Euler, Maxwell, Grothendieck & mathematical representation of cohesion of space

0:00 I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. I think he's one of the speakers tomorrow at the Griswold Hall. Okay, I completely moved to the bicycle. It's good for your thighs. It's the only way in. It's good for your legs. I mean, you know, look at me. I mean, you're going to stay very fit on all this. You're right. That's why I asked you last time. I said the answer would be no. Okay. Right. Do you want to... You'd like to share? I imagine you'd like to share. Okay. Just come in when you're ready, Bill. Okay. Let's get started. Okay. Mr. and Mrs. Marc and André have entrusted me with the honor of presenting a brief introduction to our speaker today, Bill O'Vear. I will give my presentation today in English and, with your permission, I propose to continue my brief introduction myself in English because I think that my excessive French pronunciation would be a grave proof of your amity and your understanding.

2:30 Bill Lovier was born in Indiana in 1937 and On his graduation from high school, he spent something like a year in 1955 to 1956 pursuing his first great interest, which was in engineering, an interest which has never left him, and by working on the cyclotron, which at that time was located, I think I'm right in saying in Bloomington? Bloomington, Indiana. Put forward by Maxwell concerning the need to consider many levels of precision in both physics and engineering problems, depending on the need. And those principles have had a considerable influence on his later outlook and work as a mathematician. He began his studies in the University of Chicago in 1960 under Clifford Truesdell. Indiana. I beg your pardon, Bill. I'm sorry. In Indiana under Clifford Truesdell. And worked on continuum physics, and indeed his interest in that subject, which has also never left him, was one of the major sources for his later work on the topos of smooth spaces, and indeed the whole field of modern synthetic differential geometry may be said to have been inaugurated by the lectures that he gave in Chicago in 1967 on that subject. Those lectures were one of several crucial steps Which paved the way to his great work on topos theory. In the meantime, he had moved from physics into mathematics and had completed his PhD at Chicago under Samuel Eilenberg on functorial semantics for algebraic theories, a PhD of the most profound consequences for the future shape of categorical logic, which contained many... Very powerful new ideas, not least the central importance of the category of categories for mathematics. In the following year, his paper on the elementary theory of the category of sets was published

5:00 in the Proceedings of the National Academy of Sciences, and the following year he gave his paper on the category of categories in 1965 at the La Jolla meeting. His axiomatic formulation of topos theory, which was completed in the period of 1969 to 1970 in collaboration with the topologist Myles Tierney. Grotendieck said that the notion of topos was la plus vaste qui j'ai conçu, and Bill has immeasurably extended and deepened our understanding of that notion. And the central role that it plays in bringing together geometric, algebraic, and logical notions. If I was to enumerate, still less attempt to discuss, the fertility and the depth and the range and the importance of his work over the subsequent 40 years, I'm afraid I would take up far too much of the time which you want to spend listening to him. But I will just mention two broad themes. The first, that he has consistently urged the recognition of the deep geometrical sources of logical constructions because the road to required mathematical concepts of an objective nature and of correct generality largely lies through geometry, and secondly, that he has urged the recognition of the central importance for a correct notion of foundation of mathematics That it should lead to a large extent through considerations of pedagogy and the recognition of the original meaning of mathematics as that which can be taught, it's designed to be taught. The subject he's going to talk to us about today is one which has occupied a good deal of his thoughts in recent years. The search for the correct concepts to characterize the cohesion underlying our notion of space. Over to you Bill. I understand this is a physics meeting? Well, uh... I think it's the first time I've spoken, actually, in spite of having started, as Mike said, as a cyclotron technician in my career.

7:30 So, I think what I'm saying is of interest for foundations who have some ideas about it. I will not draw quantities like metrics and affine connections and energies and so forth. What should be the nature of the environment in which these quantities are defined, since they're variable quantities, and since we objectively visualize variable quantities as varying over a certain domain space, what is the nature of these spaces over which physical, geometrical quantities could vary? I say immediately that this variation of quantity can be either of intensive or extensive nature. Intensive has been extensive, has been played a role in philosophy, has been neglected as a very much neglected through the uses of Hilbert space and so forth and so on. Already Grassmann complained about how The ubiquitous extensively variable quantities by reinterpreting them as intensive quantities in one way or another, so their famous slogan, for example, of distributions as, quote, generalized functions, unquote, which had an extremely pernicious effect that despite the great virtue of distributions per se, they are extensive quantities, basically. So there's a big... A very pernicious effect on the philosophical conception by pretending that they are, in some sense, generalized functions.

10:00 I'll come back to that, perhaps, if I have time, with the nature of variation of quantity, rather. So the reference to Maxwell, of course, he was talking about many different things when he made this observation clearly experimental in a theoretical context, namely the need for many levels. There are many levels of precision which can be cranked up or screwed up as needed. It sounds like, as I said, it sounds like an engineer's conception, but really I think it does spring from the very nature of knowledge, because our knowledge, perception and theorizing and so forth always involve the sharply dialectical feature of neglecting. Those things that are neglectable and concentrating on the few features which are irrelevant to understanding a certain phenomenon. It seems that so-called fundamental physics ignores this principle and pretends that somehow or another they're going to find the one final level. There is no final level. One can always deepen, at least philosophically. A better stance, I think, to presume that one could always deepen any theory. So even theories of numerical analysis, let's put it on that level, well, in other words, there is a, my son, who is a physics graduate student, a teacher, when he was very young, he made a computer program to describe the motion of a satellite around the Earth, but when he viewed on the screen the satellite's motion, he found that it always fell into the Earth instead of continuing.

12:30 And, of course, this program in itself has nothing to do with satellites, with motion, it's just a subjective reflection into electronics or whatever, because we give that interpretation. But nonetheless, I said, well, you forgot to program in the conservation of energy. In other words, even though the thing is totally numerical, it has nothing to do with physics per se. One can still insist that the computation preserves energy, and then of course the satellite doesn't fall into the picture. So the point about that is that even that is a certain level of theorizing. And again, it can be done in a serious, useful way or not. Everything has been neglected except a few features. And so, therefore, talking about, I could say, the plenum or the ether. People say ether doesn't exist anymore. On the other hand, they constantly use it. They just say it has much different properties than were imagined 150 years ago. So that there's some kind of plenum or ether, or maybe the ether is a kind of matter that moves in the plenum, whatever, the nature of... This planet, this space, this ether, at least should be cohesive. It has to have some kind of cohesive, otherwise, of course, also that has been denied recently in the Scientific American, where we read that space and time really are discrete without any evidence. What we observe, of course, in physics is that in the continuous world there are all sorts of very interesting discrete manifestations or aspects, but it's by no means the principle of quantum mechanics.

15:00 No coherent mathematical theory of that has ever been offered, as far as I know. Perhaps some of you know better that there are coherent theories of discrete space and time. And if there are, of course, it could be a valid level of precision. But I think that to say that it's the ultimate level is completely unjustified. So with that preface, what I'm saying is that I want to describe the kind of categories of space Which might arise at different levels and fortunately we have tools for doing that because of work done 50 years ago on the one hand by Daniel Kahn who crystallized the notion of adjoint functor out of his lifelong study which continues even now of that level of cohesion known as combinatorial topology or combinatorial homotopy theory exactly. So from that context alone, he was able to pull out this notion of adjoint functor, which turned out to apply to everything. Maybe this could be compared, actually, I just thought of this, could be compared with the fact that within the very narrow world of constructivist philosophy of mathematics, the idea that concepts are generated by pure thought, you go from one stage to the next and so forth. From that idea of moving thought came hiding algebra, which itself has nothing at all to do with that context, but it turned out to be descriptive of the nature of truth in all kinds of situations, which vary in a more physically suggestive way, geometrical way, and not just in a subjective idealist way. Very quickly, by many people, Rodendeek notably and his friends and myself and many, many other people, it was really a flowering of category theory made possible by Adjoin Funters and by that other great event of 50 years ago, namely Rodendeek's Tohoku, learned about in reading Go Demar, which was category theory.

17:30 Because, although I won't speak directly today about the AV5 abelian categories of Grotendieck, they do come out immediately from the setting that I propose, basically as the linear objects in a non-linear setting. So I'm not going to talk in detail about abelian categories or even pseudo-abelian categories or anything like that, but rather... The specific kind of non-linear background in which it seems that in most cases such things live. I think one could say, I don't have any direct quotation about this, but it seems that Grotendieck himself in the space of two or three years realized this, that the actual home of these abelian objects was non-linear, and at least the fundamental investigation proceeded about the non-linear. I should say that one has somehow the idea that non-linear is more complicated than linear, but actually you see conceptually linear is an additional structure imposed in a non-linear. I think tomorrow I will talk about some other Kinds of cohesion having to do with the monological vector spaces and combinatorial topology and so on, but of course the kind of topos of cohesion that we have studied the most since 1967 have been of a more of a smooth nature, smoother. The worlds of algebraic geometry, analytic geometry, and smooth geometry, G-infinity geometry, are the typical examples of which I want to give a common treatment.

20:00 There are many graduations between those. For instance, this is something that has never been investigated, although I proposed it. Just as one talks about field extensions and all that for solving polynomial equations, In the same sort of way, there is a whole range between algebraic geometry and smooth geometry, say, or even algebraic geometry and analytic geometry, which you adjoin bump functions. I mean, it's clearly an important example. The big difference between analytic and smooth is that in smooth you have bump functions, in analytic you don't. You can adjoin bump functions to the algebraic theory of rings and obtain an algebraic theory whose corresponding large topos should again be a topos of the same kind that I'm describing, but perhaps of interest from that point of view of finding the minimum context in which certain differential equations have a solution. The problem of finding environments where differential equations have solutions has been studied a lot. Under the name of differential algebra, but one should point out that it's in some sense inadequate because it talks about derivations, things that satisfy Leibniz's rule and so on, but differentiation equally, at least equally, involves the chain rule, so the fact that you have composition And differentiation has certain properties with respect to composition, but surely just as important, you know, in extending the realm of solutions of differential equations, as does mere multiplication of products, so I think rather than just plain rings, the real environment for differential algebra should be certain kinds of small categories, like the theory of rings, only bigger. So, anyway, so there's a range, and those are three well-known kinds of examples. Of course, there are many that have been studied for many years, precisely different versions of algebraic geometry and so on and so forth, and analytic geometry, but generically there are three types of well-known examples.

22:30 It may be necessary still to mention and underline the fact that these kind of toposes that I'm talking about are quite a distinct kind from the sheaves on a topological space, topos of actions of a group, or the topos, Petit et al. topos of a scheme, for example, all of which have the character individually of generalized space. Now a lot of the attempts to popularize with misleading disinformation about the situation that suggested that toposes were invented in order to generalize spaces in that way. But this is distinctly wrong if we look already at Brodendieck's presentations in the Cartan seminar of 1960 about analytic spaces, category of analytic spaces. He treats there as a topos, which is definitely not of the generalized space kind. It's rather the category of all analytic spaces, generically speaking. I don't think he—he didn't have the word topos, I guess, in 1960, but analogous toposes in algebraic geometry, studied by many people, were in fact called topos. The time seven years had passed and it came around to my attempting to apply Gabriel's version of Brodendieck's theory into continuum mechanics, of course I knew that it should be a topos, so the word was used then, but one has to... I've been trying to make explicit for a long time the distinction between these two classes, two extreme classes of toposes, which I...

25:00 You know, philosophically described roughly as toposes of cohesion and toposes of pure cohesion and pure variation, with the general case being somehow a mixture of the two, a combination of them. The pure variation, the original example was sheaves of sets on a topological space. There the idea is, well, it's a variable discrete set varying over a topological space as a domain. But then, something even simpler is G-sets, where G is a group. Again, discrete sets varying, but in a different way, varying along a group. Now, actually, in algebraic topology, as we know it today, it rose in the 30s from the realization that these two examples really are part of the same, because discoveries of Hopf and Steenrod and other people. amount to the fact that a good space maps to its fundamental group, i.e. if you take sheaves on the group, actions of the group and sheaves on the space, there's a morphism of toposes there, and if you pull back the point, you get the universal covering space, so the dialectic between the covering spaces and quotients of the fundamental group arises precisely from a diagram in one category. One category that includes both groups. Each of these kinds of variable sets certainly belong to the same category. Now, Groszendieck wanted to try to unite these two and successfully united these two by his concept of étendus. Do I pronounce that correctly? The étendus were defined as toposes which are locally topological spaces. There's a well-defined sense in which a topos can be locally something. And so there's a basic example is that a group is locally a topological space, namely a point. It's locally a point. So when people talk about groups acting on points, on a point, it's not really a joke because it makes sense in that sort of a context.

27:30 But really what they have to do with these sites, which are consistent monomorphisms, not necessarily invertible, as I saw pretty quickly in trying to study, for example, the discrete dynamical systems, which are actions of the additive monoid of natural numbers, which is a category because of cancellation of addition, it's a category with monomorphisms, and that topos is locally. A certain topological space or an even simpler example is the graph, category graphs. I call them irreflexive graphs in order to emphasize that they're not reflexive, completely different sort of thing, namely the idea of a graph or quiver really as a set of arrows and a set of dots and a source and target operation. If one considers that category, it's a topos. But it's a topos of the pure of the pure variation kind because it's in fact it's an etan deal in fact it's locally isomorphic to the topological space that has three points two of which are open which is you see this is the sheaves on this space these on this space are really just diagrams of this sort because you have global sections over each of those two opens you have So you have sections and you have restrictions. So diagrams of this sort, the same thing, and so this maps to the sets, S for sets, and that way, because you localize this at the arrow itself and you get this.

30:00 If you think of, of course, being locally something, it often means it's a quotient, and indeed, an etandu is a quotient of that space, which it's locally isomorphic to. The graphs are obtained from this space and then amalgamated these two points, but without amalgamating the arrows. This thing, of course, also comes up all the time in the following way, this equation of the real line where you have a non-negative and non-positive. There's two open sets so you can collapse the line into three points. But can you obtain the same thing without this topological structure, if you make pre-sheaths? This is a category here, which pre-sheaths on that category. And that category, there's no compositions in it actually, so of course it consists of all of them. And also, by the way, this is of course very important in physics to know whether a certain quantity is greater than or less than something, or equal to. So abstracting the reals in that way, you arrive at this space and taking a quotient space of that, you get this topos which is no longer a classical space, but is locally a local space. And by the way, the discrete dynamical systems also fit into this very nicely because if you interpret a graph, special graphs, where one of these maps is an isomorphism, but you come together to see identity.

32:30 What exactly is the role of that viewpoint, two points at opens? It's a three-point topological space. You view it as a quotient of the line, if you wish, where this point is preserved. What's the role of it? Well, the fact is that a sheaf on this three-point space, a sheaf of sets, involves a set of global sections, whatever that is. A set of sections on one open and a set of sections on the other and two restriction maps. The whole has to be restricted to each. Now, if you, actually the space has a total of five open sets, but if you impose the usual covering condition on that, you find that if you only look at these basic open sets, then the sheaf condition is nothing. So any diagram of that sort is a sheaf on the space. This is sort of an extended Sierpinski space. It has two open sets, two basic open sets instead of one. I mean, if you want to present it as a hiding algebra, then the two generators are disjoint. I mean, that's the axiom. Actually, the etendue, I think the etendue, again, has never been pursued, but it has a lot to do with ergodic theory, because we look at it in various sets in measure-preserving transformations. If you want to look at the orbit space of the measure-preserving transformation, well, if you think of it as a classical space, it disappears, but if you think of it as topos, it's very non-trivial. So there's analysis in any topos, as I want to emphasize, so particularly analysis in the etan-du that results as the quotient of a measurable action, probably, you know, the analysis of that topos probably contains most of what's known. So as I said, it turned out that the topos, which are locally spaces, have a modulo existence of points.

35:00 We came to realize, of course, lots of spaces don't have points. That's more or less a triviality. So if we ignore the question of whether points exist or not, a rotative topo is locally a locale. This means roughly locally topological space, if and only if you have a site which consists entirely of monomorphisms. It's easy to see that not only the pre-sheath topos, but any sheath topos based on a site consisting of monomorphisms is an eton due, because if you just take the destroyed sum of all those monos, it becomes a poset, you see. The basic idea of a locale is that you have a site which is actually a poset. The converse, I conjectured the converse a long time ago, but it took a while for Kalk and Murdoch to prove it. The point is, something I, it's natural to forget, I always forget it, the site need not be sub-canonical. So, any, any, uh, uh, Etan Du has a site consisting of monomorphisms, maybe not a site which is fully included in the topos itself. It sheaves, sheaves on, it's a sub-topos out. One which is based on way more significant generalization, I mean, groups are pretty significant, but qua generalized space, it was the Petit et al. toposes which were more exciting, more interesting, more useful, at least for people who want to prove a vague injection.

37:30 So the Petit et al. toposes are not etendus, but they do belong to another class which was actually studied later by Peter Johnstone. And again, he approached it, the conceptual approach was, well, there ought to be an adequate number of objects which are separable. To say that an object is separable means that, of course, it has a diagonal map, but that there should also exist an actual complement sub-object of the square, which is detachable in the sense that if you take the sum as an inclusion map, and if you take the coproduct, That will first induce a map toward A cross A, and you require that that be an isomorphism. Most of the time, in a cohesive category, of course this is not at all typical. If it were always true, you wouldn't be very, very cohesive. There is an otopos, a complement, a hiding complement, but the union of the two is not the whole. It falls short of being a special kind of object, a separable one. By the way, this is also called decidable by logicians, but rather that's the subjective version of it. You can decide if two things are equal, but objectively it was separable. Polynomials that don't have repeated roots, the roots are separable. If you do have repeated roots, all right, so the idea of having an adequacy of these, let's say within a given category that has sums and products, we can isolate the...

40:00 These separable objects and then the whole category, let's say this is a subcategory, a larger category of objects that are more, in general, more cohesive, then this will give rise to an adjoint pair studied by Kahn 50 years ago and expounded very beautifully in Gabriel and Ziesmann's introduction. A small category inside a co-complete category automatically gives rise to an adjoint pair, like this, but like this, the requirement of adequacy is that this should be full and faithful. A small category inside a co-complete category gives rise to an adjoint pair, connecting the co-complete category with the topos of pre-sheathes on the small category. It's adequate if and only if this is full and faithful. The general notion of cohesion that X gives you is not at all entirely separable, at least as generated as a site by objects. So there are a lot of toposes of this sort, which I still consider as toposes of pure variation. You do have it in which the subcategory of separable objects is equivalent to those that have another site. Objects are epic instead of monic. Now you have to be careful of course the e is included in the topos.

42:30 It doesn't mean that the maps and e are epic in the topos, but it means if you look at the small category in itself, epimorphicity is a property quantifiable overall. Objects. So if you quantify only over the E itself, then you have... This is a notion of generalized space, which does include the Petit Etal. This is essentially the sort of unique pathlifting idea. You have to take the connected Etal objects over your base as the site. So between two connected Etal things over the site, which means that if you have two maps between connected Etals... Which agree on some part at all, some non-zero part, then they're equal. That's exactly what epimorphicity means. We're in the category of things over a base space and we have two of these connected etal objects. Everything is a sum of connected and so these do constitute a site. If there exists at all some object, another etal object, so that these two things agree when restricted to that one, Then they are equal. So just the existence of something, thus the epimorphicity property, and of course, the Petit-État case, everything is a quotient of a sum of those, it's an adequate...

45:00 The idea of Petit-État arose, as everybody knows, because of the lack of the implicit function theorem, so one still wanted to talk about variable discrete sets over some kind of domain, and it turned out to be a good model for that. I used to also say topology, but really to think of strictly as variable sensors being a very special class, which I haven't yet said, but it should unite both of these. Let me write down the condition here. We can consider a pair FG as an equation, an equation true. True would mean F and G coincide. That the equation holds f equals g if something else. And to say that a is epic is to say that if f a equals g a, then f equals g. In other words, I can cancel an epimorphism. But monomorphism is also a cancellation. It's the opposite kind. B is monic. It says that if bf equals bg, so these are points of the domain of b. Then the points themselves will equal f equals g. Well, you see there's a common generalization of the notion of epic and monic about a condition on a category now. We've talked about the condition that all maps in a category are epic, the adequacy of the separable, or that all maps are monic, that was the adequacy in the case of A times 2.

47:30 That's what that should imply. So that's a generalization. Again, as a condition that all maps are, it generalizes because, in the case, if it happens that all maps are epic, then one of these hypotheses is already enough. Or if it happens that all maps are epic, or harmonic, the other hypothesis is by itself sufficient. This is a reasonable, this is called a Horn sentence. Sorry about the logic. Uzi recently condemned all the recent work on topos theory as mere logic, but we just found that we have to use a little bit of logic to understand what's going on geometrically. If you think of this as a property of a category, all maps A, B, F, G in the category, you have this. C to the B, again, has the same property, limits, sums. It's a reflective subcategory. Any category could be forced to be like that, and that forcing preserves finite products, what I just said, and so forth. So it's a nice, well, the reason that I mentioned the reflection of cat, cat sub zero, that there is a left adjoint to the inclusion, and that left adjoint preserves products. Ultimately, I haven't told you anything yet about what the other kind of topos, but we would like to be able to mimic this idea that if a space, if a generalized space is really an object in a suitable topos, at least it should give rise to a corresponding petitopos.

50:00 What would it mean to talk about variable discrete sets over an arbitrary space? So somehow the general topos ought to... General TOE plus x, if we localize it at an object, we should be able to collapse that to something called she's on x. This is still a research project, although I've written several attempts. So the idea would be that if I have a site for x, therefore a trivial x, I have one also for objects over x. So if I apply this forcing, I compress it down to pure variations. It sort of gets pressed down into the pure variation that it's capable of. So I stopped. For a long time I talked about Gros and Petit, which is very bad. Because it's completely misleading. It's not that we go from Petit to Gros, rather the opposite way. I don't know if this helps. Maybe that's helpful. But Colin McLarty said that my observation is, well, if you have some definition of Gros, Then grow on one point is a branch of geometry and then you can then you can then you can specialize from there maybe it's better to drop the word and it's anyway qualitative distinction rather than large rather than size this is why i i mentioned the nature of this condition it's reflexive with just apply this not to to to the sites to the site for for this we might get a good notion of the relationship between A general some notion of cohesion and variation of discrete objects over in a particular instance x to emphasize the logic here. This means if there exists an a and a b, you can always put an existential quantifier to the left due to the left adjoining this existential substitution so it becomes a property of the two f and g and then we require that to be true for all.

52:30 So we get a class of sites now. Think of these as sites. You get a class of sites which includes both the etan-du and the separably adequate, whatever. He called these actually QD tokens. Equation-decidable. Yeah, quotient. Every object is a quotient because if you take a sum of these separable things, it's clearly still separable. So every object is a quotient of decidable or separable. It's a little bit confusing because they're two quite different notions than the locals. But then looking at this, you see, okay, there's actually a still broader class. So the game for the moment becomes let's generalize this idea of this variable as much as possible without leaving that concept, without encroaching on the other end which is going to be the cohesive things. There's an even much simpler condition here that's implied by this, not equivalent at all. It's implied by this. And that is, there are no idempotents. Precisely, the construction of the vibration of points doesn't do anything. Vibration of points, just to put it in general, vibration of points over such a category is the same as the category itself. Groups don't have any idempotents, neither do partial order sets.

55:00 Well, in fact, partial order sets don't even have any endomorphisms at all, so in particular, no. But groups, we want to allow endomorphisms because of groups. And, in fact, monoids that are consisting of monos or birth epis or even monoids in more general, there must be monoids that satisfy this without being cancelable, to find an example. Monoids that satisfy this weaker cancellation law is two-sided cancellation law without satisfying either one-sided law. Anyway, so all those satisfy this very, very nice, simple condition. A reflection, still a reflection preserves products. I think in my first paper on this that was published in Bogota, Colombia, I talked about a very, very special class of categories in which, yeah, okay, so I mean, sort of half the story about what a sheaf is, is that the stocks are discrete. So it consists of variable discrete sets. However, the sheath condition is moreover something about how, paradoxically enough, these discrete fibers move continuously. Well, if we ignore the last one, it's hard to understand anyway, and just talk about discrete fibers, that's what's called unramified algebraic geometry. Unramified, yes, right. So more general than, what's the slogan? Etal means unramified and... Anyway, and then just think about pre-sheaves. So if you look at pre-sheaves on a certain category and then objects over, well, same thing, you want to single out the things that have discrete fibers, well, there's one of these unique diagonalization type properties that you can write down, which turns out to mean that you've forced every map in a given category to be hidden. So this idea of forcing every category to be idempotent, every map to be idempotent, in fact, let me start over, that's all wrong, forcing every idempotent to be the identity, got carried away with buzzwords, forcing every idempotent to be the identity, and interestingly enough, the context which I was talking about there, which has to do with generalizing graph theory, but in a very restrictive way.

57:30 Maintaining this very strict combinatorial idea about curvature zero, the site that you start with consists entirely of idempotence, well it's a monoid with idempotence, that's all idempotence, a band, but now the thing is if you take a representation and slice by it you get a little category which is the site for that, which no longer has the property that all maps are idempotence, so you can... You can kill off those idempotents again. So you start with things that consist of nothing but idempotents, and then you kill off the idempotents and you have something left, but it's because you localized in between. And this is a very good way of describing, in fact, things like bipartite graphs and B-partite graphs for various other graphs, B and so forth and so on. So this is not devoid of interest, in any case for combinatorics. I, myself, am getting more and more into some very elementary category theory that doesn't seem to have been. I think we all find that the simplest sort of algebraic calculations with composition have not yet been done. So we need to be very good at this to do that. So he's coming up.

1:00:00 So here's a very elementary thing. The vast generalization of this already, which is going to go beyond then my idea of screen sets, going beyond that. But just a category that satisfies that, if we use it as a site, take any topos, topos happen to have a site like that, they'll certainly have the following property, that there are no central item posts, topos themselves. Now, the center of any category is what? The center of any category is, you look at the identity function on it, and then the natural endomorphisms of the identity. It's automatically going to be a commutative monoid. And sometimes, often, there is no monoid. By the way, when I say no, I think Wilbacke did this. We use the idea that to deny a statement is to say it implies a trivial statement. Now the trivial statement might be false, but more typically, in ordinary discourse, it's a trivial case. So if I say there's no hidden potents, it means every hidden potent is one. Because there's a well-defined trivial case, and I say implying that is like... So, the center of a topos is the natural endomorphisms of the identity. So, it means that it's a huge thing. To every object, there is assigned an endomorphism, and that commutes with every morphism, in the sense that the one I have assigned here, it's not surprising that it should be.

1:02:30 I claim that if I have just a site, a site with that property, then the whole topos will have this property. There might be lots of... There will always be lots of idempotents in a topo, in a topo, on individual objects that have the graph of any map, for example. The map from A to B, it's graph splitting of the projection and so there's a non-trivial idempotent there which is squashing onto the graph. Unless every object is a sub-object of one, there are going to be non-trivial idempotents on objects. But the whole category is central idempotent. There's an adequate part that has no idempotent, and the whole thing has, because, you see, if you had an idempotent that was central, then it would commute with every element to find over, is there enough elements or figures, was the term used by Voltaire. People are confused by elements because of set theory, so I call them figures, since they understand geometrical figures. So if there are enough figures... These shapes belong to, these shapes have no idempotence at all, then in the whole category there's no central idempotence. They would have to commune with all the figures and that would mean it's, well, just faithfulness, like you don't even need any of this. Figures of these types are faithful even, which is much less than that, but even that's enough because if you compare the central idempotence with the identity, by means of a suitable element.

1:05:00 There are categories which do have central idempotence, and these are very interesting. Again, the obvious example is the category of points of a given topos, the calibration of points of a given topos. For example, just the sets, equipped with an endomorphism but an idempotent one. Structure is just one thing anyway, and of course that thing is central. That could be any topos or any category. This is the functor that takes either the fixed points or the orbit space as you like, it's the same thing. I give another shocking name to this. Categories, for example toposes, and an inclusion functor with a left and right adjoint which are however equal. It's a simple example where E is more structured than E0 because it has a bunch of hidden bones, a commuting hidden bone acting on this kind of a thing. I call it quality. More exactly, I want to call this a quality type, and a quality is a function that goes into such a thing as a co-domain for a quality, or correct, to speak of it that way, to pick your quality as a thing.

1:07:30 Just kind of like you, if you paint red by red, it's still red. Needs more thought. So this is an example, then, of a quality type. Maybe I should give an example. We have quantities as well as qualities, mind you, which I just didn't get to at the end. For quality, the word type usually means a codomain for something. So a type of quantity, that's why I'm tentatively calling this quality. In inclusion, the full and faithful functions of the left and right adjoints are equal. Of course that means that there's an infinite string of adjoints all equal. The ones going up are equal and the ones going up are not. So again, the reflexive graphs were a good example of an etendue, and that's also true of the discrete dynamical systems. The operations of an idempotent are a good example of a quality type. What's a good example of the kind of thing that I really am planning to talk about, which is cohesive cohesion? Well, the best example is, I think, truncated simplicial sets, or actually simplicial sets. For the first example, this of course is a natural nature, a very odd phenomenon that people who work with simplicial sets know about topos, but they never use the fact that this one's a topo.

1:10:00 As topos, it has remarkable special properties, some of which they are implicitly using, but they never write down what these properties are. There are all sorts of ideological arguments. Cubical sets are better. No, simplicial sets are better. Since nobody's ever pointed out the actual objective categorical properties that distinguish these so-called lists, debates remain on the lower level. It's special such as an example of a cohesive, cohesion of a combinatorial nature. But an even similar example is the truncated completion sets, which I call also reflexive graphs. So this is totally opposite in nature to the non-reflexive. So it means that, you know, you have... But moreover, at each point, you have a preferred loop. You may have lots of loops, but there's a preferred one, preferred in that it's preserved by the morphisms, the natural transformations in the appreciative category, preserve the preferred... Maybe it's better to start with the abstract definition. See, what I basically want to study is contrast between more cohesion and less cohesion. Which was basically Kantor's strategy which started in set theory was he looked at all the cohesive Wengen in mathematics and he said I can take an extreme case, you got the idea from Steiner, you take the extreme case where there's no cohesion at all.

1:12:30 Of course set theorists have ignored that, they've studied the cumulative hierarchy instead, but mathematics, the practice of mathematics in the 20th century. We've pretty fully internalized this idea of the totally cohesion-free sets, used them by double negation as models for, as background for, all kinds of structures, including structures that are inclined to model particular kinds of cohesion, topological spaces, differentiable manifolds and so forth that are always described as being over-structuralist sets. And, of course, when I speak about the Norwegian toposes, it's that sort of thing, except more focused. So, really, the thing is to look at contrasts. This is what really distinguishes the Rødvig toposes from the Laver-Tierney toposes. The Rødvig topos is really a pair of Laver-Tierney toposes, a property that the lower one is what they call U, close in abstract sense. There are both categories that have function spaces and sub-object classifier, but this one has more. It has split epis and it has only two sub-objects in one. It's a model of set theory. So the Rotendieck toposes are just that sort of thing, additionally structured by a pair. But then of course we should always remember Rotendieck's other precisely mathematical but philosophical discovery, namely we should always look at the relative case as well. So, after all, it's not only the case where S is modeling Cantor's idea, but even a more general topos could also be the base that is contrasted with another one as being itself less cohesive.

1:15:00 This is the solution to the problem of points in algebraic geometry because if the base field is not algebraically closed, then the base topos should just be bar sheaves on fields alone, fields only. Then everything is good. Underlying preserves, for example, the algebraic group is still a group in that topos, even though it's not when you go all the way down to sets. Because the way of going to sets from sheaves on fields is to take the direct limit of the points. The direct limit is not a filtered one, so it destroys everything. Actually, the petite topos is not such a good idea here. Topology is misleading if you don't always remember that it carries its own sheaf of local rings, which compensates, sort of recovers, and in fact, in this general process of going from a big cohesive topos to the particular topos of variable sets on an object, you find that this smaller topos picks up structure, sort of a reflection of the fact that it was born inside this big topos and you plucked it out. So it's no longer merely a case of course of that kind, but it has an internal structure like the chief of local. So anyway, it's useful to relativize even in a familiar subject like grid geometry. Not always use just the abstract sets. And so my axiomatic theory refers to such a contrast. And of course, we call these functors pi zero and points. There are several ways of measuring every cohesive space, namely look at its Cantorian points, or look at its set of components, and there is a comparison between these two.

1:17:30 Pretty space X and E is a canonical map because of the agiliness. It says every point is in a certain component. I refer to pi zero as the components, but of course I mean these are sort of indices for the components. Okay, so anyway, by a quality, I mean a functor, quality of space, I mean a functor that goes to codomain D, which is over the same S, which has also two adjoints, except this is a quality term. So these are equal. It has an analog with pi zero and points, but they're equal. Q is compatible with one or the other. So if Q is compatible with pi zero, I call Q an extensive quality. With the same D or another one, we can also consider intensive quality, which is again a functor to a D, and where D is a quality type also, but where it's compatible with points instead of with pi zero. This idea of something over S with two adjoints is not the same.

1:20:00 You could call it also an abstract quality, abstract quality type or abstract quality, but a particular quality is a functor to that from the situation where you have a genuine opposition, possibly, probably, between the two sides, but which you have compatibility in one sense or the other. So you have two kinds of, also for any possible deed. Among these qualities were the toposes I wanted to study. There are actually canonical examples. There's a canonical extensive quality and a canonical intensive quality. Probably these two are adjoints, but I haven't been able to find exactly the universe inside which is the universal property. So, in other words, given E and S, then among all the categories that are in some sense on S, that's the problem in what sense, exactly. But among all those, there are those which are adjoint string collapses, like that. And there are the functions which are compatible with pi zero. Well, somehow universal among those, there is the canonical extensive quality. And that must be what? That's got to be homotopy type. Homotopy type is such a thing. The two functions agree because when you make the homotopy category, what were components become points. Maps from one are just components. That is like the classical case and the finite cases, although not in the combinatorial, which gave rise to a lot of the complication. With pi-zero, of course, you can recover the components of a space, of course, from its homotopy type. Now, on the other hand, among the intensive qualities, again, I'm morally sure what the...

1:22:30 I mean, once you have one, you have lots of others that are composing with other functions, right? So it's not that there's only one. There can be a preferred law. So the preferred intensive quality is just, well, you look at the part where this is true. This map, if I say part, of course that's a subcategory, but it turns out that it, under the axioms that I have, turns out that it is, that this is a central geometric morphism. This is a topos again, and it's still over s. So that's what you started with, but clearly, by the very construction, the two things that are supposed to be equal in equality have become equal. That's a universal way of making them equal. Yeah, that's right. In order to get this well-behaved, I assume what I call the Nullstellensatz, which is that this canonical map for all x is epimorphic. The canonical map is there just because I have it fully included in two adjoins. I can require that it be epic. Intuitively, that says every component contains at least one point. Of course, these may be defined over coverings, blah, blah, blah, because it's not said to be a split epimorphism, it's an epimorphism. So under the assumption of this weak Null-Stone box, then the subcategory B, where the map is an isomorphism, it's not a subtopos, it's hardly ever a subtopos, it's a quotient topos, but an essential quotient topos.

1:25:00 And, of course, it's equality. So this essential morphism is equality, Q, which I call the canonical intensive equality, or any topos, any cohesive topos, which includes the notion. I haven't told you the infinitesimal setting for all this. I mean, this is all arising in analytic and differential and algebraic geometry in a certain way, which is born from infinitesimals. I won't go into all the detail, but let me just, let me just, let's consider the spectrum of the dual numbers. That space is sort of the same, sort of the same thing in all the examples, even though the examples are vastly different. Analytic and smooth are very different, but the particular space is basically the same sort of description. K could be finite or real or complex. Basically the same sort of thing. The Cantorian negation of cohesion that's relevant here proceeds in that way that we define S, all the spaces in our topos of cohesion which have the property that they have no tangent vector. Again, every tangent vector is trivial. So there's a canonical diagonal map from any space into any space to some power. Consider those for which that map is an isomorphism for this particular choice of D. This is starting from topos having an object that behaves like this one.

1:27:30 It doesn't have to be that one, but it behaves like that one. Then, remarkable things follow that this S is itself a topos automatically. It has a reflection which preserves products automatically. It's actually rather like the categories of categories of categories. There's a reflection that preserves products automatically and all sorts of other desirable properties. So this is an excellent setting for any such category. It's an excellent setting for partial differential equations of mathematical physics. They can all be defined there in a unique way. So that's the kind of S starting from an infinitesimal picture of what cohesion should do and then you may ask, well, what is then the canonical intensive quality for such bases? Well, it's the following kind of thing which I'll illustrate with the much simpler combinatorial example of reflexive graphs. Here, of course, S turns out to be the totally trivial in the abstract sense. Points are what you think they are. Components are what you think they are. The arbitrary graph has components. You can see them if you draw a picture of the graph. So those two functions are clear. What is the canonical intensive quality? Reflexive drafts, glitch points, components. That's clear. Those are drafts which consist entirely of loops. Oh, that seems even more abstract. Where are we going here? It's plugged in. This is plugged in to the starting point. So we have the two adjoints. So then we have the adjoints.

1:30:00 Which recover the points, so every graph is assigned, oh, well that's easy, the category, the graph, which just consists of the given dots and the given moves that throws away the rest. So I'm asking you to think of this as superheated material. I started with a sample, which has particles and interactions. I expand it or superheat it or, you know, whichever you keep constant. You expand it so much that the interactions can be neglected. Well, you still have something. You still know things like how many quantum numbers of such and such a kind there are at every point. On this example, there's only one number, the number of loops. But it's easy to make up more complicated common total examples where you have lots of information. But it's only the things that live at each point because you've made it rarefied. So I say that a good name for the canonical intensive quality is substance. The substance of a general object is what you see if you turn it into a gas, as Avogadro did. And then, you know, when we say that, you know, this stuff here is H2O, I'm simply saying at each point we can count so many electrons and so many things of this and that kind. So the usual intuitive meaning of substance is preserved when we heat it up, not too much, of course. So that's substance. So that suggests that the other adjoint now, which itself is an extensive quality, we should call it... Oh, sorry, that's not right. Form was the homotopy type, and these graphs all have homotopy types. In fact, all homotopy types are realized by graphs.

1:32:30 There's another quality in which we define the maps between objects as the homotopy, the components of the function space. Hooray, that's another, that's another, that's a real extension. But the left-hand side of the intensive quality is itself, itself, it's another non-canonical. If the other one really is universal, then there should be a map between them, and I don't know whether that's true or not. Whether from homotopy type, you can recover the following thing. Maybe you can't. Can you recover the, that extensive quality, which is the adjoint to the intensive one, from knowing the homotopy type? I don't know. It seems like maybe you could. It seems like there should be a universal extensive quality. Anyway, maybe homotopy theorists haven't quite found it yet. I don't know. There's certainly something like that. But concretely, what is the canonical, the adjoint to the canonical intensive quality, which is often an expensive one, even in the case of graphs, instead of superheating or supercool, so we take this graph, thought of as sort of a sample of material again, we supercool it, compress it, and what do we get? Well, instead of lots of molecules that don't interact, We put the interactions back in, but now we get only a few super molecules, and their atomic numbers or whatever are partly coming from the mere interactions you had before. There are those that are there intrinsically, but there are the additional ones. So we keep pressure constant or volume constant. So as a substance, you can think of it as Avogadro, but then the adjoining of that is the Bose version.

1:35:00 It depends on the components of the graph, but it's much more than that, because essentially you get a new graph which has only one point in each component, so all the trivial loops have been amalgamated. Let's say you start with a graph that's already connected. That's as simple as cake. It's already connected. You compress it by taking the trivial loops at every point and amalgamating them. So now you get, again, a graph which is all loops, but it's the one which is universal for maps from your graph to one that has only loops, the superheated version. And just as when there's a map from points to components, whenever you have this adjoint situation, you of course have a map from Avogadro to Bose. I guess they're souls. Or, you know, canonical intensive quality or substance into the adjoint of the canonical intensive quality, which happens to be the adjoint of substance, which is not the adjoint, the adjoint of the adjoint. The adjoint in the middle is just the conclusion. It's the thing that unites the two opposites. It's this business of super molecules. And there's a canonical map between them. The same kind of canonical math that goes from points to them. This is precisely the information that I'm trying to say. In other words, knowing the superheated picture and the supercooled picture, neither one of them tells you everything about the graph, of course. But you could, and this is like you do experiments to see if the two ends of the spectrum, due to the actual nature of the sample that you have, you'll find out something. But you can't reconstruct it, possibly. Well, there's more information that you have, which may help to reconstruct it. It may still not be sufficient, but definitely there's more information, namely the canonical map between the two, which is due to the fact that you started out with the real thing in between, which did have interactions. So here you have a thing which is just all points and loops, and it goes over to another thing like that, but as I said before, this canonical map tells you which of these

1:37:30 Electrons or whatever you call them here were really originally just interactions. They're virtual, they're virtual particles, virtual particles, virtual atomic numbers, things which are not in the image of that canonical app. So that's additional information which might be useful in trying to reconstruct the actual object. Here's where the lower of adjoint functions gets a little bit simid because you might think, Obviously, we can consider a category whose objects are maps of L to the 2, where 2 means 0. From our original, for example, the reflexive graphs, we certainly have a functor there, which is to take the triple points, not points, the Avogadro aspect of the substance, the Bose aspect of the substance, and the map between them. That certainly can be put together into one functor. So you say, ah, well, now we can reconstruct the universe from our experiments. We just take the adjoint. But unfortunately, this doesn't have an adjoint. It's constructed out of two parts, one of which is very much on the right and one very much on the left, and they are interacting with each other, but the thing put together, well, it doesn't preserve products, but it doesn't preserve equalizers. And on the other hand, it doesn't preserve sums either. So it's sort of interesting. For example, take an example from algebraic school. You have an algebraic group. Well, that's an algebraic structure based on... So if you look at its, taken in this sense, the extensive root, you will still get a group, which in fact means two Lie algebras and a homomorphism between them.

1:40:00 The sort of canonical intensive quality is in the examples of interest really the infinitesimal part, the infinitesimal spaces are non-trivial spaces, but they're entirely infinitesimal, so reflection, for example, of any space has an infinitesimal core, and the infinitesimal core of a Lie group is another kind of group which still has the same Lie algebra, although it has become... So that's, you get partially preserved global algebraics, but not even... I was going to say, do you want to draw to a conclusion in the next five minutes? Yeah, this is it. I mean, I've already actually published something about the internetism case in actual interest, so maybe this background will... Questions? Could you say a little bit... Tracking back to the presentation, to the point about the epimonic cancellation properties and its relationship to the Nolstein sets, how this relates to the way that you think of choice principles in the geometric setting, about some choice in the geometric setting. I mean, there is a connection there, is there? Well, it's something like simplificial sets.

1:42:30 The epimorphisms of the models of the learning split, and then that gives rise to degeneracies, to special theory. So while most of these Es of interest, it seems, themselves have no central, they must have some endpoints in the site, because they're not. I mean, they're putting the opposition, I'm making an approximation of the opposition between the two. The first kind of topos is generalized spaces that parameterize variable discrete sets, on the one hand, and cohesive topos, on the other hand, both over s, over some base s. By saying, well, it seems like the first kind always had sites with no impotence, at least. That's more general than the cancellation, but it's certainly easier to handle than the cancellation. On the other hand, all the examples of cohesive topos... The basic idea of degeneracy seems to be fundamental. Figures can have degenerate figures, or they might be degenerate in the sense, you know, a figure is called degenerate. If x is a figure and capital X should be A, well, if there exists a split, that'd be on it. Since x factors across, then you say x is degenerate. And if it doesn't factor across any... With epics, I wouldn't even have to mention this, it's fixed under the hidden potent, under a non-trivial hidden potent on its domain, same thing. It's non-degenerative, it means it doesn't factor through anyone. So it seems that somehow this possibility, there is that possibility, is inherent in the notion of very cohesive.

1:45:00 Simple though it sounds, you know, it means... For example, consider the projection of any non-trivial projection map. If there exist maps, then by means of graphs, you have projections. And so, you have idempotence. So if you say take the projection from R3 to R2, and you have some quantity, some figure in other quantities, well, you could ask yourself, does it really depend on those three variables? Maybe it only depends on two variables, at least it factors across. There's a potentiality of being degenerate. So although it seems like only a corner of the whole edifice of topology, it more and more seems to me like it's quite fundamental. It's almost a deciding... Now clearly you can have an adequacy of such A's, even though they're globally no central, no inner potency to define the community. So that's not the... That general thing I said before is certainly not the defining property, but certainly if you think of all the examples, well, reflection graphs, by its very nature, consists basically of two hidden components. Take any arrow and that's a degeneracy. So the geometrical structure, the way that the substance and form behave and everything flows out of this possibility of having degeneracies. I mean, you might find a site that consists of only one object, but the object can be a square of itself, and so there, it's a little bit unclear, it's always unclear if you're going to try to translate a property into what happens on a site.

1:47:30 Actually, could you probably say a little bit more about your, this idea, extensive, intensive, of motivation, because somehow it sounds a little bit strange probably, I don't understand, because something which agrees on points exactly, you can think is... You know, category of points as extension. So that would be called extensive. So somehow you describe things kind of point by point, you think, in terms of category of points. And then exactly you, as you put it, you call it just one thing. Or probably it's not that important. Why do you call it that? The idea of intensive is better understood, right? It always means intensive quantities vary on a certain domain as functions do, so the idea that a variable quantity on x is just a map from x into reals and so on, that's an intensive, because it depends on points, or even generalized points, whatever, but I mean, it depends, you see, on points. That's the intensive. The space itself can be, what kind of a quantity is it? So here it's what kind of a function or Q is it? Does it depend off just on, or by contrast, instead of functions it's measures and distributions which are extensive in their way of quantities. But again the idea there is it depends on the space as a whole. A measure gives you values. All of these terms apply to the space as a whole. It's a quantity of space rather than the quantity on space. Now, extending this idea or paralleling for qualities instead of quantities, pi zero, to be able to extract from a graph what is the number of components, you have to consider the graph as a whole. Totally, it's a whole thing. The number of components is merely a... A feature of its extent, but it is a feature that can be extracted from. The deeper extensive quality, leaving aside the homotopy, which is there too, but it is in this Bose picture, you see, that you have to think of the graph as a whole.

1:50:00 Well, first of all, to see how many components it has, which means how many points it will have after you've cooled it because all the things that really do interact with collapse into one point, but the thing it doesn't consist only of points, it has all these huge number of loops, but except you've identified two points if they are in the same, the points as such, if they're in the same component. So again, you need to know about all the... Extensive interactions in order to know which points to identify. So I think it does agree. But does it also agree with the kind of traditional, say, idea of set as extension of concepts? No, extension and extensive are maybe different. I don't know. There probably is a relation between the two. I'm not very good on philosophy. I mean, there is a relation, probably, but it's a very different thing. So the extension of a property is the part of the universe where the property is true, yes. Because there is a sense in sets, and that I think is kind of a very original thing, you just say, instead of thinking of, I don't know, a concept, whatever, a group, you just take all of these things and kind of imagine class as something. So it's just a different thing. These ideas have been long discredited. Yeah, yeah. Within a given set, within a given universe, currents in physics, currents are extensive, differential forms are intensive. Intensive variable quantities tend to be representable and, well, this is maybe a kind of representability because this L is another topos, you see, in contrast to the homotopy types, which are still Cartesian closed, but another kind. You mentioned that Grothendieck topos would be always two topos in your sense, right?

1:52:30 Yeah, over U. In fact, if you look at SGA IV, really they never mention topos. They always talk about U hyphen topos. And could be probably, yeah, I understand this present suggestion is kind of, I don't know, turned back? To this idea about avoiding taking this U is kind of fixed, but then you have this relational picture of like two E, S, so it's... The U, in other words, the U topos, look in the glossary of my book with Rosebrook on sets for mathematics. I say, well, a U topos is this, this, and this, however... Using Grotendieck's principle, if not his original definition, we can talk about U-topos, where U itself is any Laverte or any topos, and it's perfectly sensible to do so, and moreover, there are many examples where, well, really, it's much more variable properties of the functors, like pi-zero preserving products is very crucial, but it isn't true if you take the direct element and go all the way down to it. Somehow, I don't know, that's related, say, to what Skolem did for set theory, when you say, okay, you can't just say set is one object, you should also relativize to some kind of meter, set, and it's a relational thing. Consider models of set theory, which of course are precisely special toposes, where, for example, see the idea that epimorphism is split, I should have said that explicitly, this is the most common... This is the expression of Cantor's idea of negating cohesion. Because if the objects had cohesion, the math would preserve it and not every epi would split, you see. So this is the common everyday manifestation of Cantor's negation of cohesion, simply the axiom of choice. This was my question about how the axiom of choice falls into place in this geometric setting. On the point about intensive and extensive quantity types, as it were, I understand those also tend to be strongly connected respectively with covariant and contravariant form of morality, maps into and out of domains.

1:55:00 Could you say a little bit more about the connection between the covariance and the contrarian functoriality? We're going to talk about that, but we don't have time. Yeah, only so, alas. Obviously, the guy's a plant. Yeah, yes, of course. I'm sorry. Well, I'm not going to ask you to give an entire second talk, although personally I'd much like to hear it. Let's go back already. There's a traditional discussion of extensive and intensive in philosophy. It goes way back, I don't know how far. Probably an Aristotle. But anyway, it does correspond to some kind of intuitive distinction now made out that all quantities are intensive because intensive this somehow this is somehow the idea of perception you see you perceive this this and this one at a time their emotions feelings and so you somehow and since only feelings exist therefore everything is intensive to compress his argument quite a bit just as well. On the other hand, Hegel said there's no difference. He couldn't see the difference between intensive and extensive, which is odd. Actually, maybe I didn't read far enough. Maybe he comes back to this later. But at one point he says you can't see any difference. Well, the reason he couldn't see any difference is because he was considering constant quantities. In any case, it got out of the hands of the philosophers when Maxwell used the terms intensive and extensive extensively. In physics, in courses on thermodynamics, people talk about this all the time, but not in any other part of physics or other part of mathematics, even though, according to me, the basic opposition between the two is in all aspects of physics. And so I analyze it as modes of variation. You know, what is extensive or extensive in the way of quantity is mode of variation. There are variable quantities.

1:57:30 Constant quantities, you can't see the difference, okay? But variable quantities, you certainly can. And intensive or contravariant. Well, variable quantities are analyzed as varying over certain domains. Domains are objects in a category that you can change by a morphism. And if you change the domain by a morphism, the intensive quantities pull back. They pull back, and they pull back usually in a homomorphic way, right? So they preserve. Whatever algebraic operations are on that particular. So the contravariant homomorphisms, and we transform them, are the earmark of the intensive mode of variation. So this applies, you know, not only to functions in the obvious way, but also to differential forms. Or even to cohomology classes. Even though cohomology classes tend to vanish on points, they're nonetheless very good, variable quantities, and they pull back. Now, on the other hand, I analyze the extensive quantities, like volume, energy, entropy, and thermodynamical practice, the quantities of space. Well, these vary covariately. So when you change the domain space, they push forward. And they typically push forward only linearly, not homomorphically. They might preserve co-multiplication, but they don't preserve multiplication. And in particular, they are typically not just mapped for the certain domain. In the way that intensive are just mapped for the certain co-domain. are just a certain kind of thing, because they're not figures. In other words, the relationship between figures and extensive quantities is typically the Dirac's delta. You see, in other words, many kinds of extensive quantities. For example, a typical kind is linear functionals, or linear math with values and some modules.

2:00:00 So this is the Reitz paradigm for an example of an extensive quantity type, depending on v. In the smooth topos, as I was talking about, these should be thought of as the distributions of compact support, and if B happens to be just R itself, then there will be a canonical map from X to the rock delta. You can say point if you like, fine, but any figure, little x, will give rise to an extensive, particular instance of this extensive, but this is not, this is not an isomorph, typically this is a linear object, you see, this is a nonlinear, an object in a nonlinear space, this is a linear space, it's underlying nonlinear space. No, it's not, so it's not simple. At that point, the simple-minded dualization doesn't work. It's not, of course, maps from a particular thing are a covariant factor, but not a kind of quantity. It's the fact that it's quantity that, at least intuitively, you say it should be linear. Maybe that should be generalized. Another question? Well, I think in that case, we should once again thank the speaker for an unprecedentedly rich Exposition and the wonderful seminar I hope not unprecedentedly stretching certainly I think we've learned ourselves

2:02:30 We could have lunch, presumably. Yes, but I am not in good shape. I know, I can see you're not. I was just getting you a very stiff... toddy. At least let me get you that. What? A stiff, how do you say, room. A very strong room. Yes, that's it. With lemons. Little by little, little by little, little by little, little by little, little by little, little by little. I saw the review by someone who goes to the U.S. I don't know if he made it up. He said, he said, he said, he said, he said, he said, he said, he said, he said, he said, Thank you for reading your book.