Andrei Rodin

0:00 You know, why was there a lot of very good people? Because, you know, because he had just a point system. So he invented all these things. No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. The idea that you are going to make Ukrainianism worse or that you're going to, by abolishing total state-timing and direction, and in the end, I'm sorry, but what I'm All of these lectures, with a tiny handful of exceptions, which is what they've done, have sort of locked in for me. I mean, it's an extent that very intelligent people are very, very intelligent. They're just proud of themselves for the fact that they are existing in this, what they, a system which they insist must be a construction of their own excellence and their own intelligence. Just to demonstrate, they're not. You yourself have taught intellect. The media wasn't lovely. You know what exactly it is. It's a Columbus, not academic problem, quite the same learning as the Eiffel Tower, but for instance, that's my point, of course it's not, and that's what was a disaster in Russia, because they didn't try to change the system, and then it's like Gaia, that all they had to do was to sign a decree, and somehow it transforms the system overnight, the one which had been in March and was resurrected in the second year. And one which even but before has got essentialism.

2:30 There's no way, if there were a Supreme Court in Ireland, that the idea that they could simply rule the acts of freedom unconstitutional. I think that we do remember this constitution played a role in the early 90s. Yes, yes. But then from the moment that the European people were allowed to fire on the... It's a great tragedy. But you're right, you don't change the society, it's not the part of it. And you don't actually change it by just changing economics. No, I mean, of course, if you introduce the autonomous, we're talking about here, in the Russian universe, which is composed of a lot of power, you're going to get people like, to be honest with you, probably much, much worse, people like... We just want to point out that Matthew Crony could be the boss of the university, actually I think it's the case of the, well actually in the case of, in the case of, in the case of Adler. I think that's exactly what he did, because that guy who's the head of the violin is just, you know, he's kind of a linchpin. The guy is the head of the... Actually, it must have been one of his... Right now it is called Gijsdorp of Termanen. It's very close to the region. It's not fully private, but actually...

5:00 I'll try to... Did they get... Did they get... I don't know why you sneer, I'm just going to say it right, I mean, yeah. No, no, no, they have huge money, you know, they sell this kind of branch, and also they have huge informal grants. Just say something there, just briefly. You don't even have to get that close to it, I'm just speaking absolutely normally on the street, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Again, record again, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Well actually, there's some of the points that... I completely understand you don't want to get on the subject of Benabu, I mean it's psychosis, but these are actually scientific points you raised concerning the five categories that I want to ask you. I'm not excited at all about it, I mean it hasn't any value. Okay, well, unfortunately Greg's got me raided a few, crazed, crazed, and I hate to build abuses. Very sad. Oh, well, it's very good to hear you say that because I also felt really thrilled once you took out the grave of my personal attacker on Peter. But some of his points about, they've been closed. I've been to both categories. I'm subjective. Well, I rather found scientific points. OK. Right. Well, I was going to ask you... OK. I had commented... You know, by the way, he admits that it was a planted question. Do you notice that? What, Peter admits it was a planted question? No, no, no. Vandenberg says, who was reading Johnstone, he got the idea that he had to write this 50-page letter. Then, he entered the category of this innocuous professor. I told you I thought this was a conscious plant, and he's admitting that he was.

7:30 He was trying to provoke something. Very bad, very bad behavior. Anyway, his attacking Peter Pride is completely off the mark because... As I said in my email, innocently answering this question back when this started, of course, Broderick knew that the Wolbach's private products are products in the slice category. So even if your original meaning of the word Cartesian is just products, which is I think what he's saying, then of course... Are products in another category? Slightly different category? Sliced category? So should they be excluded? No. So finite limits are... Yeah. In other words, well, what I'm saying is, in fact, this thing is so basic that the localized version should not be called localized. If you want to put it that way already, locally Cartesian. Never mind that I keep Cartesian closed. Never mind closed. Just locally Cartesian. Yeah. It's called Cartesian. That seems to be fine. And since Bright had, moreover, a historical explanation, an actual content, namely that the cart solved the equations. In fact, he used the equalizer, not just the direct line. But putting in an actual point of content is significant as well. Just when he gets started attacking, he attacks all aspects, and this is completely off the table. Well, you've actually answered the very question I was going to point out to you, which was that I'm reading through it. I felt, I'm very limited in understanding, but I think I've understood enough after reading his big paper and studying the history of descent theory and how vibrant categories arose and growth groups were. It does seem to me he is making a valid point about the... That within the framework of private categories is the natural setting for studying locally confusing closed categories and of course the terminal object is. Locally Cartesian, yes, exactly. Locally Cartesian, the fact that the fibers might have internal homes in the base is very important, but it isn't yet at this very basic level.

10:00 But his point about Fryden and Shedrock was just completely off the wall. That is totally, totally true. But the thing is, Rotendijk used the word Cartesian. All of these were Cartesian morphisms in any hybrid category. So this now is a vast generalization of what we were just saying, because if you take objects over x to mean just morphisms to x, you take all the slice categories as your family and the vibrations corresponding to that, it's really just a morphism category projected by co-domain. And finally, a general vibration is the locally Cartesian globalized, let's call it Cartesian, concretely, but now a general, a general vibration, well, by analogy, even just by analogy, in that case there are certain morphisms and there are the vertical ones, which play a special role in the description, so in a general vibration it would be called Cartesian. Moreover, moreover, concretely, just by abstract analogy, vast numbers of vibrations are derived from the first one by using the pseudofunctum, by composing just with some two functions, so you could take, you could take, uh, I don't know, you could say... Pseudofunctum being Grotendieck's original terminology for what became index. So, you know, so you could take, you could take, let's say, I don't know, vector spaces in the category of B slash X. What I'm saying is that there's a two-factor which assigns to any category with products the vector spaces in it. So, you're given a vibration. All of this is viewed as a pseudofunctor, and you can compose it with that functor, giving another pseudofunctor, and then look at the vibrations that that generates. In other words, so that's the vibration of vector bundles, or in general, not any finiteness conditions or anything, but vector bundles over at the base. And it's derived from the one with the pullback. Functorality, of course, this is derived by the pullback.

12:30 As applied to the distinguished morphisms inside of the total categories of vibration, related both by analogy as well as by vast numbers of concrete examples, but with this particular one, which is that of a whole mass. If you call pullbacks Cartesian, which is Bride's minor, then you have it. So this is my explanation of how Rotendieck arrived at what John does, he takes Rotendieck's definition, he talks about product, you can only use the word product, you can only use the word Cartesian for product, and then immediately... He takes the actual point of view that Gotendieck is God. Gotendieck defined it, so anything we can derive from that is okay, but my question wasn't really Descartes at all, it was Galileo. So if you want to discuss this terminology on the basis of actual historical context... You should really call it the Galileo. Co-Galileo and co-Galileo. Yeah, instead of quibbling about who did it first. Yes, absolutely. No, I agree. I mean, that's just complete rhetoric. You could have a real one. Total rhetoric. Sorry, so I used to have a complete story about that. No, no, I'm very glad to hear it. No, no, the index category, first of all, he claims in one of his emails that index categories were invented by me. On the other hand, he claims he's never going to attack me, but he attacks viciously the index category. It's not due to me. I hate the word index. I always hated it. I would call it parameterized instead. Otherwise, I think volume 661 of the, not 666, 661, as it looks to him, looks like 666. I think it's a very fine piece of work by my colleagues Roseburg, Johnstone, Graves, Wood. It's about fiber categories, they know that. If you look at the damn thing, in the preface they give him an enormous amount of credit, some of which he doesn't even deserve. That's true. That's true.

15:00 There's one little flaw in it that he doesn't like, which is that as an expository device, instead of wrestling with all the coherence of the pseudo and the isomorphisms, Along with every category, there's a distinguished sub-category of isomorphism, and not all isomorphisms. Yeah, this is his point that they're too hung up on equivalence and that they should be looking at a slightly broader notion. They should be looking at subjective rather than just iso... subjective maps rather than just iso. Yeah, it's related to that. Related to that, yeah. But this is an expositional device, and they know it. Yeah, yeah. To make it simpler, in order to get at the real content, they introduce this which... Indeed, might turn out to be a complication. And that book is based, to a big extent, on suggestions of mine, which I made in a seminar in Halifax. And then, when I was, as you know, thrown out, Paré and then your students, Rosebrook and Wood. Johnstone and Wraith also continued the development of this basic idea. So did I in Perugia. So the Perugia Notes are largely about the same subject. They are still unpublished. However, John VandenBoo got a copy of the Perugia Notes in April of 1973. In spite of all his claims to the contrary, he actually had a copy of this and derived a lot of what he did later from my notes. There's no doubt about that. You see, the thing behind all this noise is that most of the things he stole himself. Well, exactly. It's a case of attacking for him because that's what we're defending. He stole the ideas. His contribution, as in the case of Topo's theory, was the first actual document that was circulated explaining the theory. Fine, this is an important contribution. And that's the sort of thing that he did with private categories, with the profunctors, going back even earlier, anyway, oh, fractions, categories of fractions, so, but these marvelous expositions of his, in fact, for the most part, are still not published, so he hasn't really, he doesn't publish them, but I have tried, largely, to find a category space, yeah.

17:30 He published it in the Logic Journal because he calls it foundations. So just, you know, it's great sympathy. He uses category theory. In fact, he's the one who proved the famous occasion, as you can see, to look at some of this stuff. And so, in the review, he first of all says, well, it's this and this. It's mostly a diatribe, but why doesn't he even mention simple things like... Hybrid categories are better. First of all, bullshit better because the whole point is that there's a lot of dialectic between the two. It's exactly like sub-objects and truth-value functions in a way. The fact that the two of these are equivalent is what gives all of the exactness properties of the topos and all the power to them.

20:00 And well, in the same way, in category theory, the fact that you can have pseudo-functors into cat with values on A. Or fibrations on A, and that these are basically, well, you see, again, they're sort of two equivalents, simple-mindedly so, but nonetheless, they are basically equivalents. This is the whole point. Just as people did in geometry for years, you use some kind of a map to represent families. Family is a family of fibers of the map. This is your metriconic sections. So to say one is better than the other... Just completely misleading way of thinking. But then you see the real thing is... The big issue is about stability under fallbacks. You know, when he goes on about this business about subjective, perhaps being the needed notion rather than isomorphism in some context, it's because the isos don't always preserve, you know, they don't always preserve, equalize, co-equalize, equalize on the fullback. Whereas the subjective frames of the concept and the sense there are always there. Actually, I have to think about that. That seems to me to be a telling point. That's the sad thing, he speaks out sensible, good, scientific ideas. They don't make that distinction. But I'm trying to get to the main point. Sorry. Which is that the book called Index Categories, or what I call Parametrized Categories, or the Perugia Notes, that body of material, that subject, We all realize, they realize, and I realize always, that the fiber picture is one of the two important ways to formalize or view the content, but the content is a special case, so really the issue is to get some special properties of these vibrations, which express the idea of families. It says that, and then doesn't stick to it, you see. In other words, that there's a... I mean, families on two levels. Any fibrations is a family of categories, parameterized by the base, of course, but the idea is that the fibers themselves should in some sense be... there's a fiber over one, which we can call a category A.

22:30 And the base is topos perhaps. Again, the base is not a general category. It's an extensive category or a regular category. So if we take an object I in the base, we want to express the idea that the fiber over I is the I-fold families of objects of the fiber over Y. Right. This is certainly, if you take that in a concrete sense, that's certainly a vibration, but it's a very special kind of vibration. The theory is not necessarily a discrete thing. So the purpose of this theory was to extend the Eilenberg-Kelley theory of enriched categories, which permitted you to parametrize, let's say, specifically parametrize morphisms. If the POM-AB is not a set, but topological space... Then with the topological space, i, and parameter, you just consider a map from i into om x y, and this means continuously parametrizing maps. So enrichment, in a closed category in general, is basically that idea. The base category is to parametrize workers. There are many cases where you need to parametrize objects as well. So you want smooth families of objects. What does that possibly mean? You need that even for the proper treatment of the theory of the rich categories. See, the rich categories per se are still just an abstract class of objects. Between the two of which you have a nice object worth of morphisms, yes. But it's still merely an abstract class in Cantorians. So if you want this to also have a smoothness, I call it an atlas. You can cover this thing with actual small smooth faces and do it that way, but they at 661, they didn't use that idea, but that's just a way of putting it. This is the content that we need. And Benhamou, although he mentions the idea of families, he doesn't even do this. What are the concrete properties and vibration of families?

25:00 Over, let's say, an extensive category, let's say, of a pretopos and topos as base, what are the extra properties beyond being a vibration? And this is the thrust of this theory, but she hardly touches it all. The thing Vlas asks is why didn't he consider the condition that the fiber over x i plus j is the Cartesian product of two fibers, the most obvious feature of families, that it should be the family parameterized by a disjoint sum is just a pair of families with no units. So even this axiom, he doesn't consider that. So he's not really caring about the damn content, he's just caring about... So I have that axiom in my, they consider that, and I have it in my vision. But the less trivial one, of course, is precisely descent. In other words, if you have a map from i onto j, then a family parameterized by j should be included into a family parameterized by i in a good way. So there's an exactness. There's a big deal about the descent. It's really because of the gap between affine schemes and schemes, in the sense that affine schemes is actually a bad category. It's extensive, but not a pre-topology. This is what we wrestle with all the time, why we invented extensive categories, just sums, but the sums and the quotients are not universal, so a special role must be given then to these. This is the universal effect of the epimorphism. The universal effect of the category. Or the regular epimorphism, as we call it in English, right? You're talking about the language, aren't you? Anyway, so generally, the base is not so good, so you have to isolate the good morphisms in it. And then the set is...

27:30 This is one. You're going from terminal two, aren't you? Probably. I'm sure you are. I'm sure you told me where you came in. There must be where you came out from. So I'll stay on for the last knock. No, I'm 100% certain. Well, if I'm wrong, we have more than enough time in turn to get back. But I'm 99.9% certain. So it's Axiom's life. The simple one that Glass mentioned, and a suitable version of descent, which involves more investigation, which, but then they're getting into the natural number object in the category of topos, which is race, or the race, I think, or most of them, most of them. Anyway, there are deeper, besides just those two axioms, there's a further investigation of this idea of concrete family, so to speak, smooth parameters. Fibration, whose entries are smoothly parameterized, that's the subject matter as opposed to the subject matter of general fibrations, and that's the big difference, not this difference between fibrations and pseudofunctions, that's supposed to be a pseudo-equivalence anyway, and... And I certainly take your point that discrete fibrations is... I still haven't told you the real kicker. Which is that in January of 1967, 40 years ago, I was just starting at the University of Chicago and was a visitor for a semester. We were stuck in a meeting because of a huge snowstorm in New Orleans. Stuck in New Orleans because of a huge snowstorm in Chicago. That sounds meteorologically plausible, I have to say. Oh, sorry. Got it, got it, yeah. So we had to take the train. The 14-hour train ride, I devoted almost all of it trying to convince the thick, small beta bool of necessity for a theory of family existence. And he resisted. He was against it, I told him. Yeah, yeah, yeah. The whole idea, basically. Same idea that, you know, thinking it over, thinking it over, I presented to the seminar in Halifax three years, four years later.

30:00 I went off to Perugia, but John had the seeds. Then he comes out with it as his own idea. We were in Montreal in 73. I gave him, and many other people, the Perugia notes. 74, he comes and gives a summer school in Montreal. There are 74 lectures a year after. Seven years after I deranged him on the necessity and one year after I gave him the complete set of perugia. And he still didn't get it. But the point was, it's like that also with the pro-functors. He claims to have been the originator of the... In fact, it's his name. People use his name. He heard me lecture about this in July 1966 in the Nobel World Park. And again, I spent hours trying to convince him even of the possibility or the necessity of such a theory, which he resisted entirely. And then a year or two later, one of his students is writing it up. He claims to see that... Well, I just thought you should be armed with this. No, no, I'm very glad for you to give me this. I already knew that he was a hypocrite as well as, well, I don't want to say. I don't want to dissent his level of, yeah. And, of course, the way he turns everything into a personal fact is so against that, so against science. Which is, as I say, that's pretty, because he does have, it seems to me, some interesting, well, certain points that are very well worth, you know, serious scientific discussion in the middle of all this crazy raving. It's important to get the history right. It's very interesting when he was, when he was, you know, when I was soaking up all the raving, you know, to keep it away from you, one of the things he kept on going about was, was how unimportant the history is.

32:30 Why, you know, people go on and on about the history of the subject and that, but isn't that exactly what you're doing, because you insist that the history of the subject is everybody stealing the credit which should belong to you. But, well, yeah, yeah, I did. Well, he wasn't listening to anything. I was there, but I knew that. But it's interesting, he was saying that, look, I'm a mathematician, I'm not a historian, I'll leave people to write the history, you know, I can write the history when I'm too old and seen, I'll do maths, but why would anybody want to write, well, actually, precisely because of people like you, that's why it's so important to get the history right. Well, essentially, Huzel and I were discussing this point, and I didn't bring it up. And there he was attacking the very idea of, you know, that a serious mathematician could waste their time on the history of maths at a meeting for Huzel, in honour of Huzel. And then he kept sort of going, but I've got more ideas about private categories. I've only published this, my little fingernail, you know, oh, it's still inside me, oh. Well, I guess maybe it's just business as well. Just do I. Just get the most of it. Yeah, yeah. Of course, unfortunately, as a result of having to get in touch with John, that was one of the things I meant to do last night, was to check up on the history of history. I didn't get a chance to do it. I did, however, get a chance to check, and it appears that Berkeley and Swedenborg, as far as it's known, never met. Some people have actually addressed the question. I got on to the Swedenborg Society website. I think that one does work. Yeah, it looks like it. Oh, you have to take yours out first, yeah. Oh, now it works, good. All right. Yes, yes, the Swedenborgians have asked it, nobody appears to have asked it as it were from the point of view of study of Barclay, but the Swedenborg society are aware that he was in London at the same time as Barclay, and it was speculated that they might have met, but they just say there is no evidence that they ever met. Of course, it doesn't rule without, but there's no possible evidence.

35:00 About Chateaubriand and Coleridge, which I think in some ways is an even more interesting question. I haven't yet started to do any digging, but I will. I'll try and do that before coming up with one. What did you look for in the first one, may I ask? I just did a... I just did a Google on Burt Littlebark in Swedenborg. I got the links to the Swedenborg Society website, but it doesn't tell us anything we don't already not know, so we didn't really get any further. Except that somebody has asked the question. I find that more plausible, that they could have met, because they were both very active reactionaries and conscious reactionaries, whereas Barclay, both of them seriously, exactly, exactly, I mean they had so much in common from their respective, and indeed, you know, the class interest they served, and as you say, the Chateaubriand was actually in the French Embassy in London for a period. It appears he spent, put it this way, he would have had far more opportunities in Coleridge than Barclay would have had to meet Swedenborg, so as I can see, Barclay and Swedenborg only overlapped in London for that one period, whereas Coleridge and Chateaubriand would have had several, multiple opportunities to meet. Exactly. No, no, no, no. Well, if there's correspondence, then that would be very, very interesting indeed. I don't know very much about Chateaubriand's correspondence, but Coleridge was a massive correspondent. He corresponded very profusely with almost all the leading, certainly all the leading English literary figures of his time. I ought to know whether he spoke French. He certainly spoke German because, of course, he was a great purveyor of Kant to the English, or a very distorted, idiosyncratic version of Kant, Kant with all the connections of the science of his time taken out and made into a... But I imagine he... But then Chateaubriand certainly spoke English, so they could have communicated even if Herbert Kant spoke French.

37:30 Method. Yeah. Well, wasn't Grassman? No, no, no, no, no, no. Well, I'm just going to say, let me remember, sorry, you're flying on Continental, aren't you? I'm pretty sure it's terminal C. Wait a minute. I will stay over here. It'll be on there. No, you're not going to Havana. It'll be very interesting. I'm going to Ho Chi Minh City. Ho Chi Minh and Havana are all sorts of destinations we don't even have in the US. No, true, true. But this is obviously just from this terminal, or rather from this hall. I think we have to go one hall further on. Oh my god, it's 10.20. Well, that's okay, it's still only, it's not even, it's only 29, it's only 8.40. We have masses of time. No, no, no, that means the first one they list is well after. No, but that's because that's just in this particular section of check-in. Your check-in is further up. Sure. Do you want me to get one of those luggage carts? Okay. No, I remember when I came... Got my own wheels, man. Okay, ma'am. Okay, hang loose, stay cool, stay cool. I've got to go all the way back to the hotel because the lady, well no, because I couldn't get, she was fast asleep, the lady I needed to pay in order to check out. I couldn't wake her. I couldn't raise her. That's the reason I was late leaving. Oh, I'm sorry. No, no, no. Well, it can't be helped. Now, I had my bags all packed, everything. You had these well-laid plans. I had all my bags packed and ready to leave, but when I went down to the reception and banged on the door, it was locked and she was fast asleep. Obviously. You tell me now about this. I would have objected. No, well, but that's why... You seem to, I mean, so you had these well-laid plans. Well, they weren't that well laid. You were going to leave from here. Exactly. I was going to bring my luggage with me and leave from here. Yeah, I mean, it was very intelligent. It was, but I can't do that. I've got to go all the way back because I couldn't get hold of it. I couldn't leave without paying, obviously. And there was no point in my bringing my bags all the way here just to take them back into Paris, so I left them. Now, hang on. Let's see. I'm sure it's

40:00 terminal, or rather Hall C, because that's where I met you. That's when you came in. So, in fact, Exactly, the solution. To turn them into something that... ...put them at the physical level of the paradox. Yes, yes, exactly. Because... I had no idea about that. Because Olber's paradox only arises if you assume all stars are standard candles, which has been termed one of those. Well, obviously you make very simplifying assumptions. Yes, exactly, yes. Um, but you spoke exactly about it. I had no idea. I remember, I remember John Bell talking about this. He thinks Olber's paradox is a great thing. We would have known, I think, that the problem had been solved. So, I mean, the trouble is the solution to the problem, whether Big Bang Theory is correct or not, whether Redshift is doing what they're doing or not, because a physically rational solution of the equations is a compromise. Now, that's where we had our coffee, so we were getting pretty close, I think.