of A subgroup of the disciplinary group is what we call the restitution of the cultural group. The disciplinary group is a group that comes into play in the future. We don't have anything to see in the form of a cultural group. It is a group that is going to be a kind of co-operative in the future. So, this is the main direction. This is all with which I would like to conclude this first lecture. Thank you for your attention.

1:12:30 Mathematics is an object in itself. We are not yet in the mathematical field, but it is not abstract. We don't even know when we have one, we lose it. The great promoter of this theory of connection, it was Iker Thorn, said that it was through the notion of connection accessible to aerobatics. In the first century of the 20th century, the term was used to refer to a type of object, which is now used to refer to a type of object, which is now used to refer to a type of object, which is now used to refer to a type of object, which is now used to refer to a type of object, Generally speaking, this is not associated with the system of quantum mechanics at all, and therefore it is necessary to live the chain of connections as such, such and such, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, This is certainly the main focus of this subject, and I would like to focus on this detail here, on this point here, on this point where there is this detachment between the two, but it is still that which is the main focus of the public question. The first book of Viggenerovich, which is very similar to the first book of Viggenerovich, which is very similar to the first book of Viggenerovich, which is very similar to the first

1:15:00 And so, it's interesting to say that there are terms, and at the same time they show how we can use terms, and they show how we can not use them. So, it's interesting to have both to be able to make the link with the question of the old, the old, the way it comes to me. And Reffman, well, it's certainly true, we have the definition. and associated fibers. And all of this allows us, if we don't have these algorithms, to create these fibers. And all of this, already from the point of view of the computer, we now know that the computer has the power to create these fibers. And we say, well, yes, of course, it's possible to create these fibers. It's possible to create these fibers. What I'm saying with the computer is that it's already possible to create these fibers. So the connection, basically, I'm not saying that it's a connection, but basically, when you have a space, the goal is to group these elements together. In other places, there are decentralized systems, whether or not they are linked to each other. So, it's a group. It's important, it's important, it's important. And here, everyone has a path to follow. It's not just a group, it's a group. It's a group of decentralized systems. But there is another group. It's a group of paths. It's an ocean, it's a group. In this case, the connection will establish, not in the details, but as a very general idea, the connection, the data of a connection, if it comes from a point of comparison between them. So, in contrast to the idea of homogenous spaces and the use of the principles, the co-coherence groups, the connection becomes open, or at least, we are already in a moment of co-coherence. So we have left the principles, we have been replaced by elements.

1:17:30 All of this, of course, has been in the hands of a group of fluids, because there are roles in a group of fluids that are made to understand the topology of the group of fluids, and that's all different in terms of the local-trivial character, for example, of the main space, to translate this group of fluids into the topological group of fluids, the local-trivial group of fluids. So all of this is indeed in the interior of the space, in the interior of the group of fluids. All of these things are well put together. And as we are at the same time... In the same period, we already have the notion of the Paros-Coe and De Gea, because De Gea is an invertible concept. De Gea is really a real category. If you look at it from the point of view of Einstein, it is clear that Einstein is not the same as Einstein in France. Here is the first part of the theory. So, related to that, there is the notion, I don't know what you saw, but the drawing that I had here is a point that also interferes with the mortisms, the mortisms of a fight, of a fight against another. So, we have a lot of questions about mathematics, of course, as well as, of course, operators, but, of course, it's a category, a set, on which the category is open. So, if you want, it's really inspired by the group of operators of homogeneous spaces. It's just that there's a partial group. So, since we're going to do that... There is a set of things that are given to us by these secrets, but what has already been said is practical, it is already the way in which rationality is used. So here, we have gone from this kind of animal to this kind of animal, and here, precisely, in this passage, maybe, now that we are talking about this kind of animals... On the other hand, on the equations of the equations of the equations of the equations of the equations of the equations of the equations of

1:20:00 There are many possibilities. It's simple. Even if you don't have a computer, you can use a computer. It's a kind of protocol. You can take it and put it on your TV. That's what we're talking about. The local structure now, is when you put the time on what you don't know at all. Here you have a drawing of a variety, a dream and a variety. Two cards. You draw a line of cards on the common part. And these lines of cards are spread out everywhere. All of this is a combination of different fields of study, including the field of quantum mechanics, which is the field of quantum mechanics, which is the field of quantum mechanics, which is the field of quantum mechanics, which is the Theoretically, the theory, which is still being studied in a systematic way, is the one that grew out of Schubert.

1:22:30 There is a development that becomes completely categorical, which is not necessarily categorical with problems such as relativity, quantum mechanics, but on the other hand with problems of complexion, because here he is very sensitive to the problem of complexion of the class for the varieties, it does not matter in the varieties, there is a class that is made of the class of antennas, that is, there is a problem of category, so it is rather through these notions of complexion of the class that the question of complexion for the categories will be raised. It's not the same thing, but it's clear that when you approach the questions of science, there are universal problems to solve the limits. It's clear, but it's not the same. So that's it. So here, you know, there are groups. So I'll show you again the situations of the second part. The second part, so it's clear that it's a moment where he completely starts to do things in a very categorical way. So in the second part, everything he did before, he did in the third part. The theory of lightness is a generalization of the theory of lightness, which can be called a rocket. So all the theory of lightness is a theory that has not yet been studied. Because for the moment, even at the end of this lecture, we are not talking about it. We are talking about a monstrosity of the mechanics of quantum mechanics. But if we take into account the number of variations, of completions, which I would say are based on the idea of data, if we take into account that, we have to say again that maybe we should go and see... I don't know if that would be an academic lecture or an academic lecture, but it would be an academic lecture. Historians in the end will not be able to compete with each other. So, the chronological structure, a kind of chronological structure is defined in an abstract way in the 1950s and 1980s. It's a kind of structure where there is an induction on parts. There is a euro here, but it's not really a word, in fact, it's when we make a parenthesis, a parenthesis, it's not really a word. So this is W of W, you have to say W, you have to put the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis, the parenthesis,

1:25:00 So, local structures, it's interesting because it's quite a lot of different sources of localities, of collocations. You can see that there are three different kinds of structures, but one that is more general, because they are not necessarily the union of the two. It's different, but it's not necessarily the union of the two. So there is collocation, but it's not exactly the same. This is the part where you can see the homogenous spaces, but you forget to look at the space on the graph, but when you look at the homogenous spaces, there is a quaternary, which is the homogenous spaces. So, the fourth point, which we already talked about, the fourth point is the differential prolongation of the genes. Now, the calculation of the genes is the extension of the differential prolongation of the line groups. Well, this is a very essential tool for the education of scientists, which allows them to develop a wide range of topics and debates on the regulation of the project. So here, I want you to try to repeat what I have just said. So, a small example for those who do not understand very clearly. When you have an ITM that has recommended J-U-J, U-R-V-R-S, but be careful, we are not going to include them, we are going to put them in different databases. At this point, the variety appears as an objective limit, and the tangent function appears as the limit of the tangent function as the limit of the tangent function as the limit of the tangent function as the limit of the tangent function as the limit of the tangent function as the limit of the tangent function

1:27:30 In this case, we have two topologies, T' and T'', which are called Frey's and Connes' components. Here you have an example that I just drew. T' is a topology, and T' is the topology that is given as a component by Frey. Each one of them in the topology T' has a number. And the topology T' is the sum. So you know a little the application of the square root. And so, what is important is to describe it in terms of categories, while there is a category in which objects are these little rectangles, this one, this one, and the morphemes are precisely the data that appears like this, to go from one to the other. This is just the same as when we were talking about telegraphics, which was in space-time, and we were looking at suns, and we were looking at suns, and we were looking at suns, and we were looking at suns, and we were looking at suns, and we were looking at suns, and we were looking at suns, and we were looking at suns, and we were looking at suns, and we were looking at suns, and we were looking at suns. There is a lot of information here about the causalities and the verticals of the equations.

1:30:00 It's a bit of a puzzle. This is a puzzle. It's a puzzle. This is a puzzle. This is a puzzle. This is a puzzle. This is a puzzle. This is a puzzle. This is a puzzle. This is a puzzle. Here, I show you two ways to find the correct structures. The idea is that a type of structure will be the same as T2x. T2x is not a part of x, it is a part of y. And if we want to have a more complicated structure, we have to have a log of x. Log of x. That's how we get a log of y. These are not mathematics or algebra, but they are in the form of mathematics. I don't know if you want to add something, Martin, but you talked about the structures and the hierarchies. In the case of quantum mechanics, it is a way of understanding the structure. No, no, no, no, no, no, no, no, no, no, no, no, no, no. It should be noted that, in the end, the notion of quantum physics and the notion of objective physics, in a certain sense, have the same properties as quantum mechanics. And we can see that in quantum mechanics, there is a category theory, there is an algebra theory, and then there is an algebraic theory. Well, after an analysis of the circumference of the screen, we transform the summit here into the line.

1:32:30 What is the definition of this condition? Well, every time you will give yourself an omega arrow of some sort, of some kind, There is a collation associated with omega, of the existing data, of the compatible data, of the compatible data that can be collated according to omega. So if we look at this example here, we can consider that when he made the local structures and when he repeated the same thing over and over again, he returned the same thing over and over again. Because in the structure he said that, in the end, he had put in a pursuit of the substances of the science, interacting with the differential systems and so on. Here we can see that there is an interaction that is confined to identity, that is to say, it is not as if, in the definition of destinies, the notion of what I am doing is to be incorporated. The philosophical aspect is an example. So, I don't know if this remark is useful or theoretical, but it is useful in terms of the system of logic because we are there in addition to the logic of thought. Questions? Comments? I would like to make a few comments on these questions.

1:35:00 The notion of... You talked a lot about... how to separate the pseudorubes. I wanted to talk about the pseudorubes from the ornithological ornithologists, etc. The relationship between one of the ways by which the pseudorubes are separated is the fact that a pseudorubes is a local ornithologist who is going to make a local ornithologist The lecture will be presented from a studio. So, there is this notion that there is an artist who discovered it, and so it was there, and in fact, for a very long time, he really thought that the categories were more important than the categories. Moreover, the first versions of the category, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories, the categories. It's a group of topology and a group of physics. It's interesting because it's in the category. So it was created to see if, by chance, the category of genes could not become a topological category. So the initial idea was really the topologies. It's really the foundation of this category. From the same aspects of structures, initially, in the types of functions, the question that appeared in 1959, the idea of the types of functions, is that really, simply, it is not that all the results of this group of properties are the same, they are not the same. So it was a bit of an attempt to translate categories into clear terms. It is in this article, by the way, that they write the categories for the first time. All of these categories are included because he did not do the quintessence, that is to say the square of natural transformation. But another reason for which he used the categories at the same time was that in his book, at the beginning of the book that he gave to Montréal in 1961, the idea was to designate natural transformation. So we can not just designate natural transformation.

1:37:30 At the beginning, he did not like the definition of natural information, so initially he saw that if we look for a natural information, we can take it from C to K, and if we take the category of squares from K, then a natural information can be taken from C to the category of squares from K. So, this was one of the reasons to introduce the double categories at this time. Obviously, at this time, there is absolutely no notion of matured categorism. In fact, I did not say it. I said afterwards that, once the notion of B-categories became more and more honest, that it became clear that, indeed, the double categories are matured categories compared to those of B-categories. There are also internal categories, and even for the order categories, all the categories, first there are local categories, also from the local structures, then the categories, then, because I didn't really have a need for order categories, but you didn't have, it was a little more general, if you didn't have the aggregate, you didn't have the majority, so at that time there were categories. All of this has been done before we had the structural categories, before we realized that there was a relationship between them. So, in any case, the categories and the parents are connected? That's right, yes. In fact, the categories in the data are simpler, the categories in the data are already defined very, very late. In fact, after, in particular cases, the categories are very, very late, but in fact, this is an essential point, which is that we have, in fact, the local categories, hence the local categories, and then the non-local categories. Let's talk about the invisible categories. In the invisible categories, the definition of the notion of the invisible category is completely regular. And here, we can see that the invisible category is totally regular. This is practically the same as the notion of topology or geometry.

1:40:00 So, we can see that the notion of topology or geometry and the invisible categories are two different approaches to the notion of the invisible category. There is a lot of work to be done in the academic field. There is only a second one. The second one has not been completed yet. There are two more to be done. By the way, in the study of local structures, in fact, what you were saying is that he had a complete local structure. I said that. What you said. Because what he did with the local structure did not necessarily have the sum of recalls. In fact, one of the theorems is the famous theorem of Connexion, a local structure in a local structure in the world. Still, it is about the notion of groupism. I see it as a particular case of category. Fortunately, in topological and theoretical anthropology, the notion of groupism did not exist categorically at the end of the 90s. You could write, for example, that the dream of science-fiction in Cameroon, in terms of globalism, is the same as writing, to exclude fundamental groups from pieces, it's not common, but in terms of globalism, it has a lot of analytical usage. And it was done later by Brown and Higgins, who were in the same school, and even in the same school. I would like to say that this lecture is a very interesting one, and we can look at it and see the chaos of the lecture as this passage from the beginning of the lecture to the end of the lecture, and well, we can go on and so on, but...

1:42:30 We see in the work of Copenic that we see a bit of a different passage in the buboids of any kind. And I don't know what... What do you think... Can we... Because in fact, the reasons we see such an objective in choosing the buboids, if we see them in any kind, is to say that in geometry, everything is reversible in this way. If you have A and B, if you have, on the way of A and B... In this sense, it's quite naive, but I think it's a very strong intuition. However, with this ideology, which I don't know about, but it's a very common ideology, apparently, we have more of this idea of reversibility than of... Can we say that this is still geometric? I think so, yes. Geometry aligns with geometry. Or should we talk about the important part of the lecture? I think that there is a general concept of geometry, but what I would like to touch on, is the way in which we discover it. So, indeed, So, after passing the non-invective theory to the non-invective theory, which is indeed considerable, we have to imagine that it is the other side of the equation. I'm not going to go into too much detail here, because we're not going to be able to answer that question. I can imagine, we're not going to be able to answer that question. The gestures are a good answer. The gestures are a good answer, because the gestures are what use differentials,

1:45:00 which crush the differentials. It's not an aversion, it's a crushing and a reaction. Here we are in a very different environment, because we have a very different category of differentials, which is not a component. And which is really naturally included in the situation. So, it is certain that going to different categories, I do not imagine that it is done like that at the level of the general sphere. As long as we do not have a good answer, we are in contact with the conclusion. And then, when there are particular answers, as was the case with the subject, we think about it. It's a little less, but after a while, it becomes more difficult. There is no communication between the two categories of differentials. It is not about the categories of differentials. It is not about the three or five problems of different categories of differentials. It is not about the three or five problems of different categories of differentials. I would say that the utilization of different categories of differentials is practical in this case. All of these are examples, rather than a deliberate will. In the first vein, generalization of the universe, in its local version, and in another vein, in the vein of the notion of the local, of the two parts, which is analyzed, in other words, by the group, by the group, and precisely there, it is not with the other versions of things. So, it is in this sense that it is very prior to this question of the local that it is embedded in the environment of the elective, as it is said in this book. Thank you for your attention.

1:47:30 These are the chromosomes of the work that are necessary in themselves to be able to explain, for the purpose of the class, the initial class to be able to explain, but still to be able to explain. So it's a method to search for algebraic, to analyze algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic, algebraic. These tools, indeed, are a constant variation of these tools, and they do not matter. But they know the cohomology as a placeholder, and they know that it is the common denominator. And so, they are also used in the mathematics system, and I think that this is a very important part of the work we are doing here. Yes, so if you don't speak, I'm sorry. But it's just as if he didn't do it. He did have a class of his own and all the knowledge he had in that same class, but he didn't have a class of his own and all the knowledge he had in that same class, but he didn't have a class of his own and all the knowledge he had in that same class, but he didn't have a class of his own and all the knowledge he had in that same class, but he didn't have a class of his own As we have seen today, there is a lot of work to be done in the field of higher education, in the field of physics, in the field of science, in the field of mathematics, in the field of mathematics, in the field of mathematics, in the field of mathematics, in the field of mathematics, in the field of mathematics, in the field of mathematics, in the field of mathematics, in the field of mathematics,

1:50:00 Thank you for your attention. First of all, to give you an anecdote, I was very young at the time, and I fell into the category of architecture, and I was very surprised by the results, because at the time I was in the 11th grade, and I was able to graduate in my first year of high school. When we come back later, with a more interactive and epistemological approach to the theory of categories, I would say that it was, compared to the American school, that it was the originality, and I could not deny the technical answers that were given to me when it came to the origin of the theory of geometry, which I am very fond of. I would say that at a superficial level, but also at an epistemological level, but which is clear, is that it is a critical relationship between mathematics and the subject we are talking about, mathematics. Thank you for your attention. There are all kinds of things, there are local ones, there are non-local ones, there are non-local ones, there are non-local ones, there are non-local ones, there are non-local ones, there are non-local ones. Thank you for your attention. Thank you for your attention.

1:52:30 Each time there was a subject to be constructed, you had to first re-construct the major category in which the subject becomes at least a reflection of a sub-category in the major category. This is a big mistake. It's a big mistake. It's a small mistake not to have the foundation, not to use it. There is only one way to solve this problem, and that is to use a practical way to solve a problem of this kind, and that is to use a practical way to solve a problem of this kind, and that is to use a practical way to solve a problem of this kind, and that is to use a practical way to solve a problem of this kind, and that is to use a practical way to solve a problem of this kind, and that is to The fact that the notion of mathematics has a universal role in the theory of mathematics by the composition of equations and so on and so forth, it is not completely clear what the state of construction will have to do with mathematics, for example, in mathematics, or maybe even a little bit of mathematics in mathematics. And it depends on the subject matter that we can, for example, agree that mathematics is the definition of mathematics. I would like to continue on this topic of mathematics because it is a topic that has been discussed a lot these days, so with a lot of interest, and I would like to give an excerpt that opposes the position of cosmopolitanism, the position of individualism. We understand well the position of cosmopolitanism.

1:55:00 I'm not going to go into too much detail on this, but the main thing is to place the algebraic position on the side of creation. In fact, the algebraic position is actually the algebraic position on the side of creation. And this is absolutely striking to see how the very first stages of the work are exposed to a very unique way. There is the definition of composability, which I will not go into now, but before defining the composability, we must define those that are composable. The graphic design is something that is really perfect, there is no doubt about it. But generally we associate algebra and axiomatization in this case for a plot scheme, not an axiomatic one. What does that mean? Algebra and axiomatization are generally used in this context as an example of the theory of Erine and Magnus Planck, but not in the relationship between them. Thank you for your attention. It's not really that you think so much of algebra as such, your idea was really that there was a structure, that in everything there was a structure, and at that time the idea was that the categories, if we found the right notion in the categories, it would allow us to define any structure, that's it. The structure of the research is not necessarily algebraic, it is not the structures that have the most amount of algebraic and the most amount of algebraic that we don't have yet, that's it. So, we are going to ask you to ask algebraic questions. I don't want to say anything else, otherwise we won't be able to talk about it.

1:57:30 I don't want to say anything else, otherwise we won't be able to talk about it. It was the geographer, the one who saw things, who knew the calculations, who knew two times and sometimes even five times. He didn't even know what to do with the calculations. It was someone who knew nothing. His vision of things was the vision of geometry. In the course of his career, he continued his studies in the field of mathematics, physics, geometry, algebra, mathematics, physics, geometry, mathematics, physics, geometry, mathematics, physics, geometry, mathematics, physics, geometry, mathematics, physics, geometry, mathematics, physics, geometry, mathematics, mathematics, physics, geometry, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics. There are a lot of them, I believe, and I will repeat them in the second part of the conference. I would like to remind you, with the content and the reason why I explained it to you, that it is a course of mathematics. It may have been good, but it may have been bad. The lecture was very complex and very clear. There was a confusion that was hard to understand. There were a lot of similarities between the lecture and the seminar. The lecture was very interesting because it was a national event. The students started talking about the lecture, the two lectures they were attending. And then, they were writing. I think they were writing on the table. They were writing like that, in detail. I don't know, I don't know, I don't know, I don't know, I don't know.

2:00:00 And that is to participate completely, not only to look for possibilities, but also to pay attention, to communicate the way in which you see the others, as you see them in the rest of the world a little differently. Indeed, some of the lectures, we absolutely thought they were very fascinating, difficult to decipher, but at the same time, on the same field, they gave a... You learn to do math by yourself, so you can imitate it, but not in the same way as you do it, but in a different way, that's it. I don't know if it's really the weight of the geometry, or if it's more important for us to help each other. Yes, I would like to confirm what you say, but I don't think so. I followed the courses of Rossmann a little bit before you, but they were not so... No, I'm absolutely happy that I found out that he had the courses of Van Gogh who taught me that. I followed them with a lot of interest and I have to admit that I didn't know much about mathematics. There is still a fairly sensitive difference between mathematics and Erasmus and in particular in the question of the use of coordinates. I can tell you that when I was a student of physics, it's true that I didn't know much about coordinates, but I used them a lot. And so, to go back to Westphalia, it's quite true, the feeling of discovering the causes of the problem was prodigiously interesting. I find you to be a little more severe when you say that it's true. It's true that it wasn't what you thought it was, but it brought a lot. I can tell you that. Thank you for your attention. Thank you for your attention.

2:02:30 Even the algebraic fact that there were pseudo-complementaries or something like that, it was already explained, it was not explained, it would have been explained by the internet, but it was not that. What do you do with the new technology? It is totally unvaluable.