Toposes in geometry & analysis

0:00 So, the computer is a demonstration, from a certain way, of the determinants of the atypicals, and they are used to compose geometry and analysis. You speak grammar, and you want to use mathematics. 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 sub-fields 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. But rather than proceeding in the... 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 Poincaré 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 seems to have noticed, and putting that into mathematical research or exposition, at least. 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.

10:00 Euler was not rigorous, and therefore continue these current philosophies. 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 suspected that I heard of it, too. Well, and that was... What if I hold it? Does that work? Yeah, I guess maybe. Okay. So, a particular claim made by Euler was that... ...was that a ratio of...

12:30 Everyone says, well, of course, it's nonsense. But it's a theorem. In fact, it's a good definition of real numbers. 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, see it in his, 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 there should ever mention 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, 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 part-time seminar was held, but it was just over the 60s. In any case, there's a common place, 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 counter-distinction. That has a site. Of course, the ordinary classical topological spaces, when viewed as toposes, have a site, or sites which consist of ultra-posets.

20:00 The posets have no importance. They have no endomorphisms at all of an object. 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, and so Rodendieck's first attempt to explain what that category is was his... 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. 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 Q.D. So this kind of topos, well first of all the acceptable object is one to 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 sun, 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 topos is an inductive limit of these, and that's the interesting class. Interesting for us in this regard because the teetotalic topos of the scheme 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. 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 properties. Where you assume two things originally, and two things anyway. So this is one thing.

25:00 It implies this still more general, still more general problem, that there exists a site of which there are no inner workers. And so that's very simple and then fairly well behaved property, which I think is worth studying. It hasn't been much studied. It's worth studying in this slide. The notion of generalized space, it's not simply wholesale generalization. But it does capture the relevant examples. Classical spaces, groups, Pite, Atiyah, and Tom, and so on. So, by contrast, the topos that was implicitly used in the 1960s presentation became much more explicit in later, where everyone spoke of the really risky and really tough topos. And opposite properties, namely that they have to have degeneracies. I decided, I think Herkule made one mistake, which was to call figures points. The word point is too general. There is something there, but points are too, there's something more special. So I used the term figure, which 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 is 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, of course we say that a map from one figure to another is a narrow triangle, but let's call this an incidence relation. An 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 categories, in the sense of the objects on these arrows, a category of figures and instance relations, which you could use in general, but I normally use it when you have some specified class and age 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 picture of x. Because the other side is. 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 its algebra of functions. Again, a slice category, x slash a, x. 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 algebraic function is. Again, this simple sort of definition immediately says that, well, suppose we had a morphism, a general morphism, then that would induce a functor. Or actually the morphism of discrete vibrations, because of course 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 type of discrete out-vibration. So this functor, the map of the vibration, that functor that's induced by such a thing is always continuous.

32:30 In the sense that it maps figures into figures without tearing the entrance road. So what else can continuity mean? Sequential continuity in topological spaces is a specific example of this because one can take a figure type, we call them the convergent sequence, along with a single point. There's a 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 principle apart. The fact that the point is being included at the point is the key. Incidentally, seeing that that notion of continuity is indeed an example of my general, apparently, my apparent general nonsense definition of what it is. And duly, of course, we have a homomorphism in the opposite direction between the function and algorithm. All these theorems are all cases of associativity, of course. In fact, the function in the opposite direction and the algorithm in the function next to you. Preserves action by the homomorphism of all this material, so it's not very deep in a way, and yet it counts for it. So, again, adequacy as a general concept, this was defined by Ismael, John Ismael. Adequacy means, it's adequacy with a choice of A's, a choice of a subcategory. The subcategory is... If every map between general spaces, sorry, every morphism of discrete vibrations, or if you like, every natural, every natural transformation of the set value pre-sheaves amounts to the same thing, comes from, comes from an actual map.

35:00 So in other words, the embedding that you get from the given omni-category into the category pre-sheaves on the allegedly adequate subcategory is actually full to say the least. Typically reflected. So within this general framework, general way to talk about it, how is it that, how is it that, you know, numbers of H is metesimal? Well, for that, one needs one very important perimeter reading beyond the bare notion of categories. Namely, one needs the idea of... Exponential 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. It's 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 its insulations. It has its own algebraic functions, which are called functionals, which were used for a long time, but not given a name, so that gave Frege the license to claim to have invented them in the context of truth-valued functions, but the real-valued functionals were much more profound than we knew about them. So Voltaire may be responsible for science. He says, look, you're not deciding all along, but let's see. And then, of course, his very good friend, L'Allemagne, L'Allemagne, gave his name in French, you know, Perchet, L'Allemagne.

37:30 Yeah, those two are relevant, which, yes, depend on topological record spaces. So the defining property of exponential is that... The fact of the way of having these is that you have for every pair of objects, you have another object that winds with A, but for every third object, the motorisms from X into that correspond naturally to the nuclear mass from H times X into Y. Now, so it's an instance of right-right adjoint. Con's notion of adjoint. Punctures, which was on the way to the press 50 years ago, and in particular this example which Kahn himself at the time knew about in the context of superficial sets, specifically the puncture blank of power A divided by two. So many properties of any of the, well not completely, but certainly the properties of snap spaces and all of that come from the algorithms. The point being that from the earliest days of the calculus of variation, which also according to Doudanet, I suppose, a biological deduction, they didn't do any flux 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 does not anyone care about that? Well, because one was talking about things like the problem of least descent, etc., etc., in which the variable of a 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 an interval. A variation of what? Well, of course, you might have a chosen point, and you vary that point, but that's just... So a variation is a figure of one-dimensional shape, usually inside some math space, and we can deal with that, we can deal with that because it's the same thing as a, as a function of only two variables. And this, this was the technique of the calculation of the calculus of variations from the... We get it, and Volterra made this into a definition of analytic punctuality, which basically was the fact that the category of analytic spaces has been also functional analysis was of great interest there, so it was like, so you can imagine that he exists for... Some say it's a complex analysis of everything. So, analytic work, the stage of analytic work is from how many of the stages to another, given its geometric structure. Just by calculating these geometric structures, just by calculating these things instead.

42:30 Of course, the not good thing about it all is... But you have the possibility of talking about functionals, being smooth, or being elliptic, because in this sense, it's essentially a morphism, it's domain, it's a map space, well it might be part of a map space, et cetera, you all know the modulation. Crucially, we've been contacted with something like an algebra or something like this, is that you have morphisms, for example, real-value morphisms. These domains, like the time of descent for some arbitrary period of time, 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 functional, let's say an analytic category. Functional being analytic because if I take any analytic math like this, well, naturally something like this, because I've already studied functions of several complex variables, I mean, highly conventional ones, I 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 the key to this. In some way, it's all far more simple than the whole dogma of topological vector spaces would suggest, precisely because we have no way of knowing. You can talk about open sets, you can make different definitions of open sets inside the same category with the trivial automatic properties, 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 really complex, but, so you could say that, well, this has a point, true. So if you have a map like this.

45:00 Take the inverse image of a point, a mini-space, where it's eventful or not. Inverse image of a point, I can sum that, I can call that open, with this being fixed, this being fixed. So then, of course, every math in the category is continuous, automatic, in the sense that it moves in and out of the set. 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 down the path, to say that an open set with an analytic function on it is going to make that, that's going to do this very special thing, but 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. Whereas this is automatic covariance structure given by the geometry of figures taken as 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 measure of cohesion. I promised to tell you that. I 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. Talk about quotients. So if we interpret every statement about quotients by saying there exists an x such that a times x equals b, put it back to, it's not a question of multiplication, we get something that we can be competing with and we eliminate automatically a lot of confusion.

47:30 The fact that dividing or even inverting is a non-trivial process is seen in the fact of localization of rings. That's a whole. The 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 the ratio process is to transform something into something else. Of course, very special properties, but still, it's a morphism. So, I don't see real complete ratios of infinitesimals. Well, they're going to be ratios of infinitesimals where d is an infinitesimal space. One thing about them is that they are pretty much zero. So, in other words, I'm going to define the reals as a 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 suitable ambience, the category of topos, every monoid has a universal commutative monoid associated to it. This of course is a monoid automatically, this is sometimes called synthetic. The differential geometry, because we start with nothing but an object, and we produce the algebra out of the mirror category theory of the geometrical objects, and this is highly synthetic in that sense. So we've produced them only. That's going to restrict to the multiplication of materials.

50:00 And in fact, what part is it that the infinitesimals do have one point? They have only one point. But the condition that workers have picked the point into the point, when you relativize it over arbitrary bases, as you must to explain the object of arbitrary figures A, you've got to put that as a condition. So the real side of the problem is that you evaluate the zero and then take the whole value of zero. Lambda will automatically do that, because it's the only point of D anyway, the only global point of D anyway. Another feature here is that I can take the commutative reflection of this monoid and I put the condition now. This is the nice thing about it. So this part, that is the actual ratios which preserve zero, that space, space along those. It's actually equivalent to forcing the multiplication to be commutative. Well, so that means that R itself, the multiplication, is commutative, although no function space can ever be commutative in itself. 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 Now this object, the object D, is supposed to play the role of the tangent line. X to the D is the tangent line of the inverse of X, and evaluating at zero is the bundle math. 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 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 a dot. The derivative of it, as opposed to... Well, now first, you see, in all the specific examples that we know about, that D has a concrete nature of the perspective of the human mind, and I think that one can approach it axiomatically just by purely synthetic properties of D, some of which I've already been done, at the same point, which we tried to consider. And the truth is, in particular, that R has not only multiplication... Not only this definition, but from then on, we've had the property of multiplication on R as well, sorry, addition on R as well, which makes R into a ring of, as you would hope, so I can take R as a codomain for a function algebra, and those are examples, these are the usual function algebras, or I can take it as a figure, in this case I would have paths, you see, talking about paths, the tangent, the derivative of the path, and so on. So, in particular, special recognition labels need to do is to formally re-alternate, re-alternate group objects and to reduce the extensive nature of the distributions.

55:00 So, any such category, topology group, the C or the D, has, or even a Cartesian prose category. Has a natural notion of distribution of compact support. Because the sound space is a subspace parted out by equations from a double mass space. It's a linear function. The point about the R-B attitude is that the distributions of the pitch-point sum of the two spaces must pair with the distributions of one location. So this is the end of analysis and geometry. Thank you for this categorical approach to analysis that I appreciate. 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, the kernel. 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 final 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, see, because... Because, of course, there is the Dirac-Gelton in natural math from any space into the space's distribution line, so a special case of such a key would be a natural math from x to y. I say here d sub 1 because of a sort of natural restriction. You could say, let's take those distributions of compounds and work to 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.