2 sets remarks — Map between spaces & noncommutative geometry

0:00 It's what turns out that this minimal category is much smaller than what, you know, has many fewer maps than, which, you know, which I'm saying is good because the other ones are garbage, and it's so minimal, you see, this is so minimal that Gates' theorem, I mean, again, already Chanuel had the basic, first Chanuel and me had the basic examples, but then...

25:00 Andreas Blas proved a crucial phenomenon, which is called seven trees in one. So, the typical thing, if you work with abstract sets or with recursive functions, many categories of that ilk,

27:30 once you have an object that satisfies such an equation, then it satisfies all kinds of other equations. You know, any infinite set that satisfies x squared equals x, x plus one equals x, any equations like that are all polynomial equations, they're basically all true. And this is going to be the case in any, as I say, any time you have this universal solution, universal within a certain category, but if you take instead the category containing such an object, which is universal among such categories, it can be proved that the object doesn't satisfy any more equations. See, so the tree equation that we studied, I mean, it's a, Shandingwell's case was a linear equation, which is already tremendously interesting, but negative sets is probably the slowest effort. Quadratic case is... Which is a tree. X equals one plus X squared. Which is a definition of tree. Which is one definition of tree. Actually, I came at it... Blah, blah, blah. Actually, I came at it a different way. Coding. Words and words, encoding words and words, as words, so if you choose such a code, it induces, so you feed one linear equation to the other and you get a quadratic code. Anyway, so, on the one hand, you have to work, you have to strictly work with rigs and not rings, because... I showed that from that equation you can deduce x to the 7th equals x. Just rig theoretically, but the rig theoretic proof is always equivalent to a functorial proof because each time you make a substitution you're applying a functor and functors apply to isomorphisms as well. You can think of this x equals one plus x squared as a description of the sixth root of unity, think on the circles, the sixth root of unity squared, exactly one unit horizontally.

30:00 But you don't get the equation x is one, you get only x seven is x. That's right. Because zero is a previous solution. Yes, I mean, because we're talking only about objective solutions. These must be real objects in a real category and not just abstract quantities. Exactly right. So you see, okay, this is this point on the circle. If you take the abstract circle, we'll have x to the sixth equals one. That makes no sense for objects, but the next equation does, x to the seventh equals x. And in fact, it's a provable consequence just using commutativity, associativity, and so on, and therefore it's true in any category where you have such an algorithm. Can you kind of derive an algorithmic pool? I mean, yes, an explicit calculation is vague, I mean, it gives you an algorithmic pool. That's right, an algorithm, you know, it's one of those... Not a totally mechanical thing, because the words keep getting longer and longer, and then suddenly they get shorter, you know, this is it. Is there a general phenomenon of this kind involved with sigma atomic equations, where if you used to render them positive, render them rinked, you had to keep them doing something similar? Yes, yes, yes, yes, right. Right, so, but now, okay. Whoa, whoa, whoa, whoa. I see a new word. Yes, yes, that's right, good. I've been working over the past five years with the Stashev polyhedra. But Stashev polyhedra are just, I mean, a geometrical realization of the combinatorial tree. Yeah. But then the crucial thing is, conversely, see, x to the seventh equals x, that's a wonderful... It means that you can encode seven tuples of trees as single trees. But the encoding is of a relatively simple nature because it's just deduced functorially from the basic Isomorphism and injections, projections, and so on, you see. So it's a very, very simple proof that you can encode seven tuples of trees as precisely. But then you might say, well, what about triples of trees? No, you cannot. Gloss proved in this crucial example, and then Gates in a much greater generality, is that, okay, you have such an equation, then you have the Riggs theoretic consequences, they're all going to be true, but nothing else is.

32:30 In other words, if you have an isomorphism between two polynomials applied to this figured object, then already it's provable in Riggs theory. Here's me always trying to drag it down to the elements. What does this encoding look like? It should look nice, right? There should be something pretty about the way you do it. No, no, there are various ones, but yeah, I mean it's... And it should show you why triples aren't going to work, right? Yeah, I mean, because it amounts to, okay, in terms of data transformations and so forth, it means that you, the general morphisms in the category, in the category that's generated, in the category of its products and co-products and distributivity that's generated by such an object and such an isomorphism. You start with the isomorphism, so whether you can see that or not. Of course, there could be different isomorphisms, but the basic idea is that you define a map from one of these polynomial objects into another, because the typical object will be polynomials in that basic one, that you have to, you want to know its value at a certain element. Well, you analyze it, analyze it, analyze it, only to a finite depth. And then what the result is only depends on that finite depth. Glass explains it, of course, better than I do. But to your paper I saw a mention of the polynomial factors. I mean, I've been using also the polynomial factors, the one by, you know, the polynomial factors, and so I'm curious to see what kind of, if I can pursue that kind of idea. No, there could be a polynomial factor, more or less. It's a young tableau, so it's just more or less, more or less, more or less, more or less, at least part of the combinatorics, as it is in the book by Atiyah and the book on symmetric functions by Newton, or the textbook in our committee about Atiyah and Macdonald.

35:00 I'm surprised, but I've been dealing with geometrical realisation, astership, polyhedral, of course I came across the Catalan number many times, but it never came to me that the Catalan number had anything to do with six-foot units. Computer scientists sometimes do these things. They always say it's formal. It's not. It's actually objective. Objects are calculated by maps, by projection maps, even though isomorphism was the basis case, representing cohomology by cases. Well, of course, when you mention a ring, I mean the first k0, k0 is not a ring, it's a ring. And of course, in algebraic geometry, people distinguish between an odd distinction already in the Italian days, which is one of their major contributions. Effective geometric cycles are interpositive worlds and they form a ring, not a ring. One of Shannon-Well's basic observations to see was that, well, in fact with any abstract rig, but in particular the one that arises by abstracting from a category with general multiplication, is that you could not only tensor with z to get a ring, you could also tensor with a rig where 1 plus 1 equals 1, getting a 2-rig. The first one assigns to an object an invariant inside a ring. He calls this the Euler characteristic, which it is in many cases. But the other one, you see, where you force the addition to become idempotent, doesn't destroy the multiplication at all. The multiplication is still non-idempotent. This reflects the dimensions of the objects precisely, because it's a very intuitive idea.

37:30 Suppose I have two spaces, each of which has a dimension. I take their disjoint union. What's the dimension of that? It's the maximum. The maximum, which is inimpotent, you see. The maximum of xx equals x. So your force... I mean, the usual notion of dimension is a logarithm in some sense. What is a what? Logarithm. A logarithm. We're treating the dimension multiplicatively, but it's the same information. A vast number of cases. Like case theory, you see case theory says, well, you don't decide a priori what the dimension of something is. You produce this K group where you have the values of other dimensions. So again, in all sorts of different situations, it may come out to be the natural numbers or a given category. There may be something more complicated. There may be something that's more simple. In fact, with these higher degree polynomial equation fixed points. X equals F of X, where F is a polynomial, monic polynomial, always part with positive quotients. This dimension rig reduces to three levels. I mean, you have the empty set, which is like dimension minus infinity. You have all the finite discrete sets, and then you have infinite dimension. So anything which really contains the symbol X, I mean, all that contains the objects are polynomials of X. So anything which really contains X has dimension infinity. But nonetheless, this stratification into three dimensions is crucial for understanding how the thing multiplies. Because basically, this X to the 7th equals an X business. Well, you go around the circle, but you also jump up into the next level and then back, basically. So you have a sort of three-sheeted version of the complex numbers where this stuff is going on and passing to the Euler. Passing to the ring, or the Euler curve, is essentially the collapse, which is then an abstraction. It's no longer representable by actual objects. I remember it was partly a joke, but not exactly a joke. It has some meaning. I remember that when Cator produced his very complicated proof of any complex variable, it was a very complicated proof, and I remember Cator making the joke.

40:00 I have a circle, a logical circle. I went to the Universal Corp, which means you introduce a new parameter, and each time you make a logical circle, some index will increase, and at the end you have a reasonable proof. It was intended as a joke, but it is not a joke. Contradiction, we were here and we are not here. Expanded into another parameter called time. As a picture, it's Devisage. Yes, exactly. And is it some more, I mean, is it like Rodney's Devisage? Yes, it's more or less the same. But I think Rodney took his inspiration from the proof and from his discussion by Cantor. He had first the sketch of the proof, but of course... But you think, quite specifically, the gravity of the text in which it is going up, down again, to a capital intent that it does maximum. But I think it's quite correct. It is not a mental joke. I suspect it is, do I think it was? Very much, I mean, I said that, you know, use his job, but I'm not sure that he was the author of the job. I suspect that Duaty was the author of the job, because it's more typical of the, and of course, Captain was his student, best student, best student at the time. But it is precise. What do you mean? We have been planning for years at Buffalo to have a seminar on Duaty's thesis. Uh-huh.

42:30 He never got around to it. I was in Princeton in 15, then came his thesis, not yet published, not yet defended, and at that time there was no electronic means to transmit, so people in Princeton wanted to have it, and then I visited Harvard and Mumford wanted to know about that, and so finally I received a very I was a very slick male, and even making copies was not so easy at the time, and I suppose it was Spencer who urged me to do something. Spencer was a very, very positive force, a very positive force. And then when I visited Harvard, I mean, they took me for an eight-hour long talk, an eight-hour long talk to understand all the things, maybe that they understood better in eight hours at Harvard. I remember that was, and I think Mumford, and then we went very long, and I apologize, at the end I apologize, and Mumford said, well, it was worth a cold dinner. But I wonder if Duody would do it the same way now. I mean, after his, after his bornological proof of Grauert's direct image theorem, it seems, at least to me, however you do it, it somehow involves a genuine use of functional analysis inside a topos. And so there are many, many different ways that that could be, could be done. I'm so sorry to cut you off, so we're going to have to move because otherwise Paul Carrillo is going to be left waiting at the stage. And I'd like you guys to start, so we could get the corners running, and then I guess we can. Well, yes, you know, start officially, as I say.

45:00 It is possible for me to come back to you? Okay. Thank you very much. Yes, I'm sorry. I think I would abandon the idea of this kind of, you know, relatively fun point in the mathematics work program and just talk about this instead. I must admit, I would really like to have got a discussion going about that paper about, you know, the whole issue of what, how much is contained in this, from our point of view, nearly a point. I'm taking you right up forward to the Hedwig Holt and Savitz programme, but if, you know, given what started here... Well, see how it goes. I mean, clearly, as you see, the hair has... Yeah, big hair has started, and, you know, and if this is the hair that they then want to spend the rest of their time creating, it would be an awful pity for our theorists not to get an exposure. I think we'll try to go back to that. My idea of Raptor was to cover the objective downward theory in the Lovier-Shaniel one. Yeah, I think that's fair enough. But obviously one has to hang loose. We have one really major problem, which I realized about three seconds after we sat down, which is, one, where is the car? If it's not, it's still outside John and Mimi's hotel, isn't it? Yeah, I guess. He's got the bloody key. Jesus Christ. Well, we'd better get it here fast, otherwise Leo is not going to have anybody to meet him.

47:30 You could always add some more. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. John left the other hotel in the car, according to me, 20 minutes ago, so where the hell is he? I feel like no idea. Yeah, but that could be him now. No, it's not. The thing is that, you know, we don't know where he is. It is crazy. He left about 20 minutes ago. He couldn't find the car, even though the hotel was here. With him available to input on Grothendieck and specifically on this well-known remark of Grothendieck that points are never merely points, which Pierre took as the leitmotif of his expository paper about Grothendieck and those aspects of Grothendieck's now pursued, or allegedly pursued, possibly not pursued, or unconcerned it. I mean, I'd very much like to have got a discussion on mathematics and geometry at points while Pierre's here, but if you feel that while Pierre is here, you'd rather discuss the objective number. Yeah, if you think that's a more valuable side to the use of the three hours or so that we still have left with him. Sorry, which paper were you talking about?

50:00 I mean, it would be nice to discuss it, but it is also on paper, right? Well, yes, exactly. Yes, yes, yes. Yes, it would be. Okay, well, perhaps you could run with the objective number theory, and then if there is time to come back to the Grosvenor-Devon points, would that be reasonable? Okay, we're just saying here, while you're still here, I had sort of provisionally used this morning to discuss Grotendieck's known for his remarkable points, and to develop some of the issues touched on in your expository paper, and just generally discuss Grotendieck's views on foundations of geometry. But since this issue of objective number theory has come up, perhaps you'd like to spend just the first part of the discussion developing that further. But if you can try and keep some time in reserve before you leave to touch on Grotendieck on points. Okay, the record is running. Colin and I must go, I'm afraid, and we'll bring Leo Corrie to you as soon as we can. Okay. Thanks. Colin, I feel bad. No, no, no, no, no, no. Yeah, okay. Well, of course, you know Leo Corrie and I don't. Yeah. But, you know, it's... Yeah, okay, but I am supposed to then, you know... Not otherwise. Otherwise things don't get done. No, I know. I mean, and bluntly not for the first time this week. It happened yesterday afternoon. It really hasn't happened. But particularly I'm talking about something specific. Possibly. I don't know. Well, I think... I wonder if you thought I... Yeah, but that's, you know, the thing is, he will come out with these blasé remarks, which is just what I'm having to think at the moment, which don't really bear any relation to the, you know, to the body of evidence. Well, but I need, this is something I need in my career in general. When somebody's saying, you know, he brings up something about Grotendieck, which is clearly just hasn't read Grotendieck on his particular subject. Grotendieck's a really important name, and I ought to have a view on it. But what I need to do at that moment is smile and say, what an interesting thought. Let's look at it this other way.

52:30 Yes, tactically and psychologically, I'm sure that's right. It depends on who you're dealing with. I mean, there are plenty of people robust enough, not thick-skinned enough, not to be bothered that way. Well, no, I wouldn't describe John as thick-skinned at all. I'd describe him as... Well, that's the thing. He can be very thin-skinned and at the same time completely impervious to, you know, anyway, what's the matter with all of that? Well, now you've told me that Leo Cori is a pretty organized guy with a character involved in the rest of it. I haven't got any problems. But, you know, I thought, never having met him before, that he might be a rather, you know, sensitive. Thank you very much for your time. Well, I'm sure I'll be talking with him, because we've hung out before, and I'll mention him. I won't get across. I won't. It wasn't you, it was John. No, well, OK. Don't get across. I won't be bothered to mention it. But Mimi says that John left 20... well, that would now be a problem. Oh, OK. So where the hell has he gone to? For Christ's sake. I mean, don't tell me he's got lost. I mean, for Christ's sake. He's now done that trip at least a dozen times. This is unreal. I wish, I mean, I don't know, I mean, John isn't a fan and I really regard him as such, but I do wish to God that I could invite him to join us. I wish to God I'd sort of just strong-armed on Albert, you know, overcoming his, because probably else he would have done him a lot more good in terms of boosting his general morale. Yeah, I mean, that was an absolutely fascinating discussion. Actually, even the one that you were having with Bill before, across to the cafe, and I, you know, I didn't mean to kind of cross it, but I don't think it was a good notion. Yeah, but of course, the good thing is that John is also a tremendously good friend of Angus's. They also hit it on, I mean, they also, you know, bounce off one another brilliantly, and John obviously really, well, obviously really respects Angus, without saying, and no, I think that John would have listened and gotten as much out of Angus, I think it would actually have been a...

55:00 My consternation when John couldn't make it to the NEH summer seminar was that I would have absolutely died if I'd had to put up with the pleasure and the stimulations of his company as well as everything else. The thing is, he would have also in some of these discussions, although obviously he doesn't have any background in algebra and geometry. You know, he would have been a very serious and attentive listener. He wouldn't have started fidgeting and, you know, moaning and saying, oh, this is not my bag, let's go off and have a, you know, he would have listened and very, and when he thought he had something to say, or when he saw how some of these constructions, particularly this fraction, a chance to build, might relate to his own ideas about, you know, simply the system, he would have had something, he would have had good questions, really good questions to ask. And the thing is, the best thing about John is that he does. There's a lot of help to force Bill to clarify things, to make them clearer than he would otherwise be making them, because if John Maybrick can understand it, I mean, there's an answer I can't write, which means that it's been made really clear, you know. I would like to dip in here, but this Plato paper I wrote said a lot about John and Sistine. It was a really clear statement of what I was attributing to Aristotle. It wasn't that I needed more evidence. I needed to say more clearly what I was... And the thing is, especially on the general philosophical question, when Bill is talking about the Sistine thing, John was really forcing to make these things. They're crystal clear and precise. There's one thing that I wanted to bring up after the O'Gores, I'm sorry, after the O'Gores rather, because I obviously wanted to major in mathematics and geometry while he was here, but you sent me a list of suggestions, and one was a discussion with Bill on the, I don't know if it was something, or maybe on the, it's now going to be one, one and a half days at most, that we do, you're here, you're here sharing, as to this very general principle, which you think is...

57:30 Also in Grothendieck about, you know, the way that the universities work. Do you remember you said that? Mm-hmm, yeah, yeah, yeah, yeah. Um, it's, uh, it's obviously not the... Well, I guess it is a little, it obviously is connected with, you know, the class set description, although, you know, the Grothendieck or whatever put it in those effects. Um, it's, but it's not simply that kind of course. Yeah. It's, um, and you saw the imprint of it in quite a number of books. I've noticed it doesn't work if you press the answer point. It either has to come out in the course of his own course, in whichever direction they've taken him, or it's going to come out. Unless he's speaking to somebody like Garth. This is something he said to me about one of his papers I was going to quote part of. He said, make sure you tell him about the middle because, you know, it's like all my papers. You really can't read the beginning or the end, but the middle's pretty good. So he's able to know this without forbearing to write the beginning and the end, which of course explains a lot, because all the questions I ever ask him about things that occur either at the beginning or end of his papers, and of course he memorates them, and that's not his business. I never get the second question. He has to enter it his way. He can't enter it without the person that's written about his agenda. I realize this now. This is why there's absolutely no point in my questioning. And I know he's keeping on going, saying we can't have a bit more about, you know, extensionality and choices and how it all fits within the geometry. But no, we'll get there. We'll get there. But it'll cover, as I say, as part of the background for some more specific construction. It won't cover... The only point at which he was ever, as it were, being direct, actually, which was starting from the motivation, was when he was saying that stuff on the very first day of the card game about cohesion. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah.

1:00:00 Not always, and to get it quickly, like an adventure, it can't be said. Bonjour. Oh, excusez-moi. Oh, vous avez besoin d'une, d'une... Le chèque. A rien. Ah, un de Samy. Ah, oui. C'est la belle. Michelin. Oui, c'est Michelin, Michelin. C'est en vente dans tous les magasins, tous les libraires, en... Oui, bien sûr, c'est bon. Merci. Non, pas de tout. Hum... And, uh... Can you buy? Can you buy? Yes, it's like the old Jack O'Neill part of the show, but in about some 1990s. A man told me the other day that it is possible to buy shirts with collars attached. Can this thing be true? Um, anyway, um, no, um, no, you're absolutely right. And it's quite an interesting cartoon, this thing. I find it quite interesting, though, in my head. And then, this morning... After just 10 minutes we were just looking at Bill's, you know, the left and right adjoined data types, he pretty well does it, he pretty well got the whole thing, including, you know, all the stuff about, you know, the QV12 bus, and he actually said to me, just before I started picking up on what you were talking to Bill about, about this point there, it was about, you know, mathematics and stochastic sessions, that... Thank you for your attention. The point of the group of topos is not to isolate two different topos, it's to say, we're breaking the walls of topos, and sure, we're working on a topos here, but we don't care, we'll consult this stuff anyway, and Bill, he's not against doing that, but he's saying you should organize this, you should recognize that topos are there, say what you're saying, whereas they tend to think this is, this is a matter of not worrying too much about topos, because you can work elsewhere if you want to.

1:02:30 But Bill is saying, well, but you should know in what sense elsewhere. And Goethe has a series of exercises at the end of which he doesn't exactly award himself a chocolate medal. The last exercise is, if you've done all the others, you should send a chocolate medal to the editor. He waits for you to pay him with a chocolate medal. For creating these exercises, hell, just a great big kid at heart. I'm just going to ring Mimi again. You could have had another 20 minutes. Pierre's on the phone now, so I'll call him. Thanks, Theo. Which is, again, not the least of the things which impresses me. I've had so much fun and he has been very wide about his life all the time he's been here. I mean, well, she's, her health is really very rapid and she's very, you know, how a boy the connections don't make you so. I've just lost it. Well, I'm sure she has. It looks like she's very, it's easy to say that, but all the time. What are we talking about? Almost a mile for Kate psychologically that time, I know. Well, she's certainly said to me. I'd love to be helpful. Yeah, you really were. I mean, you were. Obviously I wasn't there, but I... She said to me afterwards, when she and Dickman actually helped me bring the last of my stuff over in January this year, when I was... You know, we had a long talk on the way there. We had a nice little, we took two days, we left Bristol and we stayed in Rouen the first night and then drove down, and then they went and took the car off and went and drove around the wild valleys for a few days, but on the way down we had quite a lot of, you know, the last few years, but that came out very clearly just how much, you know, she realized how much you've gone out of your way to try and provide support.

1:05:00 It seems two and three, I think, give ten successive proofs of that theorem, using more and more sophisticated machinery. It's a really nice way. Yeah. And of course it outlines the point. And it emphasizes the machinery. Then you already actually know this. This is just a way of organizing what you already know. Sir, but can I ask a question? If you have, well, one other way to put it, a little piece of a Riemannian manifold that has zero curvature. It might as well have been beauty and end space with just the Euclidean metric. Curvature is zero. And this is Riemann's. Gauss proved it for surfaces anyway. I guess it's Riemann who made it n-dimensional. And this is in Spivakon differential? Yeah, yeah, it's copy introduction. Five volumes. Well, I didn't really even get into volume three, but I learned the parts I know, so... There isn't a kind of a condensed condition. Oh, no, no, no. Well, in the event I win the lottery, I'm going to treat myself to that, isn't it? Although, in that case, that's not the book I thought you were talking about. I thought you were talking about his calculus of manifolds. Yeah, well, that's a sweet little book, but it just doesn't cover near as much material. Well, yeah, obviously. It does. It does Stokes' theorem the really right way, where it turns out to be everybody's theorem. It's Green's theorem. It's Gauchy's integral theorem. It turns out to be everything.

1:07:30 Yes, that's the one which ends with the letter to, Stokes' letter to Thomson, I think, and then the last one is that, and if you've understood this, you're a mathematician, and that's a bit like, of course, the reader. Needless to say, I didn't understand it. It's certainly not the first five or six times I've read it, but we've got a bit of appeal now. Because, as you say, it is everybody's theorem. It's gas, it's green. I've never quite got this business of the relationship between Cayley and Cayley's theorem of the unitum. And indeed, you know, got the kind of... Are you often calls it the Cayley-Unita Grotendieck? Yeah. I'm not quite sure what he means by the first and third. To call it Unita Grotendieck is that Unita didn't really seem to ever state it. Grotendieck uses it all the time. The discussions on this have always been... I've never gone to Yoneda's papers. Peter Pride insists that actually you can find it there, but when Peter talks that way, he normally means it's not there. But I can see how to reconstruct it very easily. That's a serious topic for a short discussion of the history of philosophy. Yeah, yeah, yeah, yeah. That's interesting, I've never heard that. I was assuming it was just straightforwardly stated by him. It's always referred to as the legal evidence, even if it doesn't go back to stated. Uh, hang on. Go right the way around here. Yeah. Oh, no. That's what I like. No, it's okay. Keep going, keep going round. This is very confusing. Do they mean Sainte-Reveille of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of, of,

1:10:00 So they're telling us we're in Laval, but yes, you've got to go through it, which makes sense because Alençon and Laval are on the other side of Laval on the main road, except for Laval, I'm totally confused now because we just saw a sign that had the bar going through Laval. Oh shit! No, we're on the ring road. But if we were in Laval going to Saint-Tropez, this can only be the Saint-Tropez of Laval. No, but it's got to be Saint-Tropez of Saint-Guyte. I don't understand, because there's a sign that showed Laval with a barn. Yeah, yeah. True it. Well, I suspect it's just a perverse thing where you get downtown by leaving the... Wait a minute, well, let me follow where Hillard is on the map, and then I should be... I should be possible to tell. Here we go, Laval. Okay. Okay. Leave the one. Saint-Tropez, okay. Panic over. What was throwing me was the Laval with the sign for it, which usually means you're exiting Laval. Yeah, so it is there. Yeah, it's got a bit of Alençon and Lamar now. You basically take the ring road all the way. Yeah, that's right. I had to find the GAR, which is on... Just straight up something. Pick up GAR, SNCF, or something like that. Yes, and it should be on the right bank. Yeah. It's actually a rather pretty old city. It's got a superb chateau and medieval walls and a really magnificent cathedral and a nice old quarter, but that's all on the left bank. The right bank is where the station is. It's a bit more sort of sensory. It's a perfectly nice city, but not especially worth hanging out in the lake. Certainly not as nice as Rome. It's the Cayley theorem. Cayley says a group can be represented as a representation group.

1:12:30 I don't know what... As a permutation group. As a permutation group, yeah. You can look at the set of its elements and it acts on that set. Yeah. Each element acts on it by multiplying. Right, yes, that makes sense. Yeah. And the reason that's the Yoneda lemma, Yoneda says, well, consider all the sets acted on by this group, there's a Yoneda embedding, and it's faithful. That's the Cayley representation, is the faithful. Okay, the Cayley representation is the special case of the faithful embedding. Yeah, well, yeah. The Cayley representation looks like it's just this one group. Well, the group taken as a category only had one object. So a function from it to S looks like one set. Looks like one set, looks like one set. Okay, now in terms of finding the stations. Straight on. And I think you'll find a sign here. This is the Mayenne. Oh my goodness, this is a pretty place. Oh, it is. Yes, well, that was pretty. Not as pretty as Rhin, as I said, but still very pretty. And we're on the right side of the station, so just keep heading on along here. Yeah, keep going. No, the station is up to the right, I know. No, we've got to go along here, and then somewhere along here you take a right turn and... You take a right turn at the station. That's down to the left. So, yeah, the station is somewhere up here on the right. You've actually met Leo Kahn, right? Oh, yeah. Oh, okay, so you know him by sight, so he's learning that, right? Yeah. The intention of bringing you and Jean-Pierre, Marquis, Jessica, and Elaine Landry and a few other people together, and plus all the people in Paris. Blackson Martin, Chignac, who are not specifically, you know, historians of category theory, but you do know a fair bit of them, don't you?

1:15:00 Yeah. And Leo, and Jeremy Gray, and because Bill was going to be in Paris that week. I think it was because of the Amiens centennial, the Parisian. And Benabou had said he'd really like to come and talk about the history of category theory. When he learned that Bill was going to be there as well, he got even more enthusiastic. And he spoke to Cartier, who said yes. So I thought, well, Cartier, Benabou, Lorvier is a pretty good nucleus around which to get some serious historians of math and some philosophers of math who would like to learn more about category theory. Going through the history is the easiest way to get philosophers in there aware of the issues, so just let's have a history and philosophy of category theory meeting arranged around Bill Venable and Cartier for three days in Paris, but then of course having arranged it and got the room booked with ENS and accommodation and everything else, Bill suddenly says he can't come. Thank you very much for your time, and I look forward to seeing you again soon. I understand now. That is the Chateau, and I was turned around. We are heading in that direction. I thought this was the left bank, but in fact this is the right bank. It depends on which direction you're looking at. Usually it depends on... Ah, yes, I see. Yes, that explains it. Because it's always left or right relative to the flow of the river. On this map, the river is actually flowing from top to bottom of the map, whereas I was thinking it was flowing from bottom to top. So hence, of course, this is the... I assumed, like an idiot glancing at that map, that this must be the right bank. But anyway, that's what the meeting's about. Now Bill's pulled out, I'm just left with Benabu and Cartier, but I think it's still going to be a very worthwhile exercise. And Jeremy Gray has said that the PSHO will give it some support.

1:17:30 And although he keeps telling me that, you know, he's going to get lots of money out of the ENS for it, then I'll believe that when it happens. I don't trust any of his assurances. And I'll just about have enough in the kitty if we get what Jeremy and Gary's provided, plus the registration fee for the non-speakers to finance it, provided that bloody Avro Radar doesn't keep inviting people from Switzerland and places like that. I don't have to tell him to de-invite because there won't be enough money to pay for that hotel in Paris. There's enough money budgeted for you and for Jessica and John here and that's pretty much it. There's the guard left. Yeah, I mean, I'm sorry that Bill is not now coming. He hasn't said anything to you as to why he's had to pull out of Amiens. No, no. I was a bit worried that it might have something to do with Matthew's health, since he obviously doesn't want to say anything. In a sense, to do this, to push him. But I don't think it's for any... He's got issues with Man of Heroes now. Oh, yeah. But I don't think they're the reason he's not coming to Amiens. Ah, that is the station, that's the buildings here, the head, that's the building. Oh, yeah, that is the station. Um, no, I think it's something to do with, he's taken on some commitment, I don't know, but there are issues with it. Uh-huh. No, just going back to what you were saying about, uh, the logistics of having, uh, one of these, uh, National Ordinance of Humanity kind of thing. Yeah. There's certainly no problem about getting accommodation in the town, uh, for people. Um... I don't know how the international media... Ah, there's a bus. I bet you Leo's just on the point of getting off.

1:20:00 Well, we'll just pull up in front of the bus and see if he's... Thank you for watching. I'll check the bus timetable from the Fougere. If there was a bus to Fougere and he's not here, we must obviously assume he got on it, in which case, just drive around into the station forecourt. Just drive straight up here. I mean, since we're not stopping, we're just going to drive in front of the station and see if he's coming. But not follow those cars? Yeah, I would say follow those cars. This is very confusing, maybe... Oh, no, actually we do have to go in here. Oh, no, but you can't, because those cars are parking it. That's taxis only. No, you want to go in here where this guy is coming out of, once he's out of the way. Very confusing. Oh, maybe we're supposed to go in that time. Sorry. Yeah, I know, this does bring us around. Ah, no, it's five. What are we doing? Shit! I'll tell you what, just let me jump out for a moment. Okay. In fact, can we not just pull in here for a moment and... Well, that guy is just coming out. Just get out and look for him. Well, I think there's parking over there. Well, yes, just let's pull in here for a moment. I suspect he's long gone. Well, yeah, because you know what he looks like and I don't.

1:22:30 There's more to say, but he has to prove it's wrong. That bugs Bill in a few places, and there are times when Bill feels he gets the ideology wrong. He states something in a way that Bill feels is just completely wrong. ...meaning not stressing geometrical morphisms. Well, no, he'll say that of you did set theory without elements. And Bill insists, no, I did it without elementhood, and it's a really important big conceptual distinction. It may be, and, well, yes it is, clearly, but it's one that, in fairness, Bill himself has. I could have made clearer to his listeners for the years. ...without elements for 40 years now, so people say it. Yeah, and it really bugs Bill, but, you know, Bill partly got himself to blame for that, not having tried to make himself clear. I know that's just a question of temperament. I could understand after he explained, you know, why it was important for him to stress that, no, no, it's not without elements. Yes, Ed's had elements. He defines element on page three of his dissertation. So it's not without elements. It's without epsilonics. And element is a defined term. Yes, it's not primitive. The point is it's without membership chains. I was able to tell him some things he hadn't noticed about the early history of homology groups. You know, that's a detail. Where I met in Israel is Sir Fatih who is also invited. Yes, yes, Michel Sir Fatih indeed. He came for a lightning meeting. There was a two-day conference on Leibniz in Paris, which unfortunately I missed. I've been going down to Paris to listen to some of his... He has a seminar on the history of mathematics, after Gennari-Parcari, and he has had some good speakers at that seminar, he's had some very good speakers. He had a couple of very good talks on Tantor. Some of them have been of uneven quality.

1:25:00 The rest have been very good. He himself is always preaching to the French historians about mathematics and your standards are just an order of magnitude below that of the English-speaking world. Because he doesn't say that to do anything, you know. No, no, no. He actually cites it as, you know, go and read Corrig, read the client. This is how you should be doing your subject. So, you know, he's batting on the right side. No, it's a sign to figure it out. No, it's straight on. We've had this problem before, if you remember. No, it's definitely down the hill here. It's, um, yeah, this is definitely the right, I'm sure. Yeah, definitely. And, no, he's an interesting guy. I regret now that I didn't take up his offer to have this meeting in October at the Ars Titulari 5K. I do suspect that if it had been at the... In that case, I'm glad. In that case, I'm glad I transferred. Okay, good. I did a good deed in that case. I must admit that thought did not cross my mind, but it's... It'll be the same. Of course, if, actually, I hope you don't mind me asking this, but do you think there's any chance that in, for the October meeting, I've asked you and Pierre and Elaine the same question, do you think there's any chance that you might be able to get your department to cover any of the trouble? I'm assuming you won't. I mean, I'm budgeting, but if you did, it would be very helpful. I've got this guy, he, well, he's been tremendous for my career and he's just about to retire.

1:27:30 And he is giving a paper at an international conference in Brazil. And this is one of his complete pay-your-own-way things. Chintai Kim. No one you would ever have heard of. He's got these two manuscripts on Kant that he's been working on for 40 years. Yeah, because, I mean, the man has nothing to say. So he said, I'd pay his way to Brazil. And that's, in my department, that's the budget. I was working on that assumption, but you know, it's probably worth asking because if there's anything you can get out, I'll look and see because I'd like to know what it is. Is that a Chinese name, is it? It's Korean, actually, yeah. His father was a Congregationalist missionary, I mean, of Korean birth, and so he was very important to him in the first place. They're really rather silly because they're basically very confused ideas. Oh dear, no, he's not a... I don't even dislike the guy personally. He's actually quite the fun to hang around and have a few drinks with, but he has this extraordinary attitude which I've come across in Russian, but totally cynical, you know, he's just, you know, what is going to help me, you know, get a job in a French university? And, oh, you know, I've found this kind of weird Englishman who pays for meetings on... I don't know, he's very wild, and he, you know, so he's inviting this guy. Hey, he's forgotten to invite his head of the parliament. Oh, hey, well, I'm sorry, I'll just put that right. Obviously, he wouldn't want to invite the head of the department. That's inconceivable. He understands French academic politics, of course. He can't be such an offending item. I'm sure Hetzmann is not offended by what happened to him. And if he is, I'm sorry, but that's André's problem here. The guy is not a category theorist. He's got a very good PhD student who is. I mean, you imply the same principle. You can't invite everyone. It's crazy, just think it through. But it just seems axiomatic that I would... Oh, it must have been an oversight of Michael's part. He must have intended to invite Heinz to this. And then I have to say, listen, you know, can't be done uninvited. Or else he goes to a loony and says, oh, it's all right, we'll get some money from the NS to take this.

1:30:00 No, a loony says that they will, but they won't. And then you end up in a completely disastrous situation. He's already told me that the, you know, the budget is completely used up. There's no chance that if the budget here comes up to the money, they might be able to reimburse some of it out the next year's budget. The guy you put me in touch with that made the video, great guy, yes, he's given me copies of all of the, the, the, the, the, the film, the... I'm just really sorry I didn't get to your talk on Minkowski and Einstein. Yeah, he also told me in the end, look, I didn't know it would be a good one. He had the big names, you know. There were two Nobel Prize people over there. Yeah, you know, this guy, he went to make the movie for the French television. He gave a great talk. He's a great guy, you know. Very humble and very... Yes, I've listened, I've listened to the video, it's very interesting. There was also David Gross, but that's a guy so full of himself. What an appropriate name then. Yeah, exactly. John Stanchel didn't talk at the meeting, surprisingly. He spoke in Paris at the time. He actually gave the best talk. He gave on Einstein in Leeds about a week or four of them. There was just a little Foundations of Physics meeting at Leeds which Steve French organised, which was mainly Young. It was mainly, you know, on this theme that, you know, they're only, you know, it's quantum gravity, but it's still the structure, it's still, you know, and, you know, it's an assumption that that will be a background independent theory, so therefore this is all being structured as it's saying, indicates their position about, you know, the nature of trying to do various, but he gave a talk, well he actually gave a talk about GR, yeah, yeah, I went to record it all, yeah, for this archive, which is, as I say, my kind of... Did this guy film there, John Norton?

1:32:30 John Norton wasn't at the Leeds meeting. No, I mean, he was in Israel, but I asked if the guy was there. No, I don't think he did.