Adjoint functor

0:00 No, no, it was not a question. I had the impression, if you want, and it is interesting precisely with the forgotten pointer, that what we have is that we have a kind of unique decomposition, but at the same time when we return to the starting point, we do not have the same structure as at the beginning. We still fall back on the problems that we have faced several times. And I had the impression that this had a deep connection with the planet Earth. I think that what I see in this particular example is what we need to add. In this case, you should know... Yes, okay. Can you tell us a little bit about this? If I remember correctly, the universal word here does not have the same meaning as the universal word in the dictionary. Because when I started reading, I saw the universal word in this sense and when you explained it, I understood that the universal word in French is different. Ah ok, so it means that there may be differences between the two. This one is quite simple, but I think it's a bit complicated to say that we meet everywhere in France. Yes, but you could also say that social networks are part of the arts. No, but it doesn't make sense. He is not the one who wants to say that. No, but universal constructions are not called universal because they are not true. That's not the problem, you see. Universal constructions are not called universal because they are not true. They are called universal because each arrow in the arrow is the origin of the last energy. It comes only from the fact. That's why he called them the spheres of algorithm. I am convinced, however, I am also convinced that it has no effect to speak with the inverse, they are simpler in their original form, I think.

2:30 Thank you for your attention. The phenomenon, the fact that on the other side of the spectrum, there is something else, precisely the human procedure itself, I don't know, I don't know, but I don't know, but I don't know, but I don't know, but I don't know, but I don't know, but I don't know, but I don't know, but I don't know. Is it strictly equivalent or adjacent to a concentric tensor? Is it just a different way of saying the same thing? For example, what you give, I'm afraid I can translate it into French. No, no, it's true. The ideas are different. It's not in French, it's in the text. It's not in French, it's in the text. It's not in French, it's in the text. That is to say, a systematic way of providing objects that provide the condensates of arrows, another directive, arrows and objects. It's so much, the cards are so low that they don't work. Go ahead, in relation to the...

5:00 Yes, of course. Is that the answer? Yes, yes. So... Is it... Yes, yes, yes. Is it... Is it... Is it... Is it... Is it... No, no, it's not the same. No, no, you have adjunctions which are not representable adjoins. I'm trying to think of an example. There are examples in tensor categories, but I'm sorry, I can't at the moment think of specific ones. In the first examples which emerged historically, were they... They were the same in those examples, like in the example of the Gauss... You can build examples. Actually, I'm not... I think in... Certainly in the very early example like Freud, the first example after these homological examples that were in Kahn's paper, which were actually really quite difficult technical examples from homology, because he was a pure algebraic anthropologist, the best known example after that was Freud's construction that showed that the Galois connection was an adjoining factor. And in that case, it is, you know, the representable puncture and the junction are the same, but there are certainly examples that came up shortly after that, particularly in tensor categories, where they were clearly not the equivalent, and... The representability issue is, of course, to do with the nature of the functor categories, whether one has a full representation theorem. In the functor categories, for instance, there is certainly the case of schemes, for instance, the category of schemes, it's not representable in the category of sex. In other words, it actually falsifies the Hegelian even lemma in general.

7:30 So it's a very, it's a very delicate issue about the equivalence of records. I would have liked to know if this is what you have been talking about, but I don't think so. That the algorithms also disappear from the universe and that we talk about these kinds of dualities instead of going back and forth. That all the procedural aspects of the universe disappear in the universe itself. The place of the object in the universe or the concept of the universe. There is a kind of global configuration, as Thierry Renaud said three minutes ago, and that we already see appearing in the previous series. I don't know what he's talking about. I don't know what he's talking about. No, I just wanted to say, because on this opposition between determining the opposite and what should be and what shouldn't be, I wasn't very convinced by the idea that there was a clear opposition. And on the contrary, it seems to me, and I think Martin's explanation is quite in this direction, that in a sense, one of the problems that arises is that, because Marco said at one point, and I think it's fair, it's just that in a sense, if one operation is the opposite of the other, in fact, it's a bit the same operation. He said it like that, a bit quickly. And I think that finally, what is interesting is that we see in many examples...

10:00 We found that precisely the places where it is interesting, it is all the times where it does not work. And from this point of view, there is still a strong link between the problem of the inverse in general and the effective construction of this inverse, knowing that it is not necessarily the reciprocal algorithms. In the case of Martin, for example, you have shown it well, in essence, we do not really have a reciprocal algorithm. What we have is a problem of construction of a surplus. And what I find quite fascinating is that there is something that takes place in this type of obstacle, but we also saw it yesterday with De Morgan, who was a theater director, which forces us to rethink the underlying objects. It's interesting to compare the formal presentation of De Morgan with the 1 on p, it's a totally formal thing, and then the problem of the last one, because in a way we can see that it completely violates different concepts of what the number is and what the structure behind it is. From his point of view, I think that there is a deep link between the different problems, and I could not explain it in a clear way, but it seems to me that this question of the obstacle in the return to the sea, It's still a fairly common question, including in effective determination, in procedures, etc. But maybe not an allegorical answer, that's another problem. I would like to add something to what David said about the moments. The moments are not always as formal as they seem. It's maybe an aspect that I didn't... No, no, but it's not... I didn't mention it yesterday. No, no, no, you didn't mention it. It's formal, but there's construction underneath. And I think it's one of the characteristics... The whole idea of algebra and arithmetic is that there is the idea of quantum physics, but they don't want it to be just that. Yes, I was just going to answer that. A formal conception of the inverse would be the one that says... The opposite is just the opposite of the operation of the law, it's just to come back to the same and so in a sense it's a bit the same thing, it's just that, we can see that in Christine's examples, she said it several times, if it was just the opposite, there would be no reason in a sense to write the second stage, the fact that they write the second stage and that they do not choose the same factors, it's quite something, it's quite fascinating, isn't it? It forces us to think what is the nature of the object that is presented.

12:30 It is precisely this question. And I think that there is a problem here. The director, I would not know how to explain it in a simple way, she governs both a reflection on the mathematics and a reflection on the recyclable algorithms. What do you want to say? Apparently, you know what I mean. From the point of view of the theory... In practice, we will always have a problem to solve in the context of the question. We will solve it in the context of the question, but then we have to go back. It doesn't always work. In the case of the question, we don't always come back in the space of the question. It's the same thing in the case of the question. There are cases where the question is not the space of the question. So there are problems of going back. There may have been a lack in these days of exposure to duality in geometry, which is something we haven't really addressed yet. Well, it's still not bad. Yes, but we jumped too fast in the category and in the answers. It's good, it's good, the idea of evolution. The idea of duality is that we go back to duality and normally, if it's good, we go back to the starting point. There are cases where the visual is isomorphic at the beginning and in return is non-isomorphic, even if we don't go back to the subject. What you say reminds me of something that was said before by David and yesterday and earlier by Marco. It reminded me of an aspect of the philosophy of which I will not go into later when these words explode. Because I thought of the physical algorithm and the function that we could give it. and reciprocal operations and the way in which they were treated in tables, and also reciprocal operations, the way in which they were used in equation processing. And among reciprocal operations, one that is particularly striking is the one in which reciprocal and integral are inverse, which of course has a reciprocal function, so it plays a role in buying keys because...

15:00 There is this, but there is also a whole aspect of reciprocity, which has been mentioned a little by Benoît yesterday, since we could relate it to the reciprocal problem, and in particular to what is the ludicrous reciprocity, we could say, that is to say everything that happens in the fluid to create a problem that will be able to... This is a very important problem in mathematics. I will give you an example. During the entire 3rd millennium, the mathematics texts were divided into 7 parts. There is data on the surface and the pressure on the surface. We have the surface and the relations on the data and we will find them on the sides and we will find them on the sides. And this is really the core of mathematics and mathematics itself. These are the problems of the second degree. And in fact we can see the problems of the second degree as problems of the third degree, problems of the fourth degree, and these problems of the second degree have been solved with the help of the second degree. And that, we say, is not in the text. It's the film, it's the... it's the individual side, it's the... it's the individual side, a bit like a book, it's the same, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, there's a book, And so, in a way, there are two different things, one is to ask questions about the problems that exist and the fabrication of the problems, and the other is to ask questions about the algorithms that exist and the systems that exist. I had the impression that you didn't agree with what Desi was saying. That's why I didn't do my homework, I didn't agree. We are going to talk about something particularly tiring as well.

17:30 I also think that I had the impression to see the report that Jean-Denis wrote, that is, the fact that we were questioning the opposite and specifying the effective construction of mathematics. It may be that we are talking about practical concepts or not. The question of algorithm seems to be open to us, it seems to us that it is a stage of development. Algorithms and other people have put a lot of emphasis on the subject, so it seems to me that we should not forget that in the subject of the day. And there, perhaps, we were a little too focused on the fact that the algorithm is not so important. I don't know if it's a historical effect or if it's a mythic effect. We have followed a historical, chronological thread which, in my opinion, is still quite unified, as David said, in the problem of the effective calculation of the universe. On the other hand, the algorithm, we have lost sight of it, and I don't know if it's a question I ask you, I don't know if it's because we haven't necessarily looked at the good examples or the conditions, etc., or if, indeed, this problem tends to... I would like to ask you a question before we go into the questions, because it seems to me that we have had a very long period of time, and it seems to me that the problematic of saying the two things that we have been talking about, the problem of the current and the fact that they are connected to each other, is another problematic that falls. In general, this is the construction of the study of mathematics in the sense that it is part of the study of mathematics in the sense that it is part of the study of mathematics in the sense that it is part of the study of mathematics in the sense that it is part of There are a lot of things that I don't know. Otherwise, of course, there are a lot of things that I don't know.

20:00 There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. There are a lot of things that I don't know. And the fact of seeing everything in the reality of others, without really having to adopt the two perspectives of mathematics, and I thought we found them. So for me, I think there is something probably quite tangible in this political reality. So I understand that this is a high-level question. The conception of the universe and the evolution of the universe is written only by a person who can speak the universe in the form of the economic structure of the universe. I hope you understood the question. If you want, it's... What does she say now? Algorithms in physics can be taken from two points of view. I will repeat what I said with the two examples. Either we can say what is the algorithm of Hilbert space, as in the second part of the question. In the second part of the question, we have two algorithms. And we can say that they are not really the same. The problem of the quadrature, the problem of the tangent, the problem of density, if we notice that there is no majority in the class of the language, are there other analog situations in mathematics or is it just that? And above all, it's not so easy, Marco, because I know what I was trying to say earlier. As a result, Marco did not say that. He said that there is a general problem with the inversion and a particular problem with the inversion. What I try to say is that this is not exactly what happened in this journal, that is to say that the problem has not been treated in a general way, we have not done all the forms of inversion. We were still interested in certain aspects of the inversion that are related to the construction of the universe.

22:30 And this aspect still plays into the question, and that's what I wanted to say, that is to say that in the end, even if these two points are separated, it is not as if there was a general problem that would be that of the nation, and a fundamental problem that would be that of the United States. These two were passing, but they are still in dialogue, and after that I agree, that is to say that we can see... What I find is that there is really a big question point that interests me a lot and is related to the question of universal problems. I have the impression that in order to really have a good notion of the reciprocal algorithm, in a sense that we can have an inversion step by step. No, it doesn't make much sense. And to have an inversion step by step, we have to have a good... We need unique decompositions. We need to be something like a university. Otherwise, we'll never be here. And I think that's something that has been said many times in Marie-Josée's exhibition. That's why I asked the question about mathematics. And I think it's a rather fascinating problem because it's very rare to have unique decompositions. And we think we have some, but in fact we don't. And when we have some, we've got something. I think this is something that has emerged a little bit, I don't know how to put it, but I find it quite fascinating, from the example of Christine, we have this problem. If we really want to be able to talk about algorithms, in the sense that Marco talked about it, we really want it to be the same thing in English, exactly the same thing, and that is very important. The initial algorithm is composed of very simple preparations. Each one is inverse and we go through the inverses upside down. I will try to give an example that is not too explicit, in which we had an inversion almost by part. In the case of global use, we mix the two pieces together.

25:00 Here we have an example of a cross-sectional, step-by-step conversion, but in fact it is not used to go to the next level, but to go exactly to the stage of a global situation in the bank, where we can go to the next level, rather in a state of duality, in a life. Thank you for your attention. There is also a chapter on the use of mathematics and geometry. It is the basis of the use of the two fields of mathematics together. Thank you for your attention. I think it's very important that we focus on how we have and how we determine the value of all of this, and we can also focus on what is the question of the issue and how we implement it in the future. I think it's very important that we focus on how we have and how we determine the value of all of this, and we can also focus on what is the question of the issue and how we implement it in the future. Thank you to Dominique for the organization.

27:30 There are several people who have not been able to attend this workshop and I would like to know if it would be possible to, I don't know, to have a talk with them. What you could do, Christine, because I think it would be much easier, is to send an email at some point saying, do you have any references? You make a video, you put it wherever you want. I don't know what you're talking about, but I'm thinking about it. But I'm thinking about it. I think it would be good to put together the documents. I think it's really important to be able to look at the documents. If you want to do the photographs of the course, you can do it. I hope that you will be able to offer your own diaporamas, your original texts, your biographies, and you will be able to put together the documents that you have. I would like to remind you that I have written a book which is called... I have written a book which is called... Thank you for your attention.

32:30 But you work out, don't you? Well, yes.