Tensor categories and their role in physics

0:00 Thank you for watching this video, if you liked it please subscribe to my channel and give it a thumbs up. Thank you for your attention. I am interested in this subject. I am interested in this subject. I am interested in this subject. I am interested in this subject. I am interested in this subject. I am interested in this subject. We can't turn because we're running out of time.

10:00 If we say A and B, at the ontology level, it's really... I think it's the other side of the same problem.

12:30 If we try to imagine an atomistic image... The universe and atoms are also dual, but they are not dual, that's why we think that there is one universe and several atoms. Maybe the dual thing is wrong, but ultimately it doesn't work. There is only one way, but it is not the first way. Exactly, that's exactly why now I... that's part of the theory of general categories, and now... So, for the doctoral students... Yes, if I understand correctly, this is the direct product for vector space, and it's not exactly that. For vector space? In fact, for vector space, if we reverse, here, the terminal object is called the initial object. And in vector space, it's the same thing. This is called object zero, which at the same time is vector zero. At the same time, in the terminal, we use the product to define the product. And when it's finished? Yes, when it's finished, exactly. And when it's finished? Yes, when it's finished. Now, we have all this, and we try to... This is not enough to do something interesting. And we want to define the product. I'm not talking about the product, but in fact, for the moment, it's an empirical operation. In a more general way, we do the following things. We take our category, let's say, and we look at a bifurcator, that is to say, a bifurcator, we take two objects, like that, that's a bit of a naive part of theory, it's not a... It's a bit of a... Now you're talking about a product of two categories, in the category of credits.

15:00 Is this product unique? The product is always unique by its construction. So there is a product of categories? If it exists or not, we can take categories. And there, yes, we can say that it exists. But I see that in the paper, when people talk about Pivot-Ktobs, there is no other idea than that we take an object, a piece of object, there is not really a categorical construction here, it is a way of saying that we take two objects and we associate the third in a more or less standard way, but it is not standard, I will show you later. Here we can say that it is a functor, a category of pairs of categories. Yes, I think it is necessary to verify, but in fact there is a way of thinking about this thing, even as a functor of A in A. That is to say, it also resonates a little with the function of the function of the function of the function of the function of the function of the function of the function of Yes, yes. But it's a bit of a matter of meta-theory. We don't care about that. But I still think it would be interesting to think about functions of A in A. I think it would be interesting. That's how it's defined. That's how it's defined. By fixing one after the other, it gives a function of A. Yes, but you're still talking about it being a pair. Then I fix another and it's another function. Is there a way to say... Let's say if I write that A times A is the same, something like that. This is the major condition for this lecture, I won't talk about it because it's not part of the lecture, but I'll put it in the video.

17:30 So, there is a function at some point, and then, to give an action, what can we do about it? And it will not be here, because here it is, let's say, the construction of the media, and here it will be a little more axiomatic, let's say, formal, but what makes the difference with, let's say, the standard, when we say, well, there are operations, alternatives, associatives, etc., here we try to describe the factors, that is to say, well, the first thing, operation, we want, let's say, associativity. Or we don't want to. In fact, even in the category space, if it's infinite, it's not associative, in fact, the sensory product. That is to say, we can't just write the same, we don't want to write the same. We want to think of something else, but what other choice? And in fact, it was the very beginning, historically, of the theory of cuticories, etc. This is the problem for vector space in the table of MacLean. Well, we don't want equality, but what do we want? Isomorphism? No, isomorphism would be too weak, apparently, because vectorials in the same dimension are isomorphic, isn't it? Of course. If I talk about vector space, it's the same thing, because it's homomorphic. That is, isomorphism is too strong. Equality is too strong. What do we do? And what do we do? What do we have in the example of vector space? That is, we have isomorphism, but not just anything, but natural isomorphism. And what is natural? We can also look at this as a functor, which accuses them of something else, three times, here it's something else, and now we want some correspondence,

20:00 And this is the real motivation for all of our lectures, to talk about a notion of the category of the most important. The most important is exactly that, the notion of the most important of the most natural. That is to say, if we have, in our category, mathematics like this, then we have, for example, if we have something like this, And so here we are looking for other transformations, what we call material transformations, here I have written alpha, alpha u, u, a little bit, such as these known diagrams. That is to say, for the normal tensorial calculations, that is to say what? There is coherence in relation to basic changes. It is not only the homomorphism that votes here, but it is the homomorphism that respects the basic changes, which is what we all do, especially the basic departments. The notion of natural transformation is really... these ways of making this idea that many authors write much more clearly than in the formalism of the norm. Mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious, mysterious,

22:30 This is the definition, that is to say, we can weaken this equality. In fact, if we replace this isomorphism just by equality, ethnicity, we obtain what we call the strict category. And it's a bit another way, I mean, to present the same theory from this strict category, but otherwise... Well, we have a set of terms for our theory, but we have to give an axiom when we talk about it, because it's too much coherence. And coherence is the following. We say that if I have objects, and I put them in a parallel way, and I write a formula like that, Then, I also have the right to put parentheses in all possible ways and also to introduce the humanities as I want.

25:00 And once all this is given by our action, by composition of all this, of alpha, of lambda, of rho, it will always give us, I don't know, an initial homophysis. And the axiom is that it will always be the same as the homomorphism. Here we have an identity, because we can't definitively go beyond our particular image. And that is an axiom to be used. If I have a string like this and I insert the units and I put parentheses when I want, in all possible ways, all these things are isomorphic, and by themselves they are isomorphic, that's also good. But now there is a result, how can we deduce that? I don't want to go into it. What we call the theorem of McLean's coherence, McLean's coherence, which says that these structures can be generated by a permutative diagram, by two. It's called the action of the pentagon, now we have to put parentheses in the square, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this, like this.

27:30 Oh no, in this case, it's called algebra, geometry, algebra, geometry, algebra, well, and it's called pentagonal action, and there's the fact, but it's a bit elementary in fact, elementary in the sense that it's combinatorial. It's a bit like the arrangement of groups by a generator and a ratio. It's the same thing that if we have this indicator in the commutator, we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. It's the same thing that we have all these things. No, this is an example of what we think. Okay, but it could be... Yes, yes, it's an abstract category and we try to unite categories by structures that are, let's say, more or less similar. The structure of the pictorial space category. And this is actually called a monoidal category. That is to say, a monoidal category such that there is a small space and this axiom is empty. This is called a monoidal category. That is to say, why is this number so important? Because if we don't think of this problem, we can think of it as two categories with a single object. Here we have, let's say, A times A, there is a single object, here, yes, there are several factors of this object in itself. Yes, the functions of the alpha and the beta, of the R0. And in fact, there are mature transformations of homomorphisms.

30:00 This is the main difference between these functions. Okay? And then, we have to verify that this mature object is a very strong one. But here, it's a constructive way of doing it. That is to say, you have to make this pentagon. The second step is to make a triangle. Triangle is for unity. Unity will be like this. Thank you for watching this video. Yes, yes, if you want to do it internally, yes, of course. Well, here we are trying to get closer to the application of physics. Yes, yes, you're right. So, well, atiyah is just that. How does it work with alpha, yes? And if we have a goal, then we have the goal. That's right, we have to make it work. That's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it. In this time, why is it sometimes anti-tensorial, or anti-modal? It's quite surprising. It takes at least some time and a lot of space on the board to describe these subliminal conditions. But have you ever encountered a term of category that verifies the first part of the definition and does not verify the second part?

32:30 What was the first part of the question? Just the idea of a bifurcation with the transformation of the system. It's a long definition. I don't know what to say. In fact, you're right. But isn't that all? Yes, but that's really a very bad condition. Very bad. All of these things work, you see. Otherwise, we have morphemes in all directions, full of morphemes, scientists and all that, but it doesn't work, because that's a false prediction, but it's also a fact, and yet, we can see that it's generated by this pentagon. But if you don't have this associativity in this sense, well, scientists, some morphemes, morphemes, they don't give you anything. You can't give them anything by hand, for example. Thank you for your attention. No, no, no, I understand. Stéphane, is there an example where you have certain... because here we have a real existence, it exists, and if I understand the example, is there an example where you use it like that, but it's not really useful? Yes, yes. I just wanted to say that the other approach to mathematics theory is to start with the notion of strict category. Strict categories are things that don't really exist, in my opinion, in my opinion, in my opinion, that is to say, categories such as we have equality here, yes, and then, but in fact, it's even a bit without interest, because our interest, if we think that categories don't really exist, exactly, we don't want them to exist, we don't want them to exist. There are other theorems, especially in Marlène's book.

35:00 He shows that each monoidal category is equivalent to a certain strict category. I just want to remind you that it is equivalent to a certain category. It is not a homomorphism, but it is something. The lower categories are the things that sell. If there are two categories of A and B, they are equivalent. If there is a construction that sells, there is quantum physics. That is to say, if there is a quantum physics, I write in the order, I write in the order. If I write with parentheses, I go to the other side. For me, it's easier. Here is G and then we say that here there is a state of density and the categorization can reduce values if it exists, but here we have a mature isomorphism, i.e. it is not an isomorphism, it is more simple. This is the definition of monolingual theory, now there is another small part which is actually very important for the applications but I don't want to use it because I want to move on to this category which is very strict, it's complex, but still maybe I'll give you a definition that can survive afterwards, it's the definition of additive theory. There may be links, I will give them to you later, because they are completely different. And additives, this is also something that can be said to make a more recent category of vector space, modules.

37:30 That is to say, we just fix, let's say, a core or a nano if it is a local module. And then, for each morphism in Hebrew, and it was proposed later, it is the ensemble of morphisms in Hebrew, it makes a vector space on Korka. And in this case, we have the composition of morphisms. And in these cases, we have to add our actions to say that our theory is also bilinear in relation to what we call additive theory. And then there is another precision that we would call linear theory, that is to say that we are not going to talk about it now. Thank you for your attention. If you are given a category, and if there is a tensorial product, is it the same? Ah no, but a tensorial product is not the same at all. It's the product in the general sense. No, no, no, it's the product in the general sense, it's unique. In physics, it's unique. But in fact, these are slightly naive constructions. Here, we move on to two small gorillas, these are the important ones.

40:00 These naive constructions, in what sense? Well, we always have them as a tool. It is always a commutative tool because there is no place, it is always associative, of course. Because, you see, there is no place to put this community structure. And now, we move on to another framework. Well, in fact, a framework of two small gorillas, you can say. We look now. Of course, you can also take this standard product and... Look at them as bifurcations. Then we can define these bifurcations with similar but different properties, in another way. The important thing is that the first projection is a physical projection. In fact, we don't have that. We have two systems. The state of the two systems is a different state. There is much more information in the correlations. And in the case of polycarpions, it is absolutely not. Thank you for your attention and see you in the next lecture. Ah, and also, just a last remark about the adjective, of course, if we have this terminal object, in fact, we ask that the terminal object is the same as the initial object in the first case. Now, we can say that all the morphisms in the first case are the same morphisms. So, there is not much of that here, so it is important to speak about it. Well, but I don't want to, this is important for the examples, but not for this generalist thing that I don't want to talk about. Well, now, we still have to look at what we really gain from the general, something a little more reasonable, theoretical. Now, we will see that there are really very interesting structures that appear in it, and these structures are called the groups of 13. And that is also something that we are asking for.

42:30 Philosophy, psychoanalysis, maybe it's something like that. These are the following things. The 13th, I have a certain fixed number of strings. And now I have the right to... Well, I can say it differently, I don't know. It's like that, like an entity. That's how an entity works. And now, the group's action is... I made a 13th. I exchange, I allow forces, that is to say, I can call something like this, for example, or something like that, or something like that, or something like that, or something like that, or something like that. More complex, but that's what's interesting, that's it, and now we do composition, we do the same as before, we add the vertical to the vertical, that is to say, in fact, what makes the difference with the symmetry groups is that if I do it twice, I don't get the same result, if I do it a second time and if it's the same thing, I get something that is two times, It's really interesting because it shows that our symbolic machine, our way of writing, really counts a lot in what we do in mathematics.

45:00 Why? Because it just made me think of what we call the Kehler theorem, which is a very simple argument to show that each group is in fact a group of permutations, that is to say subgroups of symmetrical groups. To make an absolutely simple argument, we make a list of the elements of the group, then we take an element that it counts, we make it act in all the elements. And that gives us three, that gives us permutations, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here, that's here. We can't make a list of these groups of continuous information, it's something that doesn't really work, but normally we say, well, if it's infinite, it's not as simple as that, but still, it's the same thing. And here, in fact, we can look at it as, say, a realization of groups. It's the broader group, the group of rotation, simply, it's the broader group. I don't know if, let's say, René Guittard takes a serious example like this as a real one, because from his point of view, it's a real one, and of course, it depends on the fact that we write like this. It's also the thing that René Guittard talked about earlier, it's intimate, when he talked about algebra, he said to stop with algebra, because we have to think like, let's say, certain philosophers. Geometry is a topology, it's a form, and here, of course, it depends on the fact that it's written like that. If you don't know how to write in a bag, there are other things, I don't know, there are other ways.

47:30 If you don't know how to write in a bag, there are other ways. I don't know, anyway, I think it's such a deep thing, the group of tests. And now, there is also a very nice result that these groups, as we can describe them, are engendered by the interpretation of Cécile, as you can see. Generators, and the relations are as follows, and the commutative in a linear way, that is to say, it's obvious, if we change something here, the same thing we can see here, it just doesn't come to mind. That is to say, we can say, v i v j, it's the same thing, v i v j, here. I-J is more than two, it's a bit weird, okay? And after this first, second relationship, which is in fact the main one, and which is very important in physics, something that you could better understand, because we will see that from here comes what is called... Young-Baxter equation, the equation of Young-Baxter. And these are the things we want, so we can put them on the elliptic. We can put them on the elliptic, like that. Here is a kind of symmetry of this kind.

50:00 The number of nodes is fixed. If you want, it's also a matter of time, that is, we are not in a regressive situation, that is, there are always traces, we can lose them, but there are always traces that remain, we have to come back and we really have to... There is a kind of dimension of time. So, what is the group of 13? And why is it interesting in our context? Because these groups appear in a very measurable way in the monedal category, as you said. If we have, in fact, some kind of string, like this one, and now, if I want to permute this thing, in fact, there is a method, in fact, We can't just make the groups symmetrical, because we have to take into account all these diagrams, but with these groups, they don't work automatically either, but still, if we look at our other morphism, that is to say natural transformation, which is responsible for the activity, And now, I have created a BK generator, and for all these things, I look at it as an activity.

52:30 But it's also a bit the same idea as an activity, it's not qualitative, but there is... In addition, I do not claim that it is just an inversion. In the theory of representation of standard groups, it will always be an inversion, but here it is very natural. Of course, if I add here something that is square and integral, it regulates a symmetry group. We don't want that, right? And now there is also a theorem, thanks to André Gégal, which shows that if we... I don't know... We can do a triangle, but if we add a third axiom, which will be this time hexagonal, it will be more transparent. We'll see how it goes. It's simpler than before.

55:00 That is, we ask for its commune, its commune for sigma and each time also for the inverse of sigma. That is, we have two diagrams in fact. It's a bit of a way of linking associativity with... That is, we have a certain morphism responsible for the commissive. and others. It is also easy to see that this is necessary. If we can find these morphisms by the group of 13, that is, if it is the assignment of the group of 13 that I do, if it is of the same level, then the category is the same. That is, the morphism of our category does not include its image in the group of braids, not in the teaching. This is the fact of braids, that is, there is a structure, a function with a group of braids in it, and also the fact that this diagram of the Earth is six equilateral commutes, in fact two equilateral commutes, And now we can... I just want to write this equation, Young-Waxt, it's written in the following way, now we have to write it in the following way, like this, and of course there are two ways to obtain this from this, by commutation, so...

57:30 Thank you for your attention. It's 1, and this will be what? It will be V1, V2, V1, okay?

1:00:00 I turn here, then there, and then here I have to turn. But it's the same thing as V2, V1, V2. It's just something else, and what does that actually mean? There is only a physical sense, they don't understand each other.