Houzel / Houzel, finitude, quantification, deformation (contd.)

0:00 And then, in order to compare the properties of a document higher than 2, what we do is that we look at the state 1 of f and the state 2 of f to take the result of these two polymers. So we came up with the idea that the components of a document higher than 2 are going to be... Sorry, not f, but r divided by t. We have already submitted a list of the components of the two dimensions 1, F1, which is by the way a force of g, and then we will keep θ1 and g, so θ2 of g, and then take the resultant, which is a new form, but with a variable of 2. So there I can start again, my theory on the components of the two dimensions 1, which are in fact, which correspond to the components of the two dimensions 1. And then you can look for the components of the two dimensions by doing the same thing, and it will give you the components of the three dimensions of the earth and the sun. Now I'm going to go to the last rule, which is the same as at the end of this text, and which consists of looking at the rings. So the rings contain transcendent quantities and contain an algebraic quantity. A quantity of algebraic xn that will verify an equation of fxn equal to zero, an algebraic equation whose coefficients can match these transcendents. So what does he do there? Well, first he uses the fact that this ring is equal to a.

2:30 In other words, in this case, the algebraic equation for xn is associated with xn. I would also like to point out the state of what we are doing here. What we are doing is, what is the point of the circle? It does not really answer the question in detail, but there is still a fact that we have to understand. If we take any irreducible magnitude, P, in this case, I remind you that it is 3 times x1, x2, x3. Who doesn't use disk space? So, we see that this number, in this case, is included, located in relation to P, and to which I owe this algebraic quantity. So that, well, I showed you that, and it will be useful, in fact, in the future. So what I'm trying to do is, given any quantity of the type P in this case, The decomposers in these irreputable factors in O.K. In fact, the answer to this question is that for the granders who do not use discriminants.

5:00 Because there are problems, that's it. And what he does to do that, his idea is to say, well, I'm looking for the components of co-dimension 1 of O.K. since I'm looking to decompose O.K. as a product of form in O.K. And this problem, I can solve by looking for the components of them. So I'm trying to decompose this into two dimensions. I'm trying to decompose this by looking for the ideal, not the ideal in the form, I'll show you in the video, by decomposing the ideal P of x into two dimensions. That is to say that here we go into the ring n-1, which is the linear method. In fact, it's much simpler because one of the two generators of the ideal does not contain the variable of x, there is no need to use it, but we simply want to look for one. In fact, this method is not entirely new. The point is that it tries to apply it in a framework where our base contains the quantities of 60. There, we have to have each of the factors f of x equal to 1, f of x equal to 2, etc.

7:30 If we don't use discriminants such as M1, M2, etc. on all these big A1s, that is to say, F does not have a multiple factor, once we have this structure, the final result is that, because we can decompose M1, M2, O, P, because we work in a small ring, the base ring is this ring, but if we work M1, M2, O, P, the base ring will be a chord, so we have a ring that is a factor of L, so we can do a decomposition. In this case, we can find a decomposition that can be multiplied by a constant, a decomposition in factors, which are polynomials in x, coefficients in O, and once we have done that, we can show that the integral of the integral of the integral of the integral of the integral of the integral of I would like to say that this equation, when I ask if it is equal to x1, f of x1 is equal to 0, so we have that a product of f1 of xm2 is equal to P for something, so we see that P divides the product of f1, so we see that P is this product, which in a way is very simple, we have this equivalence between the two. Thank you.

10:00 Questions, comments? Is there a French translation, a French text? I don't think so, but there is someone who has been working on it for years. I don't know what to say. I don't know what to say. Before we thank the speaker again, there is a lunch buffet that is served in the hall of the IHP up to here, and we resume the conference at 2.30 exactly. We'll just have a break. I'll take you guys from here if you want. I'll take you to speak on topos, geometry, and analysis. Do you speak grammar or do you want to teach? I probably need to eat. Then I'll talk to you over there.

12:30 I'll just sit here. Roger. That's a gazelle. But I have not seen it in years now. I'll have to take her back then. So now we should see her again. It's a pleasure to follow the last lecture, because the subject of the last lecture was what originally attracted me to consulting with you is a possible improvement on it, which unfortunately still has not been carried out, because this improvement has still not been carried out because of lack of contact between different subgroups of mathematics. In any case, I was also impressed, in fact, I wrote something here, but I was impressed by the first talk today to change a little bit my topic, because of course I knew that Professor Giselle is a very respectable historian of mathematics. I think it's important to speak a little bit more about that, not about Professor Giselle in particular. The role of the history of mathematics in mathematics itself was before about the role of education, pedagogy, and so on, in the progress of mathematics. Lubaki, for example, was primarily a pedagogical project, historical, one of the historical dimensions, but led to, as everybody knows, an incredible amount of new, valuable mathematical research. To explain mathematics as it existed, you invited some people.

15:00 So similarly, there is a role for history of mathematics. My first teacher before Samuel Islander was actually Clifford Cuthdell, also a well-known historian of mathematics, who at the same time was highly critical, highly negative about the Growth of a profession of historians of science, or a profession of historians, or you could say also the philosophy of science, philosophy of mathematics, because there is this observable phenomenon now, in many cases, most of what passes as a history of philosophy of mathematics is absolutely worthless, or worse than that. It's disinformation. Continued dogmas which hold back progress, so in fact one can also observe that there is a strong correlation between those historians and philosophers of mathematics whose main product is disinformation on the one hand, and those who actually participate in the science on the other hand, and certainly again the Warren's lectures. Reminded us, in effect, of the depth to which this historian has profoundly participated in the National Club of Mathematics itself. I myself have published two or three papers on the historian. Rather than proceeding in the detailed, vigorous manner which historians should, my coming to history... Has been, in most cases, more of the following nature that started from a problem which exists now, some serious question that exists in the present, in the past, to see what really happened, how it, so I'm a great believer in things which subjective idealist philosophers might call objective idealism, but it's not.

17:30 Namely, the fact that there is such a thing as latent ideas. Or you could say, in terms of the old question, is mathematics discovered or invented? Well, the answer is yes, of course, both. It's invented by the collective and discovered by some individual, or maybe two or three individuals. At a certain point, ideas which are latent, which the collective is... Coming closer and closer to, or in fact made explicit by somebody, and those people are called the discoverers. Justly so. I mean, this is the way, this is the progress. So, trying to discern, for example, the origins, the notion of Cartesian closed category, started from reading some polemic launched by Dubonnet against both people who were terror. They claimed that Volterra could not possibly have done any functional analysis because he didn't phrase it in terms of topological record spaces. So this led me to think, well, maybe Volterra was even more important than Triesdell told me he was. Of course, Triesdell supported Volterra and mathematics, but looking into Volterra's early publications, one discovers answers to... Any questions? One discovers, for example, that it was Voltaire who finally improved the so-called Poincare level, sorry, long before Poincare, but it was exactly the same. But not only that, not only that, really gave me, gave me to explain the so-called paradox of points in algebraic geometry and many other areas.

20:00 Looking into history because of current questions, but at least finding something that nobody else seems to have noticed, and putting that into a math research or exposition, at least. Similarly, every right-minded person. There is this dogma which many histories, so-called histories of mathematics, still repeat, especially philosophies of mathematics in the current world. The current argument is that Euler was not rigorous and therefore continued these current philosophies. We shouldn't be rigorous either. We should go for beauty, not rigor. It's a bad thing. Rigor is creativity, etc., etc. Of course, a target for these attacks is often, once again, will not be. But in particular, Euler recited in a book recently published by Princeton University, he cited precisely in that book, because everybody knows Euler was not rigorous, and because everybody knows Euler was not to be famous or insane, therefore Euler should be, and this is considered the publishing philosophy of mathematics.

22:30 As has been pointed out, many of the publishers are often not very careful, rigorous, and of course I haven't read all the many volumes. One that Sini told me where to find the particular thing that I need, I suspect that I have two. I should take that off, Bill, it won't do any good. Don't move. That was pretty bad. No, I guess we're good. Okay, so, everyone says, well, of course, it's nonsense. But it's a theory, in fact it's a good definition of real numbers. Because it leads to direct proofs of some properties that we don't understand, and we'll follow them very easily from the Vatican's definition. I can't explain precisely what I mean by that, but I just want to list it here.

25:00 Again, I was left to look into Cantor, something that occurred in the present, namely, I was idly reading Cantor, I was casually reading Cantor in a train. And I noticed something that I'd never heard of before, even though I'd heard many lectures by set theorists and clients through many books by historians about the person who can't draw a circle. Something that I'd never heard of had caught my eye. The train was going to Zurich, so I immediately ran to my friend, who I hadn't seen in 20 years either, Ernest S. Brecker, a very substantial set theorist, and pointed him out to this. You can see it in his copy there as well. In fact, it was Cantor's attribution of his idea of the equivalence of sets to the geometer Steiner, not called Steiner, who said there was no mention of that. Cantor himself did it very explicitly. Brecker turned out to also be an expert on Steiner, because Steiner was a Swiss mathematician who worked in Berlin. Steiner commonly gave Hawking lectures on Steiner and so forth, but he hadn't heard about this either. So here we have a real contradiction in the present moment. This very intelligent person was an expert on Steiner, an expert on set theory, and had read these papers, but had somehow passed by because of the prevailing dogma that he mentioned. So this led me to look more closely at what Cantor said. The discovery was that he had built upon Schleier, and the discovery was something that the, again they said theorists had not analyzed, which in my view should lead to a quite reasonable answer to their so-called problem about the attending hypothesis.

27:30 A couple of other examples of his story, starting from the present. Being led to look in the past and finding something quite surprising that was not part of the common knowledge, which nonetheless leads to mathematical research. Oh, a certain kind of conclusion of this, though, because again, going back to 1960, we're back on seminar itself. In any case, there's a common place repeated again in books, papers, historical origin. You can look on the internet. There's a thing called the historical origin of topos theory, and it says that topos theory came out of the idea of generalized space, that topos are generalized spaces. And, of course, Rodendieck had the Petit et al. topos, and that was the really significant example, which, of course, it was a significant example of generalized space, but it wasn't the only origin of topos theory. And you can see it, if you actually look at it, in some of our 1960s, 1960s, 60s, 61. In any case, if you actually go back and look, you will see that a completely different type of topos was discussed there. ...which was not a generalized space, and the word came up maybe a couple, three, two years later, but I can say that through long considerations, I have revived that tentative explanation of mathematical account of extinction.

30:00 Generalized space is perhaps a topos, a rudely topos, a utopos. Of course, the ordinary classical topological spaces, when viewed as toposes, have a site, a site, which is called a toposets. The toposets have to work as a total. Immediately important, along with the classical topological spaces, were topos of G-sets, where G is a group, because, in fact, for the algebraic topology of the thirds... You see, clearly groups and spaces ought to belong in the same category because there's a map from the space to its fundamental group. You can take the kernel of that as the covering space, universal covering space. So there's a diagram that's taking place in some category in which you have both spaces and groups on equal.

32:30 So, of course, the generalization of space, but they also include groups and, as it turned out, includes quite a bit more because, uh, H-on-2s have a site consisting only of monomodels, not necessarily invertible, these monomodels. On the other hand, there's another class in which the separable objects are accurate, first studied explicitly, unless someone tells me otherwise, by Peter Johnstone. This kind of topos, well first of all, a separable object is one for which the diagonal has a complement. In any topos, it has a hiding complement, but that's not a real complement in the sense that the union is not everything. The complement, the particular complement, declares the sum. It's detachable, to say the various, various words. So, separable objects, I mean, if all objects in a topos were separable, then it would be a boon and then a special. But the objects like this in the actual, in the sense that every object in the total is an inductive limit of these, and that's the class. Interesting for us in this regard, because the eucalyptus is like that. This is basically the fact that if you take the connected eucalyptus objects, that you have two maps between them which are equal.

35:00 Somewhere, even on a very small part, then they're even. So that's saying that the map from the small part is the work of these, because it turns out that these separate mathematical things have sites consisting entirely of epimorphisms, in the sense that if you look at the site just as a category in itself, every map is epic. Of course, when you embed the site in the topos... Now both of these are good classes of categories and sites because you can exponentiate them or refer to them to take them into a new category and have such a reflection, et cetera, et cetera. These are sort of interesting categories to study as various kinds of space. And there's a common generalization which is pretty obvious. It's a sort of bi-cancellation problem. The technique of being still more general, still more general, probably, because this site is more important. That's very simple. Again, fairly well detailed. Probably, we'll study, it hasn't been much study, we'll study in this slide. The notion of generalized space, it's not simply full-scale generalization. Relative examples. Places, classical spaces, groups. By contrast, the topos of the simplicity group is simply... We came up with this later analysis property, namely that they have to have degeneracies.

37:30 So I think one mistake was to call figures points. The point is two panels. There's something there. Points are typically something more special. So I used the term figure so everybody could understand. Well, here I'd like to ask you to determine elements, which I think is very good for the time, because the idea that elements are something irreplaceable is certainly not marked in the language. To say that the window pane is an element of the window, which in turn is the element of the building. Structural elements of the living room may work, but they don't have the exact structures of the system. Monotaur is the least of this term, where it then is now, except that it becomes a set of very, let's call it, figures instead. So in an arbitrary set of sets, a figure of shape A is a map, just a map. Then if A is special, it might be a point, it might be a synopsis of a site, based on, excuse me, a code. Which would say that a mathematical next binder, from one figure to another, is a narrow triangle, but let's call this an incidence relation. An incidence relation would say, for example, that a certain curve is part of a surface, or most things can be expressed by this. So there's a triangle, a slice of that, and the object surface.

40:00 Okay, the category of figures and instances. We use in general, but I normally use it when we have some specifying class of these in mind, such as a given site or some defining property that has some content other than just another name for an area. That content is really given by the fact that it's a sub-capital area. And by the way, this simple-mindedness construction refutes a very common vicious rumor. Which is that in category theory, you can't get inside the objects. The objects are opaque, and various slanderous terms are used, but you see, it's actually the best theory of geometry of figures of incidence relations inside. Inside, that's the mean of energy. We have a picture of the A's, then we have a more complicated way of picturing the X. This is a picture. There may not be an adequate picture, but... This was my next, because again, if you're in a special class, you may want to give a special name to the maths, but given an arbitrary given domain, a special domain, of course it should be called functions. Functions have always meant something a little bit different. Sometimes we've identified world physics, transformations, math, functions. But on the other hand, function theory... He uses workisms to study special workisms, so we can take the outside, the outside of the geometrical object, this is algebra of functions.

42:30 So again, this is the algebra of functions, in the sense that these are the algebraic operations. And notice that if A has products, this includes addition, multiplication, and so forth, because A could be, for example, A prime squared, or A prime cubed, but it's really a full explanation of how the world functions. And again, this simple sort of definition immediately says that, well, suppose you had a morphism, a general morphism, Or actually, one of them is the speed of vibration, because the shape of the figure is given by such a function, because there's here the type of the function given by the speed of vibration. So this function, that function that's induced by this type of figure, is always continuous, in the sense that it maps figures into figures. Without Terry and Witten, sequential continuity of topological spaces is a specific example of this because one can take a series of categories of well-known convergent sequence along a different point, category of spaces, and then precisely sequential convergence means transforming them. Convergent sequences considered as figures into other figures without tearing the limit away from the rest of the sequence, without tearing that distance away. Back in the 40s and 30s, seeing that that notion of continuity is indeed an example of my general, apparently my apparent general nonsense of that notion.

45:00 Homomorphism in the opposite direction to functionality. All these theorems are all cases of essential timidity, of course. In fact, they go through the opposite direction. Preservative is an action by whom work is equal to theory, all of which I'm going to repeat in a minute. Adequacy as a general concept, this was defined by Isabel. Adequacy means, it's adequacy of a choice of A's. A choice of a subcategory. Subcategories are adequate. If every match between the general states, every morpheus in the discrete vibrations, or if you like, every natural transformation, except how you pre-sheathe across the same thing, comes from a matching match, in other words, if you get it, if you get it from a given, how do you get it, how do you get it, pre-sheathe on the events of the universe, is actually full of speed. So, within this general framework, I'm going to lay a problem. How is it that, how is it that, well, so that one needs one very important further ingredient beyond the very much bad words, namely one needs the idea of the exponential function. Is it half this or half this? This is truly, in a way, perhaps the most fundamental ingredient of mathematics.

47:30 Contents going back 300 years before is the idea that given two objects, the morphism between the subsets are also one object. A geometrical structure has its own figurative interrelations, like those on the algebraic functions. They're called fungals by Volterra. They were used for a long time, but not given a name. In the context of truth value functions, the real value function is much more profound than the real value function. Voltaire made this concept precise. He said, look, you had this idea all along, but let's see. And then, of course, his very good friend, Amad, gave it the name possible. And, you know, it's just shade and not volume. And, of course, his development depended on that. So in the planning process, exponentially, some of the category happiness is that you have for every pair of objects, you have another object along the page, such that for every third object, the work is from exit to that, corresponding naturally and uniquely to what. So it's an excess of right adjoining. Cons is the notion of adjoining. Bumpers, who was on the way to the press 50 years ago, has in particular this example which caught himself in time.

50:00 Bilbao, in the context of his success, specifically the function right to the power a, by the element two. So many properties of, many of the, many of the properties come from the adjuncts. The definition of what it does to the frequency of intensification. The point being that... From the earliest days of the calculus of variation, which also occurred due to the name, I suppose, of biological detection, they didn't do it in quantum analysis either. Unfortunately, they didn't do it because, right from the start, the idea was, well, what is a figure of shape and integral in a math space? Why do you want to care about that? Well, it doesn't understand. I was talking about things like the problem of least descent, et cetera, et cetera, which the variable, I mean, it's not an infinite dimension, yet in spite of that, one might argue it's variable. So what is a variation? A variation is precisely a path in a map space where the domain of the variation is a variation of plot. Well, I think you might have a chosen point and you vary that point.

52:30 There's a figure inside some masquerade. If you deal with that, because it's the same thing as a computer, this was the technique of the calculation of the calculus of derivations. In the beginning, as Voltaire made this into the definition of analytic functions, the fact that the category of analytic spaces has been a wholesale functional analysis was of great interest there. So we put it by, so we could imagine that they would use this for understanding the complex of analytic work, the states of analytic work with one analytic state to another, given its geometric structure. Thus by calculating its geometric structure, that's what he said. Of course, the not true thing about it all is... There is a possibility of talking about functionals being smooth. This is essentially a work of this domain of the mass space, though it might be part of the mass space, and so is the role of the organization. The crucial leap in continuity, similar to algebra or something like this, is that we have, let's for example, real-value mathematics. These domains have a time of descent for some arbitrary time period. Well, the objective fact that you need is some kind of smoothness. And what's the definition of smoothness? It's more of, you know, the geometries.

55:00 In other words, if I have this analytic, say, analytic category, how can it be analytic? Because if I take any analytic math like this, well, that's really something like this because I've already studied it. I know what that is, but if I plug that into my functional, I'd like to get something which is again, so that's the condition. It's all, in some ways, far more simple than the whole dogma of topological record space suggests. Precisely because we have no way of knowing. You can try to talk, you can talk about open sets, you can make different definitions of open sets inside the same category, with the trivial, what am I, the property, the normal, what am I, the continuous, and the total genesis, by having some, let's say, having some representing object, think of the sophisticates based on two possibly open sets, for example, it could be something more sophisticated, with a content for it. So you could say that, well, this is a problem. All true. So if you have a map like this, take the inverse image of a point in any space, whether it's a vector or not, inverse image of a point under some map, and you go like, oh, and you put this where you fix it. So then, of course, every map can have a discontinuity in the sense of, but still, even if I do that, I have no way of calculating what that means. I don't think that works. In any case, it's just something very complicated to try and get at the open set structure on the mass waves, knowing the open set structure, knowing some open set structure, whereas this is automatic, co-varying structure here and there.

57:30 By geometry of the figures, taken as the basic measure of the cohedron of space, rather than the contemporary structure of the algebra of functions or algebra of open sets and examples, we take as the default notion of real leverage of our integrations. I promise to tell you I've had much more to say, but I'll just say this. In teaching calculus, we make the mistake of talking about difference quotients. This is pretending that quotients exist in the same sort of way that addition and multiplication do. It's actually a much deeper matter to talk about quotients. So if we interpret every statement about quotients by saying there exists an x at a time to x equals b, put it back to the question of multiplication, we get something that can be computed with, and we eliminate our mathematical error. The fact that dividing, even inverting, is a non-trivial process is seen in the fact of the localization it brings. The whole subject is how to take a random and invert some things, pass through a subsets and so forth and so on. So dividing is non-trivial, especially dividing two things, not just multiplying something by the inverse of something else. There are many cases where you can't even get that. So, basically a ratio, a ratio of the fact that the ratio between the two has to be understood as a process by which you transform something into something else. Of course, it's very special properties, but still. So, I don't know if the real thing would be ratios of integer and decimal.

1:00:00 In the quintessence, where d is a quintessence space, and one thing about them is that they all preserve zero. So, in order to find the reals as a sub-object of the function space, the function space e to the d, are these ratios, it's a natural intrinsic multiplication due to composition. I'm going to say what the d is in a moment, but... In a suitable ambient category like the topos, every monoid has a universal commutative monoid. This, of course, is a monoid, but automatically, this is sometimes called a synthetic differential geometry, because we start with an object, nothing but an object, and we produce the algebra, the linear category theory of the geometrical object, and this is highly synthetic in that sense. So we produce the monoid. That's going to restrict the multiplication of reals. And in fact, what part is it? Interadjustables do have one point. They have only one point. The point is the point. The device that goes to arbitrary bases is what is explained in the object. Figures A. You have to put that as a condition. So the reals are the kernel of this. And then it takes a pullback. Part of these would be... The map zero to zero at a specific global point will automatically do that, because it's not the only point anyway. Another feature here is that I can take the commutative reflection and put the condition down.

1:02:30 This is a nice one. Actual ratios which preserve zero, that space, space is actually equivalent to forcing the multiplication to be. So that means that our self multiplication is true. The whole of the morphism space is always non-communicative, the S-A is one type. This is a general fact. The G to the D is non-communicative, which is part of what it is. Moreover, as far as communicative is concerned, and moreover, to follow this projection by the inverse, we see that R is actually a retract. It's supposed to play the role of a tangent, x to the d is a tangent, evaluated at zero, so you can say that r is the tangent space at zero of p itself. So first, when you look at induced maps, especially taking derivatives of arbitrary maps, this is what flows out of the d. And this retraction gives you the opportunity to write down what we would call a dot, the derivative that is opposed to. Now first, let's see, in all the specific examples that we know about, this D has the computation of the spectra of a few numbers.

1:05:00 What I'm saying is that we can approach it axiomatically with, like, fairly synthetic properties of D, some of which I've already seen before. And deduce in particular that R has not only multiplication up in this definition, but multiplication on R as well. Sorry, addition on R as well. So I can take R as a co-domain for a function alphabets, or I can take it as a figure in the case of paths, paths of a tangent, a derivative of a path, and so forth. In particular, the special commission it was used to do is to formulate the alphabets, the alphabets of group objects, and the extensive nature of them. Distributions, such categories, topos, cos, even Cartesian post-categories, has a natural notion of distribution of complex form. It was as hot a space as a subspace, sorted out by equations, from a double mass space. It's linear. And the point about R being attitude is that the distribution of the disjoint sum of two spaces was paired with distribution. Thank you for this categorical approach to analysis. Are there any questions, remarks?

1:07:30 Maybe one can say that Volterra was, I don't know, I'm not a historian, but something which is very important now is correspondence. You don't take functions from one space to another one, but you take the product, a kernel, or maybe Volterra was one of the first to use it mathematically. Yeah, I mean, you could say that, I think that the problem of distributions and generalized functions is a very bad misnomer. It's very misleading. Analysts have pointed this out. You can't simply restrict a distribution to a substance. This expression here on RxR is clearly a co-variant function of x rather than a conjugate function. But there is such a thing as generalized functions. Anyway, a generalized morphism would be a morphism from x to y. No, not x to y, distribution x to y. That's a generalized math. You see, because, of course, there is the Dirac-Gelman natural math from many states into the spaces, distribution x to y. The special case of such a P, it actually would have to be next to 1. I say here d sub 1 because of the sort of natural restriction that we need to say, let's take those distributions of compact and support. Intended to make 1 to be 1, to preserve the constants. So the thought that there aren't generalized functions is just that the distribution itself won't really play that role. Thank you. I would like to say that the campus in 1961 not only discussed some supports, but we also discussed some stacks. Discussed also stacks. Stacks, yes, yes. So we put them into the analytics. So maybe it's a good condition for the next talk. So let's thank the professor.

1:10:00 So, we did start ten minutes late, so you've only lost five minutes. We did start ten minutes late. Well, I was looking at you very much. Well, you have the last one. A little announcement before everyone leaves. At 6 p.m. there is a small party at École Normale Supérieure. For those who have attended the whole conference. Thank you very much. Thank you very much for your time. The analysis of the old notion of intensive and extensive quantity makes it absolutely clear that distributions are extensive quantities, and in the category theory framework, there are several things concerning the... Sorry, it's difficult for me to express myself in French. There are two limits and co-limits in the categories of extensive quantity and, for instance, they have contravariant functoriality, where they're quite distinct from... So his point that they should not be regarded as generalized functions, I think, is actually very sound. It's a conceptually very deep point. For me, the categorical translation of distribution will be perceived as a category. The fact that the category is embedded in the category of spaceships, you need a lemma that says that the functor of Ohm, which is a kind of bilinear form, is non-degenerate.

1:12:30 Yes, you can look at it that way too. I think you'd still have the... I still think that the point would stand that it's misleading to think of them as generalised functions, because for him a function is always, of course, an intensive quantity, a ratio of the things of which... Anyway, this is something we could perhaps pursue over dinner. Thank you for your own exposure, by the way. I'd really love to understand more about deformation, the theory of deformation and quantisation. Of course, of course. Thank you very much for your attention and I hope to see you in the next lecture.