Conversations en route FW Lawvere & Silvana Lawvere — Part 2

0:00 Look, there are appropriate levels at which we can stop, but there are, of course, in this, say, level, there have to be intrinsic reasons from the mathematical point of view in order to stay to that level. Yes. So, for example, Baez... Should have stopped at four dimensions, because he wanted to do four-dimensional field theory. He should have worked on that. But I must say, his talk was better than I expected. Because, you know, he gave a summary of one view of what he didn't categorize. Because the trouble is, these people are not doing anything. His title of categorification is an ugly word. He has given talks about that, and if he had really followed it, it would have been a bad talk, almost surely. But for some reason, he didn't really talk about that at all. In a sense, yeah. So I had trepidations about Len Day. Yeah, me too. As well as Mulvey, also. It was much better than I expected. Yeah, yeah. In general, I'm very optimistic, but sometimes I'm too pessimistic. Actually, I like the Mark Bistock. I did not imagine that he opened with, say, a trigger to your work. Yes. Not only that, he came to me earlier on and said that, Ten years ago, you were right. It wasn't only in Kenya. We had many polemical letters. We just refused to recognize what I thought was proof of life.

2:30 Sticking to a sort of obscurantist treatment, where a treatment much more intuitive for functional analysts Thank you very much for your time. Just broken open by hand pressure. Yeah. What was the point in Mahler's construction that he now altered then? Not the one that he was talking about. Not the one he was talking about. Another connection. Another connection, yes. The definition of an internal norm linear space in a topos. Yeah, I mean, he insisted, you see, that the norm of, let's say you have a vector space V at the end of a topos. All of this had to be thought of in a completely unusual way. It's a function which to every rational number assigns the subset of B, which is to be the ball of radius one-third, a ball of radius five-eighths, a word-preserving map from action rationals into certain subsets of the B satisfying certain actions so that we can think of them as balls. We use Banach space in a topos. We must look at it this way. So in other words, it's a map from Q into omega to the power of V. It's a map from V into omega to the Q. So now omega to the Q is an object independent of V. And in fact the axioms say That this map takes its values in the part of omega to the q, which are these upper semi-continuous real numbers, so this map is just a norm in the sense it assigns to each vector a real number, where the real numbers must be thought as upper semi-continuous and not continuous, and this is something that analysts could easily understand.

5:00 My whole objection was that this is making the subject totally obscure to the analysts who should be using it. The idea of a variable Banach space, a variable norm, they have considered this as something that needs to be treated, you see. But looking at it, and they are even aware of it, they're even aware of the fact that the variable norm should be only semi-continuous in the parameter, in typical, actual examples. So this mode of exposition, this is typical, you know, elitism, obstreantism, by phrasing this thing in a certain way, all the possible people who should learn and use this theory will not really want to or be able to. By this one, it's an important ideological fight, I think. And finally he admitted to turning it around. And it's just nothing but a map into upper syndicates in this real, satisfying, unusual triangle of inequalities. Simpler, I mean, the two things are actually equivalent, you see, but the point is the latter picture is immediately grasped over by any analyst who's thought about these things at all. And perhaps we can get Baez to think right about stopping the regress in his M-categories for mathematical reasons. We had a very interesting conversation with Andrew Lyle on the second day. Thank you for your attention. When I ask him to send me back the air tickets, he can send me back... Oh, so he is going to get that back. No, I thought he should not pay me yet. No, but he wanted to do it, but he will get the money back.

7:30 Oh, okay, okay, that's all right. That's okay. Okay, that's fine. I just wanted, because I wasn't sure. I need to just see my owner, if I can, to take care of that now. Yeah, Francesca told me about that. About half of what I'm saying is actually to talk about the subject of mathematics. The only thing that's really required, the most minimal idea of mathematics, that's at the Cartesian post, but it's also... And so many of these incredible barriers to learning that have been erected by analysts over the past century are due to the refusal to insist on constructing the correct reason for this category, but rather accepting whether the bad thing comes out of the wrong default setting. That would be very, yes, that would be very, very valuable to have involved in these three areas. And specifically... Thank you for your attention.

10:00 Thank you for your attention. Thank you, I'm sorry, I speak a little Italian. It doesn't even have to be that car, does it? Well, we know that there are three seats there. That's true. Or, as I say, change all three tickets, just as if we can possibly sit together. Yes, I'm sure she's got it lit. Yes, that is one of the advantages of having first class tickets. It makes them a lot easier because you don't get such crowds in them. Bernardini was telling me a little bit while you were taking a nap in the afternoon about what you had been speaking to him about in the morning. Again, largely the barriers to understanding that come from the wrong presentation of functional analysis. Because there seem to be so many different structures. Functional analysis, which is very difficult to disentangle when approached through this default category. Topology, general topology. Very difficult indeed to disentangle. I mean, bornology is fairly one. Right. But he was also saying that you'd search for him some very interesting things which he didn't follow completely because he needs to learn a lot of categories.

12:30 Excuse me? Excuse me? Is he in? Is he in? Smoking. Oh, only smoking. I think we can... We'll work it. Don't worry. We'll work it around. We'll find a way to sit in together, I'm sure. Okay. Is there a lounge car? Yeah, I know. Well, we'd rather not have smoking, so... No, we don't. No. No, no, it's okay. We'll just take those tickets. We'll work something together. I think so too, yeah. Is there a dining car? Is there a dining car on that train? Dining car. There is. Okay. Problem solved. Grazie. Grazie. But he was saying that the issue of... Limits in the category of Banach spaces. Limits, yes. Limits of Banach. Yes, limits of Banach spaces. In a category. In a category. And how... In a category. Arriving at nuclear spaces which are qualitatively different from Banach spaces. Yes, that's what he wanted to... well, that's what he was telling me about. And I, again, would like to have a better understanding of this. This is one of many topics. Yes and no. We can't actually get, the only three seats we get together is in the smoking car, which we're not going to do. But she was pretty certain we would be able to sit together when we get in the car. And if not, well, I can always, I don't mind standing, you know, hanging around. And there's a dining car anyway, so we can go there. No, you can always use the dining car. You can always use the dining car. Unless there's really a press for lunch.

15:00 I doubt it very much. So, we're in shape. We've got about 25 minutes, so all we have to do is find out which track. You'd better take your ticket, haven't you, that's going to be on the same side. That just shows the composition. That's very important if you're going to find a car, because sometimes it's... The general topology, the intuitive general topology doesn't necessarily fit that particular default category. There are a whole lot of related categories, some of which are Cartesian codes, some of which don't have Peano, Spaceville, and Kurz, and all kinds of others. But there's still, in sort of the intuitive sense, that we're taught in high school, you know, topology is about rubber sheets, and this is about deforming shapes and spaces, and this intuitive picture certainly applies to a lot of categories, not just the ones the name has been attached to. We get on the train, so then, what's that? You have to think categorically all the time. Every, you know, every aspect of mathematics, every juncture, we've always tried to disrupt the life cycle of mathematics. I'm just, just, just, really, that is fascinating. Well, perhaps they'll take that model and re-examine some of this topological group, or its non-commutative, you know, equals non-commutative geometry stuff.

17:30 Well, that's exactly what I was thinking. Hang on, let's get to the seats before we go any further. So what exactly, I'm sorry, he was saying, he was just simply saying that he felt that after listening to your talks that he had recognized that it was the whole meeting. Did he realize that the correct attitude was to think categories all the time and not just an algebraic geometry? No, very well. Not consistent, but yeah. He's written non-trivial papers in a number of fields. This is why I was hoping he was going to open up and say a little bit about what he believed. Some things are even in practice here. What? We could have breakfast here. We did already. Oh, we could have it again. Why not? Culling would. I'm surprised he's not bigger. That's how he gets to be bigger. No, but I watched him eat, I was thinking, he's not a big guy. No, I guess not. Well, he's not that tall. It's beautiful, isn't it, up on the hill?

20:00 He's not that fat. Oh, I'm probably fatter than he is, actually. Yes, good idea. Oh, extremely beautiful. Your toes are lovely. It was in, was it in Siena that you had the memorial meeting for Magari at the Certoza? Yes, Certoza Pondignano. It's a magnificent place. Yes. No, I've never been, I have been to Siena, but never to Magari. Certoza is magnificent. They have their own chianti on a hill outside, as of course the Friars did. It's a marvelous place as well. It's a beautiful meeting place. It sounds like it was a very productive meeting, too. You and Angus, I think, spent quite a bit of time talking on that occasion. Angus told me that that was at that meeting that he first really got the connection between the work that they were doing on the tasker problem and the topology program. I liked him very much. He had a good feel for it. Yes, he is a great guy. And that was where he first heard the words, you know, the natural numbers, the root of all evil, from your lips. Right. He heard them and immediately said, that's right. That's right, yeah, he immediately got the point there. We could wait a little. I'm thinking, yes. Cappuccino. And after, down, a little more. A little more. A little more. Perfect. Perfect. Perfecto, perfecto, perfecto. That's absolutely perfect. Could be absolutely right.

22:30 Let's see, where do we go from here? I'm not as well prepared as David. By the way, is it... do you say his name Davide or... Yes. Let's do that. Davide, Davide, Davide, that's how I've been saying it, but I just wanted to be sure I was saying it right. Davide. I'm good to buy the ticket on the train itself, right? Yeah, they let you do that. I think that sometimes though they do charge you a supplement for doing that. You probably have to go to... Well, I'm definitely going to bring you to your hotel because you haven't settled. Yeah. Oh, really? That's probably a long way from the station. Well, I'll tag along if it's okay, because I'm not going to be going back to Paris for tomorrow, I don't want to travel overnight on the train. I did that on the way down and it left me really, you know, I'm going to just keep staying in Milan tonight anyway. It's okay. Because only one of those bags is mine. Oh, well that's even more, one more reason why. It's okay. We've got to pack light. I'm normally pretty good at that, but because of this meeting, I brought a whole load of stuff that I didn't really need to bring, and it's never even got looked at like that. Paperwork, if I had a laptop I could have seen all of it.

25:00 He had this huge mountain backpack that he said was a library. He never looked into it, but he was prepared to do anything. I do that too. There are lots of books in it. He knows it might be there. He's still waiting for the day that they have to download... Download all the books. ...the books and the little things. Yeah, yeah, I'm still waiting for that day too. Huge. Well, it'll be a long time until the books we want to read are available like that. Yeah, well, we're probably going to have to... Tom Sawyer and the Bible or something. Well, nothing wrong with Tom Sawyer. No, but I know what you mean exactly. Yeah, well, they'll get around to it eventually. What's wrong with the Bible? Oh my God, I almost got the spoon! We will also be bountiful for anyone to use us for printing, to do laptop publishing rather than desktop publishing, as that would be useful. What was the paper that David was showing me that you had just recently written? Because I suddenly don't have that one, the one on continuum mechanics. Oh, you don't have the categorical, algebraical, continuum, micro... Definitely not. I haven't seen that one. Is that on your list of papers on your website? I think so, as revised. Thank you for watching this video. Thank you for watching.

27:30 Thank you for watching. But my impression was that you had to pay quite a lot of money even for an LCVF. Oh, I know. I see, right. I download things through MathSciNet. Yeah, did you see? Yes. JSTOR, you can access JSTOR through MathSciNet. Yeah, but to be on MathSciNet... And all that's paid by my university. Yes, exactly. But anybody in Buffalo... Yes, sure, that's the difference. Anybody in Buffalo can... Yeah, yeah, yeah, you can, but that's because you've got a university link. I haven't. And to be on MathSciNet, I think it's something like $800 for a private subscription. Well, you know our. Oh, golly. If you go through, you can go through Buffalo University. Well, I suppose technically it's dishonest, but I would be very grateful because I'd love to see that paper and a whole lot of other stuff. But I'd be absolutely killed to get onto MathSciNet. That's crucial. I've wanted badly to get onto MathSciNet. My last talk, my last report talk was... ...or a slightly improved version of some of that more in the paper. Yes, well that's certainly the most highly developed version of my original program. Well, that's the thing I most wanted to study. Oh, yes. Okay, right. No, no. Well, that's... I assume you have everything else. Actually, I don't. This is one of the problems about being isolated and not being, of course, connected with the university, as I said before. Well, no. Birkbeck, I did have. But the trouble is Birkbeck because it's so strapped for resources. Uh, they don't have those links, and, um... Okay, I don't know the site offhand, but if you put in to Google University of Buffalo and Libraries, you get the Libraries page. If there's a problem, I'll just back off and wait until I can chat to you. No, no, no. Oh, I'll give you my email, so you can... Yeah, I was going to say, give me your email. You've got mine, haven't you? Well, you will have. I'll give it to you.

30:00 Yeah, that'd be brilliant. That really would be helpful. With a very embarrassing typo on it. Disapprove. I just put this on the back. Sorry? No, you're an epidemiologist. Sorry. That's brilliant. Was that a genuine typo? It's not a mistake that everybody makes who doesn't know what epidemiology is. It was hard. That's wonderful. Do you get people coming up to you and thinking that you are a skin specialist? Oh, it's a big joke among the epidemiologists. And then I'm also a massage therapist. Which of course covers those, so they assume that's what's going on in the skin trade. And I'm sure that brings a lot of slightly off-color jokes about the skin trade. They make you put in your initials, but I think it has to be capital. Yeah, okay. I'll remember that. Thanks very much, Silvana. That really is very helpful. Thank you. Well, after everything else you've done for me already, I just... but that really is... I think this would be a rational way to proceed every day. Yeah, I mean, I would... The university pays this, you see. I'm quite happy to send a payment to the university. Anybody who does their internet access through the university automatically has... I've been trying to figure a way to get on that signet for about two years now, two or three years. Basil Heine kept promising he was going to do something about it, but he's not very good on following through on things like that. Nice guy though he is. Is that when you come to a particular review you see? Can you get math reviews over? You can get, you can get, sure. What I've been looking for was math. That's fantastic. Yeah, the link to... Oh, that's fantastic. If you come to a particular review, then it says, you know... So you can link into JSTOR through that as well, or do you have to take a separate subscription to JSTOR? Probably they pay more, but the mass sign-in at JSTOR is all paid by the university.

32:30 The British universities just don't have those resources. There's a thing now where any article, even if it's not online, you can fill out this thing and then they scan it and they send you a PDF and you can print it. It takes a day or so. Scan it. So you never have to go to the library. Okay. If it's not already online, they'll scan it. I don't know, maybe that's what the model is. And your library as well. Well, kind of electronic form of interlibrary learning, yeah. Yeah, some of us actually like to go to libraries and study and read books. Us old-fashioned guys. Well, it's not a question of wanting to, it's a question of time. Yeah, true, true. A particular section of the library, look at all the books in that section. Exactly. I suppose eventually they'll get to the point where virtual reality software will be such that you can, as it were, put on a headset and go, as it were, take down a book from a shelf in virtual reality, you know, real kind of matrix stuff, but I don't think for a long while. No, I was being flippant. Nice idea, yeah, but I mean, the root, the motive for it, of course, will be to get rid of all the librarians. However... So this article, Categorical Algebra 4, is in Math Reviews. It's not reviewed. Like they've done with many of my articles. They apparently failed before the task of finding a reviewer, so they simply repeat my own abstract there. They do that a lot. It's very unfortunate, you know, itching to find out what somebody thinks somewhere about this that you never find out. The second thing is that they have these totally incompetent reviewers who do write their own reviews. They haven't understood what is written. I've got some incompetent reviewers. Ah, I see.

35:00 The comments on the history of the Hilbert series would appear in the volumes. Yeah, I'll call back here. That's one of the ones I wanted to... That was reviewed by a competent reviewer, Anthony Michael Barr. Ah, yes, that's quite a good review. We just got one or two things wrong. Thank you, man. You know, it's a good review. It's good in the sense of actually telling you what's in the article. There's an uncanny coincidence between these two classes. Reviews that are good in that sense, and reviews that are... I wonder why that is. I think I can guess. Well, in fact, that article on the development of toposters is one of the ones I had a couple of questions about, but I guess the one I want to ask about, and I know you talked about this, obviously, in the toposters given, because I didn't get to all of them, as you know, especially not the fourth one and the last one. Yeah, I just want to test my understanding of this whole issue of the higher and lower connectivities in the spaces that are, you know, the calculus of the spaces. You'll remark this is in the appendix to the SDG paper, so the 1998 lecture notes in SDG. This is the 1998 lecture notes in synthetic differential geometry. Subtitles by the Amara.org community Yeah, I have a lot of questions about the Birkhauser one as well, about Petty and Rotorbo systems, but the, you know, the issue about applying basic topological intuition, synthetic, that we were talking about just now, before we got on the train, in Florence, in terms of the contrast between the category of sets and the categories of spaces with cohesion and variability, and obviously it applies to...

37:30 Okay, I've got a pretty good feel for the definitions and obviously what a connected object is. It's an object where the maps to the discrete space. So I understand the study of aesthetic components of the space connection with homography and homology. I just need to learn a lot more. I'm not at all clear about this last remark, but the set theory, the needed set theory is best derived from the geometry X by defining discrete spaces to be those C, those for which the ejection C in CS is an isomorphism for selected figure forms S, which are considered to be connected and which have at least have no non-trivial co-product decomposition. Now, can I, can you explain about the, well, maybe it's not a good question, about non-trivial, well, I mean, that's just unusual, that's the basic intuitive idea of the vector system, if you try to map it into a three-two point space, you don't necessarily map this to one, you try to map it to two, well, that's the definition of extensive character, you map it into two. Oh, I see, that's the definition of a non-trivial problem. Well, I mean, x is equal to a plus b, which would imply that a is equal to zero, or b is equal to zero. Right, okay. In other words, the... If you like, the functor represented by X from the category into sets, if you like to think of it that way, preserves sums. It's the wrong kind of representable functor to preserve sums in general. It preserves products, trivially, but if it happens to preserve all some sums, it's the presenting object that's affected. So this is kind of, in some cases, the precise definition, and in many cases... The other thing that's being talked about there is precisely the one I used in the lecture, I guess you'll be able to see there.

40:00 Which lecture last time? No, I missed that one, unfortunately. Well, anyway, so, a very simple idea. It's an equation. See, s equals s to the t. t is the chosen... t is this... I mean, it's the germ of all motion. The germ of motion, yeah. So it's certainly an example of an object which is deemed to be connected. So if we start from that side of the object, in particular, it has no nontrivial D, but it's much more, I mean, it has much more properties. It's having much more connected even than that sphere statement would say. But somehow it's deemed to be connected. So then we define the discrete spaces to be those S, for which S to the T equals S, in other words, every path. All of these terms are parametrized by t, and s is actually a constant. Ah, okay. So the math from s into s to the t is the inclusion of constants. But that's all there are, only constants, as a property of s. I spoke about this relation s equals s to t. It's a very important binary relation between objects in a Cartesian closed category, which, like any binary relation, gives rise to a Galois connection. So if you're given a set of objects which are deemed to be connected, t, put them in the exponent, then those s's for which that equation holds... These are the ones that should therefore be deemed to be totally disconnected or discreet or conversely. So conversely is the naive idea. You want to say that 1 plus 1 should be deemed discreet. So you apply that relation, you derive the things t that should be considered connected. So there's this Galois connection, classes of objects. And then the closed classes that are maximal, descriptive, relational, connected, and discrete. You change one, you increase one, you decrease the other. So, as usual, the story with Galois connections. But this is a kind of basic relation. In fact, as I remarked, it even makes sense for a hiding object. You see, to say that...

42:30 These are compositions of T. T implies S. It's the opposite of what we tell C. T implies S proves T implies S, and that's because you proved S, of course. But in this case, it does. T implies S entails S. So again, that's a very strong relation, very particular sort of relation between a pair of different kinds, which is sometimes valid. And it's the basis of Joyal's elegant construction of the burdening topology for which a given object is a sheet. This is one of the constructions of the SGA-4, that if you have a given topos, if you have any set of objects, you want them to be sheaths, you want the smallest subtopos, which contains certain, subtopos meaning sheaths, containing some given object. So what is the LeBere-Tierney modal operator small j, affects that, given. Let's say given one object, one object to be a J-sheet, what is it? Given the object A, what is J? So the actual formative of that in terms of twistor involves, in another way, this, the Galois connection, where that same equation is applied to the implications of the equation, which is very fundamental to the idea. There are a number of categories, both propositional and colloquial. Right, I mean it's all there in those two words in a way, but you have to... Yes, I know. There is a great deal packed into that, and that's why it wasn't a totally naive question. Taking two to be something that should be deemed disconnected, then you get a corresponding notion of connected, and that's sort of a naive notion of connected. In my work on infinitesimal motions and all this, I actually take the opposite tax, which I say is a specific realization of Cantor's program, because Cantor's idea of extracting from the collision thing and the abstract set, that's what we're doing, but in a more controlled way, because we're doing it in a way that will, by the way, will necessarily...

45:00 The bonus function will necessarily preserve finite products, always, given T, which is one of these amazing right adjoint type objects. When you define the subcategory of those F's, then there will be a left adjoint to the inclusion, which is like a pi zero, for this notion of discrete. But it will automatically preserve products because of the nature of them. Yes, yes. So, you see, if you apply this definition in algebraic geometry, for example, in other words, the topos of sheaves are finitely represented in the community of algebra over a given field. So that's the cohesive basis. And P, of course, is the famous Leibniz, Studi, Koehler object, the spectrum of the two numbers, the elements. All of these are generated by epsilon squared equals zero, or generated by the Leibniz rule of differentiation. So that's T. You take that T, which is the obvious one you're telling us, and what are the T discrete objects? So let's say among the affine schemes, there's a specter of what kind of a ring A? The topology is an ask which looks discreet with respect to these nilpotent infinitesimals. Thus exactly those rings A, but since they're all rings B, there's no derivations of A to B except zero. That's a characterization of separable algebra. And the standard topology that you have on the big topos is restricted to this one. It gives you exactly the Galois topos. Cheeks on the finite field extensions of space. This is really exactly the, what proper topology is doing Jawa, and then the pi zero, and I always knew that pi zero should preserve products there, but it's a consequence of this totally general pi zero, which goes to Jawa topology, preserves products, the naive one, the naive one based on this, the one plus one, and what you said, does not preserve. That's what I'm saying, that crucial, a crucial point of view.

47:30 In other words, even before Cantor, Galois discovered that the proper base of those was the array of geometry, and not totally abstract tests. You see, one talks about the Galois group, but... Yes, you made that point in the Unity and Identity of Agilent Opposites paper as well, I think, about the position of the Galois topos at the zero, at the kind of zero level of this hierarchy. The Galois group, in a way, doesn't really exist if you're over some other kind of topos, because it's the automorphism of the separable closure. The separable closure requires... An action of choice to sort of condense all the finite field extensions into one big universal field. Andre Weil, mistakenly, was in love with the sort of universal domain, which enlarges the, we've got the continents, and then this big, one big field, everything that's inside of it is Cartier, it's the right of way. That's a totally wrong idea, you see, because this big domain doesn't even exist. But it's based on a sort of unnatural flopping together of the field extensions. One of Galois' basic discoveries was that in the field extensions, there's more than one mass. There are all monomorphisms, but there are not as many monomorphisms. In particular, you have a monomorphism. So if you're going to try to take sort of a border-direct limit... You have to make madly choices all along on which conclusion you're going to take, so that sort of thing now doesn't even exist, because non-trivial theory can move to some other forever space. So the true version of Galois theory I claim is precisely this topos of Varshii. Varshii simply means every map is a covering, every inclusion is fields. Access and covering, that defines the group of things. Apology, excuse me, it's a... Every map of fields is a monomorphism. We're talking really about the opposite side, which is a site, which every map is an epi-morphism. But not, it's not a groupoid. Most of these maps are not inridible. I mean, the endomorphisms are inridible, but not the...

50:00 But, nonetheless, they should sort of act as though they were in a room full of sheets, it's a very limited notion of sheets, which makes the topos boolean, even though, you know, you need to start off with a groupoid. But, uh, you can start on a map where all maps are epic, but take all those maps to be covering, which, you know, in the first place, it's not like getting across the street to New York to the store every morning, you know, I work, but you do get a boolean, which is not a general phenomenon in my general setting, but in that particular case. You get this miraculous bonus that in fact the notion of discrete space is a Boolean topo, classical prejudice, you know, that it should be Boolean. The other one isn't. It's actually real life. It's actually real life, pardon me. There's a difference between Poole and Galois, you see, with the first modern geometries or algebras or whatever, 1820s, 30s, 40s, in a period of 20 years. There's this fascinating sort of reality between them. Thank you for your attention. On the other hand, if you take the Grugan algebraic classifier, which I mentioned earlier, that's not a Grugan trokos, but it contains the category of all nucleoids as a reflective subcategory, and the reflection is a Poincare 7-dollars nucleoid. Now, here's a third way to see the actual thing that Galois was studying, which was these sort of zero-dimensional spaces.

52:30 Field extent, that, of course, that drives through a Boolean of those. Even though, contrary to, it's not exactly a group. The automorphisms of this big universal field, if it existed, would be a group which determines the whole thing. In a way, that's an unnecessary complication. The thing you actually want to study is the finite field extension, a very good category that should be studied in its own right, rather than passing through this idealization and trying to descend back down again. I mean, a lot of the difficulties in Galois theory are partly complicated by this idealization and dropping back down again. You have to say then that you're looking at continuous representations in this big group, even though there's continuous representations in the screen set, as well as, you know, fuzzy stuff in zero-dimensional spaces, and in the groups, and in the end you descend back down to the finite field extension. And to what extent was one of the motives that Grokowicz had in formulating the topos concept, I mean derived from that? As he preached to me many times until I finally got it to Bradford, I must have a setting where a point can have an automorphism. A point is really a point, but you see the points are the figures in these so-called discrete spaces. And indeed they can have an automorphism. Well, indeed, that's one of the things Carter was talking about.

55:00 He mentioned that in his survey talk, and of course he mentions it in his paper too, but I think it's not in a very clarifying way, to some extent. And that of course connects up with the action of the orbits in the orbit space, doesn't it? I'm taking pericalisers for the notion of the orbit space, like in Aton D's. The action? Yeah, the action. The idea of a pi zero in the left edge line for the speed of fusion is quite broad, you see, because there are, as I say in my book that sets the mathematics, we tend to use three different names for it, depending on the actual flavor of the totals that are involved. So if we think that the big topos are sort of more active than the lower ones, then it's appropriate to think of these as urbans. The idea of components is purely becoming, giving rise to being, rather than saying, well, this point is cohesive with that point. Because one of my admissible notions takes it that way. There's a way in which that part can become another. You have to pay attention to the parametrize. It's a model of being, rather than generally being an area in which you can have become. Yes, I recall you, Alberto gave me an account of the conversation which you had with me, I think in Como in 2000, I think on this very topic. The other two words which I couldn't understand at the time are points. When you say points, it's components of points. This terminology is referring to the idea that, well, the Big Topos is more cohesive than the other ones. In other words, less cohesive, so then you have points and components. In the case where action is the main difference, becoming...

57:30 This is explained by being third, which is this phase here. And you have orbits, but you also have fixed points. So instead of points, the corresponding adjoints should really be called the fixed points. The stage phase is quite big, and they have this becoming going on, and some of the points are fixed, and those are the points that will be extracted by the right adjoint to the inclusion, the trivial inclusion. Which doesn't have any action, so the adjoint to that is the pitch part. Whereas in the case of cohesion, less cohesion, discrete inclusion, or bot off, that might be a better word than discrete inclusion. In the case you have connected components and points, or the word points somehow. The third case is, what do you think about global sections? In the third case, a global section. Variations, but in particular, you can extract the global section instead of points. See, they're calling it points. It's the same. It's homing into one. In either case, the difference is significant. It's global. Most careful theorists have sort of taken these words without thinking about what they mean. When we talk about global sections, even the case will be more appropriate to say points. I've seen no terminology that could really... Linguistics and correctness unite the three, except to simply talk about left and right. Somehow that retains a meaning which unifies very many things.

1:00:00 Even though traditional language has no word which exactly unifies it, it has different words reaching it. And of course in the case of, you know, pure variation, if pi zero exists, it's also called components. The components of the total space in this thing are very important, mobile sections. Well, they could be called global components or something. That, of course, I'm connecting up with this idea that you always have a kind of automorphism. You mean? I mean, to call them orbits, connecting with this point that you say, that gravity was always anxious to get across, but one always has a kind of automorphism. The points might have other points. Well, don't go away for them. That's right. In other words, it's very good. This first, I think, is a constant perspective. The numbers are going down to be close to Galois. That has this action point, doesn't it? You can drop stupor to that. The more abstract sense. And then you have the orbits of the Galois action. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. That's the question. Long ago, maybe 1950, a book called Galois Homology was already talking about the arbitrary opinion of Australian countries on the category of field, finite field extension, as a typical coefficient for a case of Goethe's general idea that cohomology is just the derived function of the geometric order of a star.

1:02:30 Now, why topology is kind of the boundary between German and P, because it lives, because it's German, aren't it? What she was saying, what kind of topos is general, because the idea of it is the category of all spaces is one of a certain ilk, whereas particular means you have a particular space or a particular group and you're looking at the variation or the action. On all possible discrete sets of such a particular set, hence the idea that topos was around, as you pointed out, it was around in the 70s, the topos theory was just a matter, a theory of parametrized variation over an index set, and that the internal logic of the topos plus the understanding of the parametrized... The amount of variation was all there was really to the subject, but that missed out the whole dimension of cohesion. Yeah, I don't somehow know. I mean, people were saying, you know, every model... Put those away for a second before going down. Thank you for your attention.

1:05:00 The library, sorry, the library. Well, you know, rich men don't think that there should be museums or art galleries because why on earth should the masses be allowed to enjoy all this art. We should be allowed to buy it out and keep it just to look after ourselves. People like Lloyd Webber are absolutely furious that the paintings in museums, why can't he buy them and have them for himself? It's terrible. It's the same mentality. And it grows by what it feeds on. I just want to go around and get some food. Do we have a menu? I don't know if we have a choice or whether it is menu, dill, jar, no. They have a choice here. Well, I'm not that hungry. I think the menu is probably... Oh, it's a set menu, though. It's a set price. Yeah, it's a set price. Oh, okay. Well... You choose for me. If it comes around, the Scolopoli, Polo, Boro, Salvia, I think, for me, it's all right. I'll be back at the unions. We'd better house. Hey, I'm coming in from the west of the left side. Italeto, in the west of the left part. Excuse me. No, sorry about that. This way, then. Ocupando.