Morning Discussions, incl. F William Lawvere, Pierre Cartier, Colin McLarty, Angus MacIntyre, John L Bell

0:00 I went out to give an introductory lecture in that field and I do go back to this proof, which is a wonderful proof. And now I've gotten the discovery that combining the idea of cell and this idea one could define one. But it was more abstract. It said hyper-commodity of a complex of shape. I want to be a little more concrete and I say I have a double complex which is a check-durham complex, or I could call it very complex. It has two dimensions, shape and length. But now if you have a complex of varieties, you have a triple complex to calculate your cohomony, you have a third dimension which is, let's say, suppose I have a simplicial variety, to be a little more specific, I have a simplicial variety which is often the case, a simplicial or a co-simplicial? Simplicial, of course. Then you have a third differential which is the simplicial differential. So you have the checker of children and the simple truth of children. You have a triple complex and you have the full complex. So rather than one eliminating the others, all three must cooperate. And so, that's a postman's definition of a motif, at least a date motif, which is now common in the application. And so, each object is really a complex of a different varieties. Simpli-shell variety, very often it is simpli-shell. If you know any additive category, out of the simpli-shell object, you can make complex one. And so, by ten of two commodities are three lines. You have a triple complex. This is also called multivariate. Now the game is, of course, calculation, explicit calculation by integrating. So, and then you reinterpret everything in terms of cohomology. ...of a pairing between some homology and some cohomology. Calculations I don't have to very formal. Distinguishes numerical calculations.

2:30 One of my discovery. At the end you have a certain cohomology. Let's say the zeta of 3 will be an element in some cohomology group. What makes it a number, a real number, is that at some point in each of these cohomology groups are vector spaces of finite dimension over the line. You have a natural core. If you have a natural core, You can speak of a linear form which takes positive value on the core. I take a two-dimensional vector multiplying by the square root of two. It doesn't contain any rational form except. But it divides by rational. On this, exactly one linear form on one of the aspects. Basically, it means I'm looking for identity from inequality of the form a minus b square root of two positive. If you want to have a geometric idea of what is a detecting cut, you take the rational plane with two rational coordinates, and you draw a line, a straight line. It doesn't contain any rational points except the origin. Divide the planes, and these two are played quite well, and I mean, if you want to understand what is a... But here is a continuous fraction expansion of, a simultaneous fraction expansion of more than one number, let's say one last square root of two, square root of three, then you move to a three-dimensional picture of the economic model.

5:00 And the brain, the brain, I mean, there are some algorithms to extending for a continuous fraction. And so the idea is a fully algebraic, doesn't have any mention of the real number, but if you start from this idea that the real number is a cut, In a cut among the rational number, if you give a geometrical interpretation, instead of a cut, you say it's a rational plane with a division into two half planes by a certain line, then you move object as three, etc. They are first of all purely except that at the end, they are interpreted as dedicating. And this trick is inquired, and in fact he points out you can use the integer plane. I don't make a claim for that. I mean, continuous fraction, expansion of a number in terms of a straight line in the integral plane or rational plane, it makes no difference, I'm sorry, integral plane or rational plane, except that, except that in our situation, usually you have a rational space and not a relativist, and maybe a preferred relativist, but not always, and it's not, it's irrelevant. I don't make a claim, I mean, about, I mean, those two things. I mean, my claim is... I don't know what this interpretation of identity is. Well, no. But what my claim is that you can do all the calculations about what is the data of three, there is a multiplicity of three which live in the certain cohomology group, and just at the end, the mark into the real number. And the real numbers do not have to pre-exist. They do not have to pre-exist. And so, we really keep explicit real numbers. And then, to prove that they are transcendental or whatever, you don't need pre-existing reels.

7:30 I think that would be dead. Except that this is the Shaniwell's construction of the reels. It's interesting you mention Tate, because Shaniwell claims to have been motivated by Tate. He called it the Tate reels for a long time. People started calling it the Shaniwell reels. So now he says, no, it's the Eudoxus reels, because Eudoxus did exactly this. But I have a categorical count, which you might be... Once again, strangely enough, it's bornology which comes in because, say in the case of the relation of the... You start with the integers, as Colin said, you start with the integers, and you ask... There's a notion of homomorphism of the adequate group of the integers. Which of course is nothing but multiplying by a constant, but now there is the notion of a monological homework, an approximate homework, which means that the, you're asking f of x plus y is f of x plus f of y, you take the difference of those two and you ask to be bounded as a function, so you have the approximate and then you can take an equivalence. So basically you start with a category of monological abelian groups, and then you define new hogs by this method, you get a new category, such that the, for example, the endomorphisms of the original object, Z, have now become the reals, and so on, into the vector spaces and so forth, and you're essentially just giving us a non-linear version of that, in the sense you can use some algebraic equation like this, the cube root of the... In order also to have these approximately, more logically approximate terms. What you're mentioning in Quine's, I mean, the difference is we've got this line that has no points. How do we specify that line that has no points? Quine just says subsets. So that's what the difference is.

10:00 But Quine says he reproved that log to the 18th century. And then log 2 is irrational. I think, what? I suppose I need to prove it. I certainly ought to do that. So log 2 and zeta 2 are not the same thing. But zeta 3 was a novelty. But, apparently, and Volcker reformulated them in terms of integral. So log 2 is integral, zeta 2 is a second-order integral, and zeta 3 is integral. And in my Japanese lecture... I made, I mean, I explained at great length, I mean, all these strategies on these examples, on these examples, because these examples are very elementary, and I said, and I just wanted, at every step in my lectures, I followed these examples, and that they were very concrete, but of course, my aim is to, if I can, I expect to do this for many years, but with a totally new method, which... It's very interesting, actually, this connection, this little historical thing. The fact that when Cantor and Dedican introduced their definitions, and Weierstrass too, I think, was the first way to dedicate a definition of real numbers, what they did explicitly was to introduce irrational numbers. They actually took rational numbers as given. And then somehow you feel that what you're doing with any of the irrationals was really constructed of irrational numbers.

12:30 These people were doing not the construction of the whole law, because the rations were already there. And of course, they didn't actually have to give an account. Well, I mean, all these people were well aware. There was a real difficulty in deciding for any real number whether it's rational or irrational, but the point is that their construction gave you all the irrationals without even knowing exactly which ones were, right, it was sort of implicit in the way that they defined it. They simply added all the irrationals without knowing exactly what they were, explicitly, and that problem remained. There is an idea, there is an idea which is in the mind of many people, that between the full set of real numbers and the rational numbers, there would be some, there would be a certain hierarchy, and that there are, in terms of naturalness, there are... Classes of irrational, but beyond, beyond, of course algebraic numbers, but beyond that, and the idea of motive and motivic cohomology and motivic galvanism is to get control of this, so we, what, irrational entrance, this, but then what I discovered over the years, this idea for about ten years, and, and... Step by step, all these match in a way, and I don't claim it to be the law, that at the time you build a class of reasonable number, you have to build at the same time a class of reasonable function, building at the same time a hierarchy of it, until the reasonable number in log x gives you a function which satisfies the reasonable function.

15:00 And so you have to wait at the same time. What is the solution to a differential equation that can be interpreted in different ways? It can be interpreted in a purely formal way as a power system to an asymptotic extent. I mean, in the spirit of Riemann, if you have a singular point on your differential equation, you should call the behavior near the singularity. The Riemann translates as a tangential base form. It means really asymptotically. A differential equation is not a unique one. There are many possible interpretations, and you have to play between the various interpretations.

17:30 In terms of law, I mean, you have a world of, I mean, you have a, and what you have there are different ways to interpret all these things. In my Chicago lectures in 1967, I published, about ten years later, essentially what I learned from Gabriel about the foundation of algebraic geometry. I wanted to have a foundation for continuum mechanics, actually, but as a step towards a way, a suitable, so basically, I outlined how one could take any algebraic theory and construct an algebraic geometry over that, so one algebraic theory is, of course, the theory of community of rings, another one is C-infinity functions, but I pointed out there are many theories in between, any number of theories in between. But a second ingredient is that in algebraic theories we talk about the arities of operations, binary, ternary, and so on, but that, well, certainly in terms of the general theory of monads, the object could be an arity, this is obvious, but in particular within this setting of an infinitesimally generated topos, there are sort of preferred arities which are infinitesimal.

20:00 So if one takes into it, so for example, instead of a binary operation on a space, x, which is a map from x squared into, say, x, you could have a map from x to the t, where t, this is really the tangent button. So you see the maps, the differential operators and so on are themselves. There's a form of algebraic theory which is intermediate between the classical one with discrete finite properties and the extravagant one of monad theory where any object can be in it, which is precisely tailored to just the situation you described. But I think we do have descriptions. You say there's a way to encode classes of solutions of differential equations or differential rings and the like. But the differential rings are really not a very accurate expression of what you have in mind, because you want to be able to compose these maps as well. So it should be a category, not a ring, just because when you would join some solutions, you then want to substitute those things arbitrarily in other things. Now, in differential rings, there is no chain rule, but you should have a chain rule. They capture Leibniz's rule, but you need also the chain rule. And this is inherently something about composition, about the category. It may be a very small category, but still, no one's ever taken up this suggestion, although I mention it every time. And I say, look, there are lots of algebraic theories in between here. For example, you could join the exponential function through the polynomials. And you can, you know, and people say, what are you talking about, you know, this is no interest. Talk to the wrong people, I mean, that's not the point. What I would suggest, I mean, I've been dealing, at the technical level, I've been dealing essentially with a category of so-called differentially simple things.

22:30 Yeah, okay, that's a quite convenient category. And then, using not one differential ring, but a category of differentials. Not completely what you want. It's a bit different, but at least it's some steps. But I insist on the metaphorological form. And I suppose, I mean, in order to understand what part is it, you have to have more or less an idea of all this. Because it was a simpler definition. But then, many, many definitions. Leibniz's form with r pi over four is one minus one-third plus one-fifth plus one-seventh and so on. Leibniz was rightly proud, and he claimed he had solved the problem of squares into circles by that. Not in a scientific sense, he did. He did that, but never the less. One plus x squared from minus infinity to plus infinity. By symmetry, you split the interval of integration minus infinity to plus infinity to four pieces. Minus infinity to 4, minus 1, minus 1 to 0, 0 to 1, and 1 to infinity. And by simple manipulation, the four contributions are equal. So it's just an instance of symmetry group or rather the transformation of the equation under a chain of variables. The reality of integration of differential formulas. Then you are reduced to 4 times 0 to 1. In other words, if you take x over 1 plus x squared, 1 over 1 plus x squared is 1, minus x squared is x squared, and so on, and you integrate term by term and you get exactly that, which is more or less a wave at infinity.

25:00 But it's not a real infinity. Not the same kind of infinity. And if I take an integral from 0 to 1, it's certainly no explicit difference to infinity, except in the definition of...

27:30 The integral has purely manifolds, which is a symbolic line of development, which we taste, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it, I can do it The law of sort of would make it hard, but I did not realize this form, because it's explicitly in this book on the finite difference equation. And when you write x-y intersection, of course, that you have when you select a certain subclass, and then out of that subclass you apply x.

30:00 In this case, which is one point I've tried to make, the operation of selection is just... Yes, but I never realized that for both x and y, the operations are just operations on operations. I mean, when you write x, y, and 1 minus x, etc. So, but it's, I mean, it's very often... Very often people misunderstand that. They never, even, don't mention it. Well, they're actually functions. Yeah, they're functions. This has to be true. We'll record this when we come to the day when we have a logic discussion. So, you can follow more of dealing with differential equations, difference equations, and so on. There is something fantastic which I read in the book of differential equations. At some point, he says that when he compares two differential equations, and in modern terms, they correspond to modules over the binary algebra, and these modules can obtain, in one situation, can be translated, and he makes a point. Laplace transformation could give me an explicit transformation from the solution of one equation to the other one. It's important to know that. But I can bypass. If I am rephrasing by pure algebra, showing that the model, in modern terms, the model of algebra in one situation is isomorphic, without producing an explicit isomorphic, producing an explicit isomorphic by analytical means, I can do algebra to show, which is very important to me, and importantly hard, because after every side, symbolic counter, people devoted a great deal of effort to substantiate it by not asking at all. It's not necessary. It's not necessary. That's a form. And Heaviside was faithful to Boole.

32:30 Well, I did not want to make a historical study about the influence of Boole on Heaviside. I gave it to him. I just read a book on Heaviside. I don't remember even mentioning it. He doesn't. He doesn't do it, does he? I don't remember. I did not remember any reference to Boole. I can do a great deal of calculation in this part using only that definition and the formal rules of manipulating it. For example, to multiply two integrals, change of variable, which is the Jacobian formula, or a bunch of 4, integration by part.

35:00 The definition of real numbers, that means that, let's say, suppose that I want the definition of pi as an integral to an explicit number, a certain symbol, as a symbol in a certain calculus, and I want to associate by definition into line matrices. Suppose 1 minus 1 third is 1, 5 and so on. Now it's an alternating series. So by elementary analysis, an alternating series produces alternatively an upper bound and a lower bound, produces a cut, the standard way of defining an integral. If you are using 1 over 1 plus x, well developed according to the geometric theory, you produce a series. This series is an alternating decreasing series of rational numbers, therefore, sum up to what it means, if you will, what it means that an alternating series is convergent, it means that it produces alternating upper and lower bounds, and they concentrate, and in effect they produce, okay, what do you do? Let's start.

37:30 The terms of inequality, it means that 1 over 1 plus x squared is between 1 and 1 minus x squared. That means instead of producing, if you have an alternating C but not of numbers, because it's alternating, if you truncate it at each term you produce an upper bound. Yet it's 1 over 1 plus x squared is between 1 and 1 minus x squared. By integrating, you get that the integral of dx over 1 plus x12 from 0 to 1 is between the integral of 1 which is 1 and the integral of 1 minus x1 which is 1 minus 1 third, exactly what we have here. Instead of dealing with an infinite alternating series, and this series producing alternating hyperbound and overbound, you can first of all add a set of estimates on the geometric series. To show that 1 over 1 plus x squared between 1 and 1 minus x squared can be done by pure algebra. Higher steps. Next step. Then, what you apply as principle. Okay, you have... So, now you are... A certain... You are integrating rational functions.

40:00 You have polynomial functions. You can define a rational function as a dedicated input in the polynomial. At least if you related everything to the given interval. So, you produce a rational function as a dedicated input in the polynomial. By purely finite-term means. The integral should be positive. You don't know what is the integral, but you know what it means that the integral is positive. Inequality, I mean, if you know what is positive, what it means that the integral is positive, you can manipulate inequality. And finally, you produce explicitly among the rational, not only without a few intervals, but without a few intervals. Motives, what is motif? It's just a multidimensional extension of this line of reason, and with good control. So the abstract numbers are replaced by objects? Yes. But the crucial point is a category, but somewhere you have a notion of positive homomorphism. I mean, you have the distinction between ritual, severity, and the world. Or the positive cycle and the effective cycle and the virtual cycle, which was well known to the Italian geometers. And so the positivity is really the geometrical notion of positivity.

42:30 That means that you have a linear combination of sub-varieties. It should be deemed to be called positive is the question. I'd like to get to, I don't know if you want to pursue much more on this. I don't want to cut this out, but there's a question. We can leave the sort of the motivic world and get back to just coherent cohomology. What Wendig says in his Edinburgh talk, when you calculate the cohomology in his way, you can do this by checking. This should be regarded as a secondary fact and it's very important not to take it as the definition. And I'm wondering if that relates to what you were saying about him being sort of, he couldn't stand that Serre calculated his cohomology differently. I think it's perfect for shape theory and cohomology, small differential form, many complex variables, and then the way of say is how to mimic this, doing say was imitation, mimic, mimic I mean was I think I keep pointing out to the necessity and compass balls, which doesn't in specific situations use specific means of calculating. I mean it's a different thing, and I think in a philosophical perspective. It's very reasonable to assume that you want to read a uniform and then when it's a property, and I think it's, I think any good mathematician knows that he has to compromise, I mean, the mathematical compromise innovation you don't, they are conceptual.

45:00 Each course in differential geometry, the first thing I tell the students is that there are various approaches to differential geometry and various formats. On the other hand, you have G.I.J. If you are too dogmatic, you have a left hand and a right hand. The left hand and the right hand are both under the influence of the papers of Einstein. When I was young, I was a kid, 15, 16, of course I learned the beginning of Einstein's contribution and of course I remember G.I.J. I remember I was going to the science museum and that was printed on the wall delta of g mu nu dh squared u. Display, I know the display, and there was gamma mu nu. How do you go from that? Instructed myself. I got some hints, I got some hints. The master ought to go from there. Then from that, of course, I was not afraid of writing Jimmy Hulot. And when I do calculation, I can do extensive calculation, I remember at some point I had to calculate the tensor with 12 indices.

47:30 I never really did know what a tensor was, I mean, when you do general relativity that way, you do give it a manipulation, right, and you could raise and lower a tensor index for the best of it, but on the other hand, I was not quite sure what a tensor actually was in rigorous mathematical sense, which is actually what pushed me for what it's worth, you know, into things like abstract algebra, I went right out of that area, although beautiful as it was, because I confess I didn't. I didn't really understand what these objects were. They manipulated beautifully, you saw them formally, but it was very difficult to give a definition that was mathematically satisfactory. Exactly. But I mean, completely, in my opinion, consider both sides. Oh yes, yes. And it's not as horrible as giving the means of manipulation and calculation as a conceptual one. And if you ignore one of them... Well, I think this was a mistake. This is what I think Grottenbeek is doing. Heaviside proved both. Heaviside used pure symbolism, but he was also an exponent of theory where others were opposed to it. But I think Grottenbeek's specific point here, and maybe you'll know this better than me, but I think his specific point here was of course you're going to use check resolutions to calculate these things, The topology of an affine variety is zero. Grotting's approach doesn't need that until much later when you're actually concerned with it. So that he felt that by doing it in this more conceptual order, you also, you make what should be simple look simple, and you didn't use hard facts until you were dealing with hard facts. No, I mean, I think Grotting was right to insist that we should have a conceptual framework to encompass both. But it doesn't distract us from using tools, specific tools, when we're dealing with specific situations, and in differential geometry I'm not doing it the same way. I mean, I learn quick, I learn very early to manipulate, especially at the example, because of Einstein, to learn to manipulate these tensors without knowing what a tensor was.

50:00 And then I learn what a tensor was, and I'm very relieved and very happy to know. And now, of course, when I do differential geometry, I'm not colorblind. I'm not colorblind. I know that the red, this is calculation, and I know that the blue means conceptual. But there is one more thing now, and I think it's basically the discovery of that you have a diagram. I think we can create it. And at one point, I was dealing with a tensor with 12 indices, which was a nightmare. Each long calculation to show some identity. Then I say, well, gee, I know about Penrose's way of formulating things by diagraph. I wrote my calculations diagraph, moving one string across another string, and don't want to calculate. And Penrose diagraph pretend to use code. It's not. You and you do not refer to explicit code. I mean, if you use Penrose's representation, gee, it's inside a box. And there are two strings that have a new meaning, except that when you want to do what is basic in tensor calculus, the contraction, you have a tensor which has one string below and one star above it, and then if you have a lower new accent convention, it just means that you join one lower string to one upper string, so you join them, and of course, while you are joining them, you sue the upper string to the lower string, but when it is done, you can forget about it.

52:30 Well, I mean, the one R. I like to say that my name and, okay, so, and, uh, it's non-intrinsic because when you write G, you select the coordinates. If you really believe in the dynamo, a tensor is a box, we say, in the dynamo. And the calculation of them with stencils is just sewing suitable strings and curving them and permutating them. It's purely diagrammatic. Well, and if you think of g, mu, nu as a diagram, instead of as short for a number, it would have depended on your coordinate system. I mean, it's just another graphic presentation. Exactly, exactly. So, in differential geometry, we have two possible ways. I think anyone who wants to really understand what is differential geometry and what is at stake should be conversant with the three dialects. Yes, well that's the problem with this and Thor and Wheeler's book. You know that book of gravitation. They've gone far too much over into the act. They've got this very long song and dance about trying to say how simple it is. Using the egg crates. Yes, but they've gone, I think, far too much into the... But they also show it as a machine. It's not just a box of strings. It's a machine with chutes going into it. But what you write down, it makes the actual bit of calculation, in many cases, of the exercise, in a way, much harder to do. Because you have this machinery that you're required to understand. It's admirable in one sense. But on the other hand, it was much easier to do these calculations even when you didn't know what a tensor was, you see, I mean, in fact, I know, I mean, both of us know, right, that, in fact, it didn't happen, that you should be able to retain that facility, and I don't think you actually get that facility from, I mean, it's probably weird, it's another way.

55:00 I think I did get some of it from that, but I don't know, I never, I've looked at it and I never felt, but I look palatable, I like to. To speak about the modern diver, just referring to my own personal experience, while I commute daily in Paris area while diving about 25,000 kilometers a year, which is a lot in heavy traffic, I have the advantage of living far away from the town, but yet these advantages that I have to face. But so, I came here, what I am asking Michael, for the prayer man whom I trust. It is important to know that you are not the only one in the world who is interested in mathematics and physics, and that you are the only one in the world who is interested in mathematics and physics, and that you are the only one in the world who is interested in mathematics and physics. So, which means that there are infinitely many new individuals, and they are married, or married again, if you are married. They all volunteer to legally escape the military service, volunteering to certain programs. You have to be accepted, and the price of serving for one year in the army, you have to serve two years.

57:30 Which, I think it's a reasonable bargain, one and a half year or two years, I think it's a reasonable bargain. Programs that are helpful and useful, that can be debated, but never though. And I gave you an opportunity to be not as a tourist, not as a businessman, but as a French mathematician there who at a university, Marie-Françoise Roy and her husband, but they are still very engaged into that kind of program. And now that Marie-Françoise is president of our mathematical society, it's a good mathematical society to work with. With my nephew, I decided to go around and he had a four-wheel car, but then I spent a full day, you see, sleeping, buying extra tires, learning some mechanics. Oh, when I go for 3,000 kilometers in the desert with a four-wheel car, it's likely I came to this mechanics and to have the necessary supplies. Let's say tensile calculus. It's good at some point to ignore it. My power point is that, I mean, if you have efficient tools... The efficient tools are the tools you can use even without understanding. In serious situations, you can ignore what you are doing, you can ignore what is offensive, and do good work without knowing explicitly the meaning of this concept. But at some point, when things become delicate, when you are alone in the...

1:00:00 The introduction of a good notation that you don't actually have to think about, except of course later when precisely you are in these situations when you want to make new events. Or when you discover that the good notation is thinking about it. The use of it is unconscious thinking. No, it makes you unconscious of the stuff that wasn't real. Concepts to actually be used. He wasn't saying don't think about these things, but he never suggested not using these resolutions. He does say there's a format for organizing the subjects. In fact, at this stage, as I've ruled for quite some time, I think we should try and press on a little bit before, but the 50-60 phase topics of our next two phases, I think, have blended together in the general discussion, both in the overall direction that it suggests.

1:02:30 I just wanted to make another remark about this smoothly generated topos, I mean there were three features there that, you see, I remember at the IHES I was lecturing in Rene Thomas' seminar. Third of the way through, I mentioned that, well, a very important example of all this is C-infinity. So he said, well, C-infinity doesn't really exist, basically. He said, don't you realize that an arbitrary closed set is the zeros in a C-infinity function? I said, well, yes, actually we use that all the time. But his idea was that it was too complicated. It was not generic in this sense. So somehow to get in some sense the effect of C-infinity, the typical feature of C-infinity as opposed to analytic or algebraic are these bump functions, the fact that you can have partitions of unity and things of this sort. I think that when I spoke about this continuum of intermediate algebraic theories, there should be an algebraic theory generated over the ordinary theory of rings, polynomials, functions, generated by things like e to the minus x squared, not even the exponential function as such. I'm not sure. I don't think it would be the same logical problem exactly. The main effect of C-infinity and its sub-functions could be obtained in an algorithmic way because it would be a finally presented algebraic theory and one would thereby avoid the sorts of pathologies of C-infinity that he had in mind and yet achieve the very important contrast between the sort of floppy doctrine and the more rigid one. And the other feature there was the type, you see that in pure algebra we have this notion of the adjoints of algebra, like the group and the tensor algebra and all these kind of things in Chevrolet's book. But if we expand the...

1:05:00 The visible arities of operations and tupleties of operations are just as important, of course. Basically, you want to have a mass between x to the n and x to the m, where n and m are more general than discrete integers, but certainly very, very special in the infinitesimal objects. The relation between Lie algebras and Lie groups is an instance of this so-called algebraic adjoint. Tensory algebraic group algebra, a kind of construction, becomes smoothed so that one can extract the Lie algebra of a Lie group and also go back by an algebraic adjoint. What it suggests to me is that, of course, the recent development of operaticity I mean, it has been generalized vastly already by taking some kinds of arbitrary objects. The theory, the cohomology theory, is based on... Maybe two options. Yeah, but the thing is that certain advantages are obtained by going to that level, but... You should stay out of it. But there's... Concrete intermediates out of it. Yeah, there's... There's some theorem of Lee about... Could be interpreted as a theorem about this category of...

1:07:30 Oh okay, I'll be back in a bit. That's okay. Well, we're... I thought you were talking about... Yes, I did. We always look at... Why didn't we write about Shannon? Yes, yes, yes. Is that okay? Is that okay, sir? And this is... And this is... And this is... And this is... And this is... And this is... And this is... And this is... And this is... And this is... And this is... And this is... And this is... And this is... And this is... I never heard the basic thing that he explained, virtually compositing the space into these new things, depending on the cohomology. It's almost like Hilbert McLean space. Yeah, exactly, that's exactly what I was going to say. Well, this is what I was going to ask you. This is why I was sort of harping on the theme that we have on front of us, I did say for you. You know, developing the theme of Kahn-Eidelberg with many spaces kind of within this more wider vision. Yeah, it kind of was elaborated, obviously, partly by Bredendie. But it's also like... Isn't it? Pierre was planning... I guess he was, yeah. You know, the long war just about took us, comment on it. We'll have to talk about that a little later. We should also have your comment in about five minutes if that's okay. But somehow it's a natural part of checking. Yeah. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there.

1:10:00 It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. It's all set there. Yeah. Yeah, but where are we now? You're really, really facing it. You're also doing way too, you know, this more sequential. That is true. That's what I'm saying. That's what I'm saying. Yeah. No, no, no, no. He's quite specific. He's quite specific. The theory of others largely exists now, which I mean. Rather bad. That's more optimistic than what I usually hear. But it does so without at all achieving the goals of the long run. Okay.