Morning Discussions, incl. FW Lawvere, P Cartier, C McLarty, A MacIntyre, JL Bell (contd.)

0:00 So this is the shape of your definition, yes, of a manifold that you're going to do, I mean, approximately. Well, what I discovered that by thinking about categories, you may change, you may define that if you have a category with certain objects, So, with the same object, you can read a lot, and so my idea is that we... The point is that I think we have been influenced by the point of view of abstraction. One, to define, I mean, if you follow the guidelines of Moore. To go back in, you have to define a category by giving the object and the phase full factor in the set, such that the isomorphism is exacted by the transform and that the isomorphism in the category are the ones which became a bijection into the set. And of course, that forces you, for instance, to say, in manifolds, in manifolds, I will just take ordinary manifolds, despite the variation, a manifold is something covered by charts.

2:30 I don't identify two different set of charts, but I, so my object is a scaffold, I would say it's a scaffold, and then I define properly the maps between, so my manifold is defined with a set of charts, an atlas, but then between two different sets of atlas I define various notions of maps. Now the one with respect to the atlas is the one with more art. I don't have to change that class. I mean, it comes automatically out of my head. Because they are the same, of course, now if you want, I mean, there are some objects to calculate which are isomorphic in the category, well, you may consider that they define the same manifolds, but you don't have really to define it. You don't have to give a forward definition of what a manifold is. Now, with a category of three, or whatever kind of three objects, and for Pritchett, it's the same. You well know that if you work with Pritchett, when you... You go modulo some category, invert in some arrows, invert in some arrows, you can define the sheet in terms of the pre-sheet, you take the same object but you change it up. And it's exactly the same strategy in many situations. You have some object, you have various categories of objects over the same object. And depending on the various categories, you find the various notions of anomalies. The three manifolds are analytically equivalent. They are omni-morphic, they are dipo-morphic, and then, if you want, you can speak of the algebraic class, but why to select one specific? And that's actually what a non-denial manifold is about. Then, I establish this definition. In two sentences, I evaluate it. So I really define it.

5:00 So, the notion, I mean, there is a, I mean, to substantiate my definition, I have to take into account the reasons of all of which is that this is equivalent to the definition of a constant. In one example, I mean, all examples of my definition. So, it means, well, in other terms, it means that this definition is low. I mean, in my category, you can glue all the things. I mean, if you start with this vector, you don't have to repeat. So, for example, you could start again from local models, gluing and so on, which is familiar. And I don't want to spend two weeks, six preliminaries, speaking about charts, super charts, and so on. Atiyah is only achieved with ordinary manifolding and ordinary spectrometry, but it's not very far from the definition, it's lightweight. Someday you should invite money, if you can manage it. That was one of the things I was going to discuss when you were asked, because money in a fortunate situation to be at a meeting point or two.

7:30 One was to get on school, the other took the vessel, and the combination was the people who claimed themselves to be so fortunate. I don't know why you're five years a chairman. Why do you do all that kind of thing when I'm five years my chairman? And at some point he would say, okay, well, this student, this student, he is for you because he will do nothing. I will tell you one thing. When I married him, the grandfather of mine, when he was young, his job was in the cavalry. He was very powerful. And so he would explain to me. So, when I was in the cabaret, training horses, and then they brought me a new horse. If the horse was, he said in his thing, flagada, full of steam. But what a horse! The steam out of it. Too much steam.

10:00 Just a discipline. And they have put a number of such people.

12:30 There have been others, and Charles Bettelman has some papers in the laundry for his teams, and this sort of big new star in the U.S. mathematics is Jacob Lurie, spent a good deal of his time, he's finally got into homotopy theory and might hold himself beyond, but he spent much of his time, a very brilliant adolescence, doing some little numbers. For many years, not just, you know, for a while, I mean, it's clear to me, once he got to Harvard, you're in a moment.

15:00 I don't know, it's just a very confusing concept. I shouldn't give you a term about science, I think we're all out of time. So there are a few of these people, and lesson figures are totally similar things. We're on the topic, that's okay. Okay, so the theme of categories of spaces, Cantor negated. The idea of Mengan introducing what he called Cartier-Hansel, to access infinite versions of the... Learn how to base mathematics on the idea of structures modeled in a category which was as deprived of structure as possible. These principles about choice and continual hypothesis and that law of the excluded middle, there I view them as their attempts to express this lack of structure It's important to note that the background in which you interpret the structures does not influence too much the results, and somehow this is the idea. So some of the features that the Menon were supposed to have saw in mathematics, mathematical practice, were shoved into the idea that the cohesion of objects is itself a kind of structure. So topological space, phonological space. And so forth. This is a specific kind of structure. But then there are lots of categories. And we have smooth, continuous, measurable, and so forth. What do you call it? Cohesion is the same as zusammenhalt.

17:30 Zusammenhalt, I think. Zusammenhalt, yes. Not zusammenhalt as it became a topology. Zusammenhalt as it became a topology. The zero measurement of Tuzan and Howe, that's how I knew connectedness, because the other ingredient, the need for, say, a general theory, that I, as I know, recognized by Frechet, because he wanted to recognize that the cohesion of function space and the cohesion of ordinary space have something in common, you see. There is a Hadamard. By the way, I was going to ask you, you said, when did Hadamard die, actually? Hadamard? Yeah. Fifty. He was very old when he died, wasn't he? He was very old. I met him. I met him when I was a student. He was still coming to the library once a week. But he was very old. And I remember once, too, even. So it was... It's connected with the first Schwartz seminar, so it might be 55 or 56, and one of the first Schwartz seminars was about partial differential equations, and Adama came to the library and spotted the proceedings of this seminar, which was just out of the printing office, and came to the library begging for a copy. And the librarian, who knew very well who Matamoros was, who knew also that Schwarz was his if-you, I mean, said, oh, Mr. Schwarz did not give me the list, so you cannot take it. So, just to situate the thing, so this might be 55th or 56th. I showed you before, so it died.

20:00 And I remember my first contract, you know, it was submitted by him. I mean, it was a job. I had to change the route, of course. At the very first International Congress, he was not actually present, but he sent a note to the effect that... Maybe we can actually use this set theory stuff to understand the cohesion of function spaces. I'm turning it into my language. And then there was some discussion from the Italian side saying, well, actually Volterra has already done this and Ascoli has already done this. We're starting already on this. And so it was only a couple of years later that Audemars... You know, we've gone way beyond that, I mean, but, so this was, so in particular, Frechet, Rendi-Conti, the Palermo paper on metric space, is conscious that this is, it is two apparently different forms of, again, I don't know a better word than cohesion, I could use continuity, but that's already got a specific meaning, you know? That these two forms, the forms that occur in the function spaces and occur in ordinary space are, again, each more principled. These are similar, so there must be a general theory to those both. Of course, not metric, but at that time Voltaire, Anoumaille, and Cachet. But somehow, again, the determination of, the default determination of... That became the idea of open set or closed set, in other words, the category we know as general topological spaces, which in fact has a glaring defect with regard to not only does it contain pathologies beyond curves and all that, another discussion, it doesn't permit the natural manipulation of its functions. Precisely, it should have been serving this...

22:30 So it was, as far as I can tell, it was Huriewicz who first really pointed this out, and to R. H. Fox, who wrote in 1945 the paper about sequential convergence, the category of spaces which have sequential convergence as sequences suffice to determine that. But this category is in fact what we call Cartesian clothes now. This was known in 1945. And of course that was an old concrete practice. This was an old idea of analysis. The convergence of a sequence of functions, f sub n, means that given any convergent sequence x sub n of arguments, you take the diagonal, f sub n, x sub n, this should also converge to f of x as well. That's exactly what comes out of the usual formula for Cartesian closure that's applied. So this much was more or less implicit in Cauchy. Yeah, yeah. Well, of course, he can refer to a car. Implicit definition he uses is this one. Yes. So, Box refers, in fact, to Horowitz in 1945, and then in the late 40s, unfortunately, never published, but in lectures at Princeton, Horowitz gave the definition of k-space, essentially where you replace the idea of a convergent sequence. Well, arbitrary compact phases as models, but nonetheless you achieve this Cartesian closure, and there have been many variants. Compact opened upon these things? This is a compact opened upon them.

25:00 On the functions phase, we'll again give... ...played with similar ideas in the inner work of Bobakshin. There are some drafts by him called compactor. What did he do in that in connection with uniform spaces, I mean, that was a thing that he... No, no, no, later on, I mean, he wanted, I mean, that's later on, when Govaki was discussing the integration theory. At some point, I mean, very rightly, I mean, in my opinion, pointed out that, I mean, due to the Leibniz-Markov theorem, integration over compact spaces, over compact spaces, but even... And that's the definition that Bobecki takes as a measure, I mean, by duality of various compartments in those factors. But he says that for more general spaces, the natural idea would have to exhaust the space by compact subspaces. And so you have, I mean, to glue together, I mean, many of the other various compact pieces, and he calls that compactology. And he tried to define, I mean, so a set would be measurable if it... It covers every compact subset as a measureable or immeasurable set, and so on. And that idea is resuscitated in the last two forms in the final chapter of the book, the integration of Braubach here. And this, when Braubach tried to extend his integration theory, and what to call as part... After my insistence, that should include the Wiener measure, the Wiener measure and the H-position. And for the Wiener measure, I mean, that's enough, because it's one of the things that the Wiener recombinance, that's very important. The Wiener measure sits up to epsilon on a suitable compact subset of functions. So this idea was played very small in the inflection of the math. The paper by Weiber and the general topologists pointed out that the notion of proper map and sometimes the intuitive idea that fibers are compact, from the point of view of the category of topological spaces, you must impose also closeness of the map, but in this category of compactly generated spaces, closeness is determined by intersecting with arbitrary compacts.

27:30 And therefore, if the fibers are compacted, the map is closed. All. And so you have a simplification, a definition of this basic notion in the strategy that was designed for this other purpose of moving with the punches there. Ah, it's still a good idea, I mean, to exalt the space by combating some species. Yes, yes. And as I said, in integration theory and in the way there is a general theory by, or a general problem which is an extension of formalism by Wiener on the Wiener measure, which says that, well, it's in both back and forth. In a rather general space, I mean, the measure is exalted by the compact subsets. Let's say if you have a measure which is a total mass, a finite total mass, then you can, up to epsilon, fill your space by the compact subsets. It's still very, very high. This idea was associated also by some people. The abstract measure of probability diffusion in the name of compact glass and its various notions are completely connected to that. Eugene Dinkin, Eugene Dinkin also in his books about probability of glass. So, this idea was not forgotten. It has been resuscitated in this functional analysis, functional analysis. But it's true that the K-compact spaces, I mean, were all reasonably good spaces. Yes, is this the same as compactly generated? And what is the precise definition of that? There are variants, you see, whether you assume that the models are housework or not. But essentially, these case spaces are often mistakenly called Kelly spaces because J.L. Kelly in his very nice book on topology

30:00 The point is that the set is closed if and only if its intersection with every compact set is closed. So in other words, slightly more functorially, it means that you consider the category of compact spaces, and then a given space is determined by a functor. We may even consider all maps of all the compact spaces, so we're contemplating the functor of the compact. That sort of thing. But there are many variants, you see. I mean, it's... Prox is original. In the construction in 1945, it means essentially you start with, let's say, with compact countable spaces, or even compact countable spaces of finite type. That's your model category. And the general space is just appreciative on that. Again, satisfying various side conditions for that, but the underlying structure is that you use these. More finite, more manageable, more decidable, or whatever the space is as models, and the general one is appreciated, in fact, on that. Achieved, you should be achieved, actually, with respect to some topology or whatever, but that's just a conceptual move to achieve the same as, you know, or that exposition of motif by the Wojewodzki, or Wojewodzki as the god of motifs. More or less the same idea. Gustav is a quite, quite comported guru. I mean, Archibald Fett, I.T.s and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and, and See, because of the universal teaching of the usual notion of open set is basic, there's a sort of psychological reaction that if you have to deal with a problem that involves cohesion in some way, you automatically, what I call a default, you automatically say, well, it must be a topological space, first of all, plus something else.

32:30 And this, in the case of schemes, was a very bad mistake, as Brodenty recognized in the early 70s. He went around preaching, please, everybody, forget the definition I gave before. The real definition is this. And there's a sound reason for it. I think you may have been one of the first to notice it, that if you think that a scheme, or even say an affine scheme, if you think it's a topological space with a sheaf of rings, and you forget momentarily that it's a sheaf of rings, This space in itself is a very, very bad variant, because even the simple Cartesian product of two schemes does not correspond to the Cartesian product of the spaces, and so as you said, an algebraic group is not a topological group, it should be, but it's not, and so on, so those fundamental conceptual problems are in a way tied up with this. There is a belief that on the one hand we need categories of cohesion, but on the other hand there is a default way to do it. That's why I use topological space as a step zero. That's why at some point I insisted that the scheme should be taken as a factor over the rings. Exactly. So not taking the first step of gluing rings, but as a factor over the rings. A factor over the rings. Yeah. I think this even extends into Galois theory, that the idea that you deal with this Galois theory of all finite field extensions by passing to some imaginary, intimate field extension and looking at a huge group of... And then compensating for this overextension by imposing again on it a pro-finite topology. In my view, ultimately, that's all spirits.

35:00 It's convenient. No, no, but I mean there's an environment in which all properties are clearly defined, which is just cheese on the category of finite field extensions. In some sense, the underlying set of the scheme is not a single abstract set, but it's an object in another topos, but now just using fields, needs only finite field extensions. I think every iteration, in other words... What you achieve with this continuity with respect to the pro-finance topology could more easily be achieved directly just by the concept of naturality. Yes, yes, you're certainly right. I mean, I mentioned yesterday this thing that we did in the volatility of Galois. Exactly. I mean, we couldn't have done it unless we'd taken the second form. I mean, if you'd gone and done it, that's... But just a technical point, the profiling topology isn't a very good position, the group only has one profiling. You don't have to only look at the continuous maps, it doesn't even just... I want to mention one thing about, I mean, about the algebraic closure. First of all, in nature, we have the complex... We have the algebraic numbers which are within the complex. So you can consider that you have a gas complex. And of course there is a row and series for the function base. I mean there is a row and, not only the row and series, I mean the one using, with the ramification of the previous series, the previous series, the previous series. There are models, very, quite concrete models, like the binary clause. The field of interest in algebraic geometry and arithmetics are short fields of that. So, you have a natural algebra. On the other hand, when you go to the p-addicts, you go to the p-addicts, one's got the so-called field C-P of date, which is the algebraic growth, the completion of the algebraic product. But there is one thing which is very often overlooked. Consider the finite field. What is the algebraic growth of the field is p-addict.

37:30 And it's a concrete problem that the people who try to do computer algebra, when they deal with finite fields, of course, it's a difficult notion how to encode the finite fields and the opinions on the finite fields. When everyone says, I think Q is the power of the prime, and they think FQ is the power of the prime. It's not. It doesn't make sense. It doesn't make sense. Thank you for your attention and see you in the next lecture. It's a real problem to encode these calculations on the computer. This is a real technological problem. It's just not philosophically fantastic. This is a problem in computational complexity, isn't it? Yes, and so, and so, but so, there were, now, there are four situations where half of the natural examples are natural clauses, the other half doesn't. And it's, well, when you come out of all stores, and I take in the field of all algebraic numbers, if you need an extension, I take a prime ideal. But when a prime ideal is defined only on the way to a limit, so it's, it's again the same. And so, of course, if you take an abstract field, let's say even a function field in more than one variable, how will you differentiate all the variables? What do you mean by algebraic fractions? And it's quite surprising that this is usual. We are so happy to have this general existence here. But, you know, if you look at the textbook, He's very clear, he starts to be constructive at least in part, you know. Yes, he tries to be constructive, you know. And I heard him even late, I mean, I heard a talk by von der Waaten rather late, in the 60s or 70s, where he came again to that question. So it's not a...

40:00 Well, the general algebraic existence and algebraic quotient is improvable, again. You have no real idea what these things look like. I have a question for a logician. I mean, about what is the status of the existence of algebraic clauses or the field with p elements. I mean, in what sense does it exist? Does it not exist? This is doable. This is really... But that is applicable in arithmetic. Yeah, that's true. I mean, is there a recursive model? In the recursive topos, presumably, if you have a finite... ...feel you'd be able to prove, I mean, that it has an algebraic closure. I mean, that would be the way, the sort of invariant way of presenting it. I don't know. I mean, I'm not sure whether that procedure has actually been carried out in the... ...it's quite confusing. From the standpoint of classical logic, that is okay. There's an explicit arithmetical... Let's keep it on the tiny fragment of a little bit, not so tiny, but the matter of course is quite a good example, and that's an essentially invisible entity, but it's pretty difficult to construct it as much. It would be interesting to know whether that just a finite field or the algebraic closure of a finite field is actually provable in the... You can call it the action of choice, but in another way it's just continuous variation in parameters.

42:30 In most topos, the thing just definitely does not exist. That's because the thing seems natural because you're working with this Cantorian category abstract section. But once the things are varying, then you don't get an algebraic problem. I think this was first pointed out by Rafe, as I recall. Did he publish something? I didn't lecture him, but yeah. Well, the complex numbers are not algebraically closed in some, you know, in some topics, but you know, models, you know, were found by Highlander, I mean, or whatever it was, quite a different model, yeah, of course, but I mean, there are some usual, reasonable definitions of what complex numbers are that you can generally show, even that's algebraically closed, continuous variation of parameters. Which, of course, brings us back to the general point of Bill's continuing exposition, how this whole issue of algebra and those fields fits within this wider vision of the capacities of space. Yeah, so the idea about cohesion there. So, in fact, Susanna Hout and Susanna Kahn, because we, in principle, In order to understand the form aspect, cohesion, in my view, cohesion, to analyze it, it has at least two aspects, form and substance. So form is something like homotopy, which doesn't depend so much on whether you have smooth spaces, continuous spaces, combinatorial spaces. You still have the same category of forms. And the way these forms emerge, more specifically, is precisely with help of the function space. Once you recognize that your category of spaces has this Cartesian close, has this exponentiation operation, plus just the zeroth level of Suzan and Tang, which is Suzan and Tang, in other words, connectedness, pi-zero functor, when you apply pi-zero to all the function spaces, And that way you get to homotopy groups, the homotopy category, and all this. So the sort of outer form aspect of cohesion in a given case is measured by combining the Cartesian closure with the simplest case, pi zero.

45:00 But on the other hand, the functional analysis, roughly speaking, is the other aspect where the substance is important. Of course, the homotopical result is of great interest, but you are also interested in whether things are smooth or continuous makes a huge difference there, you see. So to extract what I call the substance aspect of the cohesive space requires something else. I call it the chocolate exercise. At the end, he wants to be awarded a chocolate medal for having worked out this exercise, about ten parts. I asked him whether he was the one who, you know, various people wrote this, their DAs, and he himself was the one who wrote this, and he was still proud of it. That's right. He was bad. Admitted that he was actually proud about having done it. Now, he takes from Giroux this terminology, go into tea, which I think is a very poor terminology, because it doesn't express the fact that it's a qualitative difference and not a quantitative. But tea for what? Sites. Sites. Sites, and hence toposes. So there is, I mean, it's a common, I think it's a colloquial usage. There's a qualitative difference between these two things. It's not just that there's a method of construction, starting with the data that gives and produces the This family of petite toposes, which are sort of like sheaves on particular spaces, and on the other hand, one big topos, which is like the category of all spaces, has a subcategory, a full subcategory category of all spaces, but indeed, they're not, it's not just that they're all topos, as they are, it's very important, but they have qualitatively distinct, but this is hard to, this is hard to...

47:30 Thank you for your attention. Again, I think people have neglected this problem because instead of not being so interested in functional analysis, where there would be a huge difference, but rather in cohomology, there's a lemma that says that if you take the Grotopos over a fixed space X, and that's the one that's called the Grotopos of X, on the other hand, you take the T1, which is the sheaves on it, well, they have the same cohomology. Take a coefficient sheet and either one and transport it. All the derived functions give the same result, so, in fact, this was Giroux's description that, well, the Go one is something more convenient, pragmatic, to calculate the same thing. Because after all, the petite one, going back to this question of the stocks, typically has these horrible spaces that you can't understand, really, except in the complex space where you have formal power theories. But in general, this is very bad, whereas in the grow one, well, you can get ordinary lead groups. Everything is working there. So you can use those directly and then calculate the cohomology. So it's like a tool for calculating the cohomology, but it's not something of interest in its own right. But from the point of view of functional analysis, it should be, because it links, it combines both the finite spaces and maps between them, not just sheens on them, with the uniquely, with the unique notion of function spaces, distribution spaces, and so on. So that's the conclusion. What term would you have recommended for the...

50:00 I've tried various things, you know. One thing I tried was simply to keep the same letters and call it general and particular, giving this distinction a peculiarly particular meaning in general, but it's that sort of thing. But again, the point is, category and space, there are many particular generals. That is, they're not just smooth, continuous, algebraic, measurable. Those are some main ones, but actually there's a whole infinity of them. Simplicial sets is a good example of a growth topos. It's not a category of spaciousness. So what Cheng Ruo and I strive to do is we develop the mode of thinking topologically, but in an arbitrary category of a certain kind, not a totally arbitrary one. A category that has fiber products and disjoint, some that are disjoint, and universal and disjoint coproducts that actually one can develop the sort of topological intuition that one would probably call the general idea of shapes and all in any such category so that it's not necessary to first say, well, let's use the default category and then try to correct the errors that have been introduced. However, I did find some axioms that are true of all the examples, at least most of the examples, where you have a general category of space as opposed to the particular ones. Namely, this can be explained very easily, this pi-zero frontier. Which doesn't necessarily go to abstract sets. It should go often to the Galois topos we were discussing about before. Not only does the points functor into abstract sets, not preserved products, but neither does the pi-zero functor.

52:30 But both of them do if you go only down to the Galois topos as the equivalent underlying statements. So it's a huge, huge... But anyway, so one axiom is that the phi-zero should preserve products. Now, as you think of sheaves on a topological space, a locally connected space, there is a uniquely defined phi-zero functor, but it essentially never preserves products. It only preserves products, I think, for irreducible spaces. There are a lot of those, but it's very special. I found this one Grudenbeek also uses in his big work on homotopy theory, is essentially that every space, that is every object in the category that's called the general one, can be embedded in a contractible one, where contractible means, well of course it's connected, but indeed it's a terminal object in the homotopy category, pi zero of x to p, where p is an arbitrary object. You get one always. So that means x is contractible. Not only pi zero of x is one, but pi zero of x to the p is one for all p, which is equivalent to pi zero of x to the x, which suffices to get our p of x because the connected monoid with zero sort of acts like a unit interval so you can improve contractibility once you've got this monoid acting because that monoid of inner maps acts on any function space for all these function spaces. Anyway, so the fact that any object can be, any space can be monomorphically mapped into a contractible object is to be expected in general, but it's definitely false for the particular category. For example, there's no topological space whose categories of sheaths satisfy both these axes, that pi zero preserves products and everything can be embedded into a contractible. Now, the statement that everything can be embedded into a contractible is actually equivalent, using the internal logic of the total, to a very, very simple thing, namely that it should be a connected space with two distinct points, because there's an opposite situation where, in fact, the component structure and the point structure are isomorphic.

55:00 Every component was suddenly really reduced to a point, but it has a unique point in it. If there exists a connected object with two distinct points, then that object can be mapped into the truth value object by taking the characteristic function of one of the points, but the other point will then be mapped into false because of the distinctness of the thing. It propagates into any function space, but any function space is really an arbitrary power set, and of course anything is embedded into the power set by the singleton map. Of course you don't have to use the, it suffices to use the partial map classifier, part of the, a small part of the power set. Can be written down in the internal language. I mean, there isn't any way, I guess, just using the internal language. Of course there is. A distinguishing group or a general group in particular, right? So these are... No, no, no. Of course there is. There is. No, there is. There is. Very definitely. That's what you see. In the internal. It's not that all topos are just sort of the same. Not at all. They're qualitative in the internal. Cantor's original move amounted to saying that within mathematics as it exists, there should be these discrete infinite sets.

57:30 Topos, elementary topos, which is more or less... Then, you can define, there's a Galois connection defined by the following relation, we say that a map alpha is orthogonal to a space S if S to the alpha is the identity. In other words, that maps from the domain of alpha into S are all constant. You can't move. So this... This will define a category of all the S's, which are discrete, cantorian, with respect to, so what you need is the parameter alpha, and the thing that you, well, there are many examples, many, many examples, which works in simplicial sets and so forth, but a very, very nice example is you just take the first order infinitesimals, and take the math from a single bear point into the... The point that has a first-order infinitesimal neighborhood, just that math alpha alone, and then you take the orthogonal, the category of all these s's. So this will be then another topos automatically, which is reflective, while this reflection is the pi-zero function. So the needed, I mean, in some sense the whole contrast between cohesive and non-cohesive, which is everything is based on, is inside the same. In other words, non-cohesion is just the extreme case, but inside the same world of cohesion. So, yeah, very much so. You can see the pi-zero functor, this does an operation inside the topos, and this distinction between the categories of space where pi-zero preserves products but non-equalizers. Again, that's crucial because a typical example of non-connected spaces is gotten by equalizing maps between contractible spaces. Take the line and the line, and wiggle, wiggle, wiggle, wiggle, wiggle, wiggle, and you have equalizers as a disconnected space.

1:00:00 So if pi-zero preserves equalizers, then you have none of this, none of the interesting combinatorial spaces. There are topologies where that's true, where Phi zero not only preserves limits, but in fact has a further adjoint, and these arise in the analysis of the two qualities, extensive homotopy and intensive quality for substance, like, you know, Tom's catastrophes exist in certain spaces, if you throw away the global cohesion of the spaces that retain the catastrophes, you still have something. But that's going to be, again, a topos of this qualitatively different sort where pi zero is so damn continuous that it's actually isomorphic to points, even though, very good point. It is internal. The sets are not external. You don't even have, you don't even have, we're talking about u topos. You don't have to start with u. You extract the u from, you extract the u. The suitable version, and that's the nice thing about it too, if you start with algebraic geometry, then the suitable version of, if you're over a non-algebraically closed field, then the suitable version of discrete is this Yalwa topos, it's still got, you know, groups that come in. These are just the things, I mean, specifically, if you have this map from bear point into the first order infinitesimals, one of the things that it's... It's orthogonal to the S's. Well, taking the function spaces, tangent bundles, and you're saying, you know, that all maps on tangent bundles are trivial, that means that, say you're dealing with a, imagine that S is a spectrum of some ring, well you, this proves that this ring is separable. This is a separable extension. And therefore it's one of the, it's part of the site for the Galois token.

1:02:30 The site consists of separable products and the topology is the bar topology, so it becomes actually boolean, atomic boolean, but definitely not the category of abstract sets. So the thing that you abstract really, in some sense, comes from the thing is really the appropriate base for it. Is it the category of sets? No, it's an atomic, boolean, total, natural number object, all these elements. But it's not generated by one, unless you were algebraically closed to begin with, in which case it is the abstract set. If you were algebraically closed to begin with... That's an interesting, yeah, that's interesting. Because the difference in that case, that in both cases, I mean in algebraically closed and non-algebraically closed... You have a common basis, right, or a common construction for obtaining a sort of basis, but you always get a Boolean, you always get a tongue which has some sort of classical features. Right, right, the base has some very strongly classical features. Yes, which doesn't necessarily coincide with complete discreteness, at least in the case of non-algebraically closed case. That's very interesting. What does it mean for the old minimalist? So you construct this, but you construct it from the category of all of them. You start from that. It's like Hegel. Hegel said, we start with being, then we analyze it.

1:05:00 Rather than starting with a category of sets and then building it up. I am now. But we do that too. We start with being, then we go all the way down, and then we're in a position to climb up. A vague idea, an ever sharper idea of what this environment is within which it's I think this is the first page on Hegel. I think I understood the first page. I don't claim to understand it. Even though you know practically nothing about it, what you recognize is you have to start with it. The argument is it's a great length. You could have started with being. You could have started with becoming. No, you could have started with nothing. In the science of what? In the science of what else? In a certain sense, this is what Galois did. Reconstructing history, which I know historians pay to do, but, I mean, reconstructing it for consensus. You know, he's already working in the world of algebraic geometry. And he said, well, let's look at this special case where we have these spaces that are almost one point. And there's incredible stuff there, which is a very useful tool in analyzing the more general spaces. Yeah, this is really the opposite of what you might call foundationalism, you know, the idea of, right, well, we're going to come to that, I guess, when we come to discuss Kento and the whole, I mean, one way you can think of Kento is as a foundationalist, right? I'm inclined not to think that though. I mean, I think it's, well, I presume that's what's going to be the distortion, at least I imagine, yeah, I presume, yeah, well, part of it, part of it, anyway, yeah. The fact that he says it starts with man, you see, the fact that he never mentions it again, yeah, has misled people, but of course there was quite a bit of development.

1:07:30 Yeah, yeah, yeah, no, no, no, I, I, absolutely, and I, I, I guess that's something that will come up, you know, at length when we, when we get to, uh, that. Last session, we reached a schedule. Having a sufficiently maximatic description of these general topos is either going to T is very misleading because they're all the same size from the point of view of the topic. What they meant was something not really said to you. I mean, they didn't mean that. But in any case, the series of applications and functional analysis have been lurking there all the time. And so on and so forth, and so forth, and so forth. So, for example, within algebraic geometry, there's also a functional analysis, the space of polynomial distance, which again has not been looked at. I think there's something called the Cremona group, for example. It's not finite dimensional, but it is somehow an algebraic group. Well, it's an actual new object in this topos which partakes of the algebraic structure. There are polynomial maps into it, but it's a sort of an isolated thing that people study with it. So when the terminology of growing epitome was originally introduced, was it in the discussion screen just clearly related to the science, or was there...

1:10:00 Any glimpse of this? It was a specific site. It was a specific site. Just topological spaces. I did not participate. It's in... Maclean and Murdyke talk about a sufficiently big category of topological spaces, which is nonetheless small, and then they talk about sheaves on that and so forth. Again, it's just an exercise. So it was just Giroux's... I think part of it all is exercise. It's the most general construction of this type of thing, but it certainly is very general. But the original meaning was merely that with each topological space were associated these two topos, either the ordinary sheaves on it or... In fact, Boydak and McClain gave this example of the growth token. One of their examples is a growth token in which L'enfant maps are continuous. I mean, they use that. It's rather nice, actually, because they show how, in quite a specific case, you could realize, in this growth token, you could even realize in a rather natural way. And you can sort of see, it's actually quite easy to see quite explicitly why you're going to get that result in this, because you've got a lot of room, so to speak, to force it to be true in this particular protopos. And it's nice, it's a nice illustration. I think it's a high point of the book, personally. Yeah, there are many exercises which could mean, like I was saying, that she's considered a disruptor to the closed chess, which seems to me unpursued, but it adds a lot of content, nuclear spaces and so forth.

1:12:30 Indeed, it's one of the topics which we're going to come on to when we reach our discussion of how Brodendieck has, roughly speaking, got from 1950 to 1960. Which means we have actually improved by half a day on our schedule, which is excellent because it means, one, we have more room for discussion of further topics, and two, it means we can definitely make more of an excursion along the way. Is this an actual point at which we end the discussion, so that we can close for lunch? No, sorry, this is the old version, which you'd rather have to re-type now again, because now I think we've moved our agenda up by half a day. This is a good place to break for lunch. Would you mind the cabins, please? No, absolutely not.