Categories, Sites et Champs - lecture 2 of 3

0:00 C, D, G, we start. And so I would say that S is in C, V, is a recurrence T, J, U, T, S, of V. So we have a double I, where U is a type, and in G, it's a type of type, so it's a type of type. In general, it's radically different from the other types. So now we're going to go to the test. It's just a question. So, if there are physicists in category A, well, first of all, there are many different types of physicists. We can't admit that they are all small, unicognitive. Either the physicists are at risk of being suspended, or they are rare physicists who are confused. In my case, I don't know the technical terms, but I don't know if they are the same. When you have a physicist who doesn't have the right capabilities, you eat him, and nothing happens. I'm sure you have a good idea of what I'm talking about, but that's not what I'm giving you right now. Anyway, let's take a look at your little cases.

2:30 Now, let's take... You can find S, which is a very small number, in A, and I have U in the middle, and S, not necessarily in the opening, a part of the object. I remind you that this notation can mean... So, I'm going to define S of S, by definition. It's the nucleus of the universe. To arrive in a product, you have to train yourself in this way. What are the two natural surfaces A and B? How do I have a product in a product? To arrive in a product, you have to arrive in each term. So there are many different types of physics. I'm going to come here so you can draw my answers.

5:00 I'm going to draw my answers. So I'm going to draw my question. I'm going to call it F of F. This is the question. If you're in A, in the interior, you're in B. This is an exercise. In practice, A. In practice. It's three times the sum of the two cases. So that's F of F. So now, what do we say about the equation? The same thing, but a little faster. The physics of the particle can also be seen as a sub-category, as in the case of elliptics, as in the case of Euclid, as in the case of Newton, as in the case of Newton, as in the case of Newton, as in the case of Newton, as in the case of Newton, as in the case of Newton, as in the case of Newton,

7:30 In this lecture, I will talk about the And then, when we restrict V3 to V2, well, we have a commutative diagram. We restrict V3 to V2, we restrict V3 to V2, and so we restrict V3 directly to V3 to V2. If we don't do it by V3 or by V2, it doesn't matter. So, the commutative diagram is seen in S2S. What does that mean? What is S2S? This is an example of the union of the DI, which is an element of F of S, which is a family of SI in S of DI, so that whatever SI will be a DI interdigit, it will be a DI interdigit.

10:00 So you see, F of S, for example, you have two users, for example, D1 and D2, well, it's the same. The sections, here you have an inter-section, we call it a 2, we'll call it A. In the sections, you have S1 which is on D1, S2 which is on D2. You are in S2S. If S1 equals 2 on the inter-section, the section is defined on the reunion. So you have to come to S1 and then to S2 and then to D1 and then to D2. So now I have given you the definition of the spectrometer system. In other words, the application of S of U in S of S is a new one. And we say that S is a special, so we talk about a preceptio separatio, a preceptio.

12:30 So what does that mean? Well, maybe you already know it, I don't know if you already know it. What is a preceptio separatio? For example, you have two questions, S and T in F , and then, at the same time, U in D. So if S is equal to C, then C is equal to I, so it's the same. F in F is a monotone, but it doesn't give you the value of F, since F in the product is there, so it's not a monotone. I don't know if you remember this, but it's a separate question. That is, if two sections are locally equal, they are globally equal. So, separated, local humidity leads to equality. For biodiversity, we can have two sections, because if I had to integrate them into one, instead of locality, I would say, locally null, in the middle. So, if it's locally null, it's null. If you want to make it worse, you have to make it as bad as possible. We will see in 5 minutes the example of this equation. It's the same. You make it as bad as possible, you don't make it as bad as possible. That's how it works. But I don't know how you can put it on the ground and make it as bad as possible. So the equation is this.

15:00 The equation is 2. The condition is 1 and 2. That means that if you have a section that is open, If a question is too complex, it is a unique question in a unique way, a global question. So we can say that the existence of a local system leads to the existence of a global system. All this is done for the sake of a system of questions. Questions arise in an ecological space. Questions are made locally into questions, and they are made globally into questions. So, we have two questions now. We have to remember that there is the elimination of questions. There are also a number of different types of numbers, such as the number of the number of the number of the number of the number of the number of the number of the number of the number of

17:30 It's the inner man. I don't know why it's called an inner man, because it's not really an inner man. It's me who calls it an inner man, I don't know why it's called an inner man. So, we have to go back to the first question. In the second question, S, P, G, to keep, to follow, to do this, man, to do this. So there is a notation that is A, U, Q. and GU. It's a notation, I don't know if it's true or not, but I forgot the expression. What I call FU is GU. It's the reflection of F. We see the reflection as a direct number, but if we look at things by IU, when we say IU, it's this one. So IU means what? One day after, it's UT. So for those of you who have done tests, In a static way, they are all related to I.U., which is I.U.U. But J.U. and you are here.

20:00 I.U. is I.U. minus one. So we have three notations, if you want. You can put even minus one, because of the convergence of the two terms. So I discovered a professor. So what is that? It's a set. And that's good. And if the category is additive, then it will be a group, an abelian group. If A is additive, it retains a 2S. Well, now that I'm changing the verb, V, I can... Well, a quitter can't define the last thing. J, V, U, T, it will be a quitter in English. It will be U in English. The problem is that we mix the arrows of relativity, we mix the nortices of the category rule, and the nortices of the category go in the opposite direction.

22:30 There is no such thing as a miracle. So, you can't mix them up. So here, a small amortization. A U is amortized in Ti. So, this is a pointer. And again, J is U in English. It goes from U to U in English. U and B are here now. In the end, they are all the same. So, there is a counter of the imaginary that goes from the order of constant, of course, to the order of... So, what is the J-s counter? So, where does V go? Where does it go? It goes to O-V.

25:00 It's a counter. You send om in om. So I have a preface. In this preface, we put om rong. Om rong.

27:30 I want to give you an example before we go on. What is a map? For those of you who know, it really is a map of the origin. I have a story. I was telling you that... I was telling you that... Oh, so it's a cataclysm. I was telling you that this is a topological space. And that you have... You know, you have a green light. So I say that S is the answer to the question of U, which is the ability to calculate the sum of the two products, S and U. S is the answer to the question of A. And in particular, I also say that S is the final object. If you divide by the number, it becomes the final object. So, for example, if A is the final object, the final object will be zero.

30:00 and time economics. An example that many of you already know. It's not necessarily that it's not topology. It's that it's not topology. So, the first example that comes up here is the continuous function. We'll call it R23. We'll call it R24. Why? Well, because it's continuous, so it's not local. A continuous function is a locally constructed function. There is a much more interesting example of continuous functions with a value in Z. These are functions that are not constant. Generally, if M is in this mode, there is a particular module that is in the ring, and if the continuous functions have a value in R, M has the property of 13. So you don't have continuous functions. Well, we have to say that the functions are legally consistent. They continue to have a secret identity, that is to say, they are legally consistent. So, we must not confuse it with a set of equations. There is also a set of equations that are consistent, because there is only one, that is to say, the functions are consistent. So, even if we say that they are not consistent, it is not the same thing as the functions that are legally consistent. The functions that are consistent are not so interesting. The functions that are legally consistent are not so interesting. So, I insist.

32:30 These are the functions of a constant law because we are going to focus on the functions of a constant law. We will see three things. There are the functions of a constant law, the functions of a constant law, the functions of a constant law, the functions of a constant law, and then there is the last case, we will see, the functions of a constant law. So this is an example of a constant law. But we will see that another time. Let's see another very interesting example. That's all for now, but if you have any questions, feel free to ask them in the comments section. So let's look at some more interesting examples. We have an example of a prepesto that verifies the action S1 and not S2. So take the U at the end of the Rn, for example, and bring it all under R1. You associate the continuous functions, the Bornier and the prepesto functions, with these S1s. Obviously, if you have a prepesto of a prepesto with the big S1, you verify S1.

35:00 A sub-precept of a separate precept, a sub-precept of a separate precept, is separate. So this is a separate precept. And, on the other hand, for nuclear technology, it's not a separate precept. It's bordered, and that's exactly the case. The function, you take x equal to r, and here we really get r union of the intervals minus n, minus n, if you take the f of x function. The topology is formed on the whole interval of n-n-n-n, and not on the reunion. It's not a topology. But if you take, for example, a topological space that is very compact, and if you take it as a topology, it's not a topology for the individual. It's a topology if you think of it as a topology. If you take, for example, a topology that is open to the public, it's formed. That's it. If you take a topology that is open to the public, it's formed on the reunion. So it's a question, it's a question of religion. When you have a pre-question, it's always a question... So I said, I had to put it in the French language, I said it's a pre-question at the end of the year, but there are also questions that arrive at the end of the year, at the end of the year. So it's a question, it's a final. So it's a question of... it's not just the final.

37:30 You have to find the U, then there is U. If L returns U, then U becomes S. The topology is there, and that's the topology of the questions. And that's the topology of the questions. By the way, you have the topology of the questions, but you don't have the topology of the questions of A, B, C, and so on. So another example, another example. We have the integration series. For example, if you take the space of Lp on Rn, which is in the middle of the wave, it's not a crystal, but if you take the space of Lp on Rn, it forms a crystal. Another example, the last one before the interruption, is an example of a crystal. If they do not derive, the action is simple, it is not separated. That is to say, an example where we can be totally null, it is null. You take X, which is the P, and if you take the points at the end, the derivative of the Z is the derivation. So I'm going to take the next preceptor. In any case, X is associated with the core in the preceptor's direction. All of these things are related to the moment of evolution.

40:00 The precepts, in the case-mode, are actually the same. I call it the precepts of the cocker. In the precepts, I put the cocker, because there will be a cocker in the precepts that will not be the same. Well, I say that this precept, in the end, is not separate. In this case, it belongs to, and by definition, to U, which is completely open U, belongs to Cooke, which is null, which is completely open, which means that everything is U. In the topological space, it is complementary to zero, which means that everything is out of zero. This is the result of the differential equation. F of z equals 1 over z. 1 over z equals 0 locally for all of them, for all of them. But globally, so f of z equals 0 in f of z, whatever the index. And on the other hand, in f of i, 1 over z is not worth 0, because we can't solve the equation of f of z equals 1 over z.

42:30 If you want, you have an integral equation. It has a local solution, but it doesn't have a global solution. That's why it's called a local solution. And the solution of the square is the unit. The solution of the square is to have a global solution. But the square is the unit, so there is no global solution. So this is fundamental to understand, because it's the basis of all these things, which is the local phase of the global. Equally and not globally. It can be finished totally and not globally. The other examples we will see later are those of orientation, another example, another example that has never existed before. We have a drawing that is called the Moseley band. This is the Moseley band. I will compare the orientations. So, take this orientation here. And a little further in the Moseley band, this one, it's the same. This is also an orientation. You see what I mean? Orientation means what you will have. I move this and that and I also move nothing in the orientation. However, when I make a turn, I have the opposite orientation. The orientation system is there, but there is no bottom section. That's why you have to do the same thing. And then we do the same thing with a sum of pluses. So it will be the equations that we do every three minutes.

45:00 This is why we started from the topology and from the point of view of the homology of physics, of the topology, and so that all the topologies, let's put it this way, let's say in the real world, in a complex way, let's put it this way, by the time of the research, and the complex homology, which is the topology of the variety, which is one of the words, which says that the topology of physics, in the real world, the complex homology... And that's all about cohomology, which is interesting and surprising because it's a topological object, which is a non-local substance, which is a sort of the topology of physics. It's not that we have the different cases. The context of the different cases does not depend on the topology of physics. And that's one of the new variants of cohomology. So it's a bit of a puzzle. It's a bit of a puzzle, by the way. I don't know what to say about it. Now, let's move on. What's this? What's this? So it's not the socialist part of the problem. So, the situation is as follows. We have a piece of paper that is made of a piece of paper that is made of a piece of paper that is made of a piece of paper that is made of a piece of paper that is made of a piece of paper that is made of a piece of paper that is made of a piece of paper that is made of a piece of paper When we look for an association between S and a preceptor, we look for S-A, which is the associated preceptor, and when they are joined together, like this, they are joined together, it means that S is a natural morphism.

47:30 I called it a preceptor. The associated preceptor, after Witt, is a preceptor for Iota, and by the general rule of Newton, which is exactly what he said. I have a professor, the identity of the professor, who sends in a proposal from his sister, so we're going to, I don't know if he's aware, but I'm going to explain, I'm going to explain to you, I'm going to explain to you the concept of what we're doing, but there are hypotheses, there are a lot of hypotheses about the category. It's not very interesting, it's a lot of hypotheses. No, because as I said in all these demonstrations, we won't see them appear. But, for example, In the other terms, there are individual limits, projective limits, and exact limits. In the other terms, there are individual limits, projective limits, and exact limits. In the other terms, there are individual limits, projective limits, and exact limits. In other words, it's the same thing that we use in France, it's the same thing that we use in France, it's the same thing that we use in France, it's the same thing that we use in France, it's the same thing that we use in France, it's the same thing that we use in France,

50:00 First of all, I have an idea. For every subject, and for every movement, the idea is to say that if something doesn't work, we'll go back to what really worked. That is to say, if I take M A, I have to take S S, and then I have to take them all, because there really isn't any. So my idea is to say that if I take the derivative in S of S S, that's the first one, but I have to go back to the first one. And it doesn't work either. So, we're going to do more. So, first of all, what does this mean? You see, it's a bit... Why is it difficult? Because S2S is the objective limit. So, we're going to take the objective limit and the objective limit. And that's not at all... That's why it's difficult. So, first of all, before we take the objective limit, let's say that you have a line in U. So, U is the ratio and a ratio of S2. What is V1, what is V2, what is V3, what is V4, what is V5, what is V6, what is V7, what is V8, what is V9, what is V10, what is V11, what is V12, what is V13, what is V12, what is V13, what is V12, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V13, what is V

52:30 Instead of writing it like this, I write it like this and send it to the students. And then I say that it is conflicting. The academic subject is conflicting. This is my book. And then if you have books 1 and 2, then you have a piece that is politically correct. And it will quickly be the same with books 1 and 2. And if it's not the same, then you have a different piece, but it's not a different piece. You know the category, it's different, it's the opposite now. So now, I'm going to show you that a professor defines a framework for this category. If I take the analysis of a category, what is a professor? It defines a function. So, gs is a professor, gs is a function, so we have an analysis of f of s. So now, suppose I have a f of s1. Thank you for watching this video.

55:00 All of this can be seen in the product of F2G. In fact, I send it to F2G1. As you can see, it does not depend on the space V2. So if I send another one, if I had chosen another one, if I had chosen another V2, because for F of V'2, for V2, it is factorial. So I have defined F of S2 as F2G1. So now, if I send it to the project, I have defined F of S2 You can see that this is a very short video, but it's almost obvious, it's three years, so I wanted to define the counter, so I wanted to show that M defines a counter, where E is a counter, which means that E is a...

57:30 So if I have a counter, I can take the electric limit, so I can suppose that A is the electric limit, so I take the electric limit. So you see, it's already used that A is the electric limit and the electric limit. In the last lecture, we discussed the definition of the definition of the definition of the definition of the definition of the definition of the definition of the definition of the definition of Thank you for watching this video.

1:00:00 We have to find the counter, the plus counter, and also the identity of the identity of the identity of the identity of the identity of the identity of the identity of the identity of the identity of the identity of the identity For example, A and B. So there are several things to say. There are parts that are simple, which are A and B that are simple. So I say that if S is separated, then S in S+, is a mono.

1:02:30 I say then that if S is a cell, S in S+, is an iso. S is separated, that means F in S is a mono. No, S plus DU. S plus DU is the inductive limit for S in the inductive code of S2S. It's not a filtering inductive limit because it's not in the DU code. In fact, it's in an opposite derivative in the DU code. The derivative is filtering. So, we're going to make a hypothesis that the filtering inductive limits are exact. I don't want to give you a hypothesis, but I'm going to give you some examples. In other words, the methodologies they use, for example, are used to transform mono into mono, and so this is a mono. And to say that S is a mono in theory is to say that it is not. It's not by definition, it's just a counter. To say that a molecule of a sensor is a mono is to say that it is not. When we say that a molecule of a sensor is a mono, it means that it is unique. This is an example of an academic lecture on mathematics, geometry, algebra, algebra, mathematics, chemistry, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics,

1:05:00 And that, a long time ago, I told you, I told you on my computer, I lost the demonstration, I was in a more complicated way, in a more general way, but in this case, in a relatively long-term way, I lost the demonstration. So we're going to admit, we're going to put the counter, the counter that I have, my counter plus, I'm going to take the counter plus plus, it's called a, so a. And if you want to follow it by hand, it's quite complicated because Ks plus U is the objective limit of the objective limit and we start again with the objective limit of the objective limit. By hand, it's the objective limit. So, I'm going to tell you about the Ks plus.

1:07:30 So, I'm going to tell you that the function of these functions is very close to each other. So, the Ks plus is a function that is very close to each other. I'm going to start with a question, and then I'm going to separate it, and then I'm going to ask you a question, and then I'm going to ask you a question, and then I'm going to ask you a question, and then I'm going to ask you a question, I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it.

1:10:00 I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it. I'm going to show you how to do it. I apply the counter A. A is a counter. So if you see a mantis in it, it's a mantis in it. But G is a vessel. So to take the vessel and to destroy the vessel, it does nothing. The vessel is marked there. If A is a vessel, A is obviously a vessel. So it's the same thing as A and G is a vessel. So I made tests in the field. In each field. And as always in... This is a demonstration of the vector, the axis and the matrix on the opposite side of the water. We will show the two forms of S. So here we are in the 7 hecto. Here too, by the way, but we can put in the 7 hecto if we prefer, because both are for the hecto. And here we have a bokeum. 7 hecto is a cilium. Yes, so we have at least demonstrated the... Here, it is said that... We do not do that.

1:12:30 We do not do that anymore. We have 7 hecto in 7 hecto. There are two factors, A and Z, but we don't hear them in the lecture. We look at A as a set of factors in a set of factors. This means that the factors in the factors make no sense. First of all, what does Z mean? A factor is Z which means that the objective limit is finished and the objective limit is finished. So, the factors make the objective limit and the objective limit. What's the name of the field? I don't know, I don't know. So, you are here. So, the projectors, we can say that they admit the inductive limits, small, but I don't write small, the projective limits, and also that the inductive limits that they take, for example. Why? Because it's true. All this is true in the field. And so, in fact, in a resilient way, it's true in the projectors. Now, the category of sectors, I certainly did not say that it was the same for all of them. So, when we see the factor A as a factor plus or minus, it is always in the same order, otherwise it is not the same. I remind you that a factor is a factor, that is to say that a factor is, as I said, exact to the left, it means that it is connected to the connective limit,

1:15:00 to the right to the connective limit, and exact, that is to say, to the right and to the left. So, it is obvious... What's obvious is that IOTA-I is right-handed, because it's the other way around. It's the other way around, because it's the other way around. It's in the... Ah, yes, sorry. It's in the right-hand side. They don't look at it in the right-hand side. They don't look at it in the right-hand side. They don't look at it in the right-hand side. They don't look at it in the right-hand side. They don't look at it in the right-hand side. They don't look at it in the right-hand side. So you're right. The A-pattern, which looks like a crystal ball, but for now I will look at it as a crystal ball. The two, which look like a crystal ball, are adjacent, and so A is exactly to the right and Z is exactly to the left. So, as we know that A is simple, we can only know that the plus is exactly to the left. Because A, which is composed of a pattern exactly to the left, is exactly to the left.

1:17:30 The function of the plus is exact on the left. So here, to get to the point, it's not exactly the same. Ah, no, no, sorry. I didn't have the same answer. So, what does that mean? So, the function of the plus. What does that mean? The function of the plus is exact. The function of the plus is exact on the left. It means that whatever is u, the function of f of u... This is an example of a projective unit, a projective unit as in a projective unit. So, if I replace them with electric projective limits, then the inductive limit of the inductive limit does not change.

1:20:00 So, this function changes from a projective limit to an inductive limit. So, this function changes from a projective limit to an inductive limit, and then from a filtering inductive limit. The filtering inductive limits change from a projective limit to an inductive limit. So, let's continue. Thank you. The limit of an academic thesis is 30 commutes, and the limit of a corrective thesis is the same as in my class. Well, in the ensembles, I'll show you. But here, it's a cathegory. It's a cathegory on the cathegory of physics 30, on the exact left. So, the plus counter is on the exact left. And so, the A counter is on the left. We'll see that in the cathegories. So, that's it for now. In any case, that's it. For now, we haven't studied the cathegory of the thesis. It will come next week. What does this mean? Spacetheory, in one way or another, is the same. In reality, it's the same thing. The same thing can be pronounced in one way or another.

1:22:30 It's a spacetheory, so it's a spacetheory in one way or another. In topology, at the beginning of the field of physics, it wasn't at all like that. First, we have to say that the field of physics dates back to the 1940s, more or less. I would like to start with the rest, which is the wars, which are in my opinion in the best conditions, it could be, and I would just like to ask our parents what they did. First of all, there are the thermals, the thermals of the universe, the carapaces, the monsters, and then there is the engineer of Cartan and the spheres. And finally, physics. By the way, after that, no one has analyzed the rest. Even though physics has been invented with something. Well, no one has invented it. For example, physics has been invented with the idea that there will be an island, and that there will be something in the middle of the island. And there, there is an intuition that we call the role of the rest. Because the rest is not very important. And then, after that, there is the law of physics, which is reflected in the categorical language, which is the law of physics. By the way, at the same time, in the 1980s and 1980s, before 1978, he also made fields and bags. But for the bags, it's a bit like the rest, and it's easier to make.

1:25:00 Also, the construction of public buildings depends on the research that we have learned, and the discoveries that we have made in France. Because here, what we are doing is really... I don't know if it's the only construction, I don't know if it's the only one. However, what is certain is that there are other constructions. Here, I'm going to give you a construction with a few words, and maybe you can animate the products. Otherwise, everything I've said is enough. But what I'm saying is that it's not certain. In any case, it's in the 9th, it's a construction of the 8th century. Without these little details, you can look at these pictures and see that there are direct constructions. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. This is an example of local mathematics. So I'll give you the construction by hand, but in the case of a topological phase, 1EA equals 16, or whatever the term is. So you have a set of terms. I don't know if it's the same, but it's the same. So how do we define it? If we open up U, what does it do? We call it a section with the elements of F and U. So we'll take the applications that we have understood, In the reunion of joints, we don't take all of that. Fx is an ensemble. We are in the category of ensembles,

1:27:30 and the two on top of each other are in the subcategory of the category of ensembles. There is the notion of germ. So you take things like that with applications that cross each other. You will take all the germ of questions in each corner and you will take the applications that exist. All of this is included in the germ, and in this way, it is also included in the calvary. So, we are going to ask whether F at the point I, first of all, belongs to FI. You see, we have it in the Réunion des Moines, but you see that at the point I, you have also something that is in FI. But you see, it is only a subcode. So, you see that for all of this, today, there is a neighbor, V, open V, For example, in F2-V, the Fx must come from a certain section T in F2-V. For example, in F2-V, the Fx must come from a certain section T in F2-V. Thank you very much for your attention. I'm sorry to interrupt you, but I think we should stop here because there were a lot of questions.

1:30:00 Let's continue, and then we'll move on to another field. After this new field, we'll build the units in the field. I don't know if I'm going to be able to finish the presentation. I'd like to finish the presentation by identifying the topologies that you mentioned. I'd like to say thank you very much. Thank you very much.