1 of 6 (partial) (contd.)

0:00 I'm going to start with an index, which is here in A. When you go through the galactic equation, it's always x. When you go through f, it's always x. When you go through x, it's always x. When you go through the galactic equation, it's always x. So it's certainly a natural transformation. Now, what's interesting is to see that it's the only one. Excuse me, in the definition, f of a equals b, or f of a equals b? All of these are composed of a single element, a simple one. This is the first species. If now I take a black object in my catalog, what do I have as a species arrow? Well, I have as many arrows as there are elements in A. If I take a small arrow, I have an arrow here, because I send it, which I still call small a, but in fact it is the arrow that sends this arrow to this small a. There is one in each of these stars. In the same time, they have to put a letter in A so that it becomes a matrix. I'm going to look at what this arrow gives as a square, as a triangle. I'm going to have S, here I'm going to have S, the generic inclusion of S, which is the identity of S. Here I'm going to have O, here I'm going to have E, and here I'm going to have an element of O in E. Here I have the application, the application of small a, in quotation marks. In fact, it's L of small a, G of small a.

2:30 I look at what it gives. Sorry, excuse me. Here I have theta s, and here I have theta in the natural transformation of the equation. So here I have theta and here I have s. And I'm going to show that theta is the equation. So I look, for the moment I don't know if it's... So I know it's a commutative equation because I assumed it was, and I want to show that theta is the equation. So let's see what I do with a star. So star becomes a. So star becomes a. So star becomes a. And my square has to be commutative. In fact, an example of this kind of mechanism is that the effect of naturalness is seen thanks to a few particular objects. There are many other natural centers. The pointy center. The pointy center is a center with a point in it, an object with a arrow and a terminal object in each object. It is called a relative element. The initial element is the arrow and the final object in each object. But you have to pay attention to one thing, you can very well have elements that are not isolated. It translates into the preoccupation of the impulsivists who made the distinction between non-literate and abstract.

5:00 Okay, so here it is. I'm going to leave you with the examples. Unfortunately, I won't get to the conclusion of the session. But it's always like that. I'm predicting something and I'm going to give you some examples. And after all, the examples are just talking. You can think about them later. You can write whatever you want, it's fine. I'm going to give you two connectors, but they are interconnected. So... This is a question that means the set of arrows of the object f of x to the object y. I have an object x, an object f, an object f. I'm going to give myself an application, I'm going to repeat the time, between this set of arrows and the set c, x, and y. This is the set of arrows of x, y, and y in the category c. I'm going to ask for the time. Either it is natural, it means natural. It's g of y, isn't it? No, it's the opposite. Yes, it's G. Natural in X and Y. It means that there are two squares in the universe, because in reality, if you look, it's a misanthropism between these two functions. These are the functions of what is what.

7:30 Well, C, X is in C. But when you look at the set of arrows from X to Y, for example, sorry, the set of arrows from X to Y, well, this is something that is contravariant in X and contravariant in Y. In France, there are two converters, these two expressions here, for example, two converters that do like this, and it's these two here that are in the middle and in front. It's a bit complicated to think about, but it's not that easy. To really understand it, you have to expand all this and try to give you some examples. So, let's take a look at the time. So, let's take a look at the time. In English, it means FH1O. It means that for the size of an object... For all these things, what does a natural equation mean in a natural equation or not? In fact, a natural equation means that two factors are equal. For example, the first one is C. I don't know what category it is, but it's very difficult. This is a vector. If this vector has an addition to the right, this vector has an addition to the right, which is produced by Penrose. This is an actual projection of the axis of the axis of the axis of the axis of the axis of the axis of the axis of the axis of the axis of

10:00 The idea is to define a concept by an adjunction, but it is also to define a system for which we are close to each other, because they are sets of objects, sets of spheres of the same dimension, because the objects, I mean the sets, I mean the sets of spaces of the same dimension. All of these terms are used to define the subject of the lecture.

12:30 The subject of the lecture is the subject of the lecture. The subject of the lecture is the subject of the lecture. The subject of the lecture is the subject of the lecture. The subject of the lecture is the subject of the lecture. The subject of the lecture is the subject of the lecture. Thank you for watching this video. There is a hierarchy, there is no contradiction. But what bothers me is that in this case, each one has its own explanation. I understand better an alternative if we place ourselves in a topos, a pump, and then we can use definitions there, but without talking about the ensembles.

15:00 We can work with big groups, like for example the category of ensembles. There is no definition, except that they are degenerate and paleological. But there are topos, there are good topos. There are many other reasons. I'm going to give you one more example, so I'm going to consider the configuration of objects in a certain context, i.e. I'm going to square two variables of different types of elements in a particular context, so I have either x or y, etc. And so, once the context is established, I can use these variables, the constructs of the statements that show these variables as variables. So I called E the Gamma, the set of all the statements that have, as a variable, three other things than the ones I declared in the first place. Those that make sense in the Gamma context. These are the objects of a computer that gives me the arrows, they are the demonstrations that have been made in the past. There are a number of examples, but I'm not going to go into all of them. I'm going to talk about them one by one. I'm going to talk about them one by one. I'm going to talk about them one by one. I'm going to talk about them one by one. I'm going to talk about them one by one. I'm going to talk about them one by one. I'm going to talk about them one by one. Then, I will consider another theory, so I add a definition to my context, I have a new allegory that I don't like at all.

17:30 In fact, I have a pointer here, it's what I call the pointer of the changing of context. It means that if I have a statement that doesn't speak of x, then after having set x, my statement is not the case that I always have it. So it's always a correct statement. I don't know if it's true or not, but I say it's correct. It makes no sense. And then, if I have a demonstration that I made between two statements before I declared x, after I declared x, it's still valid. And then, of course, I have a hypothesis that is the source of the equation, etc. So, there is one such hypothesis, which is called the point of change of context. Point of change of context. Here, we have a joint on each side. We have a joint on the right, which is here. Do you want to guess what it is? If we have a joint on the right, it's the same as the one on the left. And if we have a joint on the left, it's easy to guess. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I'll justify this a little bit and then I'll stop. I've kept the E after the declaration, that in there, it's going to be E, so you see that E comes from here, so it doesn't have the occurrence of X, but F comes from the occurrence of X.

20:00 Demonstration of the form, demonstration of the form. So what is the moral of this story? It means, let's suppose that I have to show you an expression of the form, whatever X is in X, F of X, the path of the hypothesis E, I keep the hypothesis and I put it in the dictionary. And then, for the existence, I also wrote, this time it's going to be this one, it's going to be this one. Everything I'm telling you is extremely symmetrical. The joints on the left and right are comparatively more symmetrical. But in mathematics, we're going to divide this symmetry for different reasons that I won't have time to explain here. So, because of that, it's a little more difficult. I have to show you F with as a hypothesis that there is an x in x and that x is x. What I do to explain this hypothesis is that I am told that there is an x in x, so I declare it. So, first, I talk about the product and I suppose that the x that I declared satisfies the issue of the x that exists in the hypothesis of x. Yes, but there is no x, right? Yes, yes, yes, no, no, because there is no x. The problem with my projects is that I don't have a square, x is not a square, because I am in this category, I have this type of x that does not contain x, because it is a new variable, and I have an x that cannot contain x either, because I live in this context. I am making you think that x is a square, but it is a form of x, it is not the same thing, it is a form of x. Well, I'm going to stop here.

22:30 Thank you very much, I was amazed, I had put everything in my mind, I was very curious, I was very curious, I was very curious, I was very curious, I was very curious, I was very curious, I was very curious, I was very curious, I was very curious, Mr. Stratford, I could have some phone calls. I'll tell you what, are you going now to London? Are you going to Bethesda? Take that or... I'll get back to you tomorrow at... I'll talk to you tomorrow at the seminar even. There are different levels of square root and an augmentation. Then we have two vectors, C and G, from C to G, from C to DG to DG. Then there is also the fact that we have a family of objects, either C-I, which belongs to a certain number of indices, which is a family, and E-I, so C-I belongs to F-I.

25:00 I think that C is free, so we can continue to enter a projective equation, but it still doesn't work, so we're going to say it's free, on the levels that are there. So what does it mean to be free on these numbers? It means that every time I take an object x in C, suppose that f is free, so every time I take an object x in C, I have f of x. This is a chain complex, so I have fn of x in each dimension. In fact, I have a base in it, A for base. For that reason, I suppose that x is N, and that I have one by one, because I also have to use dimensions, and the Y would be Nx. In other words, there are different dimensions of life for each dimension, and there are simple variations of life for each dimension. So, basically, all the f' of f, all the f's belonging to the morpheus in C, I'll give you an example. So, there are these two. This means that the computer is free. So, we already assume that the computer is fixed on a model, and the second thing, we assume that the constructor G is fixed on a model, which means that it is the same model, of course, so that means that G is the enemy, G is the enemy which is a complete chain that is fixed.

27:30 So, if you write it with the increase, it's a fix. At this point, with these hypotheses, then, among them, the number of people is incredibly high. That's it, that's the end of the lecture. And I want to give you an example, it's very technical, it's very technical, I don't think there is a paper on it, because I have been concerned about these questions for 25 years, but lately I thought it would be nice to have a paper on it, so I started to write, and there is a title after, I have two reasons to write this paper, the first reason is that I have had close relationships with the students of the University of Lille, there is a third point in the field which is me, very often, but it happens, so the third is the point. The third point is that, always in the frame that is there, if I have written down the values ​​of these terms, we will unify the points of the range, which are here. Because the point of view of the theorem allows us to demonstrate that certain values ​​of the range are typically not there, in the cases where we can do it, they are really there.

30:00 It has brought us to a case of algebra that we may not have already known, but on the other hand, it went to the trap without any calculation. In a moment, we will see. An example, the first example is, I take the singular chains on a space of sin x, I take the tensorial product with the singular chains on the space of sin y, so here I get a complex tensorial product, a complex, already known, the tensorial product is already known, and then I take the chains, it's not singular, it's a simple chain, I don't know the topologies, I don't know the tensorial functions. I'm going to focus on the product of the two pieces of the St. Lucia chain. These are two filters in XY, so they are based on the Cataloys in the St. Lucia center, not in the St. Lucia center. They are both, they are both, they are both, they are both acyclic and linear on the same level. I'm not going to go into details, obviously, because I'm going to spend a lot of time explaining. So there are filters in both directions. But they are not in that direction anymore. The property is not really satisfied, and here we have to work a little more, it's a little more complicated than the other ones. In fact, we have an initiative in that sense, and in that sense, it is an economic application, it is the transformation of Alexander McClelland. It is the one from which... well, it is unique. It is unique and it has properties, certain properties, for example this one. If you start from C2x, C2y, C2m,

32:30 So, if you go from C to X to Y, you get an identity vector. If you go from an identity vector to an identity vector, if you go from an identity vector to a C to X to Y, you get an identity vector. And if you go from C to Y to X to Y, you get an identity vector. This is a very complicated problem. It's not just a simple problem, it's very complicated. And in fact, With the excuse that he has, it's instantaneous. And there is a certain amount of things like that that move. It's not exactly... Yes, because the model here is a delta-chain, delta-T. And there is an intuition set that is just as simple as the minimum. And if we take the complex of normalized chains, that is to say, complex by degenerate syntax. And in fact, all the syntaxes of the previous chain are all degenerate. All the n-plus-b-plus-1 subventions in the paper are all the same. It's arguments like that that make it easier to take the precaution to analyze them and to put them in the same distribution because it's not the only logic, it's not a problem. It's in the same sense that you have the transformation of Alexander Wittner, the one with which we build 4-3-3. But here, it's not at all the same. On the other hand, the same argument in the unicity allows us to show that Haydn's matrix is a section of all the choices we can make, all the choices we can make.

35:00 Of course, we argue in this case that if there is no initiative, it is in fact due to the fact that we are going to execute the hypothetical, even if it is not necessarily verified, because it is the model ... So, it's not that, it's going to fall, but it's going to die. The problem is that the models here are indeed the delta-n and that here it will give us things that are the normalization of delta-n, which are better than starting from 2n plus 1. We cannot apply the argument to show, for example, that it is the commutative flux. It means that the variables on the commutative flux are still the same argument. And here it is the commutative flux, and obviously it is the commutative flux that distinguishes the specific elements. Thank you for your attention.

37:30 Thank you for your attention. Thank you for your attention and see you in the next lecture.