Non-Abelian / AnAbelian, Abelian homology

0:00 Or, if I put them in X, they will appear again in M, the representation. So, I will be brought to work to compare the two situations. To compare the two situations for a given F, which is just an F there, and then I try to build something. That's what I'm going to talk about. And with the back of my head, I'm going to talk about the homological system, which is going to bring me back to the analysis of A, of the calculations, and I'm going to put it back to you. Yes, I understand. Your M? The M. In your presentation of homology, you talk about X and Y. No, no, no. In homology, all of that is given here. Well, ES can be made to be varied. If you make ES to be varied, then you can make it to be varied, because when you make ES to be varied, it depends on the nature of the language, but at the start, at the start, by defining one. But when you work on concrete situations, it's not given, so you have to make choices. No, you have to make choices. As I said earlier, I still have presentations, but I put my presentations on the table before, one there, one there. Thank you for your attention and see you in the next lecture. You don't have everything at the start. You didn't give it to me at the start in Excel or with Cc. No, but that's already the work of the problem, of course. Of course, yes. But the problem is to know how to exploit this work. I think it's true. I mean, the topology of mathematics is the work that allows us to have a candidate in the upper line.

2:30 For x, it's basically m, y equals m. In other words, we have to do y equals m and y equals m. So here, essentially, I'm going to have x to make a music, then m to make a part of it. It would have been simple. And here I'm going to have something else, another kind of calculation that I'm going to compare. So the goal of the game is that when we make something like the music, we make a staging. And then it's important to have a calculation of the variation. As with the classical staging of geometry and relativity, the staging is curved and each phenomenon will be represented by a curve, and then understanding the phenomenon is studied, it will be done through the bias of the dynamics. Now what I propose is that for completely conceptual or qualitative staging, staging would be a very extreme mix to compare with something else and the calculation is what is calculated. It's true that in homology, we try to find invariants, but in order to do so, you have to choose the S. So the S is given in second. What is really given is IJ, RL. And then, we will choose the S to have the answers that allow us to conclude. This is a kind of modelization with an academic lecture. These are kind of small worlds, and I represent them in the same way. I represent them in the same way. So, I have one presentation, I have another presentation, and then I can compare them. So, this is what I want to do with lectures, because, for example, the comparison of lectures is the curvature of physics. Well, the curvature is a little bit... The first step in the comparison of lectures is the construction of parallel courses. So, we are... We are out of the differential calculus, we have no more. It is no longer necessary to introduce a calculus with integral derivatives at the ordinary point.

5:00 But we have not lost, we have engraved the principle of making a comparison calculus. It will be done by limit calculus or ... More modestly, these limit calculus are first calculus that we are trying to know, which is zero. So, there is a way to determine the bonds, to know how many are how many, so here, before doing the cohomology calculations, we have already taken these bonds, and the category of these bonds, are they connected? We ask ourselves questions, in the end, that are fundamental, but on the fact that, on what? We ask ourselves fundamental questions, on a second level. The first level is the one where we conquered this topic of mathematics, for example, X, Y, Z. Thank you for your attention. We are no longer at all in the idea that the second is the code that allows us to know. And maybe we will find a kind of invariant in the infinite number of simulations. This is a very precise idea. No, it's really an interesting problem to know for a practitioner. What I want to say is that by doing this, you are not completely in the void, in the non-sense, since in this form, it is a little bit more effective. When you compare or distinguish homology and analysis, maybe we could say that the analysis would be if there was a common diagram. I don't know. I don't know what you mean. Maybe you don't know what I mean.

7:30 I don't know what you mean. Would there be... So it's a distinction between the two. No, it's not. But it has to do with the question of the commutativity of the diagram. I don't know if the diagram is commutative or not. It's already a second question. It's already a second question. So it's a dissection of the neurology and the law in relation to the situations of the diagram, or the diagram of the diagram, or the diagram of the diagram. Yes, I agree. Okay. The diagrams that are in play are comparisons, you see, the red thing here, that is to say, well, here is the first step, I can not enter an A and an equality, if it is possible, it is fine, it allows me to understand that F, O, G are not new, but if it is not possible, well, I can still put an A and then mark the B, but for one that is a natural vector, it is not the other way around, it is a comparison, and that's it, so I established... I found a way to go from left to middle, from middle to right. And when I finished that, I found a complex that already had a complicated format. It was two meters, two metros. This thing is a way of filling. It's like when we do topology, we want to fill a complex, we have a body and we fill it with cells, it's quite the same thing, there is a filling that forms a format. It is this filling that makes the cycle. That is to say, it's A, B, it's that the radius is square and commutative, is that it? Transformation of A and B, does it mean the commutative square? No, it's not commutative. The square here, the composite here and the composite there are different, but there is a natural transformation from one to the other. So in the natural transformation, there will be commutative squares, in the transformation from one to the other. Yes, there are some commutatives, but in the second stage. Simulation, the rest is the end. I would like the writing of the axiom of passage. Yes, yes, yes, I have the impression that there are a lot of them. I can just tell you, but it's not reversible, this transformation, it's reversible.

10:00 Is it reversible, this transformation, or is it maybe just isomorphism or not? Which one? It's A, B. Ah, no. No, it's not reversible. No, it's not. It's a comparison. A comparison. Subtitles by the Amara.org community Well, yes, the example I gave you is the same level, but other cases are different. So, in the field of dissemination calculations, we also have a method of replacing classical differential calculations with differential calculations of this kind. I don't think I introduced that. Because when we do the work of modulation, we have objects or robots, or observations of objects, that's it. All of this is written down, but then there must be a calculation with which we can use it, work with it, modify it. So, in the calculation of assimilation, the possibility that is offered is to replace the calculation of the ordinary equation with the calculation of the gradient of the dynamic. This operator... There are a lot of things that exist in this context, and I didn't talk about them in detail, but it is interesting to note that they exist in all the stages.

12:30 When I am in a university in several stages, I have seen a number of universities, here and there, where there are different types of students. And basically, the more we know the students, the more we have a relationship, the more we have a differential operator, the more we have a precise structure. But we have a calculation. We have a staging of relational things, of simulation, and then there is a calculation. We take the place of the calculation, and in homology, it's the same, I have a very general staging, which gives back in a particular way, but also in principle, there is a kind of calculation, which is precisely the fabrication. It is the fabrication of the category of the trees that are in time to grow, and the study of this category. What is interesting for me to understand this story with the discussion is that we come up with a stage set called the stage set, which is related to the calculations in the lecture. That is to say, I do not agree with the stage set made by Guérinot of physics in his lecture. You say therefore, you go, and there will be a stage set also according to the calculations that will allow you. That's it. And that's where it's ... I imagine that in the speech, there is a form of expression that will indeed be ... When we look at mathematics, we see very well that the calculation is played by the distribution between the actions and the rules. For example, I can put the number A, or I can put the number A, and I can say that A is not the same. In the second case, it is a tool of calculation, and in the first case, it is an expression of the process. But, for example, in the work of Francois de Méridon, there is, at the same time, a great limitation in the methods of mathematical mathematics, in the methods of mathematical mathematics, in function of mathematical methods that are completely passive, that are completely passive, and therefore, it is a part of the technique, but besides, it is certainly very productive. But now, it seems to me that with this... It's good to know that we have a very clear and very well-known department, because between the teacher and the students, we have to give ourselves the confidence and the awareness in our meditation work.

15:00 Which is not represented, which is what teaches us. Yes, it's a point. In reality, it is this relationship to those who teach us in which we teach. Thank you very much. Thank you very much for your time. The first step is to present the techniques of mathematics and physics, and above all to present the techniques of mathematics and physics. You have the correct answer in the card. So this can be found in Gabriel. And this can be found in a student of Gabriel in a text that we had already published a long time ago. A very long one. A very long one. And I had never seen it published before. So I have two things in the paper. One, a little bit that I have in me, my thesis. But in my presence, it is a thesis in fact. It is a thesis. It is a thesis. It is a student of Gabriel. I can look at x without that. That's what I call the inner edge. Does it give topology? No, not at all. No, it's not topology. There is no topology at all. It works.

17:30 It gives the edge, that is, the variation, to the inside. And then you have the outer edge in the same way. Because if you increase the epsilon and you decrease the epsilon afterwards, that's always something that contains x. And so I can make the difference again. And then finally, yes, I could have done both directly. So this is like a scale. It measures the scale. It does not mean that increasing itself increases. And so now we can try to solve equations of differential conditions such as Laplace-Zenlis-Egalité, which is a kind of differential equation. At the beginning, it is to understand that there are situations that will be modeled by the constraints and constraints. This is all simple. It is a binary relationship. What we can do with a binary relationship is that we can interact on the parts. And that allows us to make the parts decrease. I call it a regime. It's not a single relation, but a family. F is the family, and a is associated with each element of F. If a is an element of F, a is an epsilon. And then I have a family indicated by F. The second point, which is essential, is that now this family is itself equipped There is a second family, G, of assimilations over assimilations. G, for each element of G, B is an assimilation among the elements of F, which are the names of the assimilations themselves. There is an image of the height of F. There are many examples of this in elementary mathematics. Basically, all the structures are the same. An example like this.

20:00 We can have, for example, E equals F equals G. Here you have points that observe, points of assimilation. These are the masters of those who are assimilated. So an example of this is the plan. In the plan, you take the plan from F. And in the place of E, you take the plan too. And so now you will be able to say, if I take a point P in F, which is the point of view in the plan, from this point of view, what are the points of E assimilable? Well, these are the points that are aligned with the point of view. So three points in the plan P, a point P in F, Q and R in E, If they are aligned, I would say that from the point of view B, the point behind is taken for the point in front because I don't see it, okay? So this is a... So this is a regime that is of infinite height since E is equal to f and g is equal to a. But all this would not be very interesting if we had not isolated for these regimes, I call them regimes or assimilation regimes, if we had not isolated for these regimes something that is a basic axiom that translates in this case... What we present in the examples that I have just given, in the examples that I have just given, if from a certain point of view, you have, for example, if you are on Earth, you look at points a little thick, on Earth you see the Moon, then there is something that is hidden by the Moon, so it is assimilated to the Moon because it is behind. If you move a little bit, it is always the case. The principle of stability. The observation that allows to assimilate will tell me, if I move enough, well, it is still the case that it is assimilated. The axiom here is a general axiom for all regimes and in the situation I just mentioned it is called the axiom of passage. That's the essence. It is from the moment we have the axiom of passage that the theory of regimes is assimilated. If points are assimilated, there is a reason for that.

22:30 So I'm going to say it with you because there it is formally. If I take x and y as elements in E, and suppose that the formula is y here. Suppose that y from the point of view a of i, from the point of view a of i, y is assimilated to x. Well, what I'm saying is that there has to be a reason for this, that is to say, the reason I call R, small r. Small r is something at the level of a frame that guarantees that whatever J, if ever from the point of view of R, J and I are assimilated, from the point of view of J, Y and X are assimilated. Well, that means what I said earlier in the example, but we'll take it back. It means that the reason R is, for example, a small open, I say I and J are in a small open of I, in this case. And of course, what is the case of Paris? It is important to pay attention to the language of the quantifiers. The R is the function of the hypothesis of the assimilation of the three terms. This answers the question of whether there is topology or not. There is no need for topology. There is no need. The epsilons are random. And what holds the key to the work that allowed us to do topology or quantum mechanics is the fact. It is only this action that will allow us to re-do, in the future, everything that could be... So, nothing is forbidden to think about it in terms of visual, in terms of topology, but we will do it as soon as possible for abstract regimes. Regimes, that's to say... I'm not going to go into too much detail on the examples. Now, I come back to the metric regime of the plane, as an example, the curvature. In the context of the metric regime of the plane, the curvature will be able to describe itself, to analyze itself, precisely without differential calculation. So I take as a regime E is plane and F, from the point of view, is R. So here epsilon is really a number, possibly small, but whatever. So I ask that Y is an integral, and if the distance between Y and X, X is an integral. And with that, with this simple regime, I can describe the form.

25:00 So I started to do it here only on the curve. I take my drawing of the curve from the other time and I draw, I see things in a linear way, that is to say, in terms of wave propagation, that is to say, you see, on the black line which is here, in every point, and especially at the yellow point, I draw a circle, and so obviously these disks will have an envelope, so it is very easy to see that this envelope has a curve, which is exactly the radius of the curve r that I had, which we then call parallel lines, and not perpendicular, the key point of all this. The starting point is to develop a theory called the mathematical morphology. It is the analysis of shapes precisely by increasing and diminishing the vocabulary of a special form. If I call x the area that is inside the starting point, the parallel curve is just x raised to epsilon. We can take x on one side, here where I put the cursor,

27:30 which is outside x raised to epsilon. So you have a corridor. And you can calculate the magnitude of this power thanks to the order distance which is defined in the following way. But note that the definition is worth without any reference to the distance. I first define AB. It is the unit of epsilon so that A is in the increase of B and so that B is in the increase of epsilon. And then I take the inf of that and it is the distance of A. As it is an inf, you see, it means this. It is also possible to calculate in terms of the regime that exists on R, which defines the order on R. The order on R is actually a regime on R, so it is a second operator in the regime, and by using the two operators of the regime, i.e. the distance, I get R in the regime. So when I have the distance between two things, for example, here in my example, here the distance between x and e minus 1 is equal to epsilon. But what is interesting to see is that we can see the curvature, we can calculate the distance, we can go much further, as I showed in the previous lectures, we can, for example, introduce symmetry, the skeleton, all kinds of things that can be defined, first in the plane, with the distance, the augmentation, and then after, we can see that this definition works for any regime, if we know it well, so it makes this regime work. So it's interesting because if I have a regime, or in the plane regime, I can talk about the curvature, I can talk about... As for symmetry, I can talk about certain types of singular points. All this can be said in terms of origin. So, I can transport this to another example. Of course, the examples that interest me are the original examples that are linked to the discourse. And so, if there is a discourse that is analyzed in terms of modality, we will see, for example, I can talk about the basis of symmetry in a discourse. I can talk about curvature in a discourse. The geometric curvature and the metric plane, rewritten in a way in return of the differential formalism of mathematics, in the formalism of two aspects of the regime, is exportable on another level.

30:00 I'm trying to see an example of a purely geometric approach. In a way, yes, I'm not going to detail it, but indeed it is the case. You see, because here, look, for example here, I put an external envelope here. If you follow my arrow, you can see that there is an inner envelope here. But if the radius is too large, this inner envelope will have a cusp or a singularity. So, this singularity, which corresponds to a too large radius, is absolutely definable by its appearance. It is definable with the help of this regime, without any calculation. But after, the classification of singularity would not be possible. But already, the fact that there is the appearance of the cusp, like that, it is observable. Modalities. Modalities in terms of regime, because we start from the interpretation in the critical way of modalities, that is, we take a set of works that we call a set of worlds, each element of the work is called a world, we give a relationship, we call a relationship of accessibility between worlds, we call it R, and the data of all this, of the work and R, is called a frame. And when we have a frame, we can interpret it. Propositional formulas, for example, the formula of S4, we can interpret them, each proposition by a part of xf of a, the cryptic semantic gives me that the need is just xf minus epsilon, the possibility of f, you have to be careful, is xf plus epsilon, because I did not know that the relation between epsilon was symmetric. So, in the frames, it's symmetric in general, so you can't see it, but if I take a relation of any kind, I have to put a...

32:30 And so, the question of validity... So, if I have a kind of complicated formula, propositional and modal, I will be able to translate it with augmentations and diminutions, so it's something that's going to happen between the terms. You want one, plus, plus, instead of the other, necessity, necessity, plus, augmentations and diminutions. There have been a lot of people who have been developing this for the past 15 years, to have a multitude of ERs, but to make them work together, and precisely at this point, we will be able to have a complete idea of the idea of a film with an F, to say several... So I just wanted to say a word to explain the relationship between these ideas of assimilation. So you see, you have already understood that in assimilation, what comes from the curvature of the forms and what comes from the modalities will be captured in the same format. On the other hand, I also told you in the past about a relationship between... Assimilation and specular logic. So I'm just doing a little presentation here of specular logic to establish a first link and then I'll come back to it at the end of the talk. So I have a graph and X is a set of objects. So we can define on the table of decrees a graph. A creed is a set of C objects which has the property that if B is in the creed and if there is an arrow from A to B, then A is also in the creed or co-creed. So a crypt is a set of particular parts, not just any. It is a set of objects that has this property. Here I took X, a set of any object. Consequently, I can make, on the one hand, the x. So here, I did not find the b, I put a b, but it is a b. x is the maximum subscript of x, it exists. And x is the minimum subscript, it exists too. Consequently, I have... Starting from a particular particle, I can link it between two fields, a larger field below and a larger field above, which I note as x-b, which is x-th. Here is a mini-example of what I call specular logic. This is a logic in which there are propositional logic operators and operators.

35:00 And in addition, operators such as this b-th and this s-th with a number of properties. Specular logic is the logic plus the b-th of the b-th. And what are the reasons for the R in your examples? We see in your examples this epsilon, the different interpretation of epsilon, but what will be the R? Ah, the R? Well, I can't explain it now, but in the case, I come back to the point of alignment, the example of the plan, the plan of points and the perspective, well, indeed, What do I take as a reason? Well, any point on the axis. I take, I start with an assimilation. P confuses, the point P confuses Q and R. Well, I'm going to take R, P confuses A and B. The sum of the passages in the case of the plane with the point P assimilates A and B when they are like that. You can see that if R assimilates P and Q, then Q assimilates A and B. Consequently, R is any point that is on the axis. If you do an alignment, if P assimilates A to B or B to A, then you have to put R on the right side of B in relation to B. So, in the case of the plan, the action of the passage is verified because you can take it. There is not only one, by the way. There is only one reason. And this example is very different from the topological case. I'll put it at the end. So here I was at speculation and specular logic with the operator Eb and Z, which were used to talk about logic according to different points of view. I will note that this Eb and Z and the fact that it corners x like that, it is quite analogous to what I mentioned earlier with the increase, decrease, decrease, increase of z. And that's exactly why I had made, well, I had first explained the specular logic and then it seemed a little more difficult to certain people.

37:30 This is the first point, a kind of way to hold together the idea of the curvature of the shape and the idea of the First of all, I would like to remind you what the Gaussian curve is. You draw the normal at the surface, perpendicular to the point you are interested in. Then, through this normal, you pass a plane, the red plane, which cuts the surface along a line, which we call a normal section. This line, at the point considered, has a certain curve, x. You make it rotate around the normal. During this time, of course, the line will vary, its curve as well. It varies depending on the angle, depending on the angles that we pass, it reaches a maximum of 1.5. So the largest and smallest value of r, when we do a circle like that, we write them as r1 and r2. These are called curves. And their directions, by the way, are what we call the directions of the curves. But it doesn't matter where. With this r1 and this r2, I can calculate what is called the first Gaussian curve, which is 1 over r1 and r2. So, simply, that's to say that starting from the idea of the curve of earlier, we go to two dimensions. The properties of the parallel curves are easily transmitted by a formula that is a little less simple, but which is immediately derived from the calico for the parallel surfaces.

40:00 That is to say, if you have a surface and a parallel surface, you will have R1 and R2 which will obey what I said about one dimension. So you can easily calculate the case in a formula of degree 2. In one way or another, we don't know how many steps we have to do, because we need R1 and R2, so there is the other quantity, which is the average curvature, which is 1.5 of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to the power of 1 to On the surface M, you can calculate the integral with respect to the surface A element of the formula. So the theorem you know tells us what it is. And this is since, for example, I2M. I2M is the characteristic of the surface. The first way to imagine the characteristic is to say, here is the surface of the fault that I wanted to triangulate. I will now count the number of triangles, namely the number of faces. I will remove the number of bars, the number of edges. I'm going to add the number of tops, and that makes a number. A priori, each of the numbers f, e and v depends on the combination, but this combination does not depend on it, and it is called the characteristic of the n. And then it is also expressed in the form of 2 times 1 minus gamma, where gamma is the number of holes in the surface. You can imagine that the surface is made from 1. If you add ance to the sphere and gamma-ance to the sphere, you have a surface that has a gamma-hole and therefore the characteristic of 2 times 1-gamma-sphere. This is a completely combinatorial theory. It is interesting to see that from the curvature we find something that can be calculated in a combinatorial way by a triangulation. It can also be calculated, of course, in a completely different way than by triangulation, so here it is the notation with the number of bits, which are precisely the dimensions of the group, dimensions 0, 1 and 2, of the surface area.

42:30 So it seems that we are doing something more complicated, because it is not very complicated to do a triangulation, to calculate the sum, but here we have a slightly different approach, we introduce cohomology groups. They themselves can be introduced in various ways, in the theory, we can have a theory in the way of Petit, we can have a theory in the way of Dorame, and, well, isomorphisms in three different theories of homology and cohomology, what I want to retain from all this is that we find a certain information on cohomology from curvature in integral calculus. This means that there is in cohomology, in cohomology groups, There is a theory that I have not developed yet, but that I would like to point out. We can identify curvature as an element. The key point is that by the elementary method of calculation, the way of the scientific system, There are other types of calculations, including integral formulas, that allow us to access the same information on the curvature, so on the shape, not, or at least, here it is not the curvature, but it is a subproduct of the curvature. The characteristic can be qualified thanks to the curvature, that's what we know. If I can calculate the curvature, I will be able to calculate it by calculating groups called cohomology groups and their dimensions and their propositions. So this calculation is quite different, it is very complicated and very fascinating. It is good to understand all the details of this. But for today, we are just going to sit down. Maybe instead of calculating the curvature or specifying the curvature of the objects, we can calculate and specify the cohomology.

45:00 So, here we need to make a little bit of a speech, a little bit better, of course, it's quite interesting to see a lot more information, so here is a first touch, very impressive. Now I would like to come to a very formal description that I promised you, I will try to comment on it slowly. The definition, the homology that I did not tell you about, it is written by hand. By using differential terms, by isolating ourselves from closed terms, by using quotients of terms, or by combinations. Let's say we're going to define the H of M by saying what we put in H of M is a set that has a geometric meaning. So what I'm going to propose here is the opposite of the antithesis. It's to try to understand the definition of homology from the outside, that is, without putting any geometric content at all, but by trying to do the calculation process. For that, I'm going to try to explain the starting point a little slowly, and then afterwards, it will come out a little faster, it will be more formal. The first step is presented here. I suppose that A, B and D are three categories. K is a functor. I'm not saying it's a category because we've had a lot of talks about it in this seminar. So I have a functor, K, which will have to go to B, and another functor, F, which will have to go to B. The first question that arises, which is quite natural, is the question of extension. This is already an interesting question because at the same time we realize that it is not good, it is not very interesting to highlight everything.

47:30 Instead of this exact highlight, we will have the two constructions that are here and here, which we call the constructions by the particle. So this is an extremely important historical moment. The first KAN, or what we now call KAN extensions, consists of two processes, which are dual to each other, to make something intelligent, and it's not something that's going to come out of nowhere. It's something that's going to be... There are two KAN extensions. The KAN extension, which is written LAM KF, which means Lest KAN Extensions, the number of K of the data. And the other one is the right and the left side. These two processes are dual in each other. In the first process here, in fact, we make a function and you see here the little arrow, and in the middle there it designates a natural transformation which goes from the function f to the component k followed by the function ln kf in question. We don't feel in any way that this is an identity, but we put a deformation, but which goes in this direction, from f to the component k followed by ln kf. And the NKF will be such that this deformation is optimal, in a way, in a minimum sense. That is to say that if I had put another one in the place of the NKF, with a deformation, well, this deformation there, through the one that is specified, we say that it is universal. So we will say that we have raised with a deformation in the sense indicated there, universal or minimal. But since we could have chosen this deformation in the other direction, There is also another solution, a second problem, which is a minimal deformation but a deformation in the opposite direction.

50:00 Now, these two quantities, lnkf and lnkf, are called extensions. On the left and on the right, f equals 2. If we really want to make them explicitly, to calculate them, as we say, we will make calculations. We will make calculations by intuitive limits and by subjective limits. In the first case, for lnkf, we will make calculations. It seems to me that this is not the case, but there are a few essential extensions of Kahn. First of all, on the occasion of reading the book by MacLean called Categories of Mathematical Mathematics, he is enthusiastic about what is presented in the book. That is to say, the limit calculations themselves, the calculations of a joint, these things are extensions of Kahn. In fact, it is not true, but it is all the same true in the other sense, that is to say that the extensions of Kahn are also limits, but they are inverse actions. Between the notion of limits, of adjoints and functions of Kahn, indeed, I'm not really right, it's rather the usage that we're going to do, that we're going to choose to be rather extreme. The first indication that I can give you is that, proposed there as an operator of Kahn's functions, on the right and on the left, we're going to say that it's an avatar of a notion that may be more familiar than the notion of adjoints. Translatable from one to the other. As a notion of adjoints, instead of giving you examples of functions of Kahn, you can call them adjoints. Adjoint is a hybrid structure. You start from a set and you make an adjoint.

52:30 The adjoint is the same as the relay. If you want to make a better relay, the adjoint is a kind of better inverse. The adjoint on the right or on the left is a better inverse. With a deformation on the right, a better inverse with a deformation on the left. It's quite similar. If you want to make an adjoint on the left, What we are able to do is the construction of what we call a free group on an alphabet. We take an alphabet, and from there you try to form a group that goes together into a group. And from there you form two free groups, so you take two words, you have a symmetrical reality for each letter, you co-scientify all that in the relationship. The first action is to stop, if we know the notion a little. The construction of groups, we take our adjoints and then we are here with the scale extensions, we will say in a very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very Objects are the factors of A in B. Objects are the factors of B in B. If I have a factor of B in B, an object in VB, if I have an object here, I can associate something here in VA simply by composition with K. If I have a factor here that goes from B to B, whatever, I precompose it by K. It gives me a factor of A in B. In this way, we have defined a pointer that goes in the other direction, from VB to VA.

55:00 The Lanca pointer is an attachment to the left, to the Rham pointer, which is an attachment to the right. So, you see, they are both attachments, but at the level of categories of pointers. Well, the first interesting point is that this Lanca pointer is exactly to the right, it is in conjunction with the limits. I'm looking at the little figure that is in the middle, I give myself, there I explain to you the mechanism of the space, which is in the operator, then we will perhaps better understand why we do that, so there are some things that are right, I have another one that is almost the same here, you see, I have inverted the roles of R and L in relation to the bottom, so now here, I've written down the functions of m in P, m in C, x in C, and m in C. I've written down the functions of m in P, m in C, x in C, and m in C. I've written down the functions of m in P, m in C, x in C, and m in C. I've written down the functions of m in P, m in C, x in C, and m in C. I've written down the functions of m in P, m in C, x in C, and m in C. I've written down the functions of m in P, m in C, x in C, and m in C. I've written down the functions of m in P, m in C, x in C, and m in C. I've written down the functions of m in P, m in C, x in C, and m in C.

57:30 Well, I have the function Lamj, and then I have Ramj, so I already have an operator of the extension of K. And now I'm going to finish by doing the extension of K of this one, and it's a wrong thing, but I don't know. And RL is the function in the left diagram? No, no, on the left. R is a function of P to C, yes. And what does RM mean, then? RM is the function of P inverse M, which is the composition with R. That is to say, if I have a function of M in P. The same thing here, and here the situation is that with L and R exchanged is called homology, that's the name of this homology, but for the moment you have not seen neither space, so we can avoid it. We have shown that we find the same space.

1:00:00 There is something else that I want to understand first, it is the relationship between a category and a printer that counts. The homology of the x and the r is the relative homology of this data. We don't have to put j, r, etc. in the index. It's a relative homology of this data calculated in x and f. In the article, I didn't say that. This is a given homology. And this is a given... We have an extension of k on the left and an extension of k on the right. These are the limits of the projective. I like to present it like this because there is no need to know what it is, it is a definition for the moment. You have to understand that with a kind of calculation, it is written. And that's just what I wanted you to understand. After you, you can put, precisely, the interest of this kind of definition is that you can put for J, for F, L and R what you want. So if we put in there what we can think of in the logic of the class, we put other things. Well, that's the first step, the first definition, but we're going to make this definition more comprehensive now, because there is still an inconvenience.