Tensor categories and their role in physics (contd.)

0:00 There is the example of the braided category, the example that passes, it is what I have already said, now I have done the second part, what we call the eubony category, and this is the following thing, because here, if we still think about vector space, what have we done? We did some things that we produced in Zoriel, and we even added a structure that we normally do not use, The vectorial space is reduced to a few months, but we have not yet done something that exists and is important. The vectorial space is the middle of the space-dual. We are going to do this and that. It's a bit of a mess. Why does it end like that? When we say that a category is drawn, it depends on the industrial product that we have defined. Yes, yes, there is always monaïtal, we have already seen that in it, but on a given category, we have several tensorial products, and some of them are more complex than others. And you see, it's a bit of a question of what we think of a given category, because it's a bit complicated. Or else, I don't know, in terms of categories, there are a lot of categories. But there, it's not interesting, because it's a big... Varieties, so to speak... I think there's always this ambiguity when we talk about categories. Because the category of data, that is to say, the set of data, is the idea that we fix... When you want to do physics, for example, you will have the category of Riemannian varieties. You will have the category of algebras and star algebras. You will have the category of fibrins of certain types and you will have the factors between all this to define the theories possible. So what I'm asking is, each of these categories, I'm going to ask the question to see if I can represent it by the three. That's what I mean when I say category.

2:30 But just when we talk about, but I think it's good, just an analogy with ensembles, because of course for ensembles, we can also say, if you say ensembles of omens. What is a set of data? Either a set of tables, or a set of something, or... But we help the fix, preserve... And the same with the small problems. Of course, you can look at this kind of stuff, but in fact, this theory does not use these things. That is to say, we make the abstract, we build things... Precisely, when you look at the sets in an abstract way, that's where we come in. But it depends a bit, because I think that... We can help him, we can project, we can reconstruct, we can think together about the category, yes, or something that we take into account in a very concrete way, without identifying with, let's say... But when we do physics, we will be able to... But concrete too, it's also ambiguous because it's... Yes, yes, I understand what you're saying. I don't know, it must be natural. I mean, what kind of properties do I need to be able to define... Normally, it must be a question. I just wanted to say, if you do construction, let's say, it's very constructive in fact, that is to say, this idea that it's common, that it's too general, it's a bit not true, because, if you want, it's very constructive, this kind of thing.

5:00 Well, I'm not talking about constructive in the very precise sense, but you don't have to think here, it's not background with the other. But when you want to do physics. A more specific question for the tensor. For example, I believe that in fact it is transparent formalism. You see, it was to think of a tensor from coordinate systems, and then we say what is a tensor? Is it something that is abstract? Is it an object that is a little mysterious? Here, well, an advantage that we always consider to be counter-coordinate is that Using another concept, it's Tracet, I think it's Ribbon. The idea is to add the dual.

7:30 That is to say, we add an object, a star, such that we have a morphism, evolution, evolution. Here, for example, it's in the Tigris-Veltrognini. Here I just want to say that I am in my unit, here it is another sense, and now I want the following properties, and we demand that this is equal to the density, it is the first one, and this is called the action of rigidity. And that defines dual to right. And then I can do one more thing and define prime with two similar actions.

10:00 Without dual to right. And we can show that in fact... In the case of the right-hand side dual objects, they are defined like this, they are fixed and created, and there is a certain definition that the monoidal category is rigid if each object is a right-hand side dual object. And from that, we can already prove interesting things, maybe I can do it, and directly I will go to... The category of Roubani is another property of vector space, which concerns the double wall, because there is not only this, but also objects on the double wall. Here we call it a delta. The use of mathematical equations is verified.

12:30 The category is already rigid. It is called Roubanet. Why is that? I will explain it later. We can see that we can build a coronet of this type. So when I write things like that, of course, I don't write all these alphabets of type alpha, but if I want to write like that, it will be impossible to write. And that's what we call Psi. And now, if we compare Delta, Delta composition with Psi, it gives us a sort of bad influence in the web. Okay? And that's an interesting thing, which is called Thulist. And that's what we call Delta relative to a given object. And in fact, there is the other option of simple utilization of what category can be banned from this head. But what I want to present is also a symbol.

15:00 When we work with a category, we will also see the relevance of the theory of Crest and Arrhenius. In all this hope, we want to exchange objects and arrows, just in writing, but that is to say, instead of writing like this, we want to write like this, okay? I want to write something like this, and then I just don't want to write the 1, i.e. I want to write something like this, like this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this,

17:30 It's going to be something like this. It has the same thing, let's say. Now, I can really show you my dress. Yes, it's our good son. And this, I want you to see it. The opposite things, for 11 years, it will be... Well, and now we can summarize a little this rigidity action that I gave you earlier. It would be something like this. But it's easy, it's easy. That is to say, this is just this dual, dual, V, its dual, again V. And all the morphisms are all in it, and that equals density. We can make it more interesting, for example, if I write something like this, that it is equal to...

20:00 We can see that it is in fact, as Jean-Jean said, a compositification of the following diagram. Of this diagram, if we exchange, if we say the homotopic images, if they are homotopic, they really correspond to each other. Question is, because of this twist, because if we have something like theta, which represents itself, then it is homotopic to an entity, while theta is not an entity, I would say.

22:30 And in fact, all these problems, and that already, as a result, we can do a little topology here, that is to say, to look a little at it. It will not be a theory of knots, a little more general, and not even a theory of... Yes, we can think of it as a theory of lines, this kind of thing. And in fact, to really obtain this correspondence, that really each image has a certain morphism, it is necessary to replace this morphism by ribbons, small ribbons, and we can count the twists. That is to say, this theta is going to correspond to the position of the square. That is to say, we have to replace this thread by an rectangle, a ribbon, and we have to count the twists, and put them together. That is to say, we have to replace this theta, and then everything works well. And in fact, all the second parts are the same. It shows that from the point of view of literature and arts, we can make theories that are very serious. That is to say, we can make a certain translation, a very precise translation of the theory of category that is found in topology. And then the application of the theory of quantum mechanics. If you have topological fields, what would you say about them? Well, for me, it would be the same. First of all, we have the groups of permutations, then of braids, then of ribbons. I have the impression that we could continue, and it will be as interesting. I don't know. Second question. You said something like, for a braided category, Its structure is completely expressed by the structure of the string group that is under the hand.

25:00 Is it true? That is to say, if we have a stringed category and we define the string group, which is defined by its number of generators, the string group is defined by its number of generators. String theory is responsible for commutativity, it is a structure that is in it, but there is a lot more. All the tensorial products are not defined by the string theory, it is something that adds to... If you take a string theory group, it gives you a part of the structure. If you take a document group, it gives you a little more. No, we need some kind of development. It's a bit of a joke, but it's actually very important in physics, because there are a lot of structures in physics that live like that too, and that's what John Bell's articles are about. Maybe we can look at some examples from Ukraine, I don't know how to explain it, because it's really an implication, everything I gave is a bit abstract. The second half of John Bay's book is only about the groups of 13, groups of 12, but does not describe in terms of categories the situations of the 8. You can always see it as a complementary composition. In fact, deep down, it is still the structure of the two categories. The monoidal category is perhaps the best book we can do. If we just say the monoidal category, we know that the monoidal category is...

27:30 Is this the link between the categories? Yes, we can say that. We have structured ourselves. There is this basic category that we have to see. Then there is all this, let's say, automorphism, endomorphism, which defines the products. Let's say, we can do the string and their pleasures. And what really gives us the structure is... Let's say two morphisms already, natural transformation between them, and that's what holds all the things together. So it's a bicategory to an object? Yes, except there is a small problem, exactly that, that it is bifurcative in fact. But I don't know how you see that. So it's very important that you write the actions of the bicategory to an object, and we come back to the categories. To see if later, in the little specificities that we put, whether they are rigid, whether they are crossed, etc., are there subtleties that make us get out of the frame of the category of an object? I don't know. Because the question is that in fact, the categories that we have crossed are in fact the three categories of an object and of a string. And then he puts these things together, there is always this conjecture of stabilization. It's to do the math.

30:00 For example, when you look at the attempts of Newton to do this, it's diagrams like this, where you put the ones next to each other, and then you have the rules of the kind, the rules of the lines on the... There is this tendency. So, is it only a kind of illusion, or does it work? You can approach it in a different way, but it's essentially... Yes, but it's things like that, because you have the rules that make you compose... Very well, so I think that's all for today. We will return to Thursdays, the exact date will not be fixed, it will be on the 15th of January or on the 8th of January, depending on the week.

32:30 In other words, the week that starts with a father's day... The week starts with a father's day... The week starts with a father's day... The week starts with a father's day... The week starts with a father's day... The week starts with a father's day... The week starts with a father's day... The week starts with a father's day... The week starts with a father's day... André, it's a little problem for the 15th of January, it's your class in Amsterdam, isn't it? No, in fact, it ends on the 13th. Ah, it ends on the 13th, OK, I forgot. You have to look at the calendar. What's that? The semester ends on the 13th of January? The semester starts on the 1st of January. Thank you for your attention. The 7th is on the 17th, so it's Thursday, 7th, 8th, 9th, 10th, Thursday, 10th of January. It's the problem because... Colors like Stendhal, Crescent, Cormorant, Euro 5.

35:00 The Cormorant is more common. The 17th? No, but you can't. The 17th, 8th, 9th, 10th, you can't come. But if we start like that, then I'm stuck in the middle. No, but hey, you have to take a rule and always follow it, because the 10th, you're not there. Well, when the 10th is the same time. I say Monday. I don't know, maybe it's the 17th. No, it's the 18th. Ah, the 18th, so then it's the 11th. It's the same, it's the same, you're not there. The 8th, no, no, the 11th. The 11th. Ah, the 11th, no. And before 11, what would it be? What is the problem with the 15 days? I am the one who does the seminars. So if we start the seminar every 15 days, we do the seminars. I don't know if it's better to do one or two, but if it's every 15 days, we do the seminars. And if we do it only on Tuesday, the 9th, and then we do it on Thursday? On Tuesday? We start on Tuesday 9th, and then we say we'll start on Thursday, so the next one will be on the 11th and 15th, 11th and 14th, 25th. Oh, that's a good answer. Tuesday 9th, Thursday 25th, Thursday 14th, Thursday 15th. It's still 11 o'clock. And we'll try to make it happen on Thursday, but when? It would be good if it happened on the 25th? No, it's better than on May. Excuse me? It's on May. Oh, it's on May. On May. Because it's already a co-occurrence course. Okay. It's perfect.