Double categories & role in physics (contd.) / brief remarks — Morphism (& others)

0:00 So we have this notion of limits that exists in literature, in particular in the articles that I cited in Richard Erasmus and the existing theorems and so on, but there are a lot of people who have given theorems and those that are in Erasmus are not exactly original theorems, it is an article that has been published by Erasmus on this subject, on the limits, and it refers in particular to the theorems of the Australians, of John Gray as well. Why are these terms so interesting in relation to what I was just saying, and in particular in relation to this term of low plunge? Here you have seen at some point that we have constructed botany. I have not said what it was made of, you do not know what it is. Well, we can define it by a universal property, this logo here, with a filter. If you have an R4 filter, you can try to associate something that is de-universal, de-universal in the sense of adjective. In the inductive sense, if you have a fiber category, you have different fibers that are included in the fiber category, categories included in the big category, in the top. This diagram is a limit axis, describes the fiber category as the limit axis of the different fibers. On the other hand, the same fibers, you can try to group the axis in a projective way. So there is also a solution to this problem, the fiber category. In the case of effibration, I was using an inductive limit axis, an inductive limit axis. Now, if you want to make a projective limit axis for the fibers, you describe the cohomology classes, as you can see here, of the associated fibrations. Here, the first cohomology class of the fibrations is described in this way, as the projective limit axis. In any case, the remark of cohomology is described in this way, it is very ancient, it dates back to the 20th century.

2:30 I do not see that there is a variety of things in this case. No, he makes the cohomology of what he calls an operational category. We assume, as we have the example at the beginning, you have a group that operates, so you have a group G, and then you have X, a G-module, and you can look at the cohomology. All of that. You can use the G to represent this module. That's what Erisman notes. Instead of having G as a group, G as a category, instead of having a module that is simply a source of this category towards something, towards 1, towards 4, and of this animal there, you can write the analog of... On the categories, right? Or on the categories, yes. There was no general expression, there was not the whole side, the pseudo for example, but there were also categories that operate on categories but in the sense that they operate strictly according to that particular definition. Which means that we also recover, with these notions of the limit axis, we realize that if we also reflect on the waterfall construction that I told you about. There is a way, although it is a bit obscure, but it is described in the book of Reisman, there is a way to interpret this industrial construction as an academic place. So, it takes a bit of time. There is a tool that we can relax, that is to say, with which we can actually make the interoperability of mathematics. It is not as if we could do it by hand. So, I will stop there. I have finished what was asked. That is to say, to write, and to hope, to hope that we will improve. Ah, but you mentioned that RS1 was applied in the subject of relativity at the beginning of RS1. Not that. It's the abstract. Ah, ok. And then it continues to be interested in abstracts and abstracts. And from its interest in abstracts, it becomes interested in the category and all that. Maybe you mentioned that this theory seems very long to write.

5:00 On the other hand, there is a very, very popular theory in the IGA which says that if we take the limit of the category that generates the generators, then the greater category generates the generators as well. This is very, very important. What does that mean? What is it? In particular, if we take the reductive limit of the category that generates the generators, then the greater category generates the generators. As in the generator, it seems to me that everything is related to mathematics. Yes, it's related to mathematics. But it's a very deep field. It's a very deep field. Yes, it's a very deep field. Yes, it's a very deep field. Yes, it's a very deep field. Yes, it's a very deep field. Yes, it's a very deep field. Yes, it's a very deep field. The good way to understand this theorem as something trivial is to understand it in a context of accessible categories, that is to say that we are talking about categories that are described as in themselves categories of realization of the esquisse, and when we make limits of the categories of accessible categories. I don't know what I said at the beginning. No, it's not what I said, it's what you said. It's a very good point. The question of accessibility is another matter. What we are trying to do with the text is to try to understand, not by claiming that the tools we use allow us to understand, but rather by trying to have tools that have a guarantee of interest There is a kind of simplicity, a kind of naturalness to be modelled, to be made concrete in the case of the parties, that is to say...

7:30 But I haven't said that Marmar is a direct relationship with... Yes, it's a relationship in the sense that Satie and the theory of these abstractions are interesting and powerful. What they think in their case is that they have deep knowledge, right? Thank you very much for your time, and I hope to see you again soon. People, for example, Kran, but we have to deal with this in other circumstances, with Hegelian, Euclid, and so on, try to make models of mathematics that are more or less a category of things. So these people are quite good at these general considerations, but they don't have the notion, and physicists, perhaps Kran and Kran, although they are very clever and experts in a lot of things, but they don't have the notion of a natural and simple side of certain things. That's what I'm saying. That is to say that when he is going to model, he is already too worried to put all the rectangles in his model. Whereas if he would enter into a general procedure of this kind, he would put quietly what he wants, it would work the procedure, and at that moment there would be a level of abstraction, which is obviously raised for his own taste, because his taste for abstraction is great. Because you see, instead of modeling with the causality of this, he gets bored. So we can say that he creates difficulties in his text to put the axes back. As I was saying, there are arrows of causality, arrows of inclusion, there are these blocks. So now, wait, I'm going to put you in a position, but in a moment. First, this kind of machine, I come across, with very concrete things, which are defibrillators, defuncters. So, it is very clear that the passage of this description of blocks to these things has something, precisely, natural and at the same time easy, precisely. So now if they knew it, they could say that suddenly, in these blocks, they put anything, or they would be more attentive in some way to put it in these blocks, first of all to consider the blocks, and to put there things that are more sensitive, because they could separate themselves from having the problem beforehand that it is mathematically interesting. I think it's a matter of approach, a matter of method, in modeling. Maybe not so much, because...

10:00 No, I agree, but... You have to work on both. Because causality, either you apply it to points, in this case causality is perfectly fine, or you try half-finished. We have 10 minutes for this thing. We can come back, in fact, I wanted to say something absolutely simple, more general, not related to... It's not direct remarks. But I think we still have to say some things because... As long as there are questions, we may be able to continue. Yes, go ahead. We will be very quick because I think when we talk about, let's say, the introduction of physics to the category language... I'm going to look at the facts. 10 minutes, go ahead. Damn Emma, 10 minutes. There are some facilities when we talk, when we translate the relations to the little guru language, but I think we really have to be careful at this moment, because we can lose some kind of physical sense to this kind of high mathematics. Well, I would just like to point out that, let's say, relation and category and morphism are not at all the same thing, and when we say that we can easily translate, let's say, relation of causality into language of morphism, language of category, I think we have to pay a lot of attention here, because, well, there are absolutely obvious things, because what are these relations, let's say, in the sense of morphism? Abstract relation is something between A and B, it is a function in the value of truth. It is still something abstract and in addition it is something not defined for A and B. We can think of it as a concrete object but it is defined for this ensemble.

12:30 We can have the same relation between different things. There is a very concrete object, which is E for A and B, at this moment, if we say that relations of the order can be represented as categories, it is necessary, in my opinion, of course, but if we say that there is a category, however, it is not quite the same thing, and if you want, all this idea that we can describe objects as structures of relations, And another idea that we can describe what is the object of what René talked about, let's say the structure of morphism, is in fact a very different approach. It is always this thought that we have to replace A and B by several other things. In morphism, we cannot replace A and B. What we can do, if we have a, let's say, function in the arithmetic in this way, that is to say all this... I will not talk about it today, but I believe that all this thought is structuralized in the language sense of mathematics. The idea of thinking differently, it's just to say, also in the philosophical sense where we always hear, let's say, category, morphism, relational, etc. Maybe, but in the precise sense of relationship, it's not at all the case. But I don't want to talk about this generality, about causality. I believe that we do not have the right to use this lag of morphisms.

15:00 There is another thing that I did not understand well. If I understand correctly, the idea of ​​grotendic topology. If you want, from a conceptual point of view, to detach the myriological structure, the structure of parts, and the structure of covers. That is to say, properly topological. If there is only one morphism between two objects, in my opinion, it is not worth talking about hypotendic topology. There is another theory, if I understand correctly, it is called locale. Locale is a particular case and it must be done there. With all these words, hypotendic topology... The mathematicians can only correct me if there are multiple morphisms between two objects, otherwise it is trivial in the classical case, it is not at all the trouble of pronouncing them all the same. But at this point, we can really think, well, I asked the question again. There are examples of very simple morphisms, where there are only 16 morphisms, right? In the visual environment. In France, only the open and the closed are considered. But it will be a classic case in the sense that... Yes, topology, but... For example, in France, the open and the closed are considered. So an open and a closed meeting is not open. So it's not topology. Ethnic topology is a very simple example. And yet, there is only one arrow. In any case, we can think that there is a much richer category of situations in the case of... But in my opinion, it is different. The notion of relationship and the notion of morphism, which I said earlier, is an abstract thing, that is to say, it is a sort of function in truth, value of truth, and here it is something concrete. In my opinion, it may be interesting for physics, because exactly that we can think of as a physical process, transmission of information or something like that. That's the possibility.

17:30 Yes, but let's say, from a more formal point of view, if you say, well, there are two... There are two events where there is a relation. They are in effect different causes. Precisely, it is when it applies to events that the causality relationship is as simple as it would give false. Precisely, if we apply it to regions of space-time, it is no longer as simple. And precisely, we had the debate of knowing, you can say, two events are causally linked if one, apart from the reunion of the codes of the past, if it belongs to the other, it is not true or false. But there is another problem, because we have to think about what is the object of these moments. And in my opinion, it is a good question. In other words, we can formulate what meaning we can give to an identity, what physical meaning. Because, well, I think it's something for the event, not just for the point of the lectures, but for the event in its entirety. And at this point, we can think of several possible processes. We have to stop, but I think we haven't finished on the possibility. Thank you. In 15 days, then. Yes, so the next one will be in 15 days. For the 8th of February, we leave at 10.30 p.m. René, do you have a moment? Do you have a moment? Ok, thank you.

20:00 Thank you for your attention. It's easier, it's faster if I speak in English. Yes, but there is a risk. No, no, no. Your professor, is it possible for me to see your little transparency? It's possible, maybe another time. And also, can you give me his recording of yours? Yes, he gave it to me. But alas, I have a little problem with the recording. The first part is not a problem, it's perfect. But the second part, which is with the discussion, I don't understand why, but is it possible for me to talk about it for a moment? No, no, no, it's possible for you to give me a hand. You can give me a hand. Well, I'll give it to you. But you are the person in charge, aren't you? You're based in Rennes, isn't it? No, Nantes. Oh, Nantes. I thought it was Rennes. Because it's easier for me to go to Rennes. But Nantes too. I will give you a copy of the lecture and then I will give it back to him in 15 days. I will give you a copy of the lecture and then I will give it back to him in 15 days. I will give it back to him in 15 days. I hope it's possible for me to understand a little more the story, the motivations, the examples, the work of Erasmus. But you are a... But we can talk about it. OK, another time, of course. Thank you. Another time. Thank you very much. See you next time.

22:30 Thank you for watching this video. This link is the translation of the series, the repetition of the series, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, of the order, I hope one day we will be able to return to this text in the first edition of the GSI. It's far from true. Thank you for your attention.