(contd. 2)

0:00 It's interesting because it's really this notion of cones, it's open, it's more natural, in a way it's another way of talking about well-known things, on the standard, but on the other hand, this notion of duality, it's interesting, yes, when we talk about co-cones, what will it be? This kind of intuition of cones is clearer, what will it be?

7:30 But it's not the same way, we can't just do it. All the codes, it also includes all the codes. It exists, that is to say. No, not at all. We can't think at the same time in the... To build, it's obviously just standard, but if we think... The complete category is simply all the codes, it's not the full codes. In principle, eco-connes are not complete. Okay, another elementary situation is... Can you give an example of complete and non-complete? Yes, but... It's clear, it's complete, yes.

10:00 Eco-complete. Eco-complete, yes. But it's more interesting, of course, now, what is this point of view. So, first of all, what is an eco-conne in categorization? Okay, for the sum, for the product, we have already done that. If we have the three elements, we have the three elements. The product of the three is one element, it is not the problem. In fact, if there are any general diagrams, if we give ourselves any diagram, then it is easy to write the equation. Sometimes we distinguish, we say quons without this condition of the human reflex, and this kind of universal quons, we say quons. And we understand the product of all the ensembles in a diagram. And then we take the sub-assembly inside, such that for each arrow inside the diagram, the component belonging to the first assembly is the clearest. And here, I suppose, a topological instruction. Thank you for your attention.

12:30 It is interesting that here we can choose products and in the general case it is always true that a category is used. There are two things you need. You need to have all the products and inside you need to be able to give a meaning to the fact that other things are equal, so you need equalizers. What I want to say, let's say in the general case, is that it is not worth thinking about the product as a privileged diagram, because in the case of... All these ensembles, yes, it's something that happens, we see what we can do, especially in the universe, but I think it's common to see, because they are more likely to describe the products that we know in common, so I don't think it's always done that way. The idea is that the limit is, let's say, the smallest ensemble that checks that. Yes, that's it. Then there are others that are bigger, but they are trivialized by canon injections. Because if we have an ensemble that has new arrows in it, then it is enough to send the ensemble by the arrow. For example, by an algorithm. These are the main features of the product. And by definition of the cone, it verifies its properties. It verifies its properties. It's the fact that there is a commutative diagram. So you start from another Y, right? Yes, I start from any Y, equipped with an FI arrow pointing downwards. I can send it in the X and it defines it like this, by sending Y on the product element, so it's an element of the product.

15:00 So the IN component is Fi of Y, the image in AI, in the AI application. And what I'm saying is that, well, automatically, it's inside of X, because that's exactly what I asked my arrow, let's say, the diamond that I was talking about. Exactly. But that's still an advantage, let's say, in the most general case, that it doesn't necessarily have to go through products. Because if you have a diamond like that, you don't have at least a problem with the product. But... Sometimes the product is too big for what we want to do, and when we look at it from the outside, it consists of two parts inside, the collimite and the cocoon on this diagram with more inclusions. This is the intersection. The product of the three D is too big to compare with the other two. There is really a notion of completeness in topology. The limit belongs to it. These are limits that must belong to... Yes, that's right. Another thing that is also interesting to think about is the speciality of ensembles. In ensembles, if we have two functions, we have two ensembles or two objects.

17:30 If it is a terminal object, it will be the two points. And one thing, of course, always the image of a point would be a point, it would be another point, a possibility. And then, if we have a situation that we always have like this, I will ... Functions are defined point by point in the ensembles, they are sent in the same point, they are the same function. But of course, this is not the case at all in general, because we can have only one point here, that is to say, one point here, and the function is quite different. But we should not confuse morphisms with functions? Yes, if we do not understand functions in the sense of assemblies, like morphisms between ensembles, we should not confuse them. But it's interesting that we try to reconstruct things from different points of view. But that's what we thought of as demorphism. Demorphism is a circular notion. It's not necessarily a function. Of course, because category is not necessarily a category. Of course, we can use several points here, but all of these expressions are open. It doesn't work at all.

20:00 That is to say, it's also interesting, perhaps we can even try to systematize here, because to use ensembles, we can generalize. It's a bit like here, we can generalize, we can also describe it in an ensembly way, but other things. It seems to be something very special to Gréty Gouré, to say that, to say that kind of thing. Okay, and maybe the subject that I want to ask you today is the notion of exposition. The concept of function in a more precise sense than just the concept of morphism is to look at something as a whole and as a function. If we have a whole A and a whole B, We can look at the functions, the applications, and then we can think about all the functions that exist. In the particular case, for example, we can take zero in something like that. And look at all the functions from A to 0 or 1. And this is really the whole of power. It's called power. For each element, this application, he told me, participates in the whole of the world.

22:30 That is to say, we describe in this way all the... Powerful ensembles. But in the most general cases, it's probably something else. And now the question is, how can we describe this by reflection? And the idea is that we often call this the exponential object. Exponential. Exponential. Exponential. Exponential. The B is the following, if we look at it, let's say, we have to think of it as all the functions, and then we look at this product, we look at this product, that is to say, we take an element of A and we take some functions, and here there is this very rigorous function AB, let's say, there are some functions here. If there is an element of A, if there is an element of B, such that F sends A to B. That is to say, we have this kind of morphism, if we think more generally, morphism which is called evolution. But now the question is, what will be its condition?

25:00 To do this, to think of this as an exponentiation. What will be the condition of this algebra? That is to say, the idea is to introduce, by definition, this object here so that for both objects A and B given, we want to introduce this object here to make it work as... The idea is now, it's a bit in these frames, to imitate, let's say, in the theory of category, a more standard assembly motion as a nation of social exponents. And the idea is what? I take two objects, I construct in a very abstract way this object there. That's what I didn't understand, how do you start to construct that? I didn't understand. No, I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. I'm sorry. What is the element of A and what is the element of B? And really, here, it's not a matter of points, because when I say element, you see, it's produced here, you have to think, of course, in the general sense, as a limit. That's it, it's not a matter of what I'm saying, it's a point, that's it. But what can you say, except that it exists, applications like that, yes, you have to... Of course, what do we have to do? Here, it's a bit like a standard method. We take another object, whatever it is, or here we think of the night. Let's just say, for example, there is another situation like that.

27:30 And how can we distinguish between what we want and whatever? From the same type. That's the question. The answer is quite the same. It's a unique relation. That's it. It's common. It was produced with... But here, unique, it's a bit special. You can say it because you have to think of it as... It's normally like that. We can also introduce the notion of morphism products. It's really simple. If we have morphism A, B, and if we also have morphism C, E, then we have... Well, that's how I saw it. It's not a big deal. I think it's good. I think it's good. Good. And that is to say, here it works in a way... Because otherwise, of course, I would be embarrassed to play with error endomorphism, but it's not that interesting. We know that the point we just identified was here, on this side. But here, it has to be unique. But let's say... Can you make an assertion, make a sentence? So, evaluation of an agreement is not defined unless we have defined exponentiation. It's not part of the definition. How is evaluation defined? Is it defined before the explanation? No, it's just that we define, we build these things, we say, if there is a diagram like that, with its properties,

30:00 the main property is then called the universal property, that is to say, for any other. These are objects of the same type and here it will be produced by object A which is given. A and B are given. Okay? Yes, I agree. Given. And we want to define what will be B in power A. Okay. We say that it is the diagram for this object here, which is part of the diagram. With the following properties. And if we have another object, it makes this diagram, commutes with the unique morphism here. That's what I don't understand. What is it? What? That's what I... B power A as the object such that B power A cross A is the terminal object in the category of objects of the form C cross A with a morphism towards B with as arrow the morphism, the data of a morphism of C cross A in C prime cross A. It is a morphism of these terms such as the commutative diagram. Why is it true that we have the category of ensembles for the power of art? In the category of ensembles? In the category of ensembles, it would simply be like a doctrine. Really? But what does that mean? We have all the functions of A to B, that's it.

32:30 But who is it? If we have a set of C and an application of C cross A in B like that, who will it be? All the functions, the evaluation is clear, yes, it is function R that sends it to A to B. But if we have a set of C something and even an application like that of C cross A in B. How is it going to be structured? For each university, we have a... Yes, well, it's like you can see, it's kind of... And then my film, that's it, like... You can see in the sets, it's bigger than that. But that's going to happen in a unique way here. What we say is that if we give ourselves an element c of c, then we can look at the partial application f of something, and that is an application of the domain, and that is the one that we call the social element, which is the galactic component of c. Okay, we have to finish because there is a lot of time. Look, it's important in terms of organization, it will be another place, I'll give you a... It will be a good place. And what I managed to organize in terms of the room... There are only a few, let's say, 9 February, that don't have any rooms, but we'll see what happens. Either I can do it, or we'll do it two weeks later, maybe. I'll do it for the others. That's fine.

35:00 I'm sorry, but I don't know if there's any difference between the two. I think we'll see a little difference between the two. And then maybe we'll talk about it in detail. I really want to touch on these questions more closely. Mathematics is a theory of theory, but nothing more. Yes, yes, but today we had to take some time for definitions, not only for techniques, but for definitions, because otherwise we are not there.