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0:00 In other words, these are symmetries that give us the opportunity to have an optimality that is similar to that of Cartier. This is not only about the theory of physics, but also about the symmetries that allow us to talk about optimality. In these constructions, in which there is a great deal of construction that gives us these useful things. But what is this? This is a practice of counting, which is very ancient, very human. If you want to learn about it, you can go to the Ardéan Museum, which is very beautiful, the center of the world. When we share with animals these judgments of order, of organization, of percentage, of course we adjust the language, which gives us information about each other in the same way. Braav is different. Braav is a specimen. The name is the effect that temporal numbers share in two. It is the temporal number of stress which is... The judgment on which center is in fact in which sense is sometimes the most important factor. But it is true that it is necessary to add several acts of experience, of active experience, as in Kalimbaï, of the practice of the world, of programming, of counting, of the consciousness of the senses, of the consciousness of the senses, to have the experience, the experience and the concepts. All of this gives the intelligence of the organization of the world.

2:30 So, the analysis is based on this point. In France, I decided to show the evidence in my study on gold. Because the history of gold, for example, You see, it's a good order. What does it mean? Because it's discrete. For example, we put it down by languages, by generalizations, orders, phrases, and counting. And you can see it well. If we are sufficiently mathematical, we see it in general from one point to another in the discrete. All of these are elements of physics, of course, but not all of them are well-known. They are generic. For example, this is the shape of the shape of the shape of the shape of the shape of the shape of the shape of the shape of the shape of the shape of the shape of the shape of the shape. The Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pythagorean Pyth I had to do a specific drawing. You have to do it. There are 60 methods. All I tell you is that you have to do a specific drawing, so you have to participate in a relationship between the cathedrals.

5:00 And all of these give me a way to go from AY to AY, so to say, to go through all the elements. And we say, I'm going to do it for all of them, because I gave them a method to go through all of them. But be careful, it's not like that. People are going to go through all the terms. So we're going to go through all the terms. And that's the purpose, which would be good, because you know, even in a lecture, what do we do with these terms? Well, yes, there will be the Nestle-Zindach, for which I gave the introduction, but after a long time I did not admit it, so I stopped. At some point, there will be a perspective curve, with a large number of curves. So, the theoretical essence of the curve. And another remark that interests me in Hilbert is that it is a fundamental intuition, But it's still a matter of time before we get to the point where we can start with the elements of the lecture. Maybe I'll just put the base so that we don't have to start over with the elementary logic. What are we going to do? We're going to go a little bit into the second part, which I'm going to tell you about next time, in order to follow Céline's example. As I said, we have a basic logic, there are several systems, and in these traditional logic systems, we ask for the next one. But we are going to think of everything as a starting point. And we are also going to pursue Goebbels, instead of saying that a coding work is not possible.

7:30 We directly write the proof as a term in the circumstantial system, and this is called the length of the hypothesis. That is to say, I am going to associate each proof with a term. I say, if I take a hypothesis proof of A, then obviously it is a proof of B. If I take a hypothesis proof of A, then B is a proof of A. I'm going to start with B, so B-A gives me D. And then, here, I write directly in the language. If X is a possible fear, B-A gives me a possibility, then I have a fear of this arrow. How do you call it? Oh, I don't know. I don't know how to spell it, but I don't know how to spell it. B-A plus B. So we put D at the beginning. This is a mix of 32 or 40 things. We put, from the beginning, the quotations. They are done. It's a calculation. Then, as I told you, we put them in the language itself. So you see that this is the same structure as this one. It's a metaphorical structure. This passage here is something that tells me that if I have the metaphorical deduction, I can give you directly what I want in one sentence in one language. And I give you a code for the expression. So, let's try to see how this works. How do you know that A allows you to demonstrate B and C allows you to demonstrate E?

10:00 The first lecture will be held on three topics and will be presented to the audience for the first time, so that the audience can ask questions to each other and to the audience. The question asked to me by Salomon originally was a logical question, to define, throughout the 20th century, or even before, the scenes in which we are today in this edition. And it's true that today we still manage to have a vision of the tools. And it's true, like tools, I'm not talking about tools, it's mostly a tool. So I started by doing things a little bit in particular. A question that Salomon asked me is finally the negation, or the double negation. In fact, classically, everyone has seen this in their studies, classically, when we have the table of truth of negation, it is this, I have a million CE, what does it mean that the number of CE? Well, if it is true, it will be true, and if it is false, it will be false. Generally, it is limited to this, the deduction of the negation for the students. And the same for the academic, for the etc. It is clear that this kind of state does not belong to quantification.

12:30 And then the second thing that we can notice is that nobody ever uses this to reason. In any reasoning, in any demonstration, we do not refer to the state of truth. Or extremely rarely. Precisely, it will bring us this remark that we have another definition of negation. So I forget this thing. Anyway, it is not valid for the French. It is certainly incompatible. All of this is the definition of negation in terms of truth. When is the negation of E true? The truth, in mathematics, is not the most important thing. Because imagine that you have two mathematicians who work together and then one day one of them calls you and says Oh I found a great book, I found a book, what does it have to do with you? You'll say, well I've read it three times. I can't tell you anything, there's no word for it. I didn't hear the sentence. The problem with mathematicians is that they say, I found something, I have a book, and it's true, even if it's a big book, that's not what interests them. What interests them is to see the proof. In general, I notice that in the attic of the museum, when a mathematician announces something true, the first thing he says is, well, I like the proof, but I want the proof. So what interests us, as the first chief, is to prove things. When we prove them, we convince them that they are true. This led to a certain type of research, to redefine the meaning of the statement. If we think this way, we will say, well, what is it? Let's suppose that I have to demonstrate a statement that is under the influence of what I want to do. Everyone knows that, even the students, but apart from that, we're going to assume something else.

15:00 What is a contradiction? A contradiction is the proof of falsehood. A contradiction is the proof of falsehood. Why? What is the reason for being a proof? The reason for being a proof is to show that a statement is true. To convince us that it is true. To convince our interlocutor that falsehood is true. And to convince us that falsehood is true is to convince us of something impossible. Finding a contradiction is indeed a demonstration. And that's why when we want to demonstrate ourselves, we can do this, for example, we can find a normal, and the solution of the equation will start with a supernormal, a definition of the equation, which is the equation of Euler by definition, which will only imply a false. Because this way of demonstrating is the way of demonstrating an implication. So, in effect, double negotiation of Euler. So you see, here we have to understand how to do it. Suppose we are Euler, then something is impossible. So it is not possible that... And here, it is impossible, it is impossible that this is not necessarily equivalent to Euler. So for you to see a little more that it is not equivalent to Euler, we will make an analogy with a vector. But we will take for example vector spaces. So it's better to call them too. It's better to pronounce it, it's a vector space.

17:30 And then I'm going to look at a dual of Euler and a double of Euler. So what is Euler's role in the demonstrations in this seminar? These are the applications of Euler, so that's it. In fact, there is no such thing as a linear equation of the work in its duality. There is no such thing as a linear equation of the work in its duality. Well, there is sometimes, but there is nothing that is canonical, there is nothing that is universal, that exists all the time, in all cases. Which is a way of making it clear that we can undoubtedly remember the work of the work. On the other hand, there is an ideal equation of the work in its duality. It is a linear form on the duality, so I take a linear form, and of course I use it here. Because this mechanism is absolutely universal, and we will find the same in the demonstrations. But you also know that, in the opposite direction, no one has ever found the canonical application, and so this phenomenon will be found in logic. That is to say that for a constructivist, the world of the world does not involve the demonstration of the laws of the world, but in general the world of the world involves the demonstration of the laws of the world. The classical logic is something that is in addition to strictly structural rules. A little later I will define the meaning of all the connectologies and involve some of them. I can demonstrate with ... All of these are structural things that can't be demonstrated in a video or even in a videoconference. They're not structural, they're non-dominant. Yes, but here... So, are there any founders who agree with me? So, yes, there are. We have to sort out the parts together so that we can go to the right. It's a different kind of exposure.

20:00 Well, that's it. That's to try to get rid of the idea that the non-dominant world is necessarily non-dominant. We use a lot the sopatics of level 104. Yes, but there, the auto-adjoints, the auto-adjoint operators in the space of Hilbert or something like that. But the notion of homo-adjoint is a slightly different notion. There are researchers who play the role of the prognostication of Hilbert. It's the sets of arrows between two objects of a plane. I don't know to what extent, I've never tried to say otherwise. So, if I find that you have been at the beginning and that you have a double negation, it will be that it is impossible for us to... Yes, so what does it mean to know the possible? If you want, for example, it will be clearer when we will have advanced a little, especially when we will talk about existence, in the sense that it exists. It is impossible that there is no such thing, and it will not allow it to exist, that's for sure. There is another way to formulate it. There is an axiom, a non-constructive axiom, an axiom that is valid only in classical mathematics.

22:30 These are the terms that we teach everywhere and in which we do not surprise to identify a place in the work, in which we do not surprise to make choices, in which we do not surprise to necessarily realize that this is not what we do, but we do not necessarily worry about it. Of course, I do not pretend that everyone has done the same thing, that is always the case, we know what it is and the interest can play a role. But it is true that in my opinion it will still have more and more interest because it has applications. So, the action of the animation is the action of the student, if it is called a study, it cannot be called a theory. It can only be talked about in mathematics. If there is a short answer, it is good. Because we are not talking about constructives, there is the definition of Markov, etc. And on the other hand, Bischoff. Markov? Yes, Bischoff and Erich. Bischoff and Markov? I think it's the same thing. Yes, it's related, absolutely. Do you understand the differences? Yes, indeed. This is the constructive movement of the end of the 20th century. And there are indeed subtle differences between Pichot, Marconi, etc. All these things are included in the books of Trostral and Engelhoff called Constructivism. This was not really the essence of what I was interested in. What interests me most is computer science. So I'm trying to understand how he will understand maths. I am not a specialist in these nuances between the different forms of constructivism in the beginning of the 20th century. On the other hand, what is interesting is that this section here is equivalent to...

25:00 You see, I put an equivalent like this, I should put a arrow like this. Here it is a statement, but this is a meta, it is an equivalence between two statements. This is the equivalent of constructivism, i.e. the equivalence between this and this is a constructivist theorem. For constructivism, add this to this theorem, add this to this theorem, it's the same thing. And it's the equivalent of the third thing, which is actually abstract. What I wrote here, was it really abstract? The first way is to say that since I ask it and that I have said that this is a constructive group, it is certainly a problem since here I only apply the mechanism of separation, it is something that is constructive despite the fact of the deviation because we can only apply the definition of the implication. The next thing is the following. I ask them to show me their work. Excuse me, but there I thought it was the deficit that gave us money. Yes, that's the definition. No, no, what I'm saying is that the way to demonstrate an implication by supposing the premise, by demonstrating the conclusion, is a constructive way, absolutely admissible. Yes, of course. No, but I didn't understand, I asked you if that was a demonstration for an application. Yes, I asked if that way of doing it, if what is written there is a demonstration for an application.

27:30 No, it's not a demonstration for an application, it's simply the normal mechanism of a demonstration for an application. While here, there is no comparison. Why? Because I suppose my work and I denominate it, I find a correlation. So I demonstrated this. So what I demonstrated in reality is this. So what I demonstrated is this. These three things, for constructivists, are equivalent because they demonstrate themselves constructively. If this was true constructivism, it would be the same. North-E is a bit like the complement of E, in a larger sense of the word. Yes, North-E is a bit like the complement of E, in a larger sense of the word. Yes, North-E is a bit like the complement of E, in a larger sense of the word. Yes, but in a topological space, the negation of a light is the outside, the interior of a light, and the outside of the outside is not necessarily equal to the light, but larger, which is exactly what it corresponds to. Do you have an example of... I remember I read this, I can't say where it is, I don't know in which one, but I noticed some things.

30:00 To finish, I'm going to attack the categories, that's it. Try to give an argument, first give a few proper arguments to tell us what it is, what we demand from a constructive demonstration. Demand, yes. So the constructivists demand the most famous thing, which is the following thing. If we proved that there are x, y, a, r, z, e, I have to be very precise, this enunciation is essentially a closed enunciation. A closed enunciation. That is to say that there is no variable of y in it. In other words, there is only x as a variable of y. If we proved that, then we have, when I say language, I will clarify a little what I mean by language, we have a term, a term that is the registration of languages, These are the key terms that represent the A elements. We can see two different things depending on the kind of theory we are in. For example, here, with a type of data, we will say that the type of data is A. Otherwise, if we are in a frame more or less smooth without a type of data, it is simply the fact that the name C A belongs to A and is true. So, even if the data will prohibit the notation, if I put a crochet, it means that it is a function of x. It simply means that in the structure of A, there are occurrences. I'm going to replace the x with an x equal to 2.

32:30 It's a statement, a statement that contains x as a variable. For example, x equals 2 is a statement that contains x. So I replace x by 1. The point of the constructivist, when he has presented a statement of existence, is that we can exhibit the thing we have presented of existence. It's obviously a well-known thing, it's obviously what we heard from the first chief by constructivist. And then the second, the second edition, which is of the same nature. Which is exactly the same when you replace an axis by a curve, otherwise it's the opposite. Yes, I was wondering what you mean by representation. It means that a term is not a mathematical object, it is an expression of a language, it is a mathematical object. A term is a writing, so we really ask to have a writing. Ah yes, there is something that I... And in what, even if it is not a dissonance? No, not that, it is not a mathematical object, it is a writing. A term is a writing, so we must have a writing that represents... For example, whose meaning is an element of order. For example, when you write 2, you write a real one. When you write sum of infinity... No, because I would write 2 as a part of a real one. No, because I would write 2 as a part of a real one. I would say that 2 as a part of a real one. That's why, for example, a... That's a different thing. 2 as a part of a real one is a denominator. Because a 2 as a part of a real one is a denominator of a denominator. And I would say that a, there, It doesn't have to be 3M because 2A doesn't have to be R, so it's an element of an ensemble. A, of course. Why does it have to be A? What does A mean? No, because, I mean, there is a polynome, you know, something like that. That's right. Well, it has to be a polynome. It has to be a matrix or a square. It has to be of a type, you know. A simulation of a play, I call it X. I think it would be C2, right? Oh, excuse me, it makes me think. It's locked. It's locked, right? Because we don't have anything. We can only have this requirement as long as we have not declared anything.

35:00 I was telling you earlier why we have to think like that. If we have declared something, it will not work. It is very visible in the second measure, which is the same thing, but concerning the group. We have shown only one measure that we have shown. We have shown the indecisive. Here, the first one with the... The third requirement. It is to say why there is this work and not something else. There are two structural reasons. We can't prove, we can't prove wrong, so I'm going to repeat, here when I said close, in fact, it would be necessary to reason a little harder, all these things must be done in the void context, we have nothing declared, nothing assumed, because of course we are supposed, you assume that an element in a void set, you have nothing to say, so you have nothing to declare, nothing is supposed, and that's what it means to close, we have nothing declared, nothing supposed, so we can't prove wrong in the void context, for example if you ... As much as I can define it, what I mean by true is...

37:30 Wait a minute, I don't know if classical mathematics doesn't admit the principle of... We admit it, we admit it. I wonder if you take an indecisive... You admit the principle of... I know, that's what I'm saying. All right, that's why I'm talking about classical mathematics. Because in classical mathematics, the denominator is always true. All right. And a priori... Because it's an action, so I'm doing it. If it's an action, then it's good. And so we have, a priori, a table of truth that tells us when E or non-E is true. E or F is true. No, I'm not talking about mathematical physics. I'm talking about true and false too. I take E or non-E, with an indecisive E. So I'll have E or non-E that will be true. It's one of the things that titillated O.R. in 1910. It's still bizarre. I can demonstrate it, it's easy. So, it was avant-Gonella and avant-Monsage who, I don't know, I don't know, they were strong, they had a little bell. And I was talking about the idea that we don't have only three possibilities,

40:00 three desires of the infinite universe. So, because the construction of 132 is considered a connoisseur,

45:00 and as for Breuer, it was a precursor. And King was capable of Breuer and Breuer was capable of formalizing what he had. I would have asked and opposed this. As far as I know, Hedding came to make his peace with Brunner, and the subject was, well, there's all this, it's philosophy, we have to do maths, that's what I understood from the story I told you about. So, Hedding, 1930, Connemara, 1932, and Brunner, during the previous year, what did they say? They said this. So you are exhibiting a proof of E and a proof of R, so you are exhibiting a perpeculum. This is the second one. A proof is quite symmetric.

47:30 The dependent conjunction is not quite symmetric. The dependent conjunction of E and F, if E is not F, it has no meaning. If the first enunciate is A and is bordered by a line, then the sub of their A-sense can say E. But where is it?

50:00 I think you can do it in both directions as well. In the sense that you take one pair and you take 32. You will not be able to say in 32 that you have to take the red one. But this is not necessarily the case. Suppose that you are in a free context and you are not going to show us F or R. You are going to do a random by case. If it is E, you are going to show us. If it is F, you are going to show us. But the conceptual requirements are only asked in our context. Then the implication. He was saying something like a method.

52:30 A method, a theory. Now we are talking about a theory. You can see that each one has a sufficient number of algorithms. When you build a proof of F, any proof, an algorithm is supposed to work. It's not an algorithm, it's a total function. Everything that works, it stops with the result. A proof of E is an algorithm. If you build a proof of E, it becomes a proof of E. A proof of E. I'm going to put it here. I'm going to re-indicate the variable that's in it. The E of E. I belong to it. I would say that if it was an HPH, I would say no. Any good thing that can be written in words. And finally, a proof is a pair of terms.

55:00 So I don't know if I can really put a term or an element, I don't know. Why do I say that? Because in fact it's a little bit... I put that for example in a little bit of stomp, but in a way that is much more convenient. So let's say a term is a proof that... The proof of why. What does it mean? Is it a proof of E, A, A', E, A' with an E in the end? No, no, it's an algorithm. If you give me a term that contains a black element, you give it to the algorithm and it will be a proof of E, A'. So it's not related to E and A'? It's related to E, A' and E, A'. Yes, but it's an infinite E. Well, that's a real question. Yes, yes. It could be considered as a sort of infinite variable. It could almost be called a variable because it can be applied. You have, for example, formal language which obviously contains an infinite set. It's recursive. It can be applied to any element. It's an infinite frequency in the algorithm even if the algorithm is a thing that is written in the algorithm. It's the miracle of the law. The proof of something that is on the left from the proof of what is on the right. But proof of E, we don't know what it is.

57:30 I'm going to give you some examples now. It's a denunciation. The recursive definition of proof. I've given you five, but there are two others. And then, eventually, we can have axioms. What do axioms mean? It means that we have a denunciation and we take a symbol as a part of this denunciation. It's a symbol of the two probes. It's a constant. Yes, so the probes, we construct them. Yes, but we don't necessarily need them. You'll see, I'm constructing a probe with only this. I'll show you how it's done. The probe by exhibition. These are all terms that have not been referred to. So it's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. It's a bit limited. Could we just write without the press? Here, I'm trying to do it independently of Einstein's theory. No, no, because in fact it's a little bit...

1:00:00 All of this is independent of Einstein's theory. We can do this in the case of the theory of recursive functions, but we don't use Einstein's theory. Because then, at the very beginning of this... What I'm doing is reconstructing some sort of design theory. To construct the theory of this machine, we have to use Einstein's theory. For me, practically, yes, because you say I added algorithmic, but you also said I don't know what algorithmic is, I don't define it, I would like to define something from the... It's a term that we calculated, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it, I don't like it. A function that speaks of x and y. What is x and y? What is y and y? So, I'm going to write the same thing. Do you agree? In my opinion, it's not true. I think it works. I'm going to write the same thing. I'm going to show you an application. I'm going to show you an application. I'm going to show you an application. I'm going to show you an application. I'm going to show you an application.

1:02:30 This means that P proves, P proves, and at this point I have to associate, or I have a function, a proof of this. And what is a proof of this? Well, I take a given y-axis, which is something, because a proof of any given y-axis is an algorithm that is constructed from a given y-axis. And then now we have to construct a proof that exists. So here I put a pair, because a proof that exists is a pair. These are proofs of the existence of x, y, y, and q, of which p is a pair of the form a, p. So I'm going to take the first component, the pi of pi, which is something that is normal. So the second element, I'm trying to find this, the second element, it must be a proof of phi of pi of py. A proof of phi of pi of py. All of these are proofs of the Pian of P, which is a proof of the Pian of P, which is a proof of the Pian of P, which is a proof of the Pian of P, which is a proof of the Pian of P, which is a proof of the Pian of P, which is a proof of the Pian of P, which is a proof of the Pian of P, So here I have both, which is consistent with the interpretation of the H-Card, but it creates a part in which it is not announced, it is just an element.

1:05:00 I have written a function. I declared a function, I declared a function, this is a function, in fact, a function is a function that will, from something here, send a function. Now, the second problem is the Y-spectrum. It is declared here, so the length of this declaration is all that is there. So there, I have the right to use the Y, since I am in the length of the declaration of the Y. I can tell you about the second way, by the way. Do you want to see the second way? I think there's just luck. No, we'll see it ourselves. No, seriously, when we do this in a category of arrows, we see two different things, but I don't think it's going to work. So, sorry. How much time does it take to make a big... A quarter of an hour. A quarter of an hour. I'm going to try to make quick recalls of the categories and get to the definition of... Obviously, for today, you don't really know why I'm doing this. So, everyone knows what a cathegory is.

1:07:30 A cathegory is a thing that has objects and arrows. Well, it's not a big deal, it's just a term that seems to be oriented. In addition, the fact that when you have two arrows that follow, you want to compose them, it's called a new arrow. There have been arrows for many years. It's a kind of monology. By the way, a way in which we can better present a cathegory is this term. We know the notion of monology. You have A and B elements in the monoid and you can multiply them. You can multiply any element of the monoid by any element of the monoid. Everything is permissible. However, when you play the domino, you play the domino and you can't do anything about it. This, for example, is forbidden. So the difference between the domino and the monoid is that in the domino, there is an elliptic notion that says that the minerals are above. There are two categories of elements, and category is the same thing, it's a game of words, it's the same thing, it's a game of words, it's a category, it's the same thing, it's a game of words, it's a category, it's the same thing, it's a game of words, it's a category, it's the same thing, it's a game of words, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, it's a category, There are many different categories of objects and mathematical sectors known by categories. For example, what is a ring? Well, it's an alien category. And then there is the argument, we can reverse it. What is a linear category? It's a ring of several objects. And so, it works quite well because if you take, for example, we find on rings the non-quantitative books as you can see.

1:10:00 Now, a function that makes up a category, a structure, I compared that to the name of my book. It's a kind of quantum physics. If you take a galvanized gas, it's something that stays the same. I said it's a pretty neutral idea for the composition of both sides of space. I have it here, and you compose these two there, that's what we're going to eat. It's neutral for both sides. Of course, the multiplication is not supposed to be the same as you said, because it doesn't make sense to do the multiplication in the same way. But in this case, we could have clearly subdued the addition of objects, right? That's what I did there with the object. But simply, if we do that, we... We know that we still have to concern ourselves with things, it's a type notion. The starting point A is to come together with B. It is necessary that they are composed on the right or on the left with one of A and on the right or on the left with one of B. Yes, composable. That's it. Ah yes, that's it. We could do that, indeed. There are things like that that have been done. For example, we can obtain an origami alphabet. That is to say, all projectors. The things that are involved in their work. And we call that types. The angle of Karumi.

1:12:30 So, we're in the middle of it, aren't we? No, it's us, it's us. We've been talking for a long time. I know a little bit because I went to a lot of conferences yesterday. So, there's a lot of connectors. One thing that's also obvious is that we can compose the connectors, but there's a category and a category. With the use precautions, it means that it's like the set of all sets, you have to take the two precautions, it's like the set, you have to take the sum, There is a third branch in the functor category, it's quite simple, it's not very different from the ones in physics in the end, but there is a third branch that does not exist with the groups and the homologues, which allows me to focus on this one. So, for a natural machine, what is it? You are going to take two functors. What is a functor? That is to say, if I have a contravariant functor, I would put the category opposite. I will never take a contravariant functor, except if I want to, I'll talk about it anyway, but I mean... In principle, it is useless because we have an analysis of the opposite category, i.e. the category in which we progress. So here I take a function of a, b, and g. How does it look like this? It's something that we don't understand anymore. Let's see what happens. So I take an object x. Well, I'm going to play with this one. I take an object a, b, and c. And suddenly, when I have two functions, it will generate all the objects. So I'll do it like this.

1:15:00 So here I have an object a, an object a. Here, on this side, I'm in c, and on the other side, I'm in d. So here I have an object a, so I have two objects. If I take an object, I have two different objects. So a natural transformation between the two vectors F and G will be the data of the family of arrows, of the Teta transformation, of the N to the G, let's say it like that, of the arrow. It will be a family of arrows in the category of arrival, one for each object in the category of departure. So already, what we give, Teta, we can see that in fact, that's a little... The notation will be adopted later, but the reality is that it goes from object to object, from C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to C to They told us that these squares are communicative because these squares belong to the arrow. So, the naturalness of a constraint is extremely strong. I will show you this on an example. Do you know the term of a cyclical model? It is called the term of a cyclical model. In essence, it is known, but you can call it ... No, it's just a small example of what we are going to do to show the constraint that is given by naturalness. It is a very complicated example. I will start with a much simpler example.

1:17:30 The first example is very simple. The category C for objects is pairs. An arrow from Ea to Fd. This is simply an application of E such that Ea is included in E. The category is pairs. Pairs together, subsets, and the entrances between these things. The application of the set to the set. These are the following. I'm going to tell you what it is on objects and arrows. F of the pair E-A and G of the pair E-A. This is a parenthesis bubble. If you forget the sub-assemblies, well, your applications remain. You have an application B. If you have an application of order F that sends A to B, you forget the sub-assemblies. It will continue to be an application. And then, because it sends the sub-assemblies, they go in the opposite direction. So this one is good, and this one is good because you have eliminated an arrow. There are two functions, ea and fb, and you have a natural transformation of ea and fb.

1:20:00 So we are going to look at the natural transformations of the first to the second dimension. What are the natural transformations? It is to go from f of c to g, stop with the double parentheses, to g of ea. There are two sets, and it is to find an application like this. Well, here, what is f of ea? It is a, and g of ea is b. So the application is a canonical solution. If it is not natural, it is better not to employ it. What is the natural model? It is a type of arrow in the category C. So, f of 100. This is a key. I have an arrow f that comes from...