Categories, Sites, Stacks & Fields - lecture 3 of 3 (contd.)

0:00 It was a machine, a passive space, objective in its proximity and length, having an exact distance independent of the objects it consists of. The notion of rest and movement had for Newton an absolute character. A physical object is at rest if the position of the object in space does not change over time, and in the case of Earth, it is in motion. Newton himself knew very well that we can observe only the relative movements of physical objects relative to each other and that no experience of the mechanism allows us to distinguish between a fixed landmark, an immobile space in this absolute space, and a landmark called inertial or linear, in uniform translation at constant speed in the absolute space, if at all it exists. The existence of absolute space has been widely discussed, especially by Leibniz, and more recently by Ernst Mach in his book published in 1863. In practice, the mechanics chose a referential, approximately inertial, and then studied the movement of the physical objects in relation to this referential as if it were fixed. So for mathematicians now... For mathematicians, until the discovery of non-euclidean geometries, it seems that few mathematicians had clearly become aware of the distinction that we should make between the physical space in which we live and the mathematical spaces that we can imagine, thus making a theory an example of contradiction. L'espace est de celui de Witten, femme de mon prédécesseur dans ses conférences dont on a très bien parlé, et en fait d'ailleurs c'est une dictature d'élective. The main debates are based on the coherence of the two, and their independence.

2:30 In particular, the so-called fifth of these is that, at some point, we can draw a single law parallel to a common law. So this gave rise to a lot of reflections from mathematicians. I will only talk a little bit about legends, which have elements of geometry, which my predecessor also talked about earlier. He was preoccupied by this postulate and he tried to develop a geometry with a lower axiom, the sum of angles of a triangle is less than or equal to two right angles. And he showed that if the inequality is reached for a triangle, it is for all triangles and that it is equivalent to the fifth postulate of Witten. Around 1820, non-euclidean geometries were discovered. Individually by two people, Janos Bonillet and Nikolai Lobachevsky. They have almost simultaneously developed a geometry that is called today hyperbolic geometry, in which, from a point of a plane, if we have a right that does not pass through this point and which is in this plane, we can pass an infinity of parallel rights and in this sense they do not meet the right of the right. And the possibility of developing such a geometry was probably already known by Gauss, but he had not published anything of his life, we found this only in his papers after his death. And Gauss seems to have judged this possibility rather uninteresting because he dissuaded Bollier from publishing his research. Poor Bollier published his researches only as an appendix to a reissue of a treaty written by his father. Lobetzewski wrote his memoir in 1821. We asked ourselves the question of the coherence and the actions of hyperbolic geometry, and in 1868, Beltrami showed that hyperbolic geometry was just as coherent as classical Euclidean geometry, by constructing a hyperbolic geometry model in the context of classical Euclidean geometry.

5:00 And in fact, all of these have been constructed from analog models of geometry that are perhaps a little more famous than those of Newton. And in fact, the... The question of the coherence of the axioms of geometry was, let's say, resolved much later, only in 1899 by Hilbert, who proved that the consistency of the axioms of geometry was equivalent to that of the axioms of arithmetic. But it is not so much the axiomatic subject that I would like to talk about, but rather ... First of all, the generalization of the space that was created, in particular by Bronsley. Professor Witten spoke about this earlier. And Onsley was preoccupied with the properties of the projectile of the figure. He had already generalized space by introducing the right of infinity in yellow and white. He had also introduced the duality that exchanges the right and the points, the transformations by Paulette and Cypro. And he did not hesitate to consider imaginary points. They come back with nuances to make the plan more complex. And in fact, he is the great founder of modern projective geometry. So now I come to the two memories of Riemann. The first dates from 1851. General theory of the functions of a complex variable. And the second, 1854, was the hypotheses that serve as a foundation for geometry. This has really revolutionized the conceptions of space in a considerable way. In 1851's memoir, he introduced what we now call Riemann's surfaces. His goal was to transform univocal functions into multivocal functions that can be found in complex analysis. And because of this, he was led to consider the functions that were defined on an ensemble whose topology is much more complicated than that of a simple open complex plane. And also in this self-memory, he observed that we must distinguish two states in the construction of a complex space, first choose a topology, then choose a complex structure.

7:30 In fact, he will use this... he does not say the word topology, of course, it did not exist at his time, but when we read what he wrote, it was topology that matters. He will use this observation under another form in his 1854 memoir, he will use the space of a topology before defining its mathematical properties. The memory of 1854 on the hypotheses that serve to understand geometry introduces a completely new point of view. All these predecessors founded geometry on global actions. For example, by two points, they pass a single right. It's something global. At one point, we can only draw a parallel to a right-hand side, and he affirms, on the contrary, that we must start from the infinitesimal properties of space, and that it is these that, with topology, will determine global properties. So they are placed at the very beginning on a dimension n variety, where the word variety is sometimes used, well, I don't know the German, I have had access to the work of Raymond Riemann through a French translation, but he sometimes uses the word variety in other languages, and what he means is the varieties in the sense that we hear it now. He immediately places himself in dimension n, which explains how we can define the notion of dimension, how to construct from cross to cross the varieties of growing dimensions and then how, on the contrary, to bring back the study of a variety of dimension n to the study of a variety of dimension n-1 by taking a non-constant definition of this variety and by reverting the sets of values, the sets on which the function takes their place. And so, for the properties of methods, he introduces what we call today a Riemannian structure. At each point of the variety, he says that the distance between this point and another infinitesimal point, infinitely adjacent, the point he calls x and the point next to it he calls x plus b,

10:00 today we would say that the vector is given by the square root of a quadratic form. In relation to the components of dx, dx1 and x1, a positive definition and in which the coefficients g and y depend on the exponential point. I have written a formula such as the one I have already put at the base of this slide. And then he also studied other possibilities. Instead of the square root of an expression of this degree, he... He clearly looked at the fourth root of a homogeneous polynomial of degree 4 by being a polynomialist, and more generally a homogeneous function of degree 1. This is what we call today a Finschler structure, which was studied by the mathematician Paul Finschler, but well after Riemann. At each point, Riemann was fully aware that the expression he was using depended on the choice of the components. He had to find things that did not depend on the choice of the components. And for that, he introduced invariants, which today are called sectional components. And they do not depend on the choice of the total components. They are in the number of m, m-1 over 2, for a variety of m, and they correspond to the different independent directions that a plane can have, that is to say, a point like this one. And then, when the sectional curvature is constant, That is to say that they do not depend on the point considered, nor on the direction of the plane passing through this point. Riemann's approach allows us to find the geometries that we already knew, that is to say, for a null curvature, the Euclidean geometry, for a negative curvature, the hyperbolic geometries of Bollier and Noval-Chesky, and for a positive curvature, the spherical geometry. But in fact, Riemann's approach is more general. It allows us to consider spaces that are neither homogeneous nor isotropic. And Einstein will use this possibility to elaborate the theory of relativity. In the last chapter of his memoir, Riemann is interested in the possibility of using these mathematical constructions to describe physical space. And the physical ideas he presents were extremely innovative at the time. At a very small scale, the properties of space may not be correctly described by the mathematical model he has developed.

12:30 He also says very clearly that it is the physical properties of matter that are present in space that, at a very small scale, determine the geometric properties of space. This is in fact the basic idea of the theory of general relativity. Except, of course, that they had to be placed in space-time and not in space-time. So, he writes, I don't have time to say everything, I will go to the conclusion. It seems that the empirical concepts on which the metric determinations of the extent are based, the concept of solid body, that of light rays, ceases to exist in the infinite. It is therefore very legitimate to assume that the metric ratios of space in the ancient infinitum do not conform to the hypotheses of geometry. This is what should be admitted as soon as we obtain a simpler explanation of the phenomena. The question of the validity of the hypotheses of geometry in the ancient infinitum is linked to the question of the intimate principle of the metric ratios of space. In this last first question, which we can still look at as belonging to the doctrine of space, we find the application of the previous remark that in a discrete variety, the principle of matrix reports is already contained in the concept of this variety, while in a continuous variety, this principle must come from elsewhere. It is therefore necessary for the reality on which space is based to form a discrete variety, so you already had the idea that at the microscopic scale, perhaps space is more discrete and more continuous, so that the foundations of metric relationships are sought outside of it, in the forces of liaison that act on it. The answer to these questions can only be obtained by starting from the concept of phenomena, verified right here by experience, and that Newton took as a basis, so he gives a round of applause to Newton, but immediately after, he says, by adding to this concept successive modifications, demanded by the fact that it cannot be explained. Research based on general concepts, such as the study we are going to do, may have other uses than to prevent this work from being hampered by too narrow views and that the problem in the knowledge of mutual dependence of things finds an obstacle in traditional prejudices.

15:00 This leads us to another field of science, the field of physics. The object of this work was to obtain a degree in the university where he was a professor. The memoir of Riemann was presented orally on June 10, 1853 in front of a jury where there were Gauss and Dedecky. But it was only published in 1867, one year after the death of this poor Riemann. I will show you a photo of Riemann. The mathematical ideas that they contain have been developed first by Christophe Pell and by the geometrists of the Italian school. There was an extremely prestigious school of geometry at the end of the 9th and the beginning of the 20th century, with Bianchi, Bixi, Levitica. They still have 50 years. To develop from there the geometry of the differential. Absolute differential calculation, calculative and subliminal calculation, notion of parallel transport. It took them 50 years, from a 20-page memo. There were new contributions in 1917 by Hermann Weyl, who noted that we do not need a human structure to define the future. We can introduce a new notion, which is that of connection. It is enough to define parallel transport, to define geodes, and to define many other notions. Hermann Weyl was also... He introduced the notion of conformity, that is, when we can multiply a Riemannian structure by a function that does not cancel, we obtain another one, which is called conformity.

17:30 He discovered the tensor that today bears its name, whose cancellation is a necessary condition and sufficient for a Riemannian variety to be conformable to the equilibrium. It is also Hermann Weil who, in his book of 1911 on the surfaces of Riemann, for the first time used the current way of defining a variety by means of a family of cards that form an atlas, a bit like the cards on the earth form an atlas of the earth. So Cartan took the lead and he generalized the notion of connection considerably. He pushed the study very far. Hermann Weil had used only numbers. There are many types of connections, linear, refined, projective, conform, and more generally associated with one another. So his basic idea was that, in each point of the variety, to associate a homogenous space of the same dimension of this variety and tangent in a certain direction to this one and which represents an approximation of the variety, a bit like a tangent plane to a surface can be considered as an approximation of the surface next to a point of contact. So, he introduced the cohomologies, he also specifies the torsion curves and, in fact, the theory of connections will not be completed until 1950 by L.S. Mann. As for the physical part of L.S. Mann's ideas, it was extremely ahead of its time. We can dismiss in germ the great principles of general relativity and quantum mechanics, but they remained unknown for many centuries. In fact, there is still one person, it is Clifford, he saw the importance of these ideas and he wrote this. Riemann has shown that there are different kinds of lines and surfaces, so there are different kinds of spaces of three dimensions, and that we can only find by experience to which of these kinds the space we live in belongs.

20:00 In particular, the axioms of plane geometry of truth within the limits of experiments on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges He says that although the actions of solid geometry are true within the limits of experiments for finding parts of space, yet we have no reason to conclude that they are true for the very small portion, and if any help can be got thereby for the explanation of physical phenomena. We may have reason to conclude that they are not true for very small portions of space. I wish here to indicate a manner in which these speculations may be applied. I hold, in fact, that small portions of space are, in fact, analogous to little hills on a surface which is on the average flat. Namely, that the problem of the laws of geometry are not daily in them. That this curvature, being curved and distorted, is continually being passable from one portion of space to another after the manner of a wave, and that this variation of curvature of space is what really happens in that phenomena which we call the motion of matter, whether ponderate or et cetera. It's the same Glyphosate that in the physical world, nothing else takes place but this variation, the subject, possibility of the law of continuity. It's the same Glyphosate that created what we now call the Glyphosate algebras, which generalize the complexes, the quaternions, and the octaves. For example, to demonstrate the case of equalities of triangles, by moving a triangle without deforming it to bring it in coincident with the other one.

22:30 So, all the Euclidean movements form a group. This concept had been clearly recognized by the Evaluatrix of the law, which he used for the theory that carries this definition. The notion of group is imposed in geometry thanks to the work of Klein and Sokusti. Klein was German and Sokusti Norwegian. They both lived in Paris around 1870. They were friends, and I think they lived in the same building. They were influenced by Darcourt and Jordan. It must be mentioned that it is Jordan who saved, for posterity, the heritage of Galois that the Academy of Sciences had lost. The memory he had submitted to the Academy of Sciences had been lost. Maybe long-term. It is Poisson who had lost it. Félix Klein was someone extremely early. At the age of 22 or 23, he obtained a post of professor at the University of Erlangen, and his inaugural lecture, the Erlangen program, had a profound influence on the development of geometry. The idea of methods he developed is as follows. A geometry is a set, which Klein calls a multiplicity, in the current language we would say a variety, on which a group of transformations act. And the study of geometry is nothing more than the search for invariants of a group. So the most important object of a geometry is the group, and not the set on which it acts. Besides, we can construct the set on which it acts as a homogeneous space of two observations. So the group contains everything. So this idea made it possible for Atman to give a unified presentation of the different geometries in China, which are projective and also hyperbolic geometries, and to develop them. Sophie Slee was interested in the transformation groups of a variety that only depends on a finite number of parameters to be able to vary in a continuous way. Today we call them groups of life. These are the groups that have a differentiable structure and that are of finite dimensions.

25:00 His idea, in a way, is a bit similar to Riemann's. Instead of looking at global things, he thought of looking at infinitesimal things, differentiated by the parameters in the group of points. And that's how he discovered algebraic numbers. And he studied the relationships with the groups of points. They show that all Lie groups are associated with Lie algebras and that, respectively, all Lie algebras are the Lie algebras of a local Lie group. The word local, of course, is too much, but it is not Lie that has the right to distance itself, it is the other way around. It is he who gave a complete proof of this theorem, which is called the third theorem of Lie, although Lie is not the third theorem. L'œuvre d'Élie Cartan, lui, est tout à fait considérable. Il a introduit une méthode très novatrice en géométrie différentielle, la méthode du référent mobile. Il a élaboré une classification complète de toutes les algèbres de l'île, simple, complexe. Elle a été commencée par Killing. Il a découvert l'élève des espaces qui n'ont ni asymétrie ni encore beaucoup de choses. The subject that concerns us today, space, is that Tartan was very open to the progress of physics of his time and very interested in mathematics. He had a regular correspondence with Albert Einstein around the 20s, 25s, 30s. Several of his publications contain applications of physical space and the foundation of mechanics. So I will now talk about the premises of constraint relativity. Everyone knows that Maxwell, at the beginning of the 1860s, established the equations that govern electromagnetic phenomena and introduced the concept of field. I'm moving forward a little, I'm not sure if this concept is not behind in Maxwell. Anyway, Maxwell's equations predicted the propagation of electromagnetic disturbances in the form of waves at a finite speed, the same in all directions, regardless of the movement of the source. He noticed that this speed was close to the speed of light. We already knew at that time that this speed was finite.

27:30 It had been measured by Fizeau and with greater precision by Kuhn. So Maxwell understood that light was an electromagnetic perturbation and that the speed seen by these equations was that of light. But then there was a problem, because in kinematics... If a phenomenon cannot propagate with the same speed in all directions compared to any landmark, if we take a landmark in motion, the speed will no longer be the same in all directions. Physicists no longer believed in absolute space, but on this occasion they somehow resurrected it. They imagined a luminous ether that was the support of electromagnetic waves. You can see the vibration of the Earth's orbit, which is impregnable. They thought that the orbit of the Earth was the same as the orbit of the Earth. But if we measure the orbit of the Earth in different directions at different times of the year, we should be able to deduce the movement of the Earth compared to the orbit of the Earth. Now everyone knows that these measurements have been made. These measurements were made by Michelson and Morley around 1891, and they did not allow to highlight the movement of the Earth in relation to the Earth. In 1905, Albert Einstein proposed an explanation of a revolutionary matter. They clearly see that the property of light to propagate at the same speed in all directions, whatever the reference relative to the relationship, is incompatible with the absolute nature of the notion of simultaneity, and they base this theory on two principles. The principle of relativity, according to which all inertial linear referentials are equivalent. And the constancy of the speed of light, whatever its direction of propagation, the movement of the source, the inertial reference in which we are in contact. And so, he shows that on these bases, we can build a non-coherent theory, provided that we renounce the absolute nature of the notion of simultaneity, that is to say, the existence of an absolute time. And at the same time, of course, the principle of relativity, he has renounced the notion of movement and of action.

30:00 He very clearly stated, since his memory of 1905, that the concept of luminous lands is superior. I think that a little later he came back to this to please the audience, but it is a question of diplomacy. In the same year, Poincaré published a long memory in which he introduces the local times that depend on the observer. He studies what he calls the Lorentz transformations. And these are linear coordinate changes that involve the four coordinates in the first event, the three coordinates of a point in space where this event took place, and the fourth which is the time, the instant at which this event took place. So, like any transformation, a Lorentz transformation can be seen either passively or passively. This is the transformation that gives the four coordinates of an event in the referential of an observer in function of the four coordinates of the same event in the referential of another observer. And also, in an active way, it is a punctual transformation, not of space, but of space-time, since it actually mixes the coordinates of space and time. The square point shows what shape it is. In short, it determines the invariants. He has undoubtedly understood very well that local times measured by different observers are not the same, he even indicates the formulas that allow to go from one to the other. He has also noted that the transformations of Lorentz are in accordance with the property of light to propagate at the same speed in all directions, whatever the reference in relation to which we evaluate them. So why don't we ... We do not say that it is Poincaré who discovered the relativity of the string. He does not say, and it is not as clear as Einstein, that the concept of absolute time must be abandoned, nor that the concept of the Earth is subdued. The space-time has been officially named by Minkowski, and we are now talking about the space-time of Minkowski.

32:30 But in fact, Poincaré had already considered it in his memory of 1905. So the geometry of Nikolsky's space-time is an example of geometry in the direction of Phoenix Klein. His group of transformations is the Poincaré group. So the Poincaré group, we named it like that to exonerate ourselves from the point of view of Poincaré. The fact that Poincaré called space-time the relative of space, space-time is also possible, we could call space-time Poincaré. For Poincaré's group, it is simply the transformations that are composed of a Morin transformation and a translation. So, there is still a difference with the geometries that we already knew, it is that in the space-time of Minkowski, the rights are not all equivalent, the group of square points does not act respectively on the sum of the rights, we must distinguish the three types of rights of the space-time and the space-time. So, a little digression on the square points and why they remained there with the theory of relativity. Well, we understand it by reading his books on the philosophy of science. So, for him, to give the notion of simultaneity, an interlocutor, a concept, so to introduce an absolute law, the same framework as space and Euclidean, these are conventions, not physical laws. These conventions are practically revealed because they are in accordance with our intuition and they lead to a simple formulation of physical laws. He wrote, for example, in a conference he gave in London on May 4, 1912, about the new mechanics. The new mechanics is the name he gave to the theory of terrestrial relativity. 1912 is the year of his death, so something he wrote very little time before his death. What will our position be in the face of these new concepts? These new concepts are therefore terrestrial relativity. Are we going to be forced to modify our conclusions? No, sir, we had adopted a convention because it seemed convenient to us. We said that nothing could compel us to surrender.

35:00 Today, many physicists want to adopt a new convention. It is not that they are constrained, they judge this new convention more for us. Those who are not of this age can legitimately keep in the sky and not forget their old habits. I believe among us that this is what they will do for a long time to come. This is my point of view. I know that there was a quarrel about the priority to be given to Poincaré or Einstein for the theory of restricted relativity. Personally, I think that Poincaré discovered the theory of restricted relativity at the same time as Einstein, but unfortunately he did not really believe in it. He saw it as a social mathematical theory, not a philosophical one. So, now I'm talking about general relativity, the space-time of Minkowski is an affine space with a scalar force of plus, minus and minus points and in the theory of general relativity it's almost the same but it's a variety, it's no longer an affine space, so there are no more linear functions, but there is also a pseudonymian structure of plus, minus and minus points. From a mathematical point of view, it may seem a fairly simple and natural generalization, but from a physical point of view, it deeply changes the perception of space, because, first of all, There are no more privileged inertial markers, the markers of general relativity are no longer global but only local. By making possible more general changes in the markers, we enforce the principle of equivalence, which unifies the forces of inertia and the forces of gravity. And this principle explains an experiment that was known since Galileo, that is that all bodies fall under the selection of the weight at the same speed. The most important modification is that space-time is no longer a linear table in which physical phenomena do not take place without acting on the properties of this table. Matter, and more generally all forms of energy, act on space-time by contributing to its formation.

37:30 So, a word about quantum mechanics. It was created by many physicists between 1900 and 1925. Einstein himself was a precursor with his explanation of the collective effect. And then, by the way, he was a bit out of it. He didn't really know how to deal with it. The description of physical systems is very different from that of classical mechanics because the states of a system are described by the sub-spaces of dimension 1 in a complex Hilbert space. This Hilbert space is the space of the states of the system. So the physical space goes to the background of this fact. We use them to build the space of Hilbert, the states of the system, and then it also intervenes when we want to locate the system we are studying. For example, to find the position of a particle, but it is done by the operator whose true values are the possible results of the measurement. So, space asymmetry in mechanics. Relativist principles, such as time-space symmetries, appear through linear representations or, more generally, projective representations of the symmetry group in the Hilbert space of the state of the system. It is very indirectly that the physical space is lost, the concrete space is lost. The importance of the role of symmetry groups had been recognized very early, in particular by Hermann Veil, who wrote a book in 1928 on the theory of groups and quantum mechanics. Finally, I will talk about spinners. At its origin, quantum mechanics was developed in the context of classical relativism, with the group of symmetry, the group of infinite displacements. And Paul Zira presented a theory that was both relativistic and contiguous to mathematics. For this, he introduced a notion that was new in physics, that of spinors. And it is very interesting to know that this notion was not new in mathematics,

40:00 because spinors had already been discovered by Élie Cartan in 1911, when he classified the representations of the algebra of the simple. To describe the movement of electrons in the form of their spin, i.e. their momentum, Dirac tried to construct a partial derivative equation, the first order, called Dirac's equation, which had to replace the Schrodinger equation, which was known for non-relative quantum mechanics. He proposed the following form. The derivative in relation to the time of psi, which is the function of A, is an element of the Hilbert space of the states of the system, is equal to a certain differential operator of the first row applied to this same psi. For physical reasons, it imposed on the operator in the large parenthesis, in the number on the right, It is impossible to verify that its square should be equal to or minus the constant of the counter-square multiplied by the Laplacian, plus a constant that corresponds to the mass of the electron and the square of the linear vector. This relationship is impossible to verify when the alpha and the beta are scalar. But it works if the alphas and betas are square matrices with p lines and p columns, verifying the antipropagation relations. We can't do it with p less than 4, but from p equal to 4, we can find any quantity of matrices that verify this and say that with pi... I2 is the unit matrix with two lines and two columns. Sigma I is the matrix of Pauli, which was introduced a little earlier by Pauli, which is given by the following expressions. So the function of angle that comes into play is that it takes these values ​​in the space of the spinners. It is no longer a complex value, but it has p components, p equals 4 in the Dirac equation, and for a mass-limb particle, we can make it simpler, we can take p equals 2 and we can take directly to alpheize the matrix of 2.

42:30 In this case, we arrive at an equation that sinks in two and that had already been found by Hermann Weyl for the particles of zero mass, but it had been rejected by physicists because it is not invariant by space reflection. This is what I said earlier about the right hand and the left hand. In fact, some time later, physicists realized that they could be applied to neutrinos because neutrinos are not invariant by space inversion, there are left-handed neutrinos and right-handed antineutrinos. The operator of Dirac has positive and negative values, and the existence of negative values has led Dirac to pursue the existence of antimatter, which the experiment continues. The last remark is that the Lorentz group does not act on the Spinner space, but on its universal coating, which is a coating of O2. The new component is the matrix group of O2. There are many speculations. Some say that space may not have the symmetry we believe, but in reality, when we look at the projection representation, it is a representation of the Lorentz group. I will quickly go over the evolution, because I have reached the limit of my time. Well, I would obviously have to go back to 1970 and beyond, because there were still tables of mathematics. So, thank you. Thank you for inviting me. Cottenwick, it wasn't 28. Thank you. I think you wrote 28 for Cottenwick. Pardon? You wrote Cottenwick 1928, if I'm not mistaken. This is the date of birth. I told everyone the date of birth and the date of death, but as you said, it is still alive, so I only say the date of birth. And Alan Cohn, 1947, this is the date of birth. Do you have any further questions?

45:00 I just have a remark concerning Gauss. Because I think it is perhaps his vision that, well, we can guess the geometrical possibilities in Euclidean, but... I don't think it's too interesting, etc. I think it's a bit questionable. At first, we noticed that all of Riemann's work was the development of Gauss's ideas, and at the same time, it was the other kind of idea, the idea of Boyer-Lobachevsky. I think that Gauss, perhaps the first one, guessed that there are deep links between what we call geometry... Intersects of curved surfaces and geometries and all these parallel problems. In this way, we can say that he was right not to be happy with the results of Boya and even Batshevsky. That is to say, they are really two different developments. And also, I believe that when we say that Hilbert must be well-regulated in the Grundlagen, this is also perhaps his vision. These are all debatable, because all the cases of Riemann's geometry may well be axiomatized, but this vision of Hilbert does not make the difference between a model of geometry of Batchewski as a space of Riemann. Constant negative curves and artificial models such as Planck-Klein-Klein, etc. That is to say, there is another possible analysis of this situation. I am not a historian of science. What I wanted to say by saying that Gauss did not find this interesting is that in fact he had already constructed the geometry of Bohm. He did it when he was doing triangulations for geometry.

47:30 He took great care, every time he measured the angles of a triangle, to check that their sum was well equal to two degrees. He found that it stuck in the line of precision. And he imagined that we could very well also build geometries in which the sum of the angles of a triangle is not equal to two degrees. Well, he didn't know anything about that. He was so much above the mathematicians of his time that he should have had the grace to say to Romilier, yes, first of all, that he was a genius. That would have been easier. Unfortunately, he couldn't finish what he was doing because he was very young, and at the end of his life he was already very sick. There is a great historical mystery. What is the contribution of Riemann precisely to the calculation of Christoffel because we can suspect that Christoffel is a student of Riemann? He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. He spent a lot of time working on the hypothesis about geometry. No, it's in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory, in the memory,

50:00 And a cossive, which is not the best definition, but it is the data of the discovery of X and C is the data of Cij, so we work in the category, let's say yes, Cij, Qij, or X, inverted, so S is the family of... I call them open, it doesn't mean anything, they are objects of the same category. And UiJ is the equivalent of intersections. It's not a definition, it's a notation. It's a fibrous process. So, a big cycle is something like this that verifies Cij, Cjk equals Cik over Uijk. And O is commutative. There are also combinational theories, which are much more difficult to understand. I have demonstrated to you, I think almost entirely, the theory. So this is called a co-cycle. And a co-cycle, I don't know if I said that, is a combination if there are ci in gamma ui over x multiplicative I would like to demonstrate the theorem that, given that there is a cosine and a cosine, we will remember next week the two cosines, given that there is a cosine and a cosine, there is one and only one, in a sense to be precise, and that is an OX-module beam and theta-i isomorphisms of OX will lead to Ui isomorphisms.

52:30 All of this is linked to an amortism. What is unique is the data of LC and TETA-I. That's the result, it's not just LC. And secondly, if C is a cobalt, then LC is globally isomorph to OX. LC is equivalent to a right fibrillator, so it's a right fibrillator with a phase on the calvary of row 1, it's the same. And then there are several cases. There is the case, yes, but we can also think of the case of a constant phase. And even if there is a case, there is a body. And so it's called a system of cases. And a system of cases is the notion of basic mathematics. For example, when you work in C or at the origin, and you take K equals C, let's say.

55:00 So, you can't mix the variety, it's not topology, X, and the base body, it's strictly a carapace, and in the other cases, that's a body, and that's... There are a lot of local systems, and we can demonstrate that they are indexed by force. I'm not going to go into that. I think they are indexed by C, minus the origin, minus the... By C, it's like this, but... In any case, for all the alphas in there, you can associate... It's a local system, but if you want to make a beta different from a whole, I think it's the same thing. It's called Remy's model. There's a whole theory related to the differential equations. So you should have understood that, because we're going to do the same thing with a third of a plus. You have to understand this thing. You have your LC, or L alpha, on an open. The simply connected open space is the constant beam, as in this example here. On another simply connected open space, it's still the constant beam. But the problem is that we are in a category and not in an ensemble, and on the intersection they are isolated, they are not equal. And isomorphism, which is there, does not necessarily circulate with isomorphism, which is there. For example, there, they can circulate by identity, and there, by alpha times identity. And so the beam we obtain is not a beam. It is constant on each one. We can simply know that it is not constant on these three zeros. And so why? Because we do not work in sets, but in categories. And in categories, there are not only inequalities, but isomorphisms. So if we work in both categories, it is considerably more complicated because not only are there equalities, isomorphisms, but there are also two isomorphisms, equivalences. So that's what we're going to do for the whole of the next week. So we're going to start a little earlier. So that was just for the time that comes.

57:30 And we're going to do one last paragraph on beams. I will explain to you very briefly how the hypothesis that the category CX accepts products and fiber products comes about. First of all, we will always assume that and then, as always, we will take some remarkable facts from the theory and we will take them as actions. And then we start from hypotheses like that, we advance, we work, we find some results, and then we talk about the results as an action and we say it's a hypothesis. So we're going to work in the category of joint beams. So my goal in this paragraph is to define geosynthetic topologies even when we don't have this hypothesis anymore. These caps, as you can see, are the promoters of Stilisov in 7, and they are also the prefixes on X. So in 7, what do we have? The prefixes on X. All this is the same. So if I have a morphism... So I will always be in this category, at least for the first time. If I have a morphism of prefixes, a morphism in... I sometimes write PSHx, and sometimes I write Heads and Tails, which apparently are the same. They are two notations for the same thing. So, I can take the and I write a morphine like that. So, I say the following thing. Now, if I have done enough in detail, I mean, I repeat, even if I have never done it, U is a monomorphism if and only if A is a monomorphism if and only if A is a monomorphism if and only if A is a monomorphism if and only if

1:00:00 If I have an exact sequence, let's take a double arrow like this. So if L' is the nucleus of this double arrow in the pre-vessels, it's the same thing as saying that L'A, this sequence, is exact.

1:02:30 So, if this is an exact suite, I apply the function a, which is associated, it is still an exact suite. It's not equivalent, it's like this, I understand. So, a real mistake. The function a, no, the function a is exact, so it doesn't go with what I'm saying. Because I mix, I understand, I mix, it's a mistake, I mix the exact suite, mono and... At the left, at the right, Mono and Epi, they are not the same at all. Because if U is a mono, it means that whatever U, F of U in G of U is a mono, a mono whose category is the same, that is to say injective, and it implies that F of U in G of U is a mono.

1:05:00 I'll take a morphing of a beam. So, F and G are already beams and U is a morphing. So, I say the following thing. U is a mono, that's absolutely true. If only U is a mono in a pre-beam. On the other hand, U is an epic. We can also be an epimorphism. If we are an epimorphism of precepts, we are an epimorphism of beams, but the reciprocity is false, it is explained in detail. To say that we are an epimorphism of beams means that for everything open, there is a covering, etc. This is the existence of the case. So, the monos are the same thing in precepts and beams, but not the epics. So, we will give the following definition. I would call them A and B. This is a morphism in the prethesis, i.e. the x-axis.

1:07:30 So I would say that U is an epimorphism, the case, if UA of AA in BA is an epimorphism. You see, you have pretheses. Saying that it is an epimorphism of pretheses means something else. I remind you that saying that U is an epi... In CXH there is one thing to say that whatever the open U is, A in B is surjective, that is an epimorphism. But a local epimorphism, U is a local epimorphism, it means that whatever U is in CX, whatever T is in B, And whatever V is in the recurrence, there is S index V in A of V such that U of SV equals T. We can solve this locally. We want to solve the equation U of S equals T. It means that we always know how to solve. It means that we have an epimorphism of vessels. It means that we know how to solve locally. So we give a name to the pre-vessel morphism, such that when I go to the associated vessels, it is an epimorphism. I call it a local epimorphism. So I'm going to look at the properties. So you see that the notion of local monomorphism, it would not be of interest because if I translate that, In mono, it doesn't give anything interesting, because U will be a monomorphism if and only if, UA and a mono, unlike Epi.

1:10:00 So you see that the notion of local epimorphism is to intervene in the topology of X, since I took the associated effect. So I forgot to tell you at the beginning that X is a sign, otherwise what I just said doesn't make sense, and which checks the dimensions. In other words, he adopted products and fibrous products, since I talked about associated beams, so it's good that I had topology, and I constructed associated beams under the hypothesis that there were fibrous products and fibrous products, so that was the hypothesis. And so I say that now, I looked at the properties of local epimorphisms and I take that as a definition of topology, because the notion of local epimorphism depends on topology, it depends on associated beams. When you say I take the associated beam, obviously it depends on the topology on the CXC category. For example, if you put the topology on the final or initial, the associated beam is the real beam. So it depends on the topology. So in the notion of molotov epimorphism, there is the notion of topology which is a adjacent source. So what are the properties of local epimorphisms? So I say that they are stable by base change. I'm not sure to demonstrate in full what is done, it's quite long. What does it mean that they are stable by base change? So I say that local epimorphisms are stable by base change. And I explain to you what it means. It means that if I have a local epimorphism, I wrote everything down, there is an arrow here, so I understand, since U of A in B, so it is a morphism in C, which is an epimorphism there, and V of C, I will take the same notation, so I have that, I can complete, I can take the fiber product and I would call W.

1:12:30 The parallel arrow above A, W, is an epi of K. This step is a bit harder. Demonstration. Do you understand what I'm saying? What is the hypothesis? The hypothesis is that UA of A in BA is an epimorphism in SHX. This is the same thing as W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1, W1,

1:15:00 If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. If you say the following source, you fall into the next one. Thank you for watching this video. Yes, because you say it's because the beam is associated, it's a counter-reaction, isn't it? No, it's not. It's not. It's not. It's not. No, but let's drop that. What we want to demonstrate is that.

1:17:30 You have a beam epimorphism, you make a base change, it's still an epimorphism. That's why it's easy. I'll put it in the notes. In any case, it's true. So, local epimorphisms are made by base changes. So, the other is... Another term, which is also more delicate, but it's a bit too hard all that. So, anamorphism. I give a criterion for anamorphism in these chapters. So I say that U is a local epimorphism, which is only if. I take the family, which is only if. For all V that is sent in V, it is very complicated. Whatever V which is sent in B with V in C, the family, we look at the family now of U which is sent in B as this family there. So the family of U which is sent in B with U which is sent in A, which makes this quantitative diagram, is a recovery of V. So A in B is a multiple case epi, it is for all V, let's say an open B. The U-verse, which can be seen in B by vectorizing by A, can also be seen in B by vectorizing by A. So, more or less, what does this mean?

1:20:00 What does this mean? So, U is an epi-local. So, I say, the line that is there is more or less ontological. What is an epi-local? Well, this means that U-A is a morphic epi. And what does it mean that it is an epimorphism? So I remind you what I said earlier, what did I say? I said whatever T belonging to B of B, but in particular to B of V, no, it's true even, to B of V, there is a recurrence S of V, it's not the same notation as earlier, it was U earlier, it's nothing, and whatever U in S, there is S in A of U. What does it mean? What does it mean T in B of V? Well, it means that B of V is also a man in his hat of V in B by Yaneza. So, to say T in B of V is an arrow of V in B. So what do I say? Well, given that, it exists in coverings. So, it exists in coverings. Whatever the U of this cover, there is S in A of U, i.e. an arrow S like this, in which the image by the arrow of A in B is T. Well, you can see that it's the same. Maybe we'll say more, but it's more or less the same thing.

1:22:30 In fact, it's exactly the same, but it takes the other way around. So it's a translation, it's tautological, in fact, it's an emotion. It's the same thing. And for every tube of this recouplement, then, and it's a yes and only a yes, so this thing is an epimorphism, yes and only a yes, for every T that is there, it exists, a recouplement, that is to say that the family of these diagrams is defined as a recouplement, given, as I am, it's really, for every arrow of B in B, the family of arrows... So, I'm going to give you the properties of the local dimorphisms that I kept in my memory. So we have an X and a Z that verify these tables by products and fiber products. Products are two objects.

1:25:00 So I say that we have the following things that I will axiomatize after. The case of epimorphism 1. Whatever U in CX, the identity of U in U is an epimorphism 4. The epimorphisms, I remind you, loco is for the pre-thessals, okay? It's in C-H, E, C, X, hat, okay? And I remind you that C-X is sent in C-X, hat. So I look at C-X objects as pre-thessals over X, by Ioneza. C-X hat is the category of pre-thessals of the whole over X. But C-X is sent in C-X hat, okay? So C-X objects. The Ubers are preface to Ensemble. It's a bit hard to imagine. I'm so used to the Hegelian calculus that I'm fed up with it. But the Ubers are preface to Ensemble, by Yoneda. So, the identity of an Uber is a local epimorphism. What did you say? I think I got it wrong. I'm not sure it's true anyway. So, the second one, I'll verify it. C-morphism, 1, a2, a3, cv, so that's in C-x-chafaud. So I said that if 2 over 3 of u, v, and v-u are local epi, then the 3, we often see 2 over 3, we call it the 2 over 3 property. So, LE3, third word. So I'm going to check all of this later. LE3. And again, you still have A1, A2, A3, U and Z.

1:27:30 So I say that if 0U is an epilogue, so is V. And LE4 is that A in V is an epilogue. Whatever U in B, so my notation, I always write ABC in C hat and UVW in CX, so when I write that, U is in CX, so whatever U in B, then A cross B U in U is an epiphylocal. L2 is complicated because V'U'A, C'V'A'U'A. When you do the morphism, you have A1, A2, A3, U, V, V'U, and then you take the letters and the letters, and the morphism of A1, A, it's a function. If you want, you can just say yes. Why? Because A is a frontier. If it's a frontier, it leads to the composition. It's a demonstration of the probability of a frontier. It leads to the composition.

1:30:00 So now, you see, you have UA. So we're here. 1A, A2, A, A3, A. UA and VA. VA and UA. So I have three morphisms. If the two that are here are epis, the third is an epi. If this one is an epi, we will see later that it is enough for Véna to be an epi. What else? And if ... it's a general category thing, it has nothing to do with ... in a category, if you have ... it's a general thing, it has nothing to do, I think. Or at least in these hats. In these hats, X ... Why? I don't know if it's true in all categories. In any case, in these hats X, it's like the ensembles. To see that things are epimorphisms, you just have to check on whatever is U. When I apply the function, I apply that to U, it's an epimorphism. So that means I take 1A U, 1, 2, A U, 1, 3, A U, I'm going to call F and G to change. and Geron-F. This is an assembly application. We are in assemblies. If there are two out of three that are surjective, all three are surjective. Are you kidding me? So is it true in any category or not? No, but there are not many possibilities. No, but is it true in any category or is it true in the only surjective category? Surjective.

1:32:30 No, and that too is surjective. It's weird. Because I invented it, I don't know where I got it from, I don't know where I got it from, I don't know where I got it from, I don't know where I got it from, I don't know where I got it from, I don't know where I got it from, I don't know where I got it from, I don't know where I got it from, I don't know where I got it from, I don't know where I got it from, We were ready to do everything to demonstrate what we needed, but unfortunately, we had to stop there. So, the notes are not the same. If U and V are local epi, the composition is the same. This is obvious, because when I put A, the composition is two epimortices in all categories. It's still an epi. And the third... So, why? This is also in any category. I take 1a, a2a, a3a, g, ua, va, v, v, a, ua. This is a general thing in a category, it seems to me. gx, y, z, f, g, g'f. This is in an abstract category. If g'f is an epi, So, G is an epi. So, this is in the duration of G. Did you understand that you can't find it anywhere else in the world? So, 1, 2, 3 are pretty much the same. Let's look at the last one.

1:35:00 So, the last one has an equivalence. So, in a sense, this is a particular case of the first line that I announced in the demonstration. I gave you a line, local epimorphisms are stable by Schallmann-Bahn. They are stable when I take basic changes by u of Cx. It's even true when I replace u in Cx with anything in Cx. In this sense, this is the first element that I didn't demonstrate. Let's go in the other direction. In other words, we assume that I have an amortization of A in B, So, when we complete A, B, C, U in W, W is 1 and pi is the fourth. And I want to show that U is the fourth. So, I'll give you an idea. There are many things to read. Local epimorphism is a local epimorphism, that is to say, if you have AI in BI, which are local epimorphisms, then the index limit of AI in the BI index limit is a local epimorphism.

1:37:30 Why do I put the index limits in quotation marks? It's not essential, but in general, in CX, hats, we put them in quotation marks. We don't have to put them in quotation marks. If there is no inductive limit in C, then there is no need to put quotation marks. There is no risk of confusion. Putting quotation marks doesn't have an extraordinary meaning. It's just to avoid confusion. There are other inductive limits. To make the English translation, in C hats, we put... So, why is that true? Because the adductive limit commutes to the function a, and on the other hand, it transforms epimorphism into epimorphism. The adductive limit of epimorphism is an epimorphism. UIA, BIA, it's an epi. When I take the inductive limit, it's still an epi. So that's a thing I've never written. Is that true? An inductive limit is a morphing limit. It's a morphing limit. In the nucleus, yes, that's right.

1:40:00 I should have written it. I should have done it before. It seems obvious to me, but I didn't write it. And then, on the other hand, after that, we only use the inductive limit. That's the same thing as the inductive limit. So, to verify the inductive limit of the pluralism, we often think of things that are obvious, but then nothing is obvious. So, the inductive limit of the pluralism is the case, and the limit of the pluralism is the case. So then, how do we find it? Well, the trick is to remind you that B is the inductive limit of U. It's quite surprising, I showed you this a long time ago. U is in Cx, B is in Cx. So, Ws are local degrees. So now I take the inductive limit for U, which goes into B, of A cross B U, which goes into the inductive limit for U, which goes into B, of U, and that equals B. So, there is another thing that is done in exercise and that I prefer here, is that the inductive limits in these hats are stable at the base form, or the base form.

1:42:30 What this means, in the next step, is that this thing here is isolated at A3B, the inductive limit of U. So, if you look at the courses exercises, You will see that in the category of ensembles, the inductive limits are stable by base change, that is to say that in the ensembles we have such a formula. So, in these hats, we also have such a formula. On the other hand, in the category of modules on a ring, it is false. The modules on a ring, the inductive limits, even the finished inductive limits, are not stable by base change. Especially the empty inductive limits are not stable by base change. Thank you for watching this video. On the other hand, in ensembles, yes. And since it's true in ensembles, well, it's true in each one of them. What is the product? It's not the tensorial product, it's the product. The product is the direct zone, it's the same. No, the product in the energy module is not the tensorial product. It's not in ensembles. The product is an intersection. No, the product is the product. In the sub-ensembles of an ensemble, the product is an intersection. In the modules on a nano, the product is the direct zone. I agree that in the ensembles, the product is adjacent to man, so the product in the ensembles looks like the sensory product, that's what I wanted to say, because it is adjacent to man, while in the menu on a ring, the sensory product is adjacent to man, but it is also the product.

1:45:00 This formula is true in the ensembles and, by a little reasoning, in the hats. And now, how much does it cost? Well, it's A, 3, B, D, and that's A. It's a bit general in the categories of product fibrous, etc. So we've shown that... No, that was A, excuse me. Yes, A. So we showed that A in B, and since the nuclear limits are such that, as earlier I had read, the nuclear limit of local epimorphisms is an epi of A, well, suddenly we showed that this is an epi of A. So now we contemplate these four axes, but there is one thing that worries me, I ask myself. It may be the same if we take one or one of the two. So, we consider the four properties that are there, L1, L2, L3, L4. And then, we can take them as axioms now, topologies. So, now we forget everything. And, either X is imprecise and without any hypothesis,

1:47:30 unless it is small and not a fibrous product or something like that. So, a topology is the data of a family of morphisms, two morphisms in six hats, so the morphisms are called local epimorphisms, and like this family, verified, L1AM4. We will stop here. But before we stop, I would like to thank you. As soon as you ask me, I think everyone should put their name and their affiliation, it's not at all a police control, it's something related to the administration, to tell the administration that there are a lot of people who attend the classes even if they are not all enrolled in Paris. As long as you are enrolled or not, it doesn't matter, you put your name and they tell you where you are, what you do, if you are enrolled at M2 Paris, that's fine, if you are from somewhere else, that's fine too. So it's important for...