Toposes in Geometry & Analysis / Axiomatic Presentation of Einstein's Vacuum Field

0:00 So, the computer is a demonstration, from a certain way, of a particular area of the universe, and it is composed in geometry of humanity. Do you speak French, or do you want to speak English? I don't speak English. You don't speak English? I don't speak English. I don't speak English. I don't speak English. I don't speak English. I don't speak English. I don't speak English. I don't speak English. It's a very great honor to speak in honor of Professor Roussel. Even though I have not seen him for 20 years, I often think of him and was happy to see him again. It's a pleasure to follow the last lecture because the subject of the last lecture was what originally attracted me to consult with Professor Riesel on a possible improvement of it, Different subfields 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 Huzel is a very respectable historian of mathematics, but...

2:30 I think it's important to speak a little bit more about that. Not about Professor Rizal in particular, but about the role of the history of mathematics in mathematics itself. As I've spoken before about the role of education, pedagogy and so on, in the progress of mathematics. I mean, Urbaki, for example, was primarily a pedagogical project, historical. One of historical dimensions, but it led to, as everybody knows, an incredible amount of new and valuable mathematical research, which grew out, partly grew out of the struggle to explain mathematics as it existed to wider circles of people. So similarly, there is a role for history of mathematics. My first teacher before Samuel Eilenberg was actually Clifford Truesdell, 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 of mathematics, or you could say also philosophy of science, philosophy of mathematics. Because there is this observable phenomenon now, in any case, actually for a long time, that most of what passes as a history and philosophy of mathematics is absolutely worthless or worse than that. It's disinformation. In other words, it serves just to continue old dogmas which hold back progress. And in fact, one can also empirically 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.

5:00 Of the depth to which this historian has profoundly participated in the actual progress of the mathematics itself. Well, I myself am not a historian, although I have published two or three papers of a historical event. In the most detailed, rigorous manner in which historians should, my coming to history has been, in most cases, more of the following nature, starting from a problem which exists now, some serious question that exists in the present, to look in the past to see what really happened, how it originated. So I am a great believer in things which subjective idealist philosophers might call objective idealism, but it's not, namely the fact that there is such a thing as latent ideal. 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, but at a certain point, ideas which are latent, which the collective is coming closer and closer to, are 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, historical origins, started from reading some polemic launched by Doudanais against the people of Volterra, where he 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...

7:30 It was even more important than Truesdell told me he was. I mean, of course, Truesdell was a big supporter of Volterra from the angle of applied mathematics, but looking into Volterra's earliest publications, one discovers answers to many questions. One discovers, for example, that it was Volterra who formulated and proved the so-called Poincare Lemma. Sorry. Long before Poincaré, but it was exactly the same one. But not only that, Voltaire really gave me the tools to explain the so-called paradox of points in algebraic geometry and many other areas, and I hope I will come to that. So, into the mystery, because of current questions, and plucking out part of it, of course I'll be accused of taking only the parts that I liked, but at least finding something that nobody else seemed to have noticed, and putting that into mathematical research or exposition, at least. That, similarly, similarly, every right-minded person... This dogma, which many histories, so-called histories of mathematics, still repeat, and especially philosophies of mathematics of the current ilk, the current utterly degenerate ilk, is that Euler was not rigorous, Euler was not rigorous, and therefore continue these current philosophies.

10:00 We shouldn't be rigorous either. We should go for beauty and not rigor. Rigor is a bad thing. Rigor destroys creativity, etc., etc. Of course, a target for these attacks is often once again Bourbaki, but a particular Euler recited in a book recently published by Princeton University Press, he cited precisely in that way, because everybody knows Euler was not rigorous and because everybody knows that Euler got to be famous on the same, therefore we too should be, and this is considered a publishable philosophy of mathematics. As has been pointed out already, the publishers are often not very careful with what gets published. Well anyway, I claim that Euler was rigorous, and of course I haven't read all the many volumes. But that CD told me where to find the particular thing that I need, and I suspect that I've heard of it too. Well, and that was... I wonder if I hold it. Does that work? Yeah, I guess maybe, yes. Okay. So, a particular claim made by Euler was that, was that, well, of course, it's nonsense, but it's a theory. In fact, it's a good definition of real numbers.

12:30 Because it leads to direct proofs of some properties of real numbers that don't follow very easily from Dedekind's definition, for example. So I plan to explain precisely what I mean by that, and I just wanted to list it here. Again, look into Cantor because of something that occurred in the present, namely, I was idly reading Cantor, well, I was casually reading Cantor. Writing 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 glanced through many books by historians about the person Cantor and so forth, something that I'd never heard of that 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, Ernst Sprecher, a very substantial set theorist. ...and pointed him out this. He said, oh, I haven't heard of that. Then we went to the library, and sure enough, you can see it in the copy there as well. And in fact, it was Cantor's attribution of his idea of the equivalence of sets to the geometer Steiner, Jacob Steiner. No set theorist had ever mentioned that. Cantor himself did, very explicitly.

15:00 Sprecher turned out to also be an expert on Steiner, because Steiner was a Swiss mathematician who worked in Berlin, but Steiner commonly gave popular 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 is an expert on Steiner, an expert on set theory, and had read these papers, but had somehow passed by because of these. This led me to look more closely at what Cantor's discovery was that he built upon Steiner, and discovery was something that the set theorists have not analyzed, but which in my view should lead to. So there are examples of this sort, of 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. Because, again, going back to 1960, I don't know where the Park Town Seminar was held, but where did it go? In 1960? Well, it went. It was just a... 1960, wasn't it? Yeah, 60. In any case, there's a commonplace, repeated again in books, papers, historical... Historical origin, you can look on the internet, there's a, in Wikipedia, there's a thing called 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 then 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 a generalized space.

17:30 But it wasn't the only origin of topos theory. And you can see it if you actually look at the seminar in 1960. I think it was 1960, no? 60, 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 in any real sense. And it wasn't called the topos. I don't know, the word came up maybe a couple, three years later, but it had all the earmarks. In fact, it was. So, I can say that through long considerations, I have arrived at a tentative explanation of mathematical accountability. That as a site, of course, the ordinary classical topological spaces, when viewed as toposes, have a site, or sites which consist of post-sets.

20:00 Of course, post-sets have no importance. They have no endomorphisms at all of the objects. Immediately important, along with the classical topological spaces, were toposet G-sets, where G is a group. Because, in fact, from the algebraic topology of the 30s, 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, which if you take the kernel of that, it's 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 footing. Special topos is known as étendus, which are locally topological spaces, and so, of course, the generalization of space, but they also include groups, and as it turned out, include quite a bit more, because étendus have a site consisting only of monomorphism, not necessarily invertible, but at least monomorphism. On the other hand, there's another class. This is a very special class of toposes in which the acceptable objects are active. These were first studied explicitly, unless someone tells me otherwise, by Peter Johnstone under the name T.B. So this kind of topos, well first of all acceptable object, is one for which the diagonal has a copy. In any topos it has a hiding complement, but that's not a real complement in the sense that the union is not everything.

22:30 There's a complement, there should be a complement, so the square is the sum, the diagonal is something else. The diagonal is detachable, to say it in various words. So, separable objects, if all objects in the topos were separable, then it would be Boolean and very, very special. But the objects like this can be added in the sense that every object in the totals is an inductive limit of these, and that's the interesting class. Interesting for us in this regard because the key topologies are like that. This is basically the fact that if you... Take the connected etal objects, that you have two maps between them, between two of them, which are equal somewhere, even on a very small part, then they're equal. So that's saying that the map from the small part to the domain is an epimorphism, all maps are epimorphisms. Because it turns out that, that, uh, these, uh, circled and natural things have, have sites. Consisting entirely of epimorphisms, instead of monomorphisms, 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, not everything is epic, but just in its own little universe, it has the universal cancellation property. Well, now both of these are good classes of categories and sites. You can exponentiate them arbitrarily. You can take them in a unique category and have such a reflection, etc., etc. These are sort of interesting categories to study as various kinds of generalized space. And there's a common generalization, which is pretty obvious, namely sort of bi-cancellation problems, where you assume two things are original and two things are equal. So this is one thing. This immediately implies this still more general, still more general problem, that there exists a site of which there are no other workers.

25:00 And so that's a very simple, again, very complicated problem that is worth studying. It hasn't been much studied, but it's worth studying in this slide. The notion of generalized space is not simply wholesale generalization. But it does capture the relevant examples. Placid spaces, groups, Pite, Atiyah, Witten, Connes, and so on. So, by contrast, the topos that was implicitly used in the 1960s presentation and became much more explicit later, where everyone spoke of the really risky and really tough toposes, they have... And opposite properties, namely that they have to have degeneracies. I decided, I think I made one mistake, which was to call figures points. The word point is too general. There is something there, but points are too general, something more special. So I used the term figure, so everybody could understand. Voltaire, by contrast, used the term elements. In fact, that was very good for the time, because the idea that elements are something irreducible is certainly not the inherent of the language. To say that a windowpane is an element of the window which returns an element of the room and so on seems perfectly sensible to me. Structural elements of a literary work and so forth and so forth, these don't have the... So I think Volterra's use of this term was quite correct then, even now, except that because it's set here, it's called figures instead.

27:30 So in an arbitrary space X, a figure of shape A is just a map, just a map. But then if A is special, it might be a point, it might be something, it might be a... An object of a site, in which case one must be a figure. So, there's another rumor, where if you say that a map from one figure to another is a narrow triangle, but let's call this an incidence relation. Incidence relation could say, for example, that a certain curve is part of a surface or... All sorts of things can be expressed by this, so there is a category of slice category, in the sense that the objects are just arrows. You can have a category of figures and instance relations, which you could use in general, but I normally use it when you have some specified class of page in mind, such as a given site or some defining property so that it has some content other than just another name for an arrow. That content is really given by the fact that it's a sub-category of an academic category that we take. 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 how you can get inside. Because there's a geometry of figures and incidence relations inside. You can officially use the word inside of X to mean this kind of work. The way the picture is raised, then you have a more complicated way of picturing X.

30:00 This is a picture of X. There may not be that good a picture of X. Because you don't think so. This was my next... Again, if you have a special class, you want to give a special name to maths. Given an arbitrary given domain, a special co-domain, of course they should be called functions. Functions have always meant something a little bit different. Sometimes you can identify morphisms, transformations, math, functions. But on the other hand, function theory uses morphisms to study special morphisms. So it's the same theory. We can take the outside. The outside of the geometrical object is the algebra of functions in the slice category X slash A. This is the algebra of functions on X, in the sense that these are the algebraic operations. You can apply an operation theta to one of the F's and another one to the prime. 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 what an algebra function is, and again, this simple definition immediately says that, well, suppose we had a morphism, a general morphism, then that would induce a functor, or actually a morphism of discrete vibrations, because of course these... The shape of a figure is given by such a functor, just as here, the type, if you like, of the functor is given by the screen out vibration. So this functor, the map of the vibration, that functor that's induced by the functor is always continuous in the sense that it maps figures into figures.

32:30 Without tearing the entrance room, so what else could continuity mean? Sequential continuity of topological spaces is a specific example of this, because one can take a stringer type of convergent sequence along with the single point category of spaces that's generated by that, and then precisely sequential convergence means transforming Convergent sequences considered as figures into other figures without tearing the limit away from the rest of the sequence, without tearing that piece into the world. The fact that the point is being included at the point, incredibly, is the key information. Seeing that that notion of continuity is indeed an example of my general, apparently, my apparent general nonsense definition. Yes. And duly, of course, we have a homomorphism in the opposite direction between the function and algebras. All these theorems are all cases of associativity, of course. In fact, if the function is in the opposite direction and the algebras' function actually preserves action by the homomorphism, well, it's really all just . So it's not very deep in a way, and yet it is . So, again, adequacy as a general concept, this was defined by Isbell, John Isbell. Adequacy means, it's adequacy of a choice of A's, of a choice of a subcategory. A subcategory is if every math between general spaces, sorry, every morphism of discrete vibrations, or if you like, every natural, every natural transformation. The septalian presheaves announce the same thing. It comes from an actual math. So in other words, the embedding that you get from the given omni-category into the category presheaves on the allegedly adequate subcategory is actually filled and saved and typically reflected.

35:00 So within this general framework, the general way to talk about... How is it that, how is it that, you know, numbers are iterative and it doesn't work? Well, for that, one needs one very important preliminary reading beyond the bare notion of categories. Namely, one needs the idea of exponential functions. Categories that have this are often called Cartesian flows. This is really, in a way, perhaps the most fundamental ingredient of mathematical content going back 300 years or more, is the idea that given two objects, the morphisms between them in some sense also form an object. It also has a geometrical structure. It has its own figures and insulations. It has its own algebraic functions, which are called functionals. Volterra, they were used for a long time, but not given a name here, so that gave Frege the license to claim to have invented them in the context of truth-value functions, but the real-value function wasn't much more profound than we knew about then. So Volterra made his positive decisions and said, look, you have a side you don't belong, but let's... And then, of course, his very good friend, Amad, Amad, gave the name and function, as you know, crochet and automata, again, of serious development, which didn't yet depend on topological vector spaces.

37:30 So the defining property of exponential is that you have for every pair of objects, you have another object that lines with A. But for every third object, the modernisms from X into that correspond naturally to the weakly matched from X to Y. Now, so as an instance of right adjoint, Kahn's notion of adjoint functions, which was on the way to the press fifty years ago, and in particular this example which Kahn himself at the time. New law in the context of substitutional sets, specifically the function blank of power A divided by two, so many properties of any of the, well, not completely, but certainly properties, snap spaces, all of that, come from the algorithms to the definition that it does to the previously introduced notation. The point being that... From the earliest days of the calculus of variations, which also according to Doudanais, I suppose, by logical deduction, they didn't do any functional analysis either. A fortiori, they didn't do anything. Because right from the start, the idea was, well, what is a figure of shape and interval in a map space?

40:00 Why did one care about that? Well, because one was talking about things like problem of least descent. Etc., etc., in which the variable of the discussion is an infinite dimension, yet, in spite of that, one wanted to vary it. So what is a variation? A variation is precisely a path in the map space where the domain of the variation is something like entropy. A variation of what? Well, of course, if you might have a chosen point and you vary that point, that's just it. So variation is a figure of one-dimensional shape, usually inside some mass space. We can deal with that because it's the same thing as a function of only two variables. This was the technique of the calculation of the calculus of variations from the beginning, and Volterra made this into the definition of analytic punctuality. The fact that the category of homolytic spaces has been also functional analysis was of great interest there, so it went by. So you can imagine that today is really an event for some standard complex analysis of the event. So, the space of homolytic work is an interesting homolytic space. No matter, given its geometric structure, just by calculating its geometric structure, just by calculating these things instead.

42:30 But the not good thing about it all is that you have the possibility of talking about functionals being smooth or being elliptic. Because, in this sense, it's essentially a morphism that's domain. There's a map space. Well, it might be part of a map space, etc. You all know the modulation. The crucial leap in content of it, simple like an algebra or something like this, is that you have more business, for example, in real-value mathematics. These domains, like the time of descent for some arbitrary period of descent, the time to get down, the time to... It could be an integral, but it doesn't have to be, and so on and so forth. Well, the objective fact that you need is some kind of smoothness. And what's the definition of smoothness? It's a word for the association of geometries. In other words, if I have a function of this analytic, say an analytic category, function will be analytic because if I take any analytic math like this, well... That's really something like this, because I've already studied functions of several complex variables, I mean, highly conventional ones, don't know what that is, but if I plug that into my functional, I've got to get something which is, again, equally analytic, so that is the condition of each function. It's all, in some way, it's all far more simple. Then the whole dogma of topological retrospaces would suggest that precisely because we have no way of knowing, I mean, 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 automatic properties more or less continuous in the total sense sense by having some, let's say, having some representing object Think of the Saprinsky space with two points and three open sets. For example, it would be something more sophisticated and within context.

45:00 So you could say that, well, this has a point. True. So if you have a map like this, take the inverse image of a point in any space, whether it's a red pool or not. Inverse image of a point under some map, and I can call that open, with this being fixed. So then, of course, every math in the category is continuous and dramatic in the sense that we're seeing the total success. But still, even if I do that, I have no way of calculating what that means for the functions. Because that involves going back here and have a way of calculating what's going on about it. I don't think that works. In any case, it's simply very complicated to try to get at the open-set structure on a map space knowing the open-set structure, knowing some open-set structure of the ingredients, whereas this is automatic covariant structure given by geometry figures taken as the... This is the basic measure of the cohesion of spaces rather than the contrarian structure of algebra of functions or algebra of open sets, which is an example of the same thing, medicated as the default of cohesion of ratios. I promised to tell you that. I had much more to say, but I'll just say this. So 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 such that a times x equals b,

47:30 put it back to the question of multiplication, we get something to be competing with, and we eliminate automatically a lot of the confusion. Dividing or even inverting is a non-trivial process as seen in the fact of the localization of rings. That's a whole subject is how to take a ring and it inverts some things and has to open 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, which is the best kind of dividing if you can get it, but there are many cases where you can't even get that. So, basically a ratio process which can transform something into something else. Of course, very special properties, but still, it's a morphism. So, I don't see reals and d-ratios in infinitesimals. Well, there might be ratios of infinitesimals where d is an infinitesimal space. One thing about them is that they are pretty good at zero. Okay, so in other words, I'm going to define the reals as the sub-object of the function space, so the function space D is the, are these ratios, it's a natural intrinsic multiplication, which is just a composition, and I'm going to say what the D is in a moment, but in a pseudoambient category like a topos... Every monoid has a universal commutative monoid associated to it. This, of course, is a monoid automatically. This is sometimes called synthetic differential geometry because we start with nothing but an object and we produce the algebra out of the mirror category theory of the geometrical object. This is highly synthetic in that sense. So we've produced a monoid.

50:00 That's going to restrict to the multiplication of integrals. And in fact, what part is it that the infinitesimals do have one point, they have only one point, but the condition that Morphism takes the point into the point, when you relativize it over arbitrary bases, as he wants to explain to Adler, over arbitrary figures A, you have to put that as the condition, so the real sort of term is that of the kernel of this. The W is zero, and then takes the whole graph to zero. The part that leaves the D, the graph zero into zero, and of course any global point, lambda will automatically do that, because one zero is the only point of D anyway, the only global point of D anyway. So, another feature here is that I can take the commutative reflection of this monoid and I... I've got the condition now. This is the last amendment. So this part, that is the actual ratios which preserve zero, that space, the space along those, is actually equivalent to forcing the multiplication to be commutative. Well, so that means that R itself, the multiplication, is commutative, although no function space could ever be commutative in itself, whole. The whole endomorphism space is always non-commutative unless A is one point. This is a general fact, not something particular about it. So D is not commutative, but it's part of this. And moreover, that part is commutative because I said it's isomorphic to that. And moreover, the fault on this projection by the inverse, we see that R is actually a retract. A retract as a monolith.

52:30 We're supposed to play the role of the tangent model. X to the D is the tangent model of the arbitrary X. Evaluating at zero is the bundle map. So you could say that R is the tangent space at zero of D itself. Tangent space at zero of D itself. And so of course when you look at induced maps, essentially you're taking derivatives of arbitrary maps. Just what flows out of the D. And this retraction, I won't give the detailed format, but it gives you the opportunity to write down what Newton called dot, the derivative of it, as opposed to... Well, now first, you see, in all the specific examples that we know about, the D has a concrete nature of the perspective of the human mind. And I think we want to approach it as a matter of just like purely synthetic properties of D, some of which I've already been done, at the singular point, the subtraction, and so forth. And the truth is in particular that R has not only a multiplication, not just a definition, but a general synthetic property, but a multiplication on R as well, I'm sorry, an addition on R as well, which makes R into a ring of, as you would hope. So I can take R as a co-domain for a function algebras, and those are examples of the usual function algebras, or I can take it as a figure, in this case I would have paths, you see, talking about paths and the tangent, the derivative of the path, and so on, and so forth. In particular, especially what the addition enables me to do is to formally leave out the group objects. And to deduce the extensive nature of the distributions, so any such category, Toko is equivalent to the sum of the D, has, or even a Cartesian post-category, has a natural notion of distribution of compact support.

55:00 Whereas the Han space is a subspace parted out by equations from a double mass space. The point about our attitude is that the distributions of the twistor and sound of the two spaces must pair with the distributions of the two spaces. So this is the end of analysis and geometry. Thank you for your patience. Thank you for this categorical approach to analysis. I appreciate it. Are there any questions, remarks? 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 in the product, you can. Or maybe Volterra was one of the first to use it systematically. Well, yeah, I mean, you could say that... I think that to call distributions generalized functions is a very bad misnomer. It is very misleading. I think various analysts have pointed this out. You can't simply restrict a distribution to a subspace. Well, that means it's not like a function at all, generalized or not. This expression here, hom-R-X-R, is clearly a covariant function of X rather than a contravariant one like the algebra of functions or anything like it would be. But there is such a thing as generalized functions. The generalized morphism would be a morphism from X to Y, no, not into Y, distributions onto Y.

57:30 That's a generalized math, you see, because... Because, of course, there is the Dirac delta in natural math from any space into the space distribution line, so a special case of such a key would be an actual math from x to y. I say here d sub 1 because with a sort of natural restriction, you could say, let's state those distributions of compact support which integrate 1 to be 1 to preserve the constancy. So, it's not that there aren't generalized functions, it's just that the distributions themselves don't really logically play that role. Discussed also what? Fact. Fact, yes, yes, oh yes, of course. So, maybe it's a good transition to the next talk.