3 synthetic approaches to total differentials / Infinity - categorification de structures equationelles

0:00 There are many contributions of synthetic geological geometry. In synthetic geological geometry, there are many applications. Not only of this, there are other applications. There are other applications.

2:30 There are other applications. There are other applications. There are other applications. There are other applications. There are other applications. There are other applications. There are other applications. There are other applications. There are other applications. There are other applications. There are other applications. There are other applications. There are other applications.

5:00 The human structure is not part of a smooth mind book, simply a formulating of this mind book, a formal mind book in synthetic differential geometry to explain this concept in a word or two.

7:30 It is such that by this condition, everything is called to be a math. Now, in many cases, this general mapping is a finite dimension, simply this canonical mapping.

15:00 To give an introductory on synthetic differential geometry in this short time. So if you are interested in synthetic differential geometry, you are required to read these textbooks. In this paper, this is the content.

17:30 This is for the three quadrilinears, dimension 3. These three quadrilinears are dimension 1. And give a manuscript for the in-multiplic O.M. I will give an example for the in-multiplic. And two macroscopic fields are two macroscopic spheres. We denote by this notation the total of the immaculate fields on M. Given that we denote by this, the total of the immaculate fields gamma on M is this property. And in particular, this space vector denotes the notation of the symmetric curve of the set up,

20:00 which is well known to be generated by a transposition. Exchange one and part it together and fix. We define these two pieces. This is pi 1. Pi in that point is behind the third space. And this condition is for polynomial.

22:30 In the mapping, it's behind the nth tangential. The notion of the nth tangential is behind pi in that point. For this one, this tangential is the result of tangential. The result of tangential x and the result of tangential x is negative. If we store the nth tangential of pi as x, it acquires the following conditions. At the end of the lecture, there is a unique gamma-dash side.

25:00 The notion is that if we operate in the top-nose sphere, so that we have a canonical mapping.

27:30 Well, the repetitive one, in some sense, is classical because that is really just combinatorics of iterated tangent bundles. So let me concentrate by question on the two non-repetitive approaches where we gave two, one that uses mites or cubes and one that uses n speeds or accelerations or whatever you call them. These will probably be our output to formulate in classical terms, but which of them do you consider the most natural or the most useful, with cubes or with beads? Important topic of future study, these three spaces go inside, so of my approach.

30:00 The place is in the plenary session this afternoon, but now it's important that everybody who will come write his name in this paper, because we have to count how many people are going to be there. It's in a common, in a room, in a restaurant that means the rest of Poland.

32:30 Well, we are going to start to continue. We are going to have another lecture in the morning. We are going to have another lecture in the morning. We are going to have another lecture in the morning. We are going to have another lecture in the morning. We are going to have another lecture in the morning. We are going to have another lecture in the morning. We are going to have another lecture in the morning. We are going to have another lecture in the morning. We are going to have another lecture in the morning. I would like to thank Mrs. Reisman for having invited me to this conference. I'm going to speak louder. I feel like I'm going to speak very loudly. I would like to thank Mrs. Reisman for having invited me to this conference in memory of Charles Reisman, who was my first boss as a test director for my third-year thesis. And who taught me, who taught me to do the categories, so I fell into the tool and I remained a categorist until now. So, what I'm going to tell you this time, today, is the extension of a work that I did, which was published in Mécanique de l'Ocologie in 1999, on the infinite non-strict categories. After the publication of my article, Vatanine, in an article he submitted a while later, points out that the method I used could be generalized to make coherent, structural approximations. So, in the end, I only follow what Vatanine suggested and does. So, rather than coherent, structural approximations, which are not quite clear, I would rather say infinite...

35:00 When we have a structure, let's say, of dimension zero, the categorized one is to pass from the superior dimension to the first dimension, in which the equations of the first structure are transformed into isomorphisms, essentially in such a way that these isomorphisms between them are coherent. So the idea here is to make a categorization that is generally enough to involve all sorts of equational structures. Equational structures, I mean the regular ones. We have proposed one that had the advantage of being elementary, uniform. That is to say that we started from the lowest level, the lowest degree, which is the same magnitude. And we came to a structure defined by a monad. So the merit of this procedure was that we could iterate. That was the strong point of the procedure. We could iterate the procedure, categorize it, categorize it, and so on. Of course, what I mean is that when we talk about the theory of Leibniz,

37:30 by categorizing it like that, we obtain a version of the categories we know. The default of this method is that it was based on Cartesian monads, which are relatively rare, at least on the ensembles. What I propose today is another way of proceeding, which has the advantage of categorizing not only those which are non-Hermitian monads, but which has the disadvantage of not being uniform, that is to say that here you start with the algebraic theory and the equational theory, and here we arrive at a monad that will iterate the procedure. So, since we cannot iterate the procedure, we can see in practice that it can be interesting. In spite of everything, there are iterations, for example, of the monolinguals of the category of the monolinguals, and then of the category of the monolinguals, which we will continue. And so, instead, what I want to do here is an infinite iteration. All these iterations are given in blocks. And if we want to do a simple categorization, well, we stay in dimension 1. Now, a simple categorization, if we stay in dimension 1. If you will, we put in blocks all the categories of the monolinguals. That's the general idea. So how did I proceed? I start with an algebraic theory, an emotional theory.

40:00 The models of this theory are the so-called structures. We are going to categorize this infinitely. First of all, I start by doing a strict categorization. So, a story in a movie, all that, for the moment, let's go. I do something very simple. I'm going to do a strict infusion of categorization, not in the simplest way, but simply in the infinitesimal mode of the internal theory. So here we have a strict infusion of categorization, which is how to weaken, to make a weakening, so now we are really in a coherent weakening of these categories. Let's think about the categories of infinities. How to integrate them? The idea is to work on the level structure. Always work on the level structure. Because this is where all the actions, the generics, are going to weaken. They are going to pull. I'm going to give you a lecture on infinitesimal graphs. That is to say, the sets that compare them. In the previous articles, I called it an infinitgraph,

42:30 so I considered it an infinitgraph. So I considered an infinitgraph, not necessarily a relativistic one, which is the theory in the framework of the second infinitgraph. So I considered it an infinitgraph. This infinitgraph, I associate it with its trigonometric infinity, the limit. This infinitgraph, we consider it to be less than or equal to the limit. How do you call them? I have this year. So, I'm going to read it. This is just to understand what I'm saying. I have... I have... I have... And now, I'm working on the book and I'm going to make you read it. What I'm looking for is the non-strict structure of the book associated with it. And for that, This weakening will be in fact an extension, that is to say that we take the equations and we make the morphisms. I will give you the term of decryption for this procedure. So what is a decryption? How do we proceed? So already, for this, we proceed in two steps. First, we need a generalization at a level, as you can see, I'm going to start by destructuring the implicit categories strictly. Almost all of them.

45:00 So these are the three. And consider what I call the L-infinite lagma. So what does it mean when there is a L-infinite lagma? Well, first of all, it's an infinite lagma. It's the data of a lagma. So what is an infinite lagma? An infinite lagma, to put it this way, is like an infinity. Except that all the laws... But there are no more laws. We're going to... Not quite. We have preserved only the positional actions. This is a drawing to show what a positional action is. If I find it, you see, it is in the following situation. Two CDs. The first one is the name of the component. It is simply, at the positional level, as we can see here in this drawing, that the source... All of these terms are simply the root of the equation, the root of the equation, i.e. the source of the crystal. This is the kind of positional axiom. Everything else has been validated. This is an infinite number. Now, in addition to the structure of an infinite number of numbers, there are of course priorities, and of the trinity, we only remember the logarithm. Everything else is validated. So what do we remember? We give, we give each other, and in every dimension, in every dimension, for each symbol to function, an nr law, if my mass value is m, n, n dimension, so if it's an nr law, my n dimension is an nr law, if it's a product of n terms,

47:30 And the law version, which is going to be f1 plus n plus n. That's it. That means that for each n, there is a model of language realization. I just want to make a point of misunderstanding, because it's only now. You have such a composition operation that does not suppose the associativity, right? No, I don't. Do you have units? Yes, there are units. But the units do not suppose that they are all the same? No, they are not the same. They are there. But you have the source and the plus. There is the source and the plus. There are all the data, but there are not the actions. The only actions are the positional actions, as I said. The FN preserves the positional actions. The FN preserves the positions as well. Also, if we look at a drawing, for example, if we look at the sensory product, well, the sensory product of these two features has a source, the sensory product has a source, 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, 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, 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,

50:00 So we have the structure of all these actions. Why do you say that it is a unit if it does not occupy any of these actions? No, because it is a unit. Yes, excuse me, I was talking to you earlier. The x composed with the source, the identity of the source, is an x. No, I'm just kidding. So, here we are, in an object. So, we make the L-adminimums. Unfortunately, of course, for the movement, there is nothing. The non-strict structure, as we know, has passed. It is not yet there. So it is now that it will appear. So we made it appear with the concept of stretching. Here, in the context of L-stretching. So what is L-stretching? In L-stretching, we give ourselves L-infinite magma and we give ourselves a morphism of infinite magma. And in addition, the most important point, in addition, in all dimensions, there is an application of the E-N-tilde to the E-N-tilde. So the E-N-tilde is explained, E plus 1 is here, E plus 1 is here, and here we are talking about the dimension N, the world of dimensions. So what does it mean? E-N-tilde is the set of couplings of Y. There are two arrows, which are parallel and which, projected by pi, are equal to pi n of x, equal to pi n of y.

52:30 Do you see the idea here? If we look at n, the arrows are equal. If we project by pi, these two arrows are equal. They are parallel, they are equal. So in this case, there is an arrow and two circles equal to pi. It's just a data like this that satisfies the definition of the data. That is to say that the source of this is x. It's true. To project the life, to project by pi, we leave only the identity of the project. It's logical, isn't it? We can't project. In fact, it's the identity. It's all the same. We can't do anything. It's the identity. And there is an additional assumption that says that, precisely, the coherence with the identity... By identity, there is a coherence with an identity, which means that we should have eliminated identities and simply put a hook that would give him an identity. It's not related to this action. There is nothing else. There is no coherence, there is no coherence with all the boxes that signal the coherence. What will produce the monstrous structure is only the

55:00 In fact, there is a magnetic field, a particular magnetic field, there are actions in it, and therefore there are equalities, you see, strong equalities. Well, these strong equalities, I'm going to call them an extension, and I make an extension in the ISL at this point. So here I make an extension and I obtain like that a certain infinite magnetic field, T2g. All of this is stretching, L-stretching, A, G, A, G. So, we think it's in S, there's a arrow like that, because it's universal, and we construct a monad like that, T, a monad. The category structures, categorically infinite, are precisely the agents of this moment. I will take the term of the logic earlier, the T, the infinite, the categories, the categories, the categories.

57:30 The whole process, the structure first, and then the analysis. We can sum up a single free structure without any sense. So how much time do we have left? Atiyah, can you let me know? 10 minutes. 10 minutes? 15 minutes. What I would have liked to say now, maybe a little more original, In addition, we can make non-strict morphisms between these non-strict infinities and categories. A strict morphism is simply an algebraic morphism. An algebraic morphism is a strict morphism between non-strict infinities and categories. And now, what is obviously the most interesting, what we would like to know, is non-strict morphisms. In the world of the English, there is an explanation of the free structure as an infinite and infinite quantity. So here, if you want, you have terms, the equalities between terms, you can see that in the chronology, you can see the terms, and these terms then, by stretching them, You will replace the equalities, you can see that, as formal arrows between these terms. We replace the equalities by formal arrows between these terms and we go up in the construction of formal arrows and we re-insert them inside. And again, we can have formal arrows, and those that are equal, we can build new formal arrows and so on.

1:00:00 And so we go up like that, in the dimensions, to obtain this infinite structure. You can see this as terms. And so, for example, associativity, if there is an n equal to n, these are two terms, if you will, or at level 1, an r, transformed into the book, well, by whom? Here you have an equality, since you are in the strict structure. Here there is an equality, since we are in the strict structure. And so here, by stretching, you have an arrow. And as you have another in one direction, Well, you have an arrow that goes from there to there, and this arrow is not necessarily the identity, again, if you project it, you fall on an identity, and so there is still an arrow, and it goes to the other side. It's the last one. What? It's the last one. The last one. And you say that coherence comes from the universal property. That's it. Because the actions are there. The actions are there. If they are not in the infinitesimal categories, like this is the German infinity, they are there anyway. And this, on the other hand, is a categorical infinity. So there are all the strict actions. And it is these strict actions that are there, that will be lifted. In this En-infinity magma, which does not have an action at the beginning, but which, thanks to stretching, will have all these arrows, all these arrows there. All right? Did I make myself pretty clear? So, the construction depends on the language...

1:02:30 There is a construction that depends on the language. That is, if there was another... if two languages produced the same structure, I cannot answer... I don't know if it's going to be the same, if it's going to be the same thing. No, it's going to be fine. I didn't even think about it at all. I asked myself the same question as you, but I didn't think about it at all. So, I would still like to talk a little bit about morphism, because it's me, personally, who wants to do it. At the same time, it was a promotion of what I had already done. So, no one was there, and so I started. The advantage for morphism is that the procedure is the same. To consider the same, that is to say, this time I will consider an infinite categorization, I will consider an infinite case of P power 2, power 2 that is to say the morphism, which comes back to make the strict morphism of P, of the infinite categorization. So I have made an infinite categorization of morphisms. As I said earlier, I take two infinites at the end of an infinit, and I associate it with a free object. If there is a constant, we go to an infinit of one, or at least one at the end of an infinit. So I make the book. This is a morphism, this book is a morphism of infinity and infinity.

1:05:00 I'm going to consider a destructuring of the objects here. So the destructuring, of course, will use magma. So here, this is what I called the L-infinite magma morphic. To make it easier to understand, what is it? Well, it's the data of two L-infinite magmas, and then between the two, just a morphic of one. I've eliminated morphism, the strict structure of morphism. I've just kept what was left at the position of the parabola. Everything is gone. So now here, when I have L, I have L-infinity. Now, you know what I'm doing? I'm stretching L-infinity of the logarithm exactly like earlier. So it's very simple. An entanglement, it's a morphism, M, M prime, here an infinite magma morphic, here there are others, and then between the two, there are morphisms, this time, strict, of infinite magma, of L-infinite magma morphic, in the form of a commune. The commune depends on the infinite A. And in such a way that, in addition, here and there, Here and there, there is a stretching structure, a stretching structure here, a stretching structure there, without any coherence. It's normal, stretching, you see, there is already no coherence with the words, there is no coherence there. So this is what a morphic stretching is.

1:07:30 I consider the book, I say the name, I make a morphic category associated with it, then I do the stretching. And I get a new, a new infimum graph, a morphic graph, like this, a graph on a quadratic algebra. So, here, you see, there is the following thing. If I do the adrenals, here, well, you have an algebra. In fact, it factorizes, it's the same. What is a morphic between two, well, I don't know, what is the difference between two infinimums? You have a couple of infinites. Well, it's an algebra. The morphine transplit is a morphine transplit. It's a T2-algebra on the infinites under the center.

1:10:00 The infinites under the center are stacked. It's a structure of algebras on it. In such a way that it's a square. It's close to... Non-strict morphism between non-strict structures. I'm sorry I didn't go any further than that, because I just finished three days ago. It's been a minute or so. So I didn't see that you had to elaborate any further. I think there will be a very big difficulty, composed, and I think it's going to take a while. Because in the end, it's already a non-strict morphism between non-strict morphisms. I had an idea before I even started. I had an idea about the structure of the universe at the level of the universe and the cells. But I preferred to proceed differently there, because there was a kind of general approach, if I may say, we could proceed in the same way as what we had done before, but I thought it was nice to see it that way. But we should proceed in a style, in a way, where we are two, it's him, it's three, it's two, it's four, it's five, so I didn't do that. But on the other hand, there is another procedure, maybe I'll talk about it later, but for the next time. There is a question that is very morphic, but it does not say what is the role of the hook. There is a hook here, isn't there? Yes, there is a hook here.

1:12:30 Would our script of these lines be the same as the script of these crosses? There is a lot of research going on at this moment. But for now, what I want to say is that there is a lot of research going on at this moment. But for now, what I want to say is that there is a lot of research going on at this moment. But for now, what I want to say is that there is a lot of research going on at this moment. But for now, what I want to say is that there is a lot of research going on at this moment. But for now, what I want to say is that there is a lot of research going on at this moment. But for now, what I want to say is that there is a lot of research going on at this moment. But for now, what I want to say is that there is a lot of research going on at this moment. That's what I had in mind, but it's not what I had in mind. But in fact, I would like to come to the general theory that I had in mind. I started gently with algebraic terms.