Non-Abelian / AnAbelian, Abelian homology (contd.)

0:00 And so on, to see all of this, we have to know what we put in place of JLF. So, now, the goal, if you like, I'm going to try to make a little pause in the explanation, is that for a long time I was not satisfied with the definition of homology, The concept of the means of calculation, that is to say, we were always saying, we did this calculation and what we calculated like that is called homology. So there was not a definition of homology that is conceptual in advance and that tells us afterwards, if you want to concretely calculate this concept, you can do it like this or like that. For example, classical equations in quantum mechanics will take injective resolutions. So we say homology is if we calculate, we take a resolution, and then we calculate, and then the rest of it is called homology. But what is the concept? That's what we call mathematics, because it's just because it has a certain type of calculation. Of course, it was shown afterwards that it was independent of the chosen resolution, but nevertheless, to say it, it was necessary to insert it into a calculation to be made. So that's what I preferred, I wanted to have a topological definition in which this default, so to speak, is eliminated. It's not yet the case of this one, but it's the case of the next one. It's the same idea, even if it's in both. So what's the difference between P and C? Between P and C, that is, you have two P and R, and what's the difference between the left one and the right one? Well, you change the roles of the P and R. But it's just the same number, because it's two P and C. No, but it changes. First, you change here, for example. The role of L, given that L is the same data here, but written like this, because if I write it like this, I want the RL to appear.

2:30 I have RL here, so I have RMLN, and here LR, I have LNRN. I wrote them like this to prepare the eye to see what to put here and here. This is homology and this is cohomology relative to the same number of numbers. So here, if we had the opportunity to demonstrate that the homology of classical algebra can be found in a few formulas, we would have already won a little in the sense of what I say, that is to say, we would have already won that here we have a definition in which there is no resolution, objective, there is no particular presentation. Well, there is still a calculation because there is the calculation of Han and Han. So here, I must still have this operator of Han here and the operator of Han here. So it is still presented in Sheppard as a calculation. A calculation with in addition a calculation within a calculation. But maybe it's a calculation that is already more objective. So now I'm going to take these two formulas here and just try to present better what is written there. The RAM of the soul and the LAM of the soul by eliminating the recourse to these numbers. So I resume the H-star formula here and now I'm going to present them as one. There is simply a certain projection limit and the other a certain non-projective limit. So the ingredients that we had earlier. So earlier there was LR, we see better.

5:00 So in fact, I now consider what is fixed. Here is fixed all around. From this, I create a category. Each object is called like this. I consider the category in which an object is such and such, a deformation here, then from one side to the other. Now, between two objects like this, you will have a new A', B', you will have the morphisms which are given by the natural transformations that I do not write here for you, and from A to A', from B to B', you will have the natural transformations which are common with all those that are there, and that makes us a category. In order to determine an object, I can associate it with an object of the category C206. So now I have a counter of the category C206, which is defined without any calculation. If we take a complex, a small a, a small b, a small b, a complex, I'm going to think of it, in relation to the question we have in homology and cohomology, I'm going to think of it as a cycle, to be able to develop it more, to better understand it, but that's a bit of an informal idea, that is to say that it's these entities that get along with each other, these entities, small a, a small b, a small b, get along with each other through morphisms, and if we make the limit of the arrangement, then why?

7:30 Why is it so easy? It takes some demonstration. It's not very difficult to demonstrate that what I showed before is the same as this. Now we have to show that this is the same as before and that it is the same as before. Is it functor as functor obli, that is to say that you forget all these things on the left? The idea is that B would be the best thing we can put here, it's the limit. Except, of course, that there is a contradiction between the little a and the little b, they do not give the same universality. I will show you now how to find classical homology based on this, how it is located. All this is not in the paper, it is in my head for now. I have not yet articulated this, but this is it. The rest of the work would be to understand homology by passing everything to the limit here, The homology is a projective limit, no, inductive. I don't know why, maybe because it's a reversal between work inside and work outside.

10:00 How do you define a lamp? Do you send all the objects in X? All the objects? Well, each object. An object is a big thing. Yes, I understand. And it has an element that is B, so it is simply B. You have to send in the... All these objects are sent on B. But how exactly are there several possibilities? No, no, B is an element there. At the quadruple A, small a, large a, small b, large b, I associate B. B which is in the quadruple A. So it's the word. Maybe a word to see why the classical thing is here. So what I was saying here is that we have the situation at the beginning, and then we actually find the homology. When we put the category of extensions at the bottom, for example TOF, here M is a model category. Of course, if we want to localize, we have to take X, we have to take Z, we have to take an object X, we have to take Z. Instead of having M' and M'Tof, we have in fact... Yes, but now they are explicit, the choices are hidden. Now we understand that each choice will give a theory. No, because there is the choice of choice. They are all put together, they are put together in a kind of bimodule. There is a double choice that is now visible. There is a simple definition and there is a choice that is made in advance and then there is a simple limit to calculate.

12:30 And so here the choice to find ... Yes, but the choice is unique. There are several levels of choice. I will look at the analysis of the L-computer. So what is that? It is a choice in fact, but it is a choice of a way of analyzing the groups. I decide that the analysis of a group is called analysis. It is a presentation of this. But it is completely abstract. It is a presentation. And above that, there are C presentations, but they are also co-presentations of A. It's a game of calculation, it's a way of calculating the presentation of objects. Here, it's a bit the same thing, in the sense that there are spaces. The mid-levels, for example, can put things that are a little less general or complex, and put here, on the other hand, very particular spaces. And then the idea is that, indeed, if we put spaces here and here in equality, it is also a presentation because it will allow us to analyze all spaces as a... There is a collocation of symbols and other objects that are in the same place in another sense. And precisely, the idea of presentation that we have for spaces, which would then be the idea of presentation by quotient, so in both cases, it is still not established concretely, it is in a different order.

15:00 The purpose of the game here now is precisely that we establish a partial link between a first level of analysis and a second one here. The partial link is the EF. And then we try to complete this link. And that's the point here. But in a certain way, the category, the big category that I had before, which has as an object these things or these things, is a way of showing all the ways of extending the perfect link between the two calculations, between the two modes of presentation. So indeed, there are presentations all over the place, but now I don't need to execute these presentations. To know what homology is, it is a homology that will apply to one way or another. There is still a kind of reversal of the steam. So here, we can be even more precise. We can put here x, in which case the h we will obtain will be h1. I can put here exactly the length n, and I will come back 20 times here by taking the two extremities, hn and h2. So I can clearly put here a direct sum of the categories and variables and obtain the H-star which is graduated in this case. That's the idea. So now we are in an analytical environment, but what I wanted to show you is that there is a way to arrange a comparison of the two. Think of homology as something that is an analysis of the comparison between two analyses. An analysis of these objects, an analysis of these objects, and I try to compare these analyses, the way in which I presented the homology, we can use the homology as a tool for that, a tool to make this comparison between these last tables.

17:30 It is the specular logic that will present itself in exactly the same way, but obviously with a border that will be completely different. These are the two figures that we are going to see in this case here, in this case here, in the problems of transport of information in a specific logical context. I will quickly explain this context. I have a category S, the set of objects, and P, a set of objects, and then Q, a second set of objects. These two factors now give me restrictions on the sub-objects that I still note EP and EQ. If I have a sub-object, a sub-vessel of E, by restriction, it will give me a sub-object. So, I first have the idea that I have the sub-vessels of E, I have the sub-vessels of E restricted to P. And likewise, at the level of E, there is EQ. So there are no more arrows, but a set of objects. So EP is a functor, it's just a family, a set of families. So these two restricting functors at the subject level, they obviously each have an adjoint to the left and an adjoint to the right,

20:00 which I have noted, dsp, bb. So these right adjoints, precisely, I will be able to put them in the situation that is here, because a way of calculating the limit is possibly if I can put universal things in the general situation. In this general situation, given that it is just I, F, N, in general there is no universal A here, there is not a very large I, F, it does not exist in general. Precisely in the classic example of homogeneous homology, I have X here, so there is no universal thing here, but if there was a universal thing, I could take the universal here, and if there is a second universal here, I could take it here, and I would have homology here directly as an object, as a flag here. This is what happens here in this logic calculator. I take a logic operator. Simply, I take a set of sub-objects of E, P, as if it were a ball. On this ball, I take an operator, a logic connector, for example, omega, a Paris-Pierre connector, to what I have here. Now, by using these computers, in this case, by composing this by this, and by composing here by the joint of the square, I get a computer here. And now, again, Because in this context, there is a universal development. I have the same thing on the other side. What am I doing? I take an omega operation. Omega is a classical logic operation. But at the level of E. This classical logic operation, I go back to it at the level of E. E, precisely, is a pre-messo on S, so in S there are arrows. So there are coherences that are required. The sub-objects of E do not form.

22:30 We have a boolean algebra, but not a boolean algebra, because there is coherence. So I start here, on the left, from a boolean operator, and through this universal property, I bring it back to an operator that is not boolean, but which is a universal relation, in a way, of this operator at the level of the boolean algebra. And then I bring it back down to another boolean algebra, the one related to Q. So you are in a situation of transfer of information, you have S, here you have an assembly B, here you have an assembly Q, there are arrows everywhere in S, but when you limit yourself to P, you are in a calculation of P, and here you are in Q, but in the calculation of S, you establish a relationship, you extend it, which is analog, What we obtain here is a sort of homology. It's a bit like if the omega was deformed, it's a sort of homology in condition of an omega. But here the situation is quite particular because we can use the usual intermediary. With all this abstract thing that is at the end, having understood that in homology, in this form, we will be able to imagine many examples, All of these are based on an IJLR that is specific to them. In topology, for example, IJ is the topological states or groups, and LR is the calculation with the abelian groups or with the main historical modules.

25:00 To compare the two is that the second calculation can be done. An abelian calculation can be done, but a topological calculation is more complicated. So, if we stay on very general ideas, we could imagine that for a modeler, he will be able to apprehend, to give back, by saying that there is a world of things that he does not know, that he should know, which is a presentation of objects of the world, so a kind of world of music, in which objects have a certain presentation, and this presentation of objects... The objects are in X or in Y, in both. And at the level of M, there are representations. You see, you have to try to work a little bit in a metaphorical way with the example of X that I gave earlier. But the idea is that what is at the top is world not known, it is world. It is what we must hear that we do not know. And then, on the other hand, we can try to put down something that will be much more, much easier, known. As you like, when you do the jury of the groups, and then the jury of the groups, it goes well, and the homology of the groups, in a certain way, brings the unknown to the known. So it is in this very important method that I propose to you, depending on the turns at the end of the lecture, and to say, well, if you have to do a work of modernization in music, you will put at the top the presentation of your creed, of the situation that you want to envisage. In the first point of analysis, you will put something that may not necessarily be of the same kind, but that has the ability to be calculated. And then the modeling gesture is to choose an x and then to build relationships. So all this detour that I was talking about... But it was a way of concluding now, a way of saying that, well, you have the modality of speech and the composition of figures, it can be unified in this somewhat elementary apparatus that I presented at the level of assimilation.

27:30 Now, I barely sketched it, I barely talked about it, these questions of curvature are found in homology, and we might think that homology is a kind of calculation of the general essence, but there is so much more. There is no possibility to present this in the mathematical field to say that we have cohomology, it is so varied. But from the method point of view, we can still imagine. It is to be able to imagine this as a method that I find... What I said about topology allows us to depart from the specific technicalities of Hegelian topology, to see behind a kind of modeling gesture, like here, the comparison of two analyzes. And it is clear that this way we will not make the mathematical knowledge of topology progress in depth, but we will make it progress enormously, I think, on the surface. We are given goals that may not be as interesting as we would like to see, but of course it changes the whole ambition. However, this progression on the surface, which could very well be expected in mathematics, is not very important. On the other hand, especially in modernization, I think it is more important than the progression on the surface. More important because it allows us to enter without techniques or with techniques at night. In something that, for example, the two figures up there, which is the manipulation of these things from, perhaps nothing, that is, from what we want, we put in order what we want, in order what we want, and then we start to make comparisons. So doing it, or what I just told you, is a guarantee that in this practice it is not empty, since there is this particular case of thermology.

30:00 Now, apart from this guarantor, we should not exaggerate and believe that everything we are going to have to do here will have a logical sense, it is so extended that it is a sense that is probably different. Well, that's about it. Well, I have some very general questions. I was thinking about the proximity of these empirical ensembles, because in the case where the objective and the objective are symmetrical, and the objective and the objective are the same, we don't hear the calculations. So, if we don't have the objective, if the relationship... By the way, among the regimes, I had as an example, as an example I have children in my class, only, which actually correspond to a two-stage regime, the second stage is the description of the topology on the whole of the observers, and then the first stage is made up of a family of relationships which are actually reflexive, especially the reflexive ones. In all cases, what interests me in the example of general assimilation regimes, I think it is the passing action of the contribution of a kind of strong and original because it allows to have examples very different from those of the past. Nevertheless, in these examples, something must be said. So, the empirical ensembles of Benabou, they have been a little worked and generalized by Borseux.

32:30 I apologize for the Italian name, but this is an artist, an author, who has generalized a little bit in terms of mathematics and even symmetry, for example, and kept the reflexivity. So all this will be examples as well. So, in relation to that, the theory of assimilation can no longer be topology at the second level, and above all, it can no longer be very flexible. That's what I find interesting, because when we study a regime, the regime is a bit like an idea. What I expressed at the end on these configurations as a kind of vignette to modelize is that you put what you want in X, M, Y and there you have what you have, no matter how complex it may be, it's your concrete situation. On the other hand, in T, C, P, you have created something with which you know that. So here we are a bit in the same situation as in these regimes. My idea is that in a regime... What's in there are relations that are a priori necessary because I want to observe them as they are. It's not forbidden that I have between the relations of assimilation completely ludicrous relations. Because if I want to analyze, for example, in a speech, from which point of view such and such thing is said, I have to admit that from these points of view, the points of view are made up. All of these are taken at the level of the calculation by a relation of the sequence, by an epsilon. But I have to put it in the point of view of the point of view. Or, for example, the point of view of the systematic point of view of the spirit. Someone who, every time you tell him A, he tells you anything but A. Well, it will be seen at the level of A. That is to say, it can appear in a discourse. And yes, I understand that because the type has a point of view. However, this point of view is not effective. It consists in assimilating the x to anything except x, in a way, in the work of... So I don't want to put limitations on the epsilons, because the epsilons are experimental data, which are precisely what everyone is looking for.

35:00 And what interests me is not an axiom on the epsilons, but an axiom on the overall management of the epsilons. And that's the whole point. Because I need that for my theory and mathematical theory. The elements are not the same. On the other hand, there must be a kind of general stability that reigns within them, and that is the rule of passage, the continuity within empiric systems. So, there is a kind of recreativity that can be... So, there is a young woman who did a thesis on injustice. Her name is Moniz-Sassier. Thank you for your attention and see you in the next lecture. And so she made a quite interesting thesis on this, where she enriched it. So she used this thesis on the simulation of her text, but after that, she made a much more specific model for the analysis of the text that she did, thus using, on the one hand, these regimes, and on the other hand, the works of Pierre Achard, who was one of the first to decide, and now he is one of them. But there was also a social model, the universe of Beecher and so on, and it was necessary to ask ourselves, why do we think that? And Pierre Hachat and I had discussed what he was doing, and in fact, it is also a bit because of this discussion with Pierre Hachat that I brought this little model of assimilation. There is the motivation to have another model than the specular logic. At first, Pierre Achard explained what he was doing in an academic lecture with empirical instruments.

37:30 And I answered him, but if we try, if we try so, or if we try so, there is the specular logic. I explained to him and he understood because he was very well informed. And so, in his point of view, these regimes appear as an extension of empirical ensembles, and therefore, we take the same services. Well, after that, Monique Sassier, who was a student of hers in Deva, when she passed away, made a commentary on it. And she made this thesis, using the regimes for the analysis of speech. There is a book published by Larmattan, which is a very nice book to read. I don't understand very well. I don't know if there are any links between the two parts, the first and the second. There is a link, but it is not complete. There is a solid link, but we will have to develop it further. Do you think of this extension of Kahn as the case of this epsilon-up and down? For the moment, there is a link that is the following. On the one hand, there is the theory of simulation, on the other hand, there is the general homology of the two tools. For each of the two, I tried to show that there was the possibility to understand with that or to understand in that something that is both on the side of the figures and on the side of the discourses. And for the simulation, I tried to do homology too. In homology, I have other figures, but I won't go into all of them, there are many more.

40:00 And what I didn't do, but which could be done, is very interesting, because it's rather two tools in moments of simplicity. The tool of assimilation remains a relatively elementary tool. In homology, it is more sophisticated, because the way to develop it, to present it even, you know, it's a bit hard, but it's by... Here, we really have to... This is the context of the categories. Otherwise, we don't really know how to say it. In simulation, at the end, you can do it by hand. But otherwise, I think, yes, there are relationships. By the way, you also saw that in specular logic, there is a small relationship with simulation, and there is also a relationship with topology. So, I haven't finished establishing the relationship, I'm going to expose it. But indeed, it's part of what I want to support. It's not necessarily at the level of the connotation of the formulas and extensions, it's more of a gesture of numberization, in some cases, since we have... I'll give you an example of a known world. So, down there, in Sacha, for example, I don't know if you can see it, but there is a mathematical theory of some numbers, space, and we have two poles, two lines. So, is that what we have? What would we have? So, a known example... I don't know, I don't know, I don't know, I don't know, I don't know.

42:30 This is what we call the plan, which is a structure of collation and fluidity. So this is what Benioff did? Yes, this is what Benioff did. He made a kind of staging with a category of laptops. Well, not laptops, not topos, but laptops. For example, on a laptop there is a topo of music, and then in the topo of music there is a topo of music, and then in the topo of music there is a topo of music. First of all, there is a problem of knowing how the objects are placed between each other, how they turn, how they move, and all that. It has to be presented, that is to say, it is necessary to create a sub-category of topology, in terms of the environment, which is a simpler object. So we are in a situation where we have a big category. We will see how the objects are linked from the unknown. For the F meter, which is a model, as the objects are known, it is a collinear. It would be the opposite. On the other hand, it is not necessarily the same. The same is true for groups. A group has a model that I do not know. But maybe in other music, there is a part that is considered as unknown, that is considered as... We know how it works, how the objects are linked. So, a known part of the object. Here, we can talk about the unknown. In this way, we fall back into presentations of the anthropological or classical genre, which we could not forget.

45:00 But I don't want to say anything because I don't like it. For example, I considered it in... What interests me is not to apply this kind of discourse, but to try to do these things in the discourse. So, as I said earlier, the year of the discourse was the year of the assimilation, so the things that were done, In the last lecture, we created two discourses, the purple one at the top and the white one at the bottom. So the discourses in which I speak, you called them in X. Yes, so the corpus of the discourses is in X, and then in M. Anyway, I didn't analyze discourses with that, I analyzed them with assimilation. I said with 8 minutes in terms of assimilation, not in terms of logic, not in terms of... In terms of... not in terms of cohomology, but if now it's... Your thing before, you told him that it's pretty much the same thing that you did earlier on the logic, the graphic before that. No, not that one. It's a little different. You told him that it was pretty much the same thing, just at the same time. Yes, yes. The one there. It's the same thing. It's always the same thing. Yes, it's always the same thing. It's always the same method. No, but what I mean is that the... I answered earlier a question about... ...assimilation regimes, so the first part of the study, that would be used to analyze a lecture in terms of the point of view of the assimilation of... Now, I say that... I answer the question that maybe, indeed, we could imagine that this device there would also be used to do a major analysis of a lecture and... Before attacking the concrete discourses, I thought about attacking the concrete discourses of the students.