3 synthetic approaches to total differentials / Infinity - categorification de structures equationelles (contd.)

0:00 As the lecture begins, I've been given permission by my fellow organizers just to make a very brief announcement. I should explain, my name is Michael Bright and I'm the secretary of the organization for the Archives of Contemporary Mathematics, Physics and Philosophy, which is based in Brittany. And we have the sponsors of a meeting which starts on Monday at the École Normale in Paris, which several of our speakers here including Jean-Benoît Boum. I have here a couple of posters, one of which will be put on the floor outside, one of which will be stuck in here. If anybody is able to get to Paris... In the course of the next week, for this meeting, which is called The Impact of Categories, 60 Years of Category Theory in the Historic and Mathematical Perspective, and it's got, as I say myself, some very good speakers, including several people who are present here, then please let me know. That's it. Could I ask if there happens to be anybody here who was at yesterday's session? And who happens to make a recording of any of the sorts, it would be very interesting. Okay, thank you very much for your attention. Thanks. The speaker this afternoon will be Mrs. Poletina Wang, one of the first students of Charles Ersmann, who will speak on Charles Ersmann's literature in German. I would like to thank the University of Paris for inviting me to speak with you today. I would like to say a few words about the history of the University of Paris. I would like to say a few words about the history of the University of Paris. I would like to say a few words about the history of the University of Paris.

2:30 Some of them were experts, and later some of them were theorists, they talked, they were true, the first one was Colbon, and then there was Luthier, others were mathematicians, all of them were experts. And at this time, until 1942, the referents were in the decline phase, that is to say they were not totally... And Ganeshwari lived there until he was in Switzerland. He had contact with two rats, with his friend, with Hoff, the physicist of Spassky, by the way. And then there was the great rat in 1943, and Fenway Carité was deported in a bus accident. And Erascon, who came later to the faculty and the students of Bonheur to stop the students, was able to save himself, and for 20 years he was a disability director, while Rêve, who was a high school professor in Lyon, was not taken. And so Erascon and Rêve were talking, were dealing with foiling, at a time when they were bored of the university. Speakers were a bit in the clandestinity and the famous note of Erasmus of Rheims on the speakers appeared on June 20, 1944, that is, 15 days after his departure. Then, the University of Strasbourg returned to its home, Strasbourg, and there, at the Erasmus seminar, there were first very talented young researchers, including Rheims and Erasmus students, and Tom, who was very marked by... There are also mathematicians and mathematicians who, during the war, refused to return to Germany because he was considered a nobler, he refused to go there and then he understood that he could go to Europe because he had only had two wars in Spain. He was a Frenchman in 1918, he had known two wars, when he had just finished his year of these kinds of things, he educated himself very well as a European, and had a very good relationship with the mathematicians of the north. So, the young Wittsebruch, Heinbert and other mathematicians. There were Americans who came a little later.

5:00 In short, this seminar plus the lectures on mathematics was really something very interesting. There was really science being formed. Now, we only need a mail for a lot of people to know their work. At the time, you couldn't go to people, no one, to know their work. At that time, all the major congresses brought new things. And even from the 50s, even the Russians, who were not allowed to go out, came out. There was a 45th congress on topology in Algeria, there was a 50th one in Bruxelles, where Benespan introduced, for example, the connections, after the work by Henri Cartan. And Henri Cartan also made an article on the connections seen in Algeria. So it was really a period of intellectual dispersement, and then Strasbourg was a big icon of this dispersement in the United Kingdom. For example, Tom had chosen, at the time, Céline Baccarat, it was in a room where he slept in the Rue de Carismat, where he discussed every night with Rémi-Pierre Saverne about mathematics. And so there was the influence of the Rue de Carismat. Mathematics was said to be one of the oldest sciences in Strasbourg, during a great time. And I saw, for example, that the Jets came when it was not so bad, I arrived in Strasbourg in the quarter of seven, I was a high school teacher, and so what were the motivations of Westphal? It is that, first of all, the connections, it was after the Hull-Cartan, he already gave them to me during the war, he had them in the finished spaces. The introduction of hybrid space was to try to understand Helicartan, and then in this 50s article, he introduced a collection, this early article, but he still used the notation of Helicartan. So when we look at it now, this notation, finally, it works very well, we don't know why, sometimes, and it's the following year that Erispan introduced the genes.

7:30 For example, in the article on connections, we write x plus dx, where x is a tangent vector. In fact, if we take this first-order object, because if we give ourselves a gamma application of R in a value of L, well, we can say that this represents the object of a gamma application. So when we write x plus dx, it's an operation of g and not what we get with gamma. So I always take gamma as zero. And that's what represents, if you will, this x of gamma squared plus dx. And so we had to find a deep meaning. For example, when we take the... And then, when we take an application f, What is the difference between x and dx? Well, it's the a vector, we would have an element at the top. And so, it's the termination. The gess had already explained this in a more comprehensive way. And when we have a rule, when we say x plus dx, y plus y, well, it's x, y plus dx. I put the word pose, I put the word pose, so we don't have to think about it. Well, all this is the composition of the genes. It's because we take the composition of the genes.

10:00 So, it was, there was a plus to have a different nature in the study. There are people, for the first time, who... The reason was not for a tangent vector or a variety to be an elevation of an axis. It was to give a specific meaning to what a tangent vector is, to take a curve, a tangent, and then the symbol, that's it. There are a few differentials in Tiberias, and even the first one has no justification, but in reality, the writing of Cartan, when we know what it is about, it seems much more elementary, there is no need to think about it. So, the term dy, it had zero, and that we had to think about the close relationship between them. That's it for the differential geometry. There is also a collection of Zisman's exposés in the book for the honor of Lessman and a contribution in a book on the history of Algerian topology, which are remarkable, both in terms of history and mathematics. And so, among Lessman's works, there are two things that are a little less known. These are semi-holonomic Gs. I think that Radine will take care of it. A lot of collage students in Czechoslovakia, in Central Europe, a lot of people call them semi-holonomic Gs, but for a lot of people it seems a bit mysterious. And then there is something else that was mentioned. The article on terrestrial connections, it was in Bruxelles in 1950. And then the Gs date from 1951, in the years before that. All of these have been followed for several years. These are the gestures that we call the GERD or GERD-9. But in the same year, 1950, there was a terrestrial art in a colloquium in Italy

12:30 that talked about the feuilletages. And there is also a paragraph called the feuilletage of the second world. So it was before the GERD. So I'm going to talk about these two things, and I'm going to sum up the name quickly, and then I'm going to take the second word, because there, I don't know if there's something in the papers, did Mr. Asselin come on the lines of the second word? Not on the lines of the second word, no. On the lines, yes, but not on the lines of the second word. Because in the end, I wanted to show that it was a little picture of summing up the name. No, no, he didn't take that back. On the other hand, he wrote a long article on general feuilletages. Yes, but not on general feuilletages. But, well, it's... in the... in the notice of Rebs, in the book of Edelstein, he said that Edelstein was waiting for something on a feuilletage of the second world war. Edelstein wrote that on a feuilletage. And I made an illusion in the book of works written with math, and then I said, and I realized, well, I made a mistake in the book. And in a book, it's more or less the same. So, I'm going to go back to the work of Seigneur Le Dôme, to the pretext of all this. A space of, if we iterate, but it's more at the scale of a family in the world, not a family in the world of the world, but all of them. Placed, we can always consider what the applications are, the sections on them. So, you can suppose to yourself that I have a submersion of these problems. And it's easier this way to define the geosciences. So, we could consider what we call J1A, it's the space of the visual areas, of sections. Here I suppose that I have a social subject, the local sex.

15:00 So we can, with certain J1 and J1 of B and so on, we have what we call semi-e non-eulogical gestures. But it's too big. And what happens in nature when we do an advance with the self is that there is a part of the non-eulogical gestures and what we call semi-eulogical gestures, as I told you last year. The good definition is defined by an adapted section, and it is Pradine who has well read the special diagram of the Geo-Somio-Leunard and, in fact, let's say that a Geo-Somio-Leunard is a space where we have the same phototransition but where we have forgotten the symmetry and the passage from the symmetry of the Geo-Somio-Leunard to the Geo-Leunard is always defined. There is an application to this project at the EIPA, the application to the ETA, the application to the EPUB, and we have the application to the EAP by J1 of the ETA. And the monolingual GSE is the sum of the elements of the monolingual GSE, U of j squared, e of a of j squared, j squared, such as beta of u equals j of a of a, that is to say it is the close image of the diagonal. And then, from close to close, we will determine j of a of a, also by beta, j bar minus a of e, and it is always the close image of the diagonal.

17:30 And even if they cross diagonally, it will be G1 plus 1. And so, if we try to take our legalization, we realize then that these... So, the algebra is defined by primitive functions, but the semi-ironomical algebra is not. It's the same thing by forgetting the symbol. So, as an example... If I take a normality, the tangent fiber represents the normality, well, J1 is a semi-holonomic gesture, gender 2, which can be identified because if we look at a differential form, a section, it will be a differential form omega, and if I take J1 omega, we realize that we have the same probability of semi-holonomic gestures. So, this is of interest because we are talking here about the iteration of the function t1, sorry, t2 of asterisk. And then, the g of the name, if the form is correct, because the g of the name is defined by a function a1 and d a1 and cg. And when the G is the name, it means that the DAs that are here are equal to the DAs that are here, and it is equivalent to the form of the term. This is also important for landmark spaces. Next time, I can define landmarks in the name.

20:00 For example, if we take the space of landmarks of gravity, the main fiber, well, J1H. Atiyah, Witten, Connes, Hawking, and Witten, Connes, Hawking, and Witten, Connes, Hawking. And this is important in the study of g-structures because what is a g-structure? It is a reduction of the structural group L, which is a variable of dimension L, to the group of sub-objects. So the space of the first landmarks, called Hg, are landmarks that leave the g-structure in the environment. History, geostructures have not been... it's expensive to invent the word, but C. R. S. Mann, already during the war, when he was studying nuclear space, was looking for the conditions for the reduction of geostructures, and that's precisely what I have, for example, it could be the group that is complex, it could be the group that is conservative, it could be the group that is orthogonal, so it's when there is a reduction in this group, so C. R. S. Mann had... These are the conditions of the existence of reduction and this is an occasion to say that what was attributed to Chern was in reality the law. For example, it is important to say that there was a column in 1953 in Strasbourg. Chern came in talking about the infinite groups of cartons without having... You know the theories of Gershwin and so you had an oral explanation that was a little blurry, a little complicated.

22:30 And then, in the meantime, I went through the text and in the text I detailed the theories of Gershwin and then it gave me an idea. Atiyah called this structure. But the thing was, we call it the infinitesimal structure of Hegelianism, which was by the way the title of the text. Well, now hold on, I forget to say things. It was constantly, for your introduction, it is a subject that has a more complex structure than complex. So, when we have a structure, in the first year, we have enough possibilities. But, what is the extension of g, of h ? A good definition would be to make the semiologic g-structures intervene, to make it possible, because G1 has this problem, so G1 to G will be a main library because it is a structure called H2 to G. And what will be the set of two-dimensional landmarks of the g-structure? It will be something called H2G, which is the intersection of H2, the space of all landmarks. With H2, because all the markers are of the same name, and H2, sorry, H2G is the intersection of H2, the space of all the H2 markers, that is to say defined by the H2 genes, with H2. Well, in general, if we have a structure that is not very biblical, this will not be a possible hybrid. And precisely, we say that structure theory is integral when this is the main theory and that the projection of H2G over HG is equal. So, the theory of structure theory and extensions is very important, and we need to apply it if we use the G-Solidarity.

25:00 But what is important in the theory of Euler's equations is that if we take, well, this is a fibrous fiber, so the fibrous fiber is torqued to the associate. So there, what we are doing in an article is a little bit fastidious to demonstrate, but the result is good, isn't it? So this, the associated vector space is the fullback of the fibrin, linear applications, the pi asterisk, in the vertical of the common fibrin, of the tangent fibrin, of the tangent fibrin, and then it is the conjunction of the two fibrins, so we have something that is This is the plan of the Q-linear projection. I'm going to put it in the paper because on the linear projections,

27:30 the pi asterisk tm, the vertical film is the core of the projection pi on these applications. This is a set of applications. All of these are important because we can find sections, there is always a existence of sections since we have a algebraic system. And when we talk about connections, for example, already in number 2, we will have applications. If I take a projection in the usual sense, it will be extraordinary. J1E is the space of sections of E above L. So if I give myself this model in S, and then now we can take J1 of J1E, well, if we take a section, we can extend this, we have the application J1S, which goes from J1E to J1E. We will go from E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to E to

30:00 If this is the value of the value of d, we will say that the connection is equal to 1, and we will be able to make the differences with the help of relaying, since we have affixed spaces, and this will give us a place for the connection. This is how we can introduce the connection. And if this is the value of the value of d... This is equivalent to a given section, in each corner, a plane element transversal to the fiber. Well, to say that this proposed application is of value in Jouer, It means that this field is completely integral when we define a transverse plane. And the obstacle is the curve. It is important that it is curved. Null and locally integral. So, finally, we can say that the bioluminescent gestures always meet in nature. They are objects that must be given. And now I will move on to the distribution between the two. In the article on objects, it does not mention the element of contact defined by objects. It did not say that it was the same thing as the element of contact in the usual sense. Under the element of contact, space tangent, a tangent space of a variety, as well as in all tangent spaces. I think there is a lot to be said about them.

32:30 So, what is the difference between the two? Well, we consider the dimension L and the set of elements of contact tangent to dimension P. We have a space in the genre of everything, which is the space TXM, which is of dimension L, and then we consider the sub-spaces at each point, the sub-vectorial spaces, and then we consider those that are of dimension P, P-Component. So this is what we call the variety of global contact elements, and it is a fibrous space, so the fiber and the Grassmannian, we will call it G, and the whole is a vector space of dimension P, and a vector space of dimension N. So here, these are what we call the elements of the first order. We can consider Cp of Cp of L, that is to say the variety of contact elements tangent to the variety Cp of L. So there are projections that have a contact element of the original zone, so I am in Cp of L, and there we also have a projection that is not necessarily tangent because an element of contact

35:00 A projection can give an element of contact of an inferior dimension. I'm not going to give you a number, but it's an element of contact of an inferior or equal dimension. Well, what is defined in the Riesmann's article, but we can't write it because it was done before he wrote it, is simply the space of an element of contact We consider Cp bar 2, which is the reciprocal image of the diagonal, if both are equal, then we will only take those which are projected on it. This is what Einstein calls an elongated contact element and those which are projected on a contact element. And then it comes back to the reciprocal image of this diagonal. So, to have a diagonal, we will only take this line, that is to say those which are projected on it. This is often a contact element of dimension P, because the others are of an irregular dimension. And this is what we call a contact element of the elements. And a distribution of contact elements, this will be the element of Cp in Cp . And this always exists because we have a projection that is unfiltered in the space-time. There is a section, and that's what we call a field of the second order, and this field of the second order, we say that it is completely integral. Each section corresponds to an integral variety, that is to say that all the elements of the second order belong to this distribution, which is equivalent to saying that it is a removal of Cp in C2.

37:30 Considering these two facts, what does symmetry mean? It means that an element of contact of the first order on a variety is an element of TPM, that is to say, of RP in the variety M. We take the object of the first order, we have a certain number, for example the origin, This is an application of an application of an application of an application of an application of an application of an application of an application of an application of an Tp over Np. Because at the source, when I locate the line Np, where I have Np over R, well, we don't change the image. So that's the element of contact between Tp and the genes. Tp is the space of the genes, the space of Rp in N. So this is the space of the elements. And the elements of contact between them, The two M's are the linear group of the two M's.

40:00 The group of the inversible g's that are made up of the two M's. Well, that's it. That's the cycle of life. Whereas what we have in general is, honestly, in the space of the semi-linear g's, which are made up of the LPG group. So, this language of the element of the superior order, we prefer it to the superior order, well, it was linked to the gesture of Newton, which was not marked since the article on the second order dates from 1950, and it is a kind of theorem, and then as an example of low distribution, well, it is the totally illusory variances, A human connection, a constant curvature. Total geodesic surfaces. Human variability is not generally total geodesic. And if its curvature is constant, it is total geodesic. When we remove the elements, we have integral variability. And this is an example of the qualities of the universe given to the universe. For example, projective spaces as well. But the distribution of these terms always exists, whereas a connection, if it is of a certain value, then the value of the value is homogeneous and it is not always there when it is of a certain value. So these are the two aspects. I don't know if there is anything else to say. We can also define the qualities of... Well, I'm going to give you two. Again, it's a bit complicated to find examples that are not examples of geophysical totality, and it would be nice to have others who know us. So, what I'm going to say is a small part of the work of Respa, but which is less known. And for those who don't know us, it seems like something mysterious,

42:30 The work of a mathematician was called Morin, and Morin was something that was called the gesture of a man, but the air did not see it, so it did not seem that Morin was a genius, so in fact it was already the gesture of a man. I do not know if you remember, you were in France at that time. Ah, you were in France? Yes, in France. But this is a simple idea, we iterate, but not that we iterate, it's just too big, and you have to take, and it's just what happens if you take fibers in space of g, there are things that are symmetrical, that is, we leave them on the other side. And precisely, by interaction, we always obtain extensions. What David Levenham said, well, sometimes it can happen. We may have time for a very brief question. Thank you. Well, in this case, we will separate into two groups for the rest of the afternoon.

45:00 But then I was more impressed by Harrison. He was very at ease with life. He was relaxed in his seminar. Then he invited us all to dinner and he was a gentleman. His intervention was not the answer to anything. And he also had this very intensity in his eyes. And he was a very charming, very charming person. In the way to the restaurant, he would talk to me about contrarians and things like that. He also asked me about the mathematics, about the lecture. I mean, he was fairly generous about it. Probably not independent of all these qualities. Of course, I have said before that a great couple of years ago, a mathematician was Martin Einstein, which is no big difference to Einstein, and so on, and Schlesinger, on top of all that, which was, and it was, I think, a natural consequence, started to be in charge of Harvard. I have a young, interesting and beautiful woman by my side, who does not come here. So, I make a point to come to this conference because I really remember those days and how she welcomed me as a very talented person.

47:30 So, now I would like to talk about two-figure calculations. In a way, I was thinking that because covenants, the core of the chemical space, form a category which is not filtered. This shotgun is usually bypassed or motored. It becomes filtered. But there is an interest to do things without it. This will be the example in which the interest of this is. This is a joint work with Andrei Layal at Ross State. I have been worrying about the problem and not getting anywhere. When I told them the problem, they were immediately interested. I was spending two or three days in Montreal talking about it. And it must be linked to Witten. Then I did the hard work. But the collaboration is essential. So, first, let me remind you what is a filter type.

50:00 SGA for Gaussian. Means you have A and B. Then they resist. You see. Side of the filter. And PS2 is the following. If I have E but A is equal to B, then there exists, and here the problem is that it's the same U. If A is equal to B, then these two are made and you take it from this. That's the important fact about this set of numbers. These actions are a little bit redundant, the way I write them, but I think we said that redundancy is not seen, you have to look for actions that really describe the situation, I mean, don't worry about whether they are redundant or not. You cannot do the empty category as well, too.

52:30 And then with FCO, we have filters. Now, what is a two-filter? Apparently, you all know because I support it. This is a mix of technology and mathematics. You see, the two-filter category, to begin with, is a two-character. And FCO is designed as a filter. You use two objects, you can go whatever way. If I have two parts F1 by 2 times, then I can complete it, but not in a commutative diagram that is an isomorphic double set. The second action, which is backwards on the one-dimensional case, says that if I have Two W-cells, gamma 1 and gamma prime 1, meaning I have studied the M1 and G1 and completed in two different ways, can go further with a single W, a single W small and W prime small, in such a way that these two W-cells become equal. And here the important thing is that the W is the same for gamma 1 and gamma prime 1. Now this, I've got two figures to have here, because these actions, they are not strong enough, the important theorem and the basic theorem I'm going to tell you now, and so practically it's easy to understand the world in this way.

55:00 You see, alpha means if you... It means that the gamma 1 followed by alpha is equal to the gamma prime 1 followed by beta. It's meaning that even two cells, if you go further away, they become equal. Now the actual actions have to be simultaneously winning. It means that you have a 1 and g1 and a 2 and g2, two of them. Then there exists the same thing you want. And the same for F2. I have gamma 1 and gamma 2 related. I have a pair gamma 1, d1, and then equalize it later on by alpha and beta. Of the same W, the diagram comes up the same. It's the same way you pass from dimension 1 to dimension 2 in the notion of field. And I think that at the moment, at the moment I think that they are on the right, there are more particular actions that allow you to think like this and bring you to the right thing. So you see the importance of, you want to approach that if you have a two filter, a two category, with three and two sets, it should be a filtered category.

57:30 In a category, F2 is backwards. But you see, in the notion of filter time, you have the same U. It would follow from F1, but from FW. It's a category, so that is a trivial discussion from you. But U doesn't have to be equal to D. So I need the second action, because then I take A, the identity, the identity for A. This is T1 and this is T2. Then I can complete this and you see with the same U and V, shows the plot that U has to be equal to this and I will obtain the whole strength of this. So in a way, besides the fact that when you want to create the theorem you have to use the whole strength, it shows you that this... It's essential, because if not, you would not have a filtered category in the case of this entire system of truth. So, this is the notion of filtered character. Now, for example, which is the, I mean, the theorem that opens the door to all the applications and all the components,

1:00:00 filtered coordinates of sets, is that if I have sets, they land here. A final set is in my direction that we have to rescue in the two-dimensional case. So, first, once I have those actions, you have to construct the category that is going to be the collinear of the system indexed by such a category. So the quicker actions are in line. The same way to construct the co-limit of a filtered system of cells, B2, PS2 over there is not necessary. Only this PS1 is enough. You need PS2 to plug this over here, but the construction of the co-limit using jars is enough to help this.

1:02:30 It only means of the two filtered system of calculus. And we already knew that A is very difficult. I was using email, I was not using tech, and so I had to put a fancy A. And the convention was that every fancy letter was a capital letter, duplicate. We have to keep going. So, the construction of L is the following. First, the question of notation. G is 2 functor. I am assuming 2 functor and not 0 functor, because I want to be sure that the proof I indeed are correct. Ross Street tells me, no problem, this all goes away, it all goes the same way for 0 functor. I am not so sure. So, for education is enough to... So, we assume 2.0 and the proof I have are for 2.0. Now, this front of G, if I have a margin in G of A, then from G into A, I can write, I mean, you should work with comprehension or use of knowledge. This gives, in the category G of A, defines this macro-conformation, x-hat, so the notation I write the x going into the other direction because on the opposite side.

1:05:00 So the objects from G of A can be written afterwards from G into x. This will help a lot the intuition of proofs. But it's rigorous notation. Now, we define the category A of E. You have to say which are the objects and the arrows, because it's going to be a category. Now, the objects are the pairs, each pair, with x and g of n. These are the objects. It's like a rocket helicopter. So I can write the objects as arrows from g to n. Three arrows, because the arrows, I want to give you three arrows divided by an equivalent of an edge. The pre-order from an object X in G of A into an object Y in G of B is just an arrow in G of C, and an arrow in G of X and Y is a side, a triple, U, side, and B, where side is an arrow from U X into D in the category G of C, and there is an equivalent relation of the pre-orders which looks the same as the action. I would say that two arrows, x and y, exist in the environment of the same alpha, beta, as it is seen in the diagram. Meaning that in the category G , the two arrows corresponding to this diagram are equal. So two arrows are equivalent, if folded away, they are equal. It's more or less what you would expect.

1:07:30 So, we have this construction and it says that this hydroxide on the left, if you follow the balance alpha, then you have an equivalent carbon. By the definition of the equation, you do the homework and it comes out. This is an equivalent relation. I mean, you have to show the density. This also, once you know the statement, is the ischial part. So, actually, this time the truth, we have not too much problem, but says that if you have an arrow, a piece of wire, and you have a W, an arrow from C into W, and an arrow X into D, then passing this same thing, you get an arrow, another arrow from A to B, which is equivalent to the first one. This is the rules. So once the composition of average has to be defined, we get the calculator, and you see that there are two variables, c and f, from x to y and from y to z.

1:10:00 I use the, not the diamond one, but the silver one, and the diamond, the U of A, I complete into diamond, silver, and diamond. And then over there you can see which is the composition of the parameters. It's like a large circle. This is the composition of the parameters. And this, the next composition... It's a little bit more problematic. I mean, you actually do have to do something not trivial at all to manage to show that the proposition is well defined, that if I have a different centric of the other, psi and psi prime, eta and eta prime, and a different gamma and gamma prime, well, the thing that you get, you manage to do that they... The algorithm is equivalent. It is independent of the algorithm. So, composition is not dependent on the algorithm. Only it is the action of three, two filters. And I associate it by the identity score. This is not very difficult, so we have to recapitulate. I mentioned that what I wrote down, you can see in the written argument, is not... I mean, the statements of the fields are true, but... The proofs are not there. And the actions I put, they are not enough. I have to, the actions, correct actions, to put the statements that I want to put there. So the practice published is not quite correct. Okay? That is no problem at all.

1:12:30 Now, the other problem tells you that this construction, L-A of G, is because you are looking for it. Which is rather clear, I mean, if I have an object in G of A, lambda A of X is just X A, the pair X A. If I have an arrow, then you have this arrow in A A of G, because I have an arrow side from X and Y. So I have units, units into B, which is an object in G of A, and units into A, which is an object of G of A, and I have this arrow in the end of G, because I go to B to do the identity, and the identity map is to decompose units and units into B and arrow in the end of G. This is a cellophane, and it uses an equivalence of capital. I take the factor capillary, or two-factor capillary, the energy hits, or any hits, composing with this cellophane, I get the cellophane from these two hits, and this is an equivalence of capital. This shows that the construction is what it's supposed to be. The system of categories is indexed by two filtered categories, which is not a filtered system, but nevertheless we have a construction that is going to have the good properties we need.

1:15:00 So, which is not filtered, we cannot construct the collimity with the good properties, but nevertheless we can construct the collimity. This is the first theorem, and this is the second theorem. To prove this second theorem, you need the full force of the action, the full strength of the action. Meaning this will be the theorem, the relative action, the correct action is LL, that means the quality of the system g to the i. If you have I, it is a finite category, so each category G of A, you can take the counter-category from I, and then with the analysis, take the collinear category from I to the I. The appearance of the distance... You see the objects which are on the image, system XI in GA, with maps which are in GA too, but the general object here is XI, XJ. But the maps are not, these are different, are in a category associated to each transition from Y to G, which transition from Y to G, you have a map in A, A of G, by definition of maps in A of G, you have an object associated to A, and a map in the category G, A of A.

1:17:30 You have to show that every object of this form is isomorphic to an object of that. Clearly stated what you didn't have to show to get the statement. So this is the first basic and important theorem. What does it mean to be an equivalence? Three statements, essentially subjective and full. And each, the proof of each one of these statements is in the, in somehow the pretenom of their awareness has to be done. And as a technical, I mean, it is convenient to prove first the few levels that will help you to organize the flow. So, the first lemma tells you something more about the notion of two filters. You see, it says that if I have a system, a family of double cells, gamma i, In the ocean, you have gamma 1 and gamma 2 and gamma 3, but here you have a one family, a giant family, gamma 1 and gamma 2, and then you can go through the way, in this way, and they are all becoming one, which follows from there.

1:20:00 It's not enough. If you have two kinds of families, each one within the family, all of them become equal in 2007. But the two families become equal with the same C, the same U, and the same V. So you can equalize the final family of Gamma L, but if you have another final family, you can equalize it identically, both of them, with the same C in one V. And let me provide further information on the construction of them. By the action method, one of course, they want to imply PS2, the important action that will imply theta, and not only 3 theta, in which you have, if you have two, a pair of arrows, a pair of arrows, you can close it with a double cell, but with the same W, W, W, W, W, W, W, W, W, W, W, W, W, W, W, W,

1:22:30 This arrow here and this arrow here, which I go from A to C and B to C, is independent of I. This one is the same for all I and this one is the same for all. This is essential. So with these lemmas, you start working and I am a little bit of, well, a lot, a lot, maybe not. I'm causing a lot, maybe somebody with more experience in two cells is less careful. But we essentially prove the three things you need. To prove that this essential is regenerative, it doesn't need a chromosome. Only the personal line cannot think that that's a pretty loss given before. It doesn't need the fourth or the fourth. But to prove that it's faithful, you need the action step two. If it's faithful, the action step two is vacuum, right? If you have a one-dimensional case, of course, faithfulness is accurate too, but full is important and it follows from axiom F1. If you have a one-dimensional case, full, isomorphic. It's not logic, but it's semi-somorphic. But they are irrelevant. That corresponds to injecting. So injecting, you need axiom F2 in the classical case. To prove full, you need that much. So, with this, a corollary, and this is the following, and with this, I finish. If you have a system of categories indexed by a filter to category G, but each of the categories of the system has finite limits,

1:25:00 Meaning that the handkerchief takes the value of the category of categories of final limits and the construction technology has final limits and the commons, the inclusions in the commons, preserve those final limits. ... and equivalence of categories but already to final limits and preservation of final limits. This follows as an easy corollary of the previous theory. Which taking exponentials to all present the construction. And this is essentially what we need. We need the inverse limit of the two-fingered system of topology and inverse. Inverse system of topology is your power and then you can argue. I've worked on that, I've worked on a lot of projects, but always for a difference. And we can get a new pair of applications that will actually work. As I said, I think that this construction is mainly relevant for strong-shape theory everywhere in which you use as an indexing the category of covariance, which is not filtered, but is still filtered, and you don't want to take things modulo or modulo, you want to keep things without quotient by the modulo. Then this construction can be a lesson 1b for questions. So, for all the three categories, you have a theorem that flat computers, although they're smooth categories of elements, are filtered.

1:27:30 Flat computers correspond to double-digit categories, which implies that filtered code is commutative. Do you think there's an analogue in two dimensions? Well, I don't think about, I don't currently thought about that, actually. I mean, this is, the construction I'm getting is a capital, which may injects it. So, for the proof of the system of this unit limit of time, I'd only use the concept of flag, the classified concept of flag time that I've been talking about. Now, here, of course, you see, let me answer one question. If you look at the action, the action of BF2, it says That given two cells, you can complete it with an equality. This gives you the notion that if the notion of a cell starts to grow, it opens the door, and if it does stay in equality, which is a very specific thing, then you have, this is the starting point for a mathematical system. It's somewhat crazy enough now to think about, of a, which will be the end category, not corresponding the weakness to the end step of... The notion of freedom and the corresponding construction of justice. I'd like to send the speaker again to reserve the questions for the two people I have for the next talk.