Morning Discussions

0:00 It was a joint venture of Heidelberg, Zabrokin, Skarsgård, and the Taekwondo Hall, that's the community here in Heidelberg, Gabrielle, myself, and a few other people. And it was an intensive meeting, an intensive meeting in the 60s, and so it's when the very age of us, when we were just finishing... And I don't think you have, well, you have to understand the jobs that we are appointed with. I forget the exact French, but it's supposed to be that Verge was the translator, as Verge has just written up. Presentation of the right categories. At the time when Brody was complaining, I finished it. But you know that, well, Verge kind of, well, let's do his math. And showing out glimpses of the various reasons. But the published version won't really replace Cartel-Islandberg, but it was meant to, and we did all of that.

2:30 There are three chapters, my seniority is to give me a description of the derived category, but not of the derived category. And it doesn't compare to the derived category, because it's really the non-linear. By taking the category of resolutions and just converting scenarios, but it's not a calculus of fractions. You make it a calculus of fractions by first doing the identification and then inverting arrows there. The class of arrows you're inverting is not as nice if you don't form the equivalence relation first. Yeah, yeah, yeah, exactly. Okay, so the inverting is not exact.

5:00 Now, you first go from complexes with ordinary maps to homotopic classes of parts within complexes. And that way you have a calculus of fractions in the now-standard sense, is that everyone's just going to define it or whatever. All you had to do was invert the arrows, but it's not a nice class. But the fractions, instead of being just binary, instead of being words, are just binary, because you can bring it in one major pass to the numerator. Homology equivalences, homology, algebra, physics, and this is what they do also in the non-linear theory where they introduce vibrations, co-vibrations, trivial and all this, where again the main step is said to be inverting certain arrows, but you go through all this first. But I think it's still a really good observation. But I think we still came to a solid end of the door for it. And I remember just when Serre gave his set of lectures, that was first a set of lectures at Collège de France, but then he published his account in the Lecture Noodle, and I remember when that was published, Locke was visiting us. He said, or he took, as usual, emphatically, I mean, he took the new book by saying, now I can burn my copy of Foundations of Archaeology by André Bey. André Bey was not so happy. But this idea of... One reason that occurs to me, I mean, the point to derive category is to avoid taking spectros.

7:30 If you do, you'll have to take Speckholz. That's the very point you said about his oppressed youth. The commonwealth don't... But Sero's not interested in opposing Speckholz. He likes Speckholz. I could imagine it was this complicated spectacle with a good later version about the Riemann rock. It was devoted to the K-group. And this idea that means that the K-group, yes, and that the... Yes, but the... The point is that there are two presentations. You can say that K, K0, group represents a midsole. You first define it by generator and variations. You say to each sheaf or to each module associate a generator, and each successor will give you a variable. But there is another point with complexes. You can say that each complex is finite necessarily to define an element of the K0,

10:00 and that two complexes define the same element of K0, if and only if they are isomorphic into the item. It's more or less the same idea. So, to me, the definition, you take the isomorph, this is the archetype, and then the product in the k-not is immediately obtained, which is an extension of what I said before, from the tensile product. What k-nots are called is Atiyah. Even homology or some other space can be calculated from the k-not, which is one of the best known results of Atiyah. It's even easier, so here we die in Capuchin. To me, always, but it's exactly the same. Isolating us, when I said in the beginning I start with a delta function which has two pieces, and I want to be uniform, and the interesting fact is that you built and you made a coordinate, and then, without mentioning this method, you buy it.

12:30 It's a spectral sequence connector, this coordinate. There are many ways, why don't you...

15:00 Sometimes you have simplicial resolution and co-simplicial, and sometimes simplicial versus co-simplicial at the same time, whatever you say. We have an expectorant sequences, not in a good region, expectorant sequences in this quarter, in this quarter crossing, but the point is at the diagonal, you see. That's when you have, instead of you've got these, they are crossing your square, you've got zeros on the left. Yeah, yeah, yeah. Now I mean, when you climb along the, you know, infinitely they come forward and get slow, get slow in this calculation. I'm sorry, sir. No, I mean more, more the right way. Yeah, yeah, that's fine. But you have to identify the various commas. It's a painful process, a painful process to identify them, because the situation is very complicated, and to identify the problem. But what we hope is that once you have, you will find immediately a good formula to prove their transcendence. I mean, people so far, people in particularly mind, they play with complicated formulas. So far they have not been able to find out.

17:30 Giving a cohomological interpretation. At the end we will be able to say, OK, take this forward. It's cheating. Practically mind that people will, if we speak of a lot of cohomology, they will not understand. And we quickly get to defining real numbers by possibly without points in the integer plane. Motives are not harder than that. I mean, well, sure, particular problems can be. But thinking about motives leaves you with a simple If I can make a suggestion, we've seen a very good illustration of how this rich cohomology, which was developed and then transferred from the analytic into the algebraic categories, is also now being used to attack very subtle issues in number theory, and this might be a suitable place to break and then after lunch go back and look. The way that it was also seen particularly in connection with the relation of both the verbal and verbal captions to have these deep consequences for the understanding of how logic fitted into the geometric picture. Going back to Topos and to Verdi's presentation in La Jolla, which would allow Bill to take up the story from there, would that be a good way of proceeding? In which case, let's go for lunch now. But to switch those off, we thought that the next item to take was precisely to perhaps break away from the and look directly at the toposphere as they're crystallized in the Logier-Tenney axis, i.e. seeing the logical structure as naturally falling into place within the framework of the algebraic geometry.

20:00 And in connection with that, I was particularly going to suggest that Phil and indeed Colin might want to say a little bit about the relationship of the local and global coverings to the logical structure, the organisation of coverings and the decidable solubility conditions, one of the things that picks out the set of set-like topos. And perhaps talk a little bit about the relationship between the projective and injective limits and permanence in connection with choice and intentionality. Obviously other people will have other... I mean that's a specific thing but I think more... Well it's one of the things that we have to... Before you got here this morning, that was one of the things that I had to say very much that we wanted to discuss. Sure, but I think one should go to the difference between what the school, Gordon Deegan, what they were looking for. Yes. Well, that actually was one of the things I hoped we'd cover this morning. And what Bill, you know, this one was actually looking for. And that's really what made the difference, of course, in terms of the... You were looking for something, right, different, or at least in terms of your own motivation. Anyway, that's what I'd like to see emphasized. Well, for the first approximation, that's in fact exactly what we had said at the beginning of the morning session that we would try to discuss. To a rough approximation, I think that's a fair summary of what we said we'd try and tackle this afternoon, so yes. I think that'll bring us back to McLean again, to some extent. No, I mean, who was interested in founding logic, you know, and then some more general framework.

22:30 Yes, and one of the things which perhaps would like me to lead into that discussion would be for Bill to tell us, as it were, what your first reactions were when you were here in La Jolla with this, as I understand it, what was the first expose of Topos Theory 2 categorists. And how soon you saw it as connecting with your own earlier program for expertizing the category of sex without using membership. I'm not planning to give autobiographical details here, but I think after all it might be appropriate because I've just discovered that... The only published biography of myself is totally in error in nearly every line. Yes, and I think the autobiographical details... Mainly the three paragraphs in MacLean's just published book, which seems to be confused recollections of an old man who consults his notes. Well, if the biographical details then naturally lead to conceptual development, which they do in this case, by all means, yes. So it was, I guess you could say it was from electrical engineering that I started with amateur radio and went to physics at the university as a cyclotron technician and planned to study a major in physics but decided soon on that the mathematical level of the physics courses in the mid-50s was not up to explaining things clearly and therefore I switched to mathematics. In any case, I was from the beginning under the tutelage of Clifford Truesdell, who had interviewed me for a scholarship when I was just still in high school and was impressed by the fact that when he asked, what is the trajectory of a stone thrown into the air?

25:00 If you're expecting me to say parabola, I said it's an ellipse, one whose focal points are in the center of the earth. I've heard that one before. I always thought, when I got positive comments from teachers, I always thought they were exaggerating. They were overreacting because this is really only one little thing, right? But it's still a good story. Actually, I was thinking about it again, you know, just recently, because it means that you have some fourth degree infinitesimal, up to a fourth degree infinitesimal, it looks, it does look, it's more, it's not just up to second degree, it's up to fourth, because it's quadratic. So, anyway, so, but, so I started to study under the guidance of Truesdell, but... Contrary to what McLean says, Sammy was not in Bloomington. He had long ago left, three or four years at least, long before I even entered the university, so I knew nothing of Sammy at all. It's not that he taught me categories. So I learned the categories eventually from, actually his name was Ernst Schnepper. He was an algebraic, a geologist, and a mathematician. So one of his courses, he mentions homology. Did you already live from Zorn, who was in Bloomington? Surely he's still teaching. Well, okay, let me say, so I already became, in effect, a graduate student after three years of taking only grad, you know, I had the Van der Werden course and I had general topology and functional analysis, various courses in functional analysis. And I had a teaching assistantship. I mean, this was, you know, it was to support my family that, you know, as an undergraduate I was given a teaching assistantship and so on. So I had an office, which was right next to Zoran's office, and Zoran was a wonderful, you know, gadfly. He would often come over and say, what about this, you know, what about... If a function is one everywhere except zero, what is its value at zero? He thought there should be a rule of inference which would yield this result.

27:30 This kind of thing, it's always sort of playing around on the edges, so it never could be made quite rigorous, but nonetheless it was very, very penetrating. And also in Truesdell's lectures he would sometimes attend and make remarks like this. There was a theory of mixtures, see a very elaborate theory of mixtures, different kinds of substances mixed together and look what they do and you know, so Zoran says, have you ever actually mixed a mixture? That's sort of funny, he was quite a character, I really enjoyed having him there. And also I must say both he, jumping ahead, and Bernays, Bernays is really weird, they both had There was really an interest in k-theory, for example. They felt this kind of thing is really the next wave. They weren't in any way limited to the narrow definition of logic or something. So, one of Truesdell's favorite students was Walter Noll, who was in a certain circle with Society for Natural Philosophy, which I remember. He was for a long time basically the god. We've been hired in Grotendieck in the sense that everyone followed his notation and terminology without any deviation and which, because he himself would change it from time to time and everybody would shift as well, but it was Walter Knoll, you know, it was out of respect for his fundamental contributions. He was He was claimed, I think they dropped this claim after a while, he was claimed to have solved Hilbert's problem. One of Hilbert's problems was to axiomatize physics. And so physics is a broad thing. There are many different aspects. So, Noll had certainly made a big step toward axiomatizing the continuum mechanics part. So anyway, I was impressed, very impressed by the way, which on the one hand, Noel would describe these physical concepts like the body and the sub-body and force and motion and all these things, very clear concepts.

30:00 But then, in order to implement this, one had to invoke, you know, Cauchy sequences, sequences of Sobolev spaces, atlases, a whole lot of presentation, a whole lot of presentation as well as particular determination of the, or what I came to call the degree of cohesion of the spaces, you see. So I felt that really there should be a way to express this kind of physical idea. In a way which takes account of cohesion in general without committing itself to continuous, smooth, analytic, algebraic, combinatorial. So I wanted to provide a language which is powerful enough to express all those things and even indeed powerful enough to, by its own internal properties, distinguish between them and yet to go as far as possible. In order that, it's an old idea, right, that through abstraction you actually come closer to everyday intuition, because these concepts, physical concepts, are close to everyday intuition, whereas the particular mathematical machinery were under so-called foundations for it, or not, close to it. So this was the basic conviction and program that I started with. When I heard about categories, I thought in a reckless manner to change my whole career from having begun a graduate career in one area to completely switch to a different advisor, different place, and so on. So I chose Sammy as a destination because he was the one with the most joint authorship. There was Carton-Eilenberg, sorry, Carton-Eilenberg, Steinrod, Eilenberg-McLean, Eilenberg-Moore, and so there was a whole sequence, so obviously this is the guy who is really the moving force in category theory, which is what I want to learn. Very fortunately and fortuitously... Clifford Truesdell and Sammy Ireland were close friends, not because their mathematical interests were very close, but because they were common art collectors and that sort of milieu, their second lives, were much more closely connected, and so Truesdell simply phoned up Sammy and said, well, here's this crazy guy, you can take him, and Sammy said, okay, so it was just a very bizarre way to get, you know.

32:30 No entrance exam. Again, I felt a bit guilty. Why should they do this? But anyway, so I did that. I learned quite a bit from Stanley when I finally got there, of course, but it was not so. And then, as I later did, Truesdell himself had taken time off from his graduate studies in mechanics at Brown University in order to study logic. Even considering to commit himself to the study of logic by going to Princeton and taking notes for Church, which later became his book. Because of these struggles with foundational questions and category theory, I decided to go to Berkeley and study with Tarski and... No one can remember? ...Dana Scott and William Crane and so forth. The name I knew was Tarski and Scott. So there's again something McLean says in this completely mixed up book of his that he came to Columbia during my first year there and that's when Sammy told him to talk to me because Sammy thought that what I was doing is probably Well, it was dubious. And so, McLean talked to me for one hour and then he decided that this would never work and told Sammy so. Now, at that point in the book, McLean says, LeVere lost his graduate stipend and was forced to move to Berkeley. I don't think that's true. I think my stipend, my meager, meager stipend, would have continued all the same. Sammy was not so easily swayed. But the bad thing, really bad thing that came out of that interview is the slogan, you see, which he claims, he claims that Sammy told him.

35:00 And in any case, he, McLean, is the one who publicized it far and wide as being nonsense, namely, do sets without elements. Some mysterious way to do sets without elements, you see. I never said that. I said sets without elementhood. But the elementhood relationship was giving, again, far too much furious structure to the things that we wanted to simplify. So, it took me a year, and I probably still haven't succeeded, especially with this new publication, to overcome this one slunga. So many people think that's what it is, it's just a... Otherwise, I remember that some of you often, in discussions, would say, I mean, let's try to make a proof, you know... Yes, yes, yes, I understand this, but I mean, it's... It was quite consistent. Well, without elements in the sense of going down to the nitty-gritty of the individual elements. There were generalized elements always. There were always generalized, so-called generalized elements. I always advocate that the word generalized gets used too often. Well, but that's what happens. There are no elements. But I don't think that's going to change. Well, maybe. The very word element, of course, is supposed to connote some kind of smallness, right, as opposed to something more general. Anyway, so at Berkeley I met with considerable opposition, except for Dana Scott, who reluctantly accepted, step by step, some of the things that were advocated, but also Kreisel. So Kreisel made a point actually talking to me. He drove me out to Stanford and back to Berkeley and would come and we'd have these discussions. So later, many years later, it turns out that he'd been on the phone to Gödel telling him, at least according to Dana Scott, telling Gödel about this. So Gödel and Berenice were corresponding actually.

37:30 About this question, without naming any person, but it's the same month that I was writing my thesis, February 1963. Okay, so, anyway, so I did finish the thesis, and then the thesis, of course, presented the idea of using category of categories as a framework for those discussions. It was slightly inelegant, but it had a small error in it. Yes, and it's never, you know, it's a small error. Well, the inelegance is worse than the error. The error was easily corrected, but I haven't republished it because I feel, you know, it should be much better to put out there. Although I think that's the framework in which categories unconsciously work, even if they claim, in church, that they are doing neurogrammatics or something. It's clear that that's not true. So the practical framework is really a category of categories. So then I suppose the real turning point was meeting Gabriel at his seminar because he lectured about essentially the Groves-Aristi topos, topos of algebraic spaces and analysis So, you know, I saw right away that this was the right kind of framework that I wanted, except, you know, to make it better it should be axiomatic and so on. So infinite decimals were really already present. You can see that I have infinite decimals already mentioned, well, it's later actually, but in Gabrielle and Day, Missouri, you know, we don't do any groups as a... Yeah, right. So that sort of idea of a formal infinitesimal, or at least of nilpotent infinitesimals, was already present in work at the time when you first came. Yeah, I mean that book is an outgrowth of the fact that a bunch of that material was presented in the seminar. But there's not too many lies about it. André Weil presented to Bobacki a foundation of infinitesimal calculations of geometry using a kind of new potent element as far as 53.

40:00 And he has a summary among his collected papers, four pages of summary, which he presented in the Eosman seminar in Strasbourg. But a much longer draft which was presented to Bobacki was a long draft. There was much more in Kahler's Italian papers, of course. I mean, the idea went back at least to Studi around the turn of the century. I think somehow it couldn't be properly exploited without the categorical point of view. Just the idea that domains and codomains are definite. If you don't have that, then it all sort of dissolves. That was 1965-66, the seminar on January 1, 1966. Do you remember that date? That's the date which Dana Scott wrote in large, had his nice handwriting at the top of the page where he introduced Boolean value models. There's a well-known book on that. At any rate, this was really, in some sense, even though I didn't fully understand it, this was clearly the signal, you see, that because, you know, this is a small, logically small difference in some sense between the Boolean algebra and what you get by reducing its modulo and ultrafilter or something of that sort, you know, it meant that the previous treatments by logicians had always been tied up with these. Well, you saw it, but neither...

42:30 Well, I mean, I would have done it because they wouldn't know about boolean value models in one hand, and Paul Cohen wasn't into any of this stuff. He wouldn't have noticed it either, although he admitted it. Scott says in his preface to my book, which is a bit stark in the design of what people thought, that he admitted that the real ideas had come essentially from Bob Solomay. And Solomay, I mean, he was the one who first had the idea of summing up forcing conditions to get a little bit of algebra from them. At least that's what Scott says. Yes, that's what that's what he actually said. It was always called the Scholar Scholarly method, although Scholarly never wrote anything about it. No, that's right, that's right. But it was Scholarly who had the original idea and that was... He could have been the one who would have understood. Yes, that's my point. Yes, that's the point. He was the one with actually that kind of, that sort of background in a way. I know the day-to-day is amazing, but... You know, after all, he'd come from algebra and geometry, and so, I mean, he'd done his thesis of Riemann-Roth theorem, and he was a MacLean student, etc. He was a MacLean student, and he was accustomed to the world by key mathematics. Exactly, exactly. But what you're saying is that it's not clear that even he saw so clearly that this was going to fill in a fragment of the Bigger Geo-Mathematics. Oh, absolutely. But he never did. He never did, but he was the one, I mean, any of the logisticians were going to, that would have been him. But obviously these issues of credit are... If we can get back to the old exposition of... Cohen was actually on algebra and geometry at Stanford. I know, but Cohen wasn't in, Solovey became a logistician, Cohen never was a logistician. He went back after, he breezed in, did an amazing work, and then got out of it. Yeah. Sure. Yes, of course, I can understand the line of the importance of... Cohen wasn't interested in logic at all. He was interested in solving problems, and he's such a... For him, he was exposed to quite different... In other words, that year in Berkeley was definitely not wasted.

45:00 Yeah, yeah. Nor was the trip to the Molfac. It was incredibly... So then came... McLean always wanted to hire me. He decided after all that I was alright. You know, there was a whole story about my thesis and how he accepted it in San Diego. That's a well-known story. But anyway, he wanted to hire me at Chicago. So I said yes and oh but could I please have the first semester leave of absence because there was a gathering in Zurich of people working on triple theory now called monad theory and I wanted to participate in that so he said okay so fine so so then but when I went to Chicago it was with a definite program in mind with Marshall Stone because Marshall Stone had developed an interest in these matters. And so I corresponded with Marshall Stone about the idea that the Scott-Solovay model of subsensory were really special kinds of... So he, you know, you can understand he loved all that, so it was he who was really going to greet me when I got to Chicago, except that he wasn't there for a time, and so as a result I had very few discussions with him, but this was, this was, I went there with this program, you see, of combining, let's say, combining Scott Sullivan with... and so forth, with the ultimate aim of simplifying the understanding of physics. So this whole program was present there. McLean again says in his book that he gave a course on mechanics and that's what inspired me. He knew that wasn't true. In fact, it was a joint course. I had proposed it. We were both teaching it. I mean, I'm really, I'm reeling with, I'm surprised, you see, that I got this manuscript, the degree to which the inaccuracies in this book of McLean. Peter, I have a question. No, I'm just... It was long after he left for Chicago. Oh, yeah. Yeah, that is... Yeah, yeah. Just as it was long after Sammy left Indiana. No, no, but the influence of P was deferred in Chicago.

47:30 Oh, I suppose so. I just didn't recognize it as such. Your arrival in Chicago with this program already crystallized in your mind was 1966. Thank you, yes. The last day of 66, December or December 30th. You see, he said that, Bill, that does say, of course, the logic, the Cohen effect was felt enormously. I mean, you know, as Dana says in his preface, you know, sort of like real weight on it. My vote fell. That was when the bomb fell. Frankly for logicians, I mean, they've been thinking about the continuum. It was a specific problem, the Hilbert's first problem, I think, was the third problem, anyway, it was a major problem and nobody had made any progress on it, really. Ah, there have been some by Shecklinson. You know, there were various limited resources that have been, you know, proved. And then an analogician comes along and does it in a very, it looked very specific. I mean, you know, it was incredibly ingenious, of course. I mean, he was an extraordinary, he was a brilliant mathematician. It was very difficult to see what the general, to extract what the general significance of it was. Particularly because, well, intuitionistic logic was apparently. Why? You start with a classical model, you end up with a classical model, and somehow there's the actual methods that you're using. In between it was intuition. Yeah, yeah. Very puzzling. Or rather, hygiene. Yes, okay. But the rules were... And it was extremely puzzling. Now, Emily, you weren't attracted so much to... Well, I tried with all those things before, you know. But the bridge was really... That was the bridge. Oh, yes. To the general picture. Ah, good. No, it was incredible because it really revealed that these models were not these sort of super syntactical things. Exactly, exactly. The core of it was a sort of obsession, a refinement of this. He went out of his way to make it syntactical. We need not read it at once. It could have been less frivolous, could have done a slightly different thing. Kern is on record since then on several occasions of defending others.

50:00 A lot of mathematicians did this, there was this feeling, he went and ventured into logic, even maybe at that time, he had to be very careful. It was Abraham Robinson's first formulation of non-standard, you know, the way he does the whole, it's belts and suspenders, he uses tight, you remember. And then when Marcia Markover came along and simplified the whole thing and made it more freewheeling, you know, rather than this. You had to be cautious. Watch if you were supposed to be coctelious about the details. Right. And I think, I think... Well, if you have to mention all of it. Yeah, sorry. Sorry, I mean, I was in Jerusalem in 64 when you were going to announce... Oh, yes. Announced formalism 64. That's right, that's right. At least the idea. Well, it wasn't a crime. It was still derisive. More firmly. And again, of course, I played with that, you see. And again, you see, much later I realized that there's very little in it that's not in, what's his name, the German guy? And take the power with respect to that, you've got all the content there. The fact that the logic is interpreted in that. So if something, you know, if the interpretation were not conservative, you would detect it already there. So following this by the choice of an ultra-builder, making it a point, making it pseudo-constant, is really almost spurious for any independence result. But there's exactly the same feature there as in the independence group. It's really just a Boolean value. Well, I think things are variable. So I got this whole philosophy of variable sets, not just from that, of course, but from the word general sheaves. In fact, can you say a little more about exactly how that crystallized and the different sources of that? Well, I mean, again, the Grotendieck theory, as presented first by Gabriel, and of course I studied Grotendieck as well, and all these things, I mean...

52:30 There's a variable space, parametrized by the base, and just everywhere, everywhere in geometry and analysis and physics and land logic, you have this phenomenon of structures that are variable, that behave like structures, and they're all the same. So the question is, and you see it all the time, there's a very general answer to the question, what kind of a universe are they living in while they're varying? In the sense that a vector space interpreted in the category of sheaves over a space is really a bundle over the space, but it is a vector space from the point of view, if you put yourself inside this topo, it's merely a vector space, but you interpret it externally and you see that it's moving. It's variable in this sense. No, it's an internal motion. It's an eternal variation, not comparing different models. Not among different models. Varying in itself. Right. And that's when I used to say variable sets. I had to make that precise because the immediate idea is when you're changing from one set to another. One set is varying in itself. This is sort of a typical. But later I realized that this is not quite accurate. Roughly speaking, the Gros topos are having a quality which I would call instead cohesion rather than variation, and that the general topos have both these aspects, cohesion and variation, and that they're sort of pure cases of both, and so on, so that, well, throughout the 70s, I talked about variation. You did, you did. Very effective, which is a very important thing, but, you know, so in other words... For example, remember Anders Karp, under this influence, he was saying, well, the girl with the risky topos consists of variable sets because the ring of definition varies. Well, that's true, it's varying, but the content that one is expressing that way is really the algebraic style of cohesion, more than, you know, unless you pass through a parameter space, say you're doing it over the integers or over a field.

55:00 The way that the ring is doing is really just tying things together to give a model of cohesion more than just the variation in some more precise or narrow sense. So developing that whole dialectic is still underway, but it's still a case that, crudely speaking, it's all variation. It's controlled, smooth variation. Except for the Boolean-Calvinian case, which is random variation. The fact that it's Boolean means that you can specify something here and something on the calculator without worrying about any boundary conditions and you've still got a good thing. I mean, just in that sense. You can only take something and it automatically has a complement that will fit with it whatever the shape, right, of the initial, of the sub-object. Right. This is the condition of the total splits as a full product. Well, not a full product, I mean a product if you've given any sub-objects of one and there's this complementary MD that's a product of two. The topos is a product of the two, which really, well, as a category, it's a product, which in the category of topos, it means it's the sum, so it's just the sum of the, but if every sub-objective one has that property, then you have the Boolean situation, rather than, you have no boundaries, no class. Sorry, Tom, you wanted to possibly say it another way, did you? In other words, in the Boolean case, you know, there are certain natural joints. Actually, there are almost none in the case of where, well, you know, we have a topological space in which you have no cloaking sets at all. In general, you're always going to have these rather large boundaries, and in the cases where you can't, in other words, it's like a jigsaw, where you actually have two things, one there, one there, and there's no way of fitting them together in general.

57:30 And there are very few cases where you can fit them, and there are other interesting intermediate cases where the Kloppen sets, there are some Kloppen sets, there are some complemented sets. Those are exactly the parts of the jigsaw that you can sort of get together, but in general you can't, but anyway, that's the sort of vision I have. Actually, this talk of boundaries, I'll just actually make it precise, because I was amazed last week, my colleagues who mainly work on topos theory, at most one of them had realized that the lattice of sub-topos or topos is co-hiding. The internal logic, if you ask what is the logic of topos, they're two totally different questions because given any topos, then the fixed one, the sub-objects of an object form a hiding element in the sense of the infinite unions distributed over finite intersections, so there's a kind of implication operated on sub-objects, whereas if you look at the lattice of some topos, it's the opposite. A and not A intersect in the boundary. There's a definite notion of a boundary of a subtopos, even though there's no notion of a boundary of a sub-object within a topos, because they do these elementary calculations as you come back to this, the actual... Kearney and I worked out this, Kearney and Kearney worked out the aspect that's extremely simple, that the real deep topologies, the notions of covering, just boil down to one simple endo-map of the truth value, and that is satisfying three simple actions, and one easily calculates that this forms the hiding algebra, you see. Yeah, yeah, yes, yes, yes. Covering operator, local operator, whatever. But then, you see, and then these parameterize the sub-objects. So it's a striking situation that even though these are large categories, there's only a small number of sub-categories of the type in question, and even they're internally parameterized. However... Then they stop and say, well, okay, that subtropos is from a hiding album. It's nonsense because it's a reversed correspondence.

1:00:00 The more coverings, the fewer sheaves. The more sheaves, the less coverings. So it's actually a co-ident. So that means that there is a boundary for any subtropos, which is another subtropos. Totally unexploited, as you see. No one has really explained it. There's an immense amount of structure there which has not been delved into at all. The theory is of direct significance for model theory because, as Grogendieck again pointed out, essentially these topos are classifying topos for positive theories. So if you considered model classes for positive theories, Then there should be a boundary model class. Distributed lattices among something else, well maybe it's the boundary of another variety of them. So this kind of thing, there could be very simple examples where this comes up, but it's not, no one is carrying it out. And perhaps the nucleus of the logic of contradiction. Well certainly in the beginning. I'm not a paraconsistent, you know what I'm saying? No, no, no. It's a curious, anyway. We should discuss that later. It's an important fragment. I mean, why is it that the open, you know, I mean, the hiding algebras, well, they used to call them the, what was it called, the co-hiding algebras were called Bavarian algebras. Yes, although, actually, that's... We're still in the way, but Brauer thought that Heide was getting things from the wrong hand. Yeah, okay, but he never... It has nothing to do with Brauer. It was Dana Scott, I'm sorry, it was Tarski who published the so-called Brauerian analysis. And Dana Scott explained it this way. That Tarski had this fixed idea that algebraic varieties are described by saying f of x equals zero. And so he turned everything upside down formally. He was talking about hiding algorithms. Formally, it looks like co-hiding, but he always thought of it as . Because f of x equals 4, as far as the course of logic, you care about f of x equals true. So if you take f of x equals true, it's, yeah, that's quite a rotational tango there.

1:02:30 I think Brett already meant to do it. Well, it's just his name was floating around. Yeah, yeah. Tarski knew the formal idea. He never used the term. Well, that's another subject. Yeah, he never used it. No, no, no. Tarski worked, of course. Tarski and McKinley. Yeah, all that stuff by Tarski and McKinley, they didn't use it. They did very fundamental work in representation of these things. You know, and biases, or records of biases of open steps and all that stuff. I got it. Berklee, of course, is well aware of that. He comes to the ground a lot, that's right. Oh, you're right, yeah. But those are, I think so. In the lattice theory, in the lattice theory. I thought they were called regenerators or something. Is that what the term was used? I wouldn't swear to this because I remember I did a reading course as an undergraduate on these things. I certainly was reading Berkow, but I think I probably was reading other papers, certainly, where there were Paul Breuer letters, maybe by Montaio or something like that. But there certainly were Paul Breuer letters as long as Berkow. That's true. There was a serious byproduct of this compute, which was that those people who were perfectly in position to do so never even noticed there was this boundary operator. In particular, it was not the boundary operator that had satisfied Leibniz's product rule. Hiding logic turned upside down, but then it has this completely different significance, you know, in other words... That's what some of your friends down in Australia don't understand. Yeah, well, Grand Priest is my first PhD student. You know, they think if you... Well, actually, the topos theorists have the same problem. They think, well, since the topology is a form of hiding algebra, then so is the subtopos. You know, there's the question if you... Okay, if you have this abstract lattice, fine, you can turn it upside down if you're going to have lattice. But if it's embedded in a larger category as being, you know, the sub-objects... Well, then there's one way of looking at it as material, so to speak, and the other is purely formal. The one that's compatible with actual sub-objects' inclusions is quite, you know, it's distinguished in certain ways.

1:05:00 And so in those cases where it's colliding, then you have an intrinsic notion of boundaries, not some additional structure. And it does satisfy and slide its rules. But nobody ever noticed that until we around the world. Only a few years ago, I noticed that you have this Leibniz rule, and it's just amazing. It's a very powerful rule. You can start calculating stuff, all of which could be translated back to intuitionism if you wanted. I don't know what it meant, because again, the thing is that the closed sets in space are somehow much more tangible than the open ones. That's quite true. They have their boundaries. They have their boundaries. You can intersect them and so forth. They're bald. There's a definite end to them. But, of course, in the case of the internal, one striking thing, and I like this as an exercise, you know, in local set theory or top, you know, internal language or the top, is that, you know, Can't be, you know, a co-hiding algebra, unless it's Boolean. I mean, you're really, you're very, you're stuck in that. Exactly. It's a nice little exercise. It's a very beautiful... But then if something is not true, it doesn't mean... No, no, the thing is, also for sub-objects, because there are many topos. For example, any pre-sheaf topo is over a moon and... Where the individual sub-object classes are actually co-hiding, but they're naturally hiding, but they're individually co-hiding. So I have a lot of insight out of the fact that, well, even though they're not totally natural, they are preserved by some orchid. Yes, but then in that case one needs a more general framework that you get to hiding the migratory rule. For example, relative to Cartesian product, for that it's sufficient to know that the projection maps, which are very special maps, you see, the projection maps substitute into the...

1:07:30 But then in that case, I always felt that one needs a more general, flexible, formal framework, which transcends the internal language, of course, in order to do justice, to make all those calculations at the same level, without having to talk about something external. But the other end of Australia, they have this generic answer, lax. See, so in the same sense that the Hankin models, the functors of preserved products, but not exponentiation, nonetheless they must therefore compare the exponentiation, so, you know, so you can have homomorphisms of, say, colliding algebra that preserve union and intersection, don't preserve the difference operation, but of course there's a reduced comparison. I don't know how Hankin really discovered something there, I don't know. I don't know what philosophical basis he had for it, but it's a constantly recurring property that you have these, once you get into the two-categorical world, punters can preserve certain things, but then if those things themselves have adjoints, those adjoints won't be preserved. But there'll be a comparison of all the same, so you don't lose control completely. That was Hankins. You know, something that combines, you know, something whether you make the same kind of calculation, you know, which is similar to the type of calculations that one makes in the local, you know, in the internal language. Where they look the same, after all. Boundary calculations don't look any different, of course, from closed sets. Yeah, yeah, who cares? And you sort of do it in set theory, but set theory is, because you essentially, what they said is Boolean, and so you're just using Boolean properties of complements. But formally, the calculating with open sets and calculating with closed sets is the same. Look, in the set theory case, it's because you're... As long as you're staying inside one space, but you... Yeah, you need a fragment that makes it look like one space. Well, it'll never look like one. No, no, no, but I mean, there is such a thing as possibly a formal theory of the two-category topology. It's not a formal theory, I think. But Bain and Boo actually makes moves in that direction right back in 73 or so.

1:10:00 Yeah. So, I mean, again, that's... I mean, that's where the logic of contradiction would live. That's right. That's right, because you have inside each one, you have the hiding logic between them. You see, notice you have, by the way, you have existential quantification. Given the geometric morphism, it can be factored into a co-monadic and a full inclusion, so this is... This basically gives you a notion of image, which will behave, which satisfies the rule of inference for existential quantification of, you see, thinking that the logic is in the derivus of subtropical, in the narrow sense, and then you can substitute, of course, again, not preserving difference, but preserving union intersection. And there's an adjoint to that, taking the image. So you have exit potential, but apparently not universal, okay. So it's relatively... Even this, you see, even this tin can problem, it's in the Kobo 90, where there's a long article called something about the future of topos theory, where I mentioned this Verdi-A's talk on the beach in the Hoya Inn, it's from 65 years. Engler and I were in the gluing section. We said, oh, this can't be found. He started off by saying this is about foundations. We said, oh, no, this is not foundations. It's based on set theory and external set theory. It's not axiomatized. So, in perfectly acceptable French fashion, threw down the chalk and abandoned the lecture. Because he'd been insulted. This was incredibly stupid on our part. Very narrow, very narrow. I'm sorry about that. I mean, there's nothing more to the story in that sense, but later we, very soon thereafter.

1:12:30 Anyway, anyway, this is the Trin-Cann problem, which I described in the short, the short, the Leibniz rule for the co-hiding boundary and certain pre-sheath topos. So, every pre-sheath topos has this co-hiding structure, and within a given object. The boundary formula, the Leibniz rule, is to evade colliding algebra, which is a property. It's a consequence in the rules of entrance of colliding algebra. The tin can, this is when I teach calculus, you see, I talk about the tin required to contain the beans, you see. So the boundary of a cylinder, which is itself a product, a Cartesian product, an object, is the union of two parts. Namely, one of the parts times the boundary of the other, which is the two ends, and on the other hand, the opposite, so we take the circle times the... So it's a line that shows whether it has Cartesian product in it rather than intersection. We're actually talking about sub-objects of a bigger product. See, I'm thinking of taking the disk out of a box. We're inside a bigger product. So products are just intersections. In fact, it's hard to call it a cylindrical algebra, the idea that you're substituting along projections to get these special sets called cylinders, but then their intersections are self-strategic products. So these are the result of substituting along the projections. Tim Kahn for me, the Leibniz rule for Cartesian product, and the topos that already has both kinds of them, you just have to know that substituting along a projection preserves, well, the co-hiding knot, because the boundary is just A and not A, right?

1:15:00 Well... So the knot is... So that, I actually, the condition on... Small category C, so that if you pre-sheet on C, this formula is true, for which the Leibniz formula is true for the coherent boundary and so on. And it's kind of amusing because it's just that a map can always be factored, but not in the image way, like the graph of a map, factoring it through the Cartesian product. More exactly, if you have two opposed maps, say A to B and B times A, then in A times B, you have the graph of one map and the graph of the other map, but the graph always retracts onto one factor, so really you've got two other photons on this third space, A times B, such as you split them, and then you compose the components of the spirit in the wrong way, so that is, you take the projection from one and the injection from the other one, then you get your map. So any math can be expressed that way. Clearly, if you have products, because I just told you how to do it, use graphs, you see, but equally well if you use co-graphs, if you have sums, you use co-graphs to get the same result, but there are a lot of small categories that have neither sums nor products, but where this kind of factorization is possible. This third object is necessarily a bit bigger than the other, so you get an infinite. Even starting from the smallest thing, you get an infinite sequence of objects equipped with idempotence. And it turned out that the Simplitial sets, sorry, Delta, the Simplitial category, has this property. And so the tin can property is, so after Simplitial sets, really it's about topologies. It's like they're always in the frame. In this boundary formula, there's a crucial intuitive... 90s in here, but I think, I'm sure what we all like to hear is that quantifiers achieves, and they've been in the 1970s because, well, I mean, it was a sort of general thing that I was going to mention.

1:17:30 On one hand, the contribution from Boolean, Valiant, Mullen, but also from Gabriel. But on the other hand, the other input from your emphasis, of course, on adjunction, I mean, you, the notion of adjunction, of course, was, you didn't discover it, but you exploited the notion more than anybody else, in areas that nobody, you know, where you went boldly, where nobody had gone before, and in particular, of course, the logician, and the point that I was still astonishing first-year logic students with, well, you know, operations or adjunctions. Now, of course, this was your idea, and one coming, I guess, from working, early working, just in your thesis, of course. There was some kind of confluence of these two notions, and they had to quantify, you know, you talk, you talk quantifiers and sheaves, and there was... Some kind of a fusion, if you like, of these notions in discovering, of course, the axioms, developing the axioms, you know, for the elementary topics. I think that's where we're going to get to. Right, so, okay. I went to Chicago to talk to Stone. Actually, Stone had a student. The idea that the student was going to develop Boolean-valued models, that would be Coppola's, the student of Marshall Stone, Poulsen, and if they don't know him, because he scoffed at the fact that when I moved to New York, the student came along with me, and then one day, most of the books from my life were really scoffed at, all over the earth. And he was missing too. I could probably conjure up his name, but he didn't do what Marshall Stoneman hoped he would. David, what was his second name?

1:20:00 Chutton, you say? I don't know. Thanks. Yeah, so anyway, so again, as I said, contrary to the Saunders story that I was inspired by his course on mechanics, which was actually a joint course, it was not a bad course, but the thing was that these ideas about putting, applying Gabriel, what I learned from Gabriel basically, to this project of having a simplified framework in which continuing physical problems could be discussed. It just sort of gelled, you know, and I was giving a series of lectures, so in May of 1967, I gave three lectures where I expanded essentially the paper that was eventually published 12 years later in Denmark. Yes, yes. But again, that paper is essentially what I said. I mean, Ken McLean for some reason thought this was terribly wrong. And I put in comments, later comments, but they're clearly marked in the paper, so if you just carefully excise the things, you'll have three stars, and then you'll have essentially the original. He did give me his notes that he'd taken in Denmark, to compare this very thing on, and I just, I couldn't see what to do about it. No, I couldn't either. But Robert Dubuc was there, and he had his exact notes. So during this controversy, he reproduced them and mailed them all around and said, my exact notes, I've been ministering on French, my exact notes from Le Verre's lectures. There are a number of people there, actually. And Koch, of course. So part of it was the adjuncts, that's right. In other words... Okay, now just take specifically the notion of function space, which is absolutely crucial to all this stuff, as Rewitz recognized. So, you see, in the 40s, at least, Rewitz incited the Fox, and then he gave his own development of phase space, which is what it was.

1:22:30 So he was, what he wanted was exactly the property of Cartesian closed category, except in the concrete context, so he didn't put it that way. But then when you come to Connes' paper, Connes was, and even still is, an active researcher in combinatorial topology, in the Simplicial Cess and so on. So in his paper on adjoining functors, he points out that for Simplicial Cess, there is this internal hom. Yet he spells it out, but it's a concrete example. You can give the definition of this, you know, in terms of elements, and then verify that it's adjoint. I noted, well, adjoints are unique. That's a general theorem. So therefore, you can use it as an axiom. And of course, if you set things up properly, everything you want to do follows just from adjoints, a really interlocking system of adjoints. I'd already pointed this out about the quantifier and some before that. What's the term? I'm not sure. Just remind me. Can you use the term adjoint? Oh, yeah, sure. And that would suggest that McLean... That's a great paper, adjoint polite. But what I mean is if we didn't use the term, it goes back to presumably the idea of adjointness in functional analysis, right? Well, at least that was what McLean says. Oh, yeah, that's... But why did he use that? I joined operating. McLean had to give some speech at an anniversary of Marshall Stone, 60th, 70th, 80th, probably 70th or something. And so he asked me what he should say about Marshall. How does Marshall Stone fit into the development of a categorical point of view on things? Funkin' Algebras and Stone-Shank Impacts. All that sort of thing. Stone duality, which by no means limited to Bonac II of Luhmann Algebras, which logicians seem to think was actually about continuous functions and so forth. So I said, well that's clearly, and that was one of the topics that I'd been teaching in Trudell's course that helped me discover the idea of algebra without naming exactly that. So I told him this.

1:25:00 Stone inspired the notion of adjointness, explained that way. McLean carried this into the lecture hall and it turned out that adjoint operators, you see, you write it as . . . Yeah, yeah, yeah, yeah, sure. . . . and interproduct notation for Ham was, well, it is fairly common. So, but I think that's completely misguided. I mean, it's practically, you know . . . It's incited various students to try to concoct some way that adjoint operators are adjoint factors. None of them have ever found anything. So it was actually kind of a misleading thing. So, it's not a term, but then in that case, the term adjoint was simply some general con- Well, but again, the term adjoint- But again, the notational similarity might have been what he was trying to prove. There's no- Surely- That's not necessarily wrong. It's just a very inadequate explanation of the role of stone, of the role of stone, and much more profound than just transmitting from the tissue. One thing which it strikes me might be helpful in providing orientation, including, as it were, to the wider audience, would be if you could just run over again what you were saying to Angus briefly during the luncheon call. ...about the QD objects in the topos in relation to the connectedness of the sub-object transifier and this relationship between the decidables. In other words, what you refer to in the Kano lecture as the unity of S-U-D objects. And in relation to this misperception that's propagated, propagandized for by Lucy, that's your... And here is work. It was just as where the applications of topos-theoretical ideas to logic, whereas in fact, as you made clear this morning, the whole point was to see the logic as already being there in me, wasn't it? Yeah, well, since we knew that modifiers were adjoint, the crucial point in this thesis, you see, I wanted to axiomatize the category of categories. I knew that the functor category is a crucial operation. In fact, it could be everything you think of.

1:27:30 Part of a function category, if not a whole one. But then, to treat that axiomatically, adjoinus presented itself. I think that's probably, that may be the first time that adjoinus was used as the defining property of an axiomatic system. With this program, which I call categorical dynamics, and already there, of course, there's the observations all together. There's a suitable category of cohesive spaces, and you can also construct categories of dynamical systems over that, and you know, these kinds of formalisms are all change, the change of time, you know, if you go from linear time to circular time, this was a construction that, say, Smale used in a special case, but not ever pointing out that it's just a con-extension, it's a change of... You know, internal categories, or going from discrete time to continuous time is also an adjoint. So all these kind of different things were, you know, I mean, obviously in each case it's a tremendous amount of particular information, but at least there's a uniform conceptual framework, which, you know, which anybody can understand, quote, unquote, you know. We're doing that, you see, because it's a universal... The universal smooth time dynamics were given discrete time, but I'll see all that sort of clear, it's just a matter of making it. So I recognize that it would be necessary to more fully exercise the notion of TOCOs as a framework for the categorical dynamics. In the article, in the talks, I mentioned the TOCOs as a... At the same time, in order to have this sort of conceptual simplicity, I realized one would have to, you know, completely re-do the definition that they had, so that's where- Well, it was a sub-object- Well, you already told me, at least as far as I- and I think this is- Well, I'm going to- Yes, sir. I'm going to explain. So, again, I sort of made the next step.

1:30:00 There's a power set. Oh my god, a power set. Cantor should rise up from his grave. There's a power set in every Grotendieck topos and at the same time the characterization of it is elementary. Now, since we've gotten to that point, It surely was the input of large art, you know, that led you to see that, or you and Tierney and you to see that. What I mean is it didn't come from, you know, we really weren't interested in it. Anyway, a subjective object classifier was something that may have been implicit in what Groton was doing. Okay, let's back up, let's back up a bit. That was the purification of it, so to speak. No, the original desire was that, well, okay, some kind of general logical background. ...completely planetary algebraic system, because I thought that would be an assistant to my conceptual understanding. So the fundamental construction of the associated sheaf, well first of all the definition of sheaf, and well for the definition of a Groton-Deke topology with respect to which you can define it, so that whole sequence of requirements, but it starts from the associated sheaf as... How do you construct the associated sheet? Isn't that a huge, direct limit over a class of coverings and then you adjust it twice and isn't it all that? Isn't it all this inventory stuff? Well, no. And then, okay, but you were doing all that to reflect into the sheath. What's a sheath? Isn't that something that satisfies the universally quantified hereditary blah, blah, blah, blah? No, it's just an equation if you do it right. And so, basically, the first idea was partial sections. That any object has a space of partial elements. We might call it that, but if you're going to get any sheath topos, these will be... For the appreciative topos, these will be partial sections. So the idea of a sheaf is one for which the partial sections whose domain happens to be a covering, in fact, are equal to the object itself, in other words, the elements.