Euler, Maxwell, Grothendieck & the Mathematical Representation of Space

0:00 Thank you for your attention. Okay, are we ready to start, ladies and gentlemen? Actually, I don't think we can. Gentlemen, actually. The ladies have all left us. They're smarter than we thought. Well, no, they're as smart as I thought they were. Any concepts of correct generality needed for the deepening unification of mathematics and for the solution of its most significant problems are laid through geometry and particularly perhaps through algebraic geometry, but into the rest outwards, into reshaping our understanding of the notion of logic, a perspective very different from the view of all foundations that we have inherited for the last 50 years from the more profoundly than our next speaker, Professor Recitation of his achievements.

2:30 But I will say that I'm very proud indeed to be here to introduce. And one of the has been what may well be able to give us part of it. We had one of those famous buffalo snowstorms and I didn't arrive until well after the hour of the talk. People came from Harvard to hear you. I'm 13, but I still haven't met them. We've postponed the storm this time, in your honor. We've postponed the storm, in your honor. Keep me here. It's due to start the moment you finish speaking, Bill. 85 categories are a... You could say that for a few years Rodentich realized that the typical 85 categories

5:00 actually arise in a more basic way from a non-linear category with a linear structure. The reason I cited Maxwell here, I'm not giving you an elaborate mathematical connection with Maxwell, but just the spirit of his at one point, that physics one has to many levels as needed, experimental. But also of some physical sets, physical sets, and again, space is represented for many purposes, but for example, for numerical purposes, calculations, the finite element method, which means essentially you want to use a combinatorial model of space, chosen, of course, to approximate a more basic non-combinatorial model. And so the appropriate intensive and extensive quantities which vary over those. Specifically, you might take linear structures, again a non-linear category of quantity, or more precisely, the quantities of interest are always variable quantities, essentially variable quantities.

7:30 Variable quantities are always in one way or another modeled as... Quantities that vary over some kind of domain space. Domain space might be very abstract, spectra were mentioned as domain spaces for conscience, but even more abstract, the conceptual division between the domain of definition and the possible values preserved in nearly every, the distinction between intensive and extensive, which I mentioned after the papers on this, if you didn't look up. Essentially... This is an old philosophical distinction which has largely disappeared from general discourse, except that because Maxwell made such extensive use of it, and because it's a practical use in thermodynamics, it is still a construct. I guess in engineering it's used in a way similar to dimensional analysis, that an equation cannot be, cannot have an equation be intensive and extensive. Both sides of the equation have to be extensive. So it's a way of basic hygiene on your equation. Exploiting the fact that the intensive quantities, like pressure and differential forms, these intensive quantities all depend contrarianly on the direction of the measures.

10:00 And this distinction actually persists, or normally persists, but actually directly carries over to the right categories. They're connected by the fact that usually you can multiply intensive times extensive and excessive, or extensive and module and intensive, which poses a division problem. If you give them two extensives, they might have the ratio. The best we've done on torsion and electrolyte is to try to refute some of the standard cliches that are brought. One cliche is that it won't be invented topos as generalized spaces. Of course, there were some important topos that were generalized, but they just weren't made of these. Still is, the Atiyah topos are clearly a kind of generalized space, that sort of thing, but a very special kind of topos. Even before that, in 1960, Grudnick had treated the category

12:30 of all analytic spaces, which we now recognize. That kind of category has much generating impact. I used to say that topoids are not general variations, variation in sets, but that applies best to the generalized space. A sheaf over a topological space is a descent that varies between a general set and a non-general set. In terms of what was said before by Colin, another example is the category of G-sets or G-groups. From the point of view of definition of structure, two sorts of things do fit into the same category, and in fact there are maps between them to express some of these algebraic topologies, so that the G's on topological space, the G sets we're dealing with, are both examples of special kind of topological space. Trying to combine these, we find that they follow the atom nu, which are two poses which are locally topological. There is a general notion that topos may not be of either of these forms, but it is located in the first form, Etan Du. Well, it turns out that the second one is attributed, as I suppose, to locally one point. So you're often here at Penrose talking about a point, talking about this kind of topo.

15:00 It turns out that Etan Du were a bit more general than emergent beings originally thought. All they have to do is pay attention to the definitions. The direct description of the site turns out that there is no closeness that is described by means of sites that consist of auto-morphisms who are categorized as morphisms. So, for example, all the ejected endo-morphisms are natural non-morphisms in a category like that. The category on that is the NH-undue percent. T.H. Howe's scheme, for example, is not like that. It does belong to another class studied by Johnstone, which he called Q.D., indecidable. I think he more descriptively said something that's adequately simple in the sense of algebra. Anyway, it's secular in the sense of algebra, which means it refers to an object with a property. If you consider the diagonal mapping, if you would square, there is a 90 problem. There's always something that deserves the name of the problem, but here what they're asking for is the actual, the only part that comes with it, diagonal, and together they literally make up everything.

17:30 The rest, you see, is not, is the, you know, you might think from the picture that the diagram is closed and this is an open, in fact, that's the general situation, this is something very special, and these things are just literally put together as if the world were gooey. It doesn't mean the world is gooey, it's not only that it is, it's not only that it's gooey, but that's the idea that a circleable object, adequately circleable means that you have a... Talking about topos E, you can look at the subcategory of all the central objects and it should be adequate in the sense that every object is determined by maps of E at the moment. It turns out that these topos, in terms of sites, they have epi-sites, described by a site which is a whole category that involves maps. It does not mean that these maps are epimorphic within the topos, but just in its own universe. So the quest to explain what kind of topos can be viewed as generalized spaces, then at this point the next step is often to unite these two. I don't know if there's history in the way, but I run with the idea of a cancellation property like Ebbing and Mono, but a two-sided one. Cancellation property means that you can conclude that two things are equal, given that you know in this country.

20:00 But in this case, it does require two things. We want Ax to be equal to Bx, and we have to say that those two things together should apply. This is a weaker cancellation. Mono is the fact that, say that F is mono, it's to say that just F-A is F-B, or to say that X is F-E, same thing, but without the F, so those are, those are, and of course, if you want to insist on positive logic, you can quantify it, but I want A and B, I don't care which X and which F I use to do it, anyone. So it's a category, this is a property of the whole category. So if I have a site, this property, then there's a two-quote of sets on this, a notion of variable set, and this is a quote generalized. Well, actually there's something even more general, even simpler, which is notion of modes in the site. This is still a question of the site. So if I put this condition on a site, it's easily verifiable, puts the condition, and it's able to tell geophosis, she's going to touch the site, kill geophosis, or whatever it says.

22:30 Now you might say, well, what about the geophosis of the hole? Many people feel comfortable with the troglodytes, the site descriptions, often times the description of the whole troglodytes, except for Japan, if you have any questions about the presentation. So, I kind of leave it out of the concept, I'll talk about it. Still, a much weaker condition, which is no centralized troglodytes. What is the center? The center of the troglodytes, a little closer to the category, Of course, it has an identity endofuncture. And the identity endofuncture has its own endomorphisms of natural transformation. That's called the center of any category. For example, the natural endometrisms of the identity functor. It's always a computer moment. It might be trivial. It might not have any purpose. When I say no central influence, I'm saying that any natural, any work is in contact with any function. And why does that, why does that follow from this, for example? Well, it follows from this. In other words, if you have, if you have a site with no influence at all, then you're right. The bottom is blocked by the, uh, the bottom is blocked by the, uh, link. Not a very good thing to do. So you can easily verify that an exercise, suppose you had specialized to attend to more than every single object, it's natural to commute to every single morphing.

25:00 You suppose such a thing, but on the other hand, you have a subcategory of objects that have no input at all, and the subcategory which is inadequate, the test transfers to the other side, and all of these can be done. I'm not saying that there are still generalized spaces, but I did this really in order just to emphasize how far from generalized spaces to following headings do I have, you know, I'm really getting philosophical in order to define for you what equality is. Oh, we heard the word quality. What does it do? Well, in general, in the context of geometry, definition, you know, first of all, it's kind of a quality type. And when you call it a quality type, it's a topos defined over another one, whether it's a north or a south one, with the property that the two adjoints, the central one, we have the left adjoint and the right adjoint, the same, let's say, points after the global section of the topos. Thank you very much for your time.

27:30 One of these is phasors. These are much less cohesive and can be extracted from certain points, and one set of points is a phasor, and the other one is a set of components, a sheet one. But in many cases, this is a basic object of study. For a quality type, we want the outrageous addition of the two opposites. Using that terminology of points and components, every component contains exactly one point. Contains, well, because you will always have a natural map of my lures going around on a string, which tells you, given the weight, which component is, and so when I write this equality, you know, it's perfect. But I want to call this equality type, and of course the, the very simplest possible example of this, like, when you take the steps, which are acted on by a single rhythm book, you know, acting on a single rhythm book as the property, that's what you get. There are lots of points, there are six points in particular, and that's what this, like you say, the clumps are the same in number, the left kind of relationship, left or even a bit more, I'm trying to say it, I'm calling it quality time, naturally.

30:00 But notice that essentially this whole situation is given by central literature. Quality type is the total equipment that happens to spread out the two androids are nothing but taking the fixed points and taking the components, but they're the same, which is technically important. Did he, did he, yeah, no, here, yeah, here, on this, he is the structure, but it's also an endomorphic. Good, good point. Whenever you have a commuter that is locally active, look at the category of actions that each element of the component also serves as an essential in the work of the study. This is a quality type. Quality type implies basic roles, non-basic roles, on very, very far and huge sums. So my aim is to talk about the other kinds of children, which is neither, but which we can measure with help of quality types.

32:30 Generalized spaces, of course, are... Extremely important and the general program is that we would like to think that we can, given a space in a cohesive topos, extract, we can plot from it topos that are varying just over that object, whatever that means. So these things don't at all disappear, they're just an aspect of the analysis. Confined as a category in the 1950s by Eilenberg and Zilder, the superficial sets have become a standard object of study in algebraic topology. In fact, most algebraic topologists have more or less the geological viewpoint that everything of interest to them can be calculated on the basis of that one. That is done by a complicated source of geometric realization and singular complexity. Although some people say, well, you should use cubicle sets instead. And I say, no, we want semplificial sets, not other cubicle sets. The point is that these are both like several other candidates for the purpose of mathematical objects. And so properties which make them desirable should be written down mathematically, but that's never been done. I have a whole secret list of twelve properties that semplificial sets have that no other topologist does.

35:00 And people use these properties implicitly all the time without making an explicit sense of those debates. Anyway, when Heidelberg and Zilber introduced, they started by the very first line that says, you should be aware that these spaces are not determined by their points. There are a lot of these that have only one point. These combinatorial objects, they function as spaces. You have intensive and extensive variable quantities inside these compotes. You have harsh truth values as in any of those developments. It has its analog there. It has a unique analog there. So I mentioned the 1960 treatment of the B category of all analytic spaces. Since 1967, people have been doing quite a bit of work on C infinity. There is not only a topos approach to Penolso, and of course to algebraic spaces, the category of schemes that was mentioned is embedded into two or three topos as important, say Tau or Ischievich.

37:30 This segment, continuum, precision, and when we develop the theory of electromagnetic fields or continuous bodies, weighted functions, or the principle, these could be seen in any one of these levels. Be sure to understand to what extent such theories are independent of a level, and at the same time understand what is the differences between the levels. We change them. So this is my basic goal is to try and explain what are the general features of space. And again, space in the sense just of the background plenum on which the physical quantities are described. Like that, about the physical quantities. It is intended that the objects can serve as domains for trying to refute some more myths. In category theory in general, the objects are opaque. You can't get inside the objects. She worked without elements, et cetera, et cetera. All this is periodontology. Any category, when you can ask it why, they've got the rest.

40:00 Category three people said that category three must first have the theory itself in the most basic theory. The only point about that, but one has to get used to the fact that, as was already pointed out in the 1950s, the kinds of maps that are used as elements in some narrow sense. It may not be as narrow as typically Kantorian sets, which were designed to be Kantor himself, to be a negation of all cohesion. That philosophy is seriously, by the way, and he's right in any of my comments, is true, but the Septimists have never really pursued Kantor's original suggestion, they've instead pursued to tolerate it. Everyday properties that we give to the Cantorian sets are that every one of the sets has sections, which implies that the logic is Boolean and sub-objects are secondary, but moreover that there aren't any sub-objects involved either. So those two actions added to Coppola's theory is basically Einstein's set theory for all, for natural numbers. For everyday mathematical purposes, this is the difference between non-cohesive, non-variable sets and the other interesting ones, but these are interesting too because they serve as a factor. Well, in particular, if this is true, and of course much more generally than that, but still, if this is true, it means that if you have two maps, x and y, and if they're different, then there exists

42:30 A map from one to x is quite easy to find by looking at, where one is a determinable object A. But again, as Eilenberg and Zilber find out in 1950, this property is immediately lost when you go to the social status. So, on the other hand, it's not really lost because what you typically have is a few objects A that can still serve this separating role. And maybe if you like, there's one among them. But they will not, one of them will not suffice to separate. I found in Volterra, C.T. Davies, a very good terminology, he uses the term element, the element of X to A. From our point of view, that's just nothing else except a map made of X. The only thing is that A belongs to some special class. So they could be points, or they could be curves, or they could be... And I think that's quite consistent with everyday use of the term element, because elements have become identified by the set of points, and I'll take this out of your mind, but instead I use the word figure, so a figure of shape A, NX, just a map, except that we have distinct categories.

45:00 And then we have the subcategory A. Not only do we use the terminology to shape figures, but we have some condition that says that these are adequate to use, or maybe even to draw attention to that, to follow each other. Now, there is a dual terminology. We say that a function is a map with a special co-domain in the function. There are not just these, and again, the finder might not have co-adequacy, co-separation, typically, not necessarily the same thing, maybe another something, which is co-adequacy for, in that category, that is a very complicated space in a simple way, and that's typically what sort of projective space doesn't have functions, I mean, it doesn't have any consequences. There are enough figures that reflect the shape. On the other hand, both of these figures are not that much. Now, not that you have these, but there is a category given an object x in size t. There is a slice category with an object over x in which the morph of the unit is a triangle, and I call this incidence relation. This might be saying, for example, the point belongs to a certain surface, and it might be saying that the path passes through another kind of figure, so they think that both the membership and the inclusion are special cases of relations.

47:30 Both Deficium and Bonnard, by the way, use one notation for both also. So, not only can you get inside an object, but you can get inside an object in a very special way if you have this category, which is called the geometry. And the geometry consists of figure-initial situations, just like specific types of geometry that work. And of course, in order to picture this geometry, the first thing is to be able to picture the object as A. You have a way of doing that. And in principle, the picture of the general action is a complicated combination of, for example, in sequential sets, we can take the, not the point, but the basic arrow, the basic triangle, and so forth, and in general, it's a complicated mixture of objects of all those different shapes, but the way they fit together is specified in categories and effects. On the other hand, they do exactly the same thing for functions, again, as it should be, and again, category A, instead of belonging to, as a word for this, we could say here, depends on, F depends only on G, and you can calculate F knowing G might be something, namely some theta.

50:00 So, actually, the great emphasis is placed on... Belonging to relations of a set theory tradition perhaps obscures the fact that equally important in mathematics is dependency. Does that depend on... So the alphabet kind of splashed out over here. So the algebra functions. The functionality of the system functions and the morphisms of the operations of the algebraic theories. And of course these things are contra-variably factorial. You change space. And you get some of the algebraic functions which preserve the type of structure. And this is a homomorphism in the sense that it preserves all these things. Now, first, these are algebraic theories that arise without any emotional presentation. I just arrived because you have this category. On the other hand, you might like to present these algebraic theories. Basic operational equations like condition multiplication, if those have to be valid, and you can do that, and it often turns out, as a matter of fact, that the whatever degree of co-ethnicity might have been enjoyed by A inside of B, you can even get a lot of it with less. For example, in a continuous category, continuous basis, If you take any of these complex numbers, real numbers and the like, you have all continuous functions that operate in, say, algebra, meaning those are the algebra operating and homomorphism preserves them. But, as you can restrict, let's say, the rest of the polynomials, it turns out that those weaker functional numbers are still different, collaterals, so there's no representation beyond them. In the same way, you can simplify...

52:30 You can simplify the figure shapes. For example, Paul and I have been using paper on the theory application of categories, showing that you can always assume that in the category of figures, there are no endomorphisms. If you're given one, you may have to spread it out a lot, but then you can sense that there are no endomorphisms between the types to serve the same adequacy. This is the basic framework. Now, we come to the idea that when mathematical structures and mathematical objects are structured, There are other structures, well, structures that are in a certain background. Where is the background? Well, we can find it all here, by following Cantor, who was in the great murky world of Reagan, meaning in variable sets or spaces of various kinds, and that great murky world he extracted, he negated it really by saying, let's consider things that are absolutely known to each other, and then... That seemingly futile exercise is used by everybody else for the next century, including as a background for structures also, not direct structures, yes, but also geometrical structures, models of particular kinds of buildings replaced by a number of different things to be performed. To follow that idea here, let's say we want to assume Cartesian codes, that's it, you know, topos.

55:00 This one is hard to watch, on the one hand, because starting off from the category of topology is amazing, which would be a default model, which would be difficult in achieving it, but on the other hand, the functional analysis and geometry itself is always forcing on us. For Ray, which in 1949 finally achieved a category, a lecture in Princeton, unfortunately not published, for Ray was invented the so-called case-based compact, case of compact, compacted space, which was building on the original idea of topological spaces, but negating it, because the original idea of topological spaces is one that gives no function, actually. Function was valid in the Sapiensian space when the primary structure is the contrarian function of the monodocus. He turned it around and made a huge, huge category of figure shapes, namely all compact spaces. But with that he succeeded to achieve the Cartesian probability. And this is reported in the paper by David Gale. For some time it was confused. People thought K spaces must be Kelly spaces because Kelly... He gave a beautiful exposition of this in his book, without bothering to mention the two great works, the two great works of Bonhoeffer. It's actually a great work. So basically in the same year, we had some special sets. We also had, for the first time, a real case basis.

57:30 But the necessity for this was back already to Bernoulli, Volterra, Audubon. The morphisms from one space to another themselves constitute a space. What does it mean they constitute a space? Well, it means they can serve as a domain of variation for both intensive and extensive quantity. And it means that they have their own cohesion, the same ilk the other spaces around it have. So in order that the, if you like, the function holds on it. From this you see instantly that the analysis of the map space is a function. The problem is what is functional and what is unfunctional, right? But at least that tells us exactly what kind of conditions are satisfied, namely all of these should transform into other good things, because they are. All right, so assuming this, assuming this, I want to define part of particle. S is the subcategory of all those objects which are perceived by all the objects in A as discrete.

1:00:00 This relation, the Cartesian subcategory, between two objects, is the canonical math of the diffusion constant. It's an invertible. There is a Galois relation that generates a Galois connection. So, there is the S, which is already Galois, but it's by construction. On the other hand, we can go back and, if we like, we can find the bigger A. All the A's are the same thing with respect to all the S's, because then we have the idea of how much pair there is in respect to this. Both basic concepts should be that if you have a balanced pair, then every object is determined by its figures considered as something we can cast. What does that mean? This is a subcategory, but I want to assume it's co-reflective. And that means that to every space, there is a maximal, I should write that, I'll just call it points. To every space, the points of every space. So this is in modern terms what Cantor did in 1850. Starting centuries with Cantor, all the different, in the 5th century, all the sets of stars that we have, they all could be used at the same level. So he's totally abstract. He says he got the idea from Steiner. Oh, I do know a lot about this. Steiner.

1:02:30 Steiner. Oh, yes, yeah, yeah.