Am session: FW Lawvere, P Cartier, C McLarty, A MacIntyre, JL Bell

0:00 Yes, they would be. These sessions of discussion are, and would be submitted for editing by you. Ten. Okay. Sharp ten. Right. Okay. Sorry. The 11th of June, 2005. A discussion between Professor Pierre Cartier, Adil Oviangis-McIntyre, and Colin McLarty. In which John Bell will probably shortly join. On a topic which we have loosely headed Eilenberg, McLean, McLean, Eilenberg, and if I just say very briefly before I make myself scared, what we thought might be a useful The framework for the discussion would be if Colin perhaps could lead off by making some general remarks about the historical background in homologic theory in the 40s, out of which Eilenberg and Maclean's collaboration began, and the direction in which it took the subject in the United States. The shape in which homology theories arrived in the hands of those who developed it in the next decade, particularly, obviously, in the Grokendijk School. And then, having provided some history, I was going to ask Bill and Pierre Cartier if they'd like to say something about the way in which... Eilenberg-McLean spaces move from being a very important part of the conceptual standard of homology theory into a central position within the development of category theory itself, and particularly as to the way in which Bill sees Eilenberg-McLean spaces as fitting within the development of the understanding, making explicit, of the general notion of categories of space. Is that a reasonable... I don't have a lot to say about that. The aspect of it that I think could be stressed more than commonly is, is Lefschetz's role that Eilenberg and McLean, their first paper on it was written as an appendix to a book by Lefschetz, that Lefschetz wanted homology explained, picked these two as the people most apt to do it. That Leschitz had encouraged the whole problem of the cohomology of the solenoid in a sphere, or the complement of the solenoid, because he thought this might push the subject.

2:30 He said, here's something that we can't do now, we ought to be able to do it. And by figuring out how to do it, we'll maybe live with it. And the other direction is, talking with Mike earlier, I think, it looks to me, but you'll know probably a lot more about this, that there's a real divergence between how MacLean sees group cohomology and how Serre did especially, and that Serre was real important in both. Yeah, yeah, yeah. I shouldn't say you can say Sarah, I mean Carter. As far as I remember, Sarah did, but I don't think they were very much involved. What about the general developments in homological algebra in the 30s and 40s that led up to the discovery of the great spaces? Well, I mean, in some sense the key discovery brought about the recognition of the importance of categories. Well, cohomology was a functor. This was more or less known, even if not formalized. The concept of representable functors was not so well known, and probably, I mean, how could you predict that cohomology would be representable? This is not in any way obvious in the concept, at least to me, that it could be, but it turned out that there were these spaces.

5:00 At the very moment that category theory was being invented, it also became clear that you needed abstract categories because the representability is only in a homotopy category, so that if it hadn't been for this development, one might go on believing that categories have faithful underlying set filters and this sort of thing, which many of the structures considered had had, but here was a glaring example that was definitely not the case. It was really a major qualitative change, almost simultaneous, I mean within two or three years it's exactly the same. It was very rapid. I would like to make a comment. First of all, I remember I attended a carton-seminar on the Wehrmacht-based basis. I was very young and I don't claim I understood everything, but nevertheless. And I remember my surprise when Sir, in a café discussion, Mentions that the Kπn were the thing to represent the cohomology. I don't think it was, I did not study carefully the original paper by him, but it seems to me that, I mean, there are these papers by him on the homology mode 2 of the Kπn. And I think that, at least, maybe it was not right in there, but I think the great importance of the fact that the Kπn were representing the cohomology found to... I mean, he appeared clearly in the work of Sayer in the Commentary on Mathematics at MITC about the Steyr-Watt squares. And then, I remember a discussion with Tan, I mean, he insisted that, well, his emphasis was slightly different. Sayer wanted to consider cohomology operations, which is, that is, natural representation for one cohomology factor, but I don't find it. The Steyr-Watt squares are operations in the cohomology mode, too. So they are functorials, they should be considered as the endomorphic or the cohomology functor 1-2. And then they remarked that since, at least that was a crucial remark, since the cohomology 1-2 is represented by some space,

7:30 calculating the cohomology of Hilbert-Markin spaces. Amounts to be exactly the same as calculating and describing the cohomology of a nation. This paper in the commentary is one of the strong points. So I don't know exactly who realized. Well, the idea of representative order factor was not so clear at the time. Until, well, I think, until Gordon put it. Certainly, but this was really the idea, that the cohomology of the hydrodynamic spaces corresponded exactly to cohomology, and I think that's one of the strong points. And I think that material is another example, I mean, it's about the classic mind spaces. And already in Borel's thesis, I mean, when he, BG, I mean, it was clear that BG represents the frontal of the principal G-bundle over the principal G-bundle. So, but again in the homotopy category. Homotopic classes of X into BG correspond to principal bundles of over X with group G and already in the construction of Borel of the classifying space the homotopic point of view is very prevalent because he says one definition I like of B and BG is treating a little bit but not too much, take a point, take any action of a group or a point, you think nothing serious can happen. Take the orbit space of the group G acting on a point, that's BG, provided you were not in the category which is the presentable of the mass set, but you know a point is a set, and then you get exactly the Bohrian construction. You take a point, you represent it by a contract, then you let the group G act on this contractible space, but in a way which is free, which is a new thing, and where resolutions lurk behind.

10:00 So I think there were two homotopic categories, and the representable factor began in two different, slightly different, the island air market spaces and the climate spaces. And the classifying space is more or less a discovery of Ramon Borell after many others, but he was the one who crystallized it. And the cohomology of the Heidelberg-Macklin spaces was started by Sayre in his paper about cohomology and what to do, and then developed by Heidelberg. But of course, we all know that Sayre and Gorey were very, very close at the time. And when was Gorey's thesis? Well, it was defended in Paris at the time. Really, the work was done in the 40s. Gorey managed to visit lectures at Zurich. Oh, yes. And then he wanted to go. But after the war, I mean, in Paris he was a little disappointed to realize that Le Ré has lost part of his interest. But then, in my prehistory, I mean, I appear on the mathematical scene in 1950.

12:30 I became a student at École Normale, and on the first day of École Normale, I attended, that was the Monday, and I went to the 14th seminar. I was, I suppose there were not so many students, first year students, to go to this. He has given lectures in this year about the cohomology of groups and axiomatic presentation of the cohomology of groups. And that's where I learned about the cohomology of groups, from Heidenberg. And that was my personal recollection. I mean, my first year as a student, as an economist, was illuminated by it. That's the year he visited Paris. Two things that you mentioned. This concept of the cohomology operations, which itself is independent of the theorem, that is to say, for me, this was the inspiration of my thesis, that the concept is mainly a natural transformation, a natural structure, a natural algebraic structure on a function, even one which doesn't look representable. All of this is a well-defined concept, so you say this was due to Serre. I think so. I mean then, so there's that general concept, and then in the context where you have representability, then you can say to compute it is the same as computing the cohomology. But I think... Or whatever, but it's necessary to have it as a concept in order to make the identification. Exactly, exactly. So in some cases it turns out to be it isn't representable, but it's a direct limit of representability. Or something of that sort. In his first visit to the beginning of the day, and almost immediately, Carton published Notier d'Estimaux, and he gave a more or less axiomatic description.

15:00 One of the main properties is a functor. I think this is still not widely recognized, you see, the values of some functor that you have computed. All of this is set. It turns out to have some intrinsic structure. So of course you discover particular things, like the square, and these are very important if you develop them, but the opposite, the generality, the fact that the generality is tremendously limited by the naturality, so that there's actually a definite thing, often a set, of all the possible operations. And you have to remember that the example of cohomology operations that I learned about from Sammy in Columbia, this was one of them. And one of the first people, like Borrell himself, was about the impossibility of some vibrations. And it's used, it's deeply, I mean, it's used. Borrell was among the first people to recognize, well, still on himself, but that he liked it. By putting the stenoid operation in general, you have an extra structure of the cohomology, and that you can, the cohomology group, the naked cohomology group, or the Beckley number if you want, do not distinguish between certain spaces, but if you take into account the stenoid operation, you can distinguish between them, and you can also prove that some maps cannot exist, some continuous maps cannot exist, because they would not respect the stenoid squares. And all these things, I mean, there is a short note by Borel, Borel and Say, one of their first publications, where they show the impossibility of something else.

17:30 And they were very quick to recover. But then, to me that's so, of course. And the same, I think there was an analogy with the work of Borel on the classifying spaces. But nevertheless, these ideas that the cohomology is not a neat object, that it has operations of its own, and that these operations are controlled by the naturality, was an idea which spread among in the world of Heidelberg, Guantan, Sayer, and Boyer. But even the fact that it's also a natural structure, you could imagine... You construct it just as you said, and you say, my God, it's a good product. And a good product, and a good product. Yes, of course, the multiplication. But this, I think that was a quite important idea. But historically it seems that people understood very well at the time that there was, I mean, there was a lot of structure on the common school. But the point of view that all came from natural, of course they were natural, but the idea that they really... The origin, is really the naturality, was not so well understood at that time. And I think Serre, in his paper about the cohomology of mathematics, is the first who really put it in the system. Despite the fact that Serre himself was not so fond of categories at the time. Even today, he is not really so fond of categories. He knows how to use them, but he has never been so fond of them. No one important paid attention to them. One used them, but that's it. You were told, buy it there. Buy it there, yes. That's what you could put on. Well, I say he's a no-nonsense man. Can we go back a little further and explain the origins, the specific origins of the emergence of the concept of Schiff?

20:00 I mean, did LeRae, what was that specific problem that led to him developing the concept at the time he did? Is there a simple answer to that question? There's a simple answer. You're probably wrong. The problem was to avoid getting the Nazis recognized. He was good at applying that. No, no, no. That's what he said very often. No, more practical, more explicitly. Leroy, and I think, well, I recommend reading the recent paper. It's a historical paper about what, but he has some comments on his mathematics. Sorry, last issue, I just got the issue yesterday. It's the other day before. And so, last issue, sorry, it's in the mathematical twist. Right, yes. Call on mathematical twist. Yeah, I know. And Peter Michaud and two collaborators. I mean, Peter Michaud is in Austria. I'm very happy that he was the one who really promoted. And he had a diplomat, but he had some diplomatics from his government and he was getting some travel, more or less, officially. Without visa restrictions. Well, Wien is the center of Mitterrand. Of course, but he views. But this explains that Lurie started topology in his work with Schauder and the Schauder fixed points, Lurie-Schauder fixed points. The emphasis for Lurie at the time was to use fixed-point theorem to prove existence theorem in analysis, which is still evaluable to.

22:30 So they proved, I mean, Lurie and Schauder proved their fixed point with the idea of using it to prove existence for partial difference. But then, in this setup, you are not working with a space but with a map. And Lurie was interested in developing invariants not for cohomology, invariants not only for a space but for a map. Which I realized clearly in the end. Well, I knew that by the end. And so Lurie was trying to invent, I mean, his paper is really not about the cohomology of spaces, but about the cohomology. He says, I think that, I don't remember the title of the paper, but the emphasis is on maps. He wants to build a cohomology theory, a homology theory for, he always says homology, but for maps. And I think the origin of shift is that when you have a map in the lower spectral sequence, you consider a map and then you consider over the base space, you consider the various fibers. Then what you have as information over the map is that each fiber has its own cohomology. But the fiber depends on the port in the base, and so you have a cohomol and, in a very natural situation, the cohomology of the fiber may change for most of them, which was already known by Hans Hoppe, when the base-base is not simply connected. There is an action, I mean, even if your space is a vibration, even if your map is a vibration, if the base space is not simply connected, you have a natural action of the pi 1 of the base space over the cohomology of the pi. And that was already more or less known. And so, I think, motivated by some examples from functional analysis, Lurie wanted to understand what happens when the fibers do not have all the same, not the same, all the fibers do not have the same cohomology.

25:00 What is more general, what can happen? And so the shift is a shift idea occurred by the fact that you want to organize, you have this map, on the base you have the various cohomology of the various fibers, each one associated to a bone, and you want to organize these things in such a way that you can calculate the cohomology of the total space in two steps. First of all calculating the cohomology of the fibers. And then reorganizing things into sub-objects over the base, and that's a Schiff. And then calculating the cohomology of this new one. And I think that was the motivation. Now, about the definition of Schiff, I mean, there was much hesitation about the definition of Schiff at that time. For Le Ré, a Schiff was a functor associated to its clause. And also LeRey had a different idea, the idea that Carter explained as grating. LeRey was strongly motivated by the analogy where he wanted to have an extension of the Derham theorem for calculating the cohomology. Of course, the Derham theorem was well known at that time, especially in the... The algebraic object, which is the differential force, with the exterior differential, which was well understood in the end. But you have some more structure that was discovered by Lorentz. Each differential form has its own kind. The smallest clause sets. And so the axiomatization, and Lorentz realized that if you try to analyze the proof of... You have these objects, it's purely algebraic object, which is the differential form global, because we are working with smooth compact manifold growth, at the time you don't have to distinguish between local and global.

27:30 Then, but you have these extra structures that each differential form has its kind, a certain growth set, which satisfy obvious action. If the carrier of a sum is contained in the union of the carriers, etc. And then, one main thing is the so-called Poincaré's lemma, which says that if you take a differential form with a close enough... A differential form would support around a certain point. If the support is small enough, then the differential form, if it's closed, it's a combined line. So local triviality of cohomology. And then Leray developed a theory where you have a generalization of that. So for space, he wanted to associate to any one. In modern terms, the aim was to associate to every space, to every space in a certain category, a DG, a differential gate, an algebra, with a notion of speed, not yet achieved. And that's what, in his Harvard lecture, Garton called a gate. So this is more or less a dual notion of a gate. In the context of, say, the C.N. Kennedy case, namely that every section... All of these terms are restricted to something global, so it suffices to consider the global thing, but then in turn they have this support, which is more or less the complement. Yes, because what you say, I mean, so the shift occurs dually. You take, let's say, you take the differential force, and you take, you want to have the section, the section of a certain shift over an open set. Then you take the open set, you take the closed complements. And now you take the global differential force infinity and you quotient it by the form which has a support outside. Outside is a dual motion. And so the notion of grating is really dual. In a sense, a grating is a description of some sheaf which is a quotient of a constant sheaf.

30:00 I mean, in modern times, exactly that. You consider shifts which are quotient over constant shifts. It's exactly that, exactly that. So, but, and then, Laurier was not, I mean, Laurier did not consider shifts per se. I mean, that was one of his motivations, this generalization of the Durand theorem. I think which is still an active area of research I mean to have in general spaces and especially I've been working for many years on infinite dimensional integrals and I've here the manuscript of my book which is due to in the fall and so and in infinite dimensional spaces there's still a calculus of variation if you want to apply this idea to the calculus of variation it's still it's still an active And then he redeveloped a huge generalization of the De Haan theorem. Then he wanted to understand the cohomology of a map. And then gradually he realized, first of all, the established situation when the factors are all the same cohomology. And then it came to exceptional fibers and other situations, which when you have group action is very natural, you have a group action and you look at the open spaces and such. And then it allows that, I think, associating directly to the... Now in the base space you take a cross set and you take the counter image of the collection of fiber above this cross set and then you take the global cohomology of this. And then you have a collection of groups associated to the closed subset of the text. And then it took some time, and there was some hesitation, and I think Carton and Sayre, if you look at the various editions of the Carton Seminar, they have some variation in the definition. And I think it's only the book of God among which everything is straight and to the definition that we have. For a while there was some hesitation because if you are motivated by many complex variables, of course in many complex variables sheaf appear in a slightly different form. What you have, you have the local power series expansion. Exactly. So you have the fibers of the sheaf. The fibers are relatively concrete in this case.

32:30 Yes, the fibers are relatively concrete. The sheaf itself... In French, we do not distinguish between the stokes and fibres, the stokes of the sheaves. But then, in many complex variables, the stokes are there, before you have the sheaves. In the Caton seminar, he more or less took as a definition that a shift would be a collection of Dunkelstokes glued together with the topology of the union, which is a literal space. Which is again a representability if you like. As a factor, a sheaf is representable by a subspace which is a total other space. It's another representability. Yes, both left and right. Yes, yes, yes. But then, in the literature this was called the Lazar definition. The name of Michel Lazar, as an aside, Michel Lazar was... I had a close friend in my youth and he had a very sad life. His family, I mean, he was in Vichy, France, and he lost some of his parents. And more than that, he had an elder brother who joined the French guerrillas and was killed. Since his brother had joined the communist guerrillas in France, he went on to remember that in France it was... A very tense situation because we had two different guerrillas, the FFE, which were more or less under the control of the war, and the FTP. You don't distinguish Jewish girls? And there is also the Moy, the Moy group, which was part of the communist guerrilla. The Jewish group and the brother of, I suppose the brother of Michel Azar was in the Moy.

35:00 Moy was a Jewish group. We joined the communist guerrilla. I'm not so well trained in that. These are not very happy days. But then you have to understand that at that time I lived at the end of 1944, in the summer of 1944, in a region where we had four armies. The retreating German army, the US and British army in the south, so 10 kilometers north of us, the US and British forces 30 kilometers south, and in between the communist guerrillas and the non-communist guerrillas for two months of no man's land. He was the mayor of a small town, but you were in between four different armies. It's not an easy thing. So Michel was our closest brother, and that's why he joined the communists after the war. He was a devout communist member, like many academics in France at the time. And then he developed his mathematics, beautiful mathematics. His best paper is about periodic groups. He developed beautiful mathematics, especially free groups and combinatorics of groups and things like that, and periodic groups. And I collaborated with him on various things. And he was one of the most promising persons. And he participated like every bright young mathematician of the time. And he is the one who is credited with the exact definition of a shift as a collection of stocks. Putting the topology, what he discovered is that putting the topology of the union of stocks took all the structure out. And for a while, Cartan converted to this approach. And it's only later that, it's only later than, I think that, maybe it's Godemont who's... Partly under the influence of Searle and Goethe, but it's Gordimer who really, I mean, put the standard in this book.

37:30 It's very legitimate for me. Yes, for many people. So, but Gordimer was not really an outsider. Gordimer's work at the time was mostly on legal representations, infinite analysis. And for him, Schiff was on the side. But it's just to tell you the spirit of the time. That means that the French mathematics, especially the Paris mathematics, was very united under the guidance of, as I mentioned earlier, and we are all your students. And that was a quotation from Arnault, Vladimir Arnault. The first time that Arnault visited Paris in 1965, that was, there was a retirement, that was a time... Cato retired from Econorman to move to what became Orsay University and then there was a small party at Econorman and Vladimir Amnol was the first time visiting France and during the speeches, I mean at some point, I mean Cato made a speech in his familiar tone, I'm surprised, I mean I had the people who organized this very happy day to call on my former student but I think many people. And then, Vladimir D'Arnault took it back. But, Mr. Cato, in Moscow, we are all your students. It's Arnold and it's Cato, if you know what. It's difficult. The story is difficult. So, just to say so, but in order to understand the development of all these ideas, I'm not taking a science, I mean, I'm just a scientist. In order to understand, at least in Paris, not... my point of view is from Paris. They opened all these ideas. I mean, you have to consider that Carter was really the boss and that everyone was under his influence and that most of the most active people were official members of the Bovacki group at the time and so the discussion which began in an open seminar would continue in the close meetings of Bovacki and the book of...

40:00 Gordon Marlborough-Chief was commissioned by Bourbaki. Bourbaki said, OK, we have to consider Schiff someday. And according to the rule of Bourbaki, we selected the one who was not the expert to write the first draft. And since Gordon was not an expert in topology or in Schiff, I mean, we are in a discussion, inside of a discussion, someone who is not an expert, too. So, Godemont volunteered and started it seriously. And finally, for various reasons, the public, for various reasons, Bobacky did not publish an account. And then at some point, we said to Godemont, well, if you want to publish it under your own name. Many things occur like that. I mean, Bobacky commissioned people, non-expert people in the region to write a draft about a new subject. And eventually, I mean, the plan shift, the plans of Bobacki shifted and these things would not be part of the Bobacki treatise, but we decided, okay, take it out, take it out and publish it and tell your whole name. So, just to tell you about shift. But that's an important remark. I mean, French mathematics at that time was organized, I mean, the central figure in Paris was Cartier. Later on came Schwarz, with a different and slightly different emphasis, and when we discuss Gottendieck's functionalities Schwarz came with a different emphasis, I mean more analysis than topology and geometry. But then there were two ports of French mathematics, and in the late 50s they were dominating. At that time you would have to be a student of... I was a student of Carton, or a student of Schwarz, or a student of Bose, which was my case as a student of one of the students, not four, but I mean I participated to various seminars, to the Schwarz seminar, I gave lectures about quantum details at the Schwarz seminar, and I gave lectures in the Carton seminar.

42:30 All these things were closely knit, closely knit. I always say, I mean, despotisme clair, I mean, how do you put it? Dysphonic... Democratic centralism. No, no, no. It's a quotation from the 18th century. Yes, I know. Despotism clair, which means that a king is a strong person, but he has an enlightened mind. An enlightened despot. An enlightened despot. Enlightened despot. Enlightened despot. Enlightened despot. And the French mathematics at that time was under this rule, this dual rule of Gaetan and And it was a time where you did not have many, I mean, many discussions about whom to recruit and whom to appoint. Carton and Schwartz would retreat two days in some place in the summer and decide everything. Because everyone was a student of both of them, at least one of them. And they would decide, okay, well, this one we will, by God, we both have got to recruit, etc. Everything must die. And we were, I mean, we were happy that, so, these rulers were enlightened people. He made some mistakes, everyone made them, and they were deemed to be rather acceptable in the historical situation. Okay, but so, just to understand that, so, and to come back to Michel Lazare, unfortunately, Michel Lazare, I mean, was still under the impression of the Nazis, and gradually he lost confidence in the Communists, like many of us. So, not too long. I think it's in 1956 when the Russians invaded Budapest. But many of us, many of us. Until 1956, I mean, I think 1956 was a turning point. When you realized that the Red Army was invading, you could swallow many things, even unhappily, but not the rest of them.

45:00 His life had no more meaning because, I mean, he was a communist because his brother, his elder brother was an anarchist and then he lost confidence in communism and he was in great despair, very, very, I think one of the most talented mathematicians, one of the most talented mathematicians. And they're very important, they've had a second wife. Sayre, I think as long as he was in contact with Sayre, I mean, that was good because Sayre was really, I mean, a mover. There was very quick to understand new ideas, that's something you should get. And he was really our, not our father, Carter was a father, Serre was an elder brother, the wife was, and so that's about the way Schiff, and Schiff himself was a little bit, I mean he was more fond of his approach, and he was more or less fooled by this situation to introduce Schiff, and it's only later, after a thought, I think in 1946, after he came back to France, that he really developed the common law of sheaves, which was immediately taken up by him, and there are various steps until, until, and I think the final, it's really, it's really Eilenberg who shaped the final definition of sheave common law, as you see it in the...

47:30 And you see clearly that in the seminar of 1551, the first part of the seminar is an exposition by Heidelberg of the cohomology of groups in an axiomatic setting, giving axiom for any cohomology theory. I remember when I came for the first lecture, I was 19, not even 19, I was not even 19, and I came to his first lecture. We defined an homology theory to be to someone who was just out of Euclidean geometry. So, it's fair to say then, that it was really the... it was a sheaf of groups, essentially, that emerged. Through the, now with several complex variants, that led to the idea of the general sheaf. In other words, was that the direction? I mean, sheafs were sheafs of good for many years. Yeah, and then the notion of sheaf, well, what they called a sheaf in the, I can't remember in Godemont's account, it's a long time since I looked at it. He actually, the notion of general sheaf is already in the U.S. Oh, yeah, yeah. It was published in 69 or something like that. Yeah. Much later. Much later. But the notion of general sheaf was then, really gradually, liberating itself, if you like, from the notion of a sheaf of groups. Yes, yes. For many years, a sheaf was a sheaf of groups. Whatever definition you took of a sheaf… And there were more or less equivalent definitions. It wasn't the definition of a sheaf of goops. And I remember the suspicion of the people when one began to discuss, I mean, the sheaf of non-goops. I would say exactly the same, sheaf of non-goops. Sheaf of non-goops. And there was some reluctance, there was some reluctance. I think it's really good that he convinced us. So when was the theorem proven? The theorem that you find often at the beginning of counts of sheaf theory, which really does require the notion of general sheaf, the sheaf is a espacetal, and the equivalence between that and the idea of a pre-sheaf, of a sheaf defined in terms of pre-sheaf to a covering condition, well when was that? That theorem, that really does require...

50:00 The notion of sheaf, of non-globe in a sheaf. When did that emerge? I think it's Godemont. That was Godemont. I think it's Godemont who clarified completely the difference between the two, to the best of my knowledge. I don't know exactly who put it in the discussion, but it's a written exposition with preliminary drafts in the Baubach. Right. I mean, Godemont took it in 1956. 1956. It came out shortly before the Tohoku paper, but it refers to the Tohoku paper. That's right. And he has an appendix, he has an appendix where he takes into account. But of course the term... Also the Comonad. Yeah, and the Comonad. It was motivated by... Grotendieck's aim was the following. I take, I mean, Carton-Eilenberg homology called Archibald. I take the Carton seminar when Eilenberg and Carton and Sayer want to develop a cohomology for shifts. But the cohomology, the expositional cohomology, there are many restrictions and many adopt constructions, and Goethe-Nick was the first to really understand that there was something common to them. But if we take a sufficiently abstract construction of homology, collage, algebra... That's why he hates more principles. Yes, yes, and so that's... because certainly Gatteau and Heidelberg had in mind, I mean, the analogy between cohomology of groups, that means, and he found tools. But they could not get rid of some restrictions. They could not get rid of some restrictions because they did not have, I think, the idea of injective shift.

52:30 That was a new idea brought by Godemont. And in the book of Godemont, it's just in the appendix, which was written after Godemont. Not yet published, but written is so complete. And also the idea of comorbidities. In the Grote-Dixer correspondence, they even talk about injectives meaning something more general than what we now call injective, because they don't think that what we now call injective, or at least Serre doesn't, will work for sheaves. Yeah, exactly. It means effaceable. Injective loosely means somehow effaceable. And Buchsbaum told me that as soon as he saw a Grote-Dixer toke of paper, he was just kicking himself because he said, this is exactly, in hindsight, what anybody should do. This is the bare construction of injective modules done for sheaves. But no, but it wasn't even asked in the seminar. No, no. In the Cato Seminar this question was never asked. And the idea that there should exist injective things wasn't. But you have to understand that it's, I think it's partly a personal duality. Because homology, resolution, when you, resolution is historic. Projective out of three is also too many, but not more. It's not a divide. Existence of a projective resolution is more. But the bare construction of an injective resolution costs much more. And for many years, I mean, people considered only projective resolution. And if you look at... Garton, Heinenberg, homological algebra, you will see that the projective resolution are more or less prominent, even if they put formally, I mean, they say, well, if you have a projective, you have an injective, because you have the duality, general principle of duality, and that was really in the mind of Heinenberg, I mean, but injective were considered as rather awkward objects, and it's only after the work of Buxbaum and Sayer and T.T. Barger, Baruch, Lorenz.

55:00 This parallel exists in the logic, of course. The languages and the projected resolution. On the other hand, the models, to prove the existence of models in general, you need Zorn's lemma in the same way that you need it to be injected. And I suppose that's also because I think the Bayer construction uses the Zorn's lemma. So, I mean, projective resolutions are very constructive. I mean, you can explicitly make them very explicit. Injective resolutions, well, not so much. And so that's why Rothendieck defined an injective shift, well, the definition is not the same, but proved that there exists injective. But by mimicking the Bayer's construction using the strong lemma. And if you look at the exposition by Godemont, he makes a point to make things explicit without the strong lemma. He has an explicit, very explicit concept. Well, it's a case of ideals inside. That's the starting, the inductive step. Yeah, yeah. I mean, now the point, yes, but the point... It does depend on Zorn's language. Yeah, yeah, you can. You can't prove it without Zorn's language, I think. Well, without the accident choices. No, no, no. You can, you can. I mean, if you look at... No, but... Well, that's a subtle point. I mean, you first... The way it's explained in Godemont's book is the following. You start with a module over the ring of integer z. Then q is injective and q over z is injective. Out of that... Dualizing into q-mi-z. Dualizing into q-mi-z. I mean, that's a perfect duality. So you have an explicit construction of injective. An explicit construction of an injective for a module. Okay, by dualizing these two things. You dualize twice. And then, and then, then, so you have a duality with two things to construct it and to prove it is injective. I'm suspecting that the latter still requires. Maybe, maybe, maybe, maybe, maybe. Well, it exists. The construction does not require. I thought it was great. You're right, you're right. But maybe to prove that it is injective requires.

57:30 Yes, because a lot of those injective things are actually very close to being equivalent to the action of choice. Divisible abelian groups, all that stuff has been shown to be... And then what you have. There were two things discovered by Godemont. First of all, by a change of ring of scholars, you can go from one ring to another one. The duality functions by the adjoint function between the various adjoint functions between different categories of models. So if you have the construction of the Z, you have the construction of the ring A. And then there's another step which is similar to that. You go from rings and groups to shifts. Which uses the basic idea of Gaudemont that if you consider a sheaf as a collection of groups associated to the open set, then you put the stalks by going to the diagram. Then you have the collection of stalks. Then now you put a new sheaf over an open set. You take a function which takes at each point its value in the store but without any consistency relation, any continuity relation. A very, very, very hectic, I would say, a very hectic shift. So the product, so the section of an open set is a product over the point of the open set of the base. That's a new shift. And that's, and the old shift injects that. And then you do that. Which is not yet, it would be relatively injective, I mean, with respect to some adjoint factor it's injective, relatively injective with respect, but it's not yet injective, if you consider the stokes as modules over some ring, it's not yet injective. And then you do the same trick. So you do the trick of building injective in various categories using adjoint factor between the categories.

1:00:00 And so the idea is that you have a certain model category, the Z-module, where everything can be made explicit. And then by using various functor, adjoint functor, making various categories. And that's explained in great detail with great care in the book. And Godemost, which is repeated, and after that people were convinced. I mean, that was... There must still be traces, I mean, some places we have some traces. Yeah, I mean... Before forming these products or something, there must be... The definition of an objective is that there exists an extension, and this is a non-precise existence and there is no explicit definition of an objective. The point is that, I mean, when you want to give a dual situation with three modules, you select the basis, and to define a map, you just have to define a map on the basis, which also implicitly requires the actual problem, but very implicitly, very, very implicitly. So, yes, I suppose that the chief weapon was the birthplace of the standard exposition. This is the one place where all these concepts are completely clarified and put. Except that the point of view that a sheaf is a fountain is over the category of open set is... Unless it's a slag, it's still something. Yes, yes, yes. It's not explicit in the Gordon-Walsh approach. Of course it is there, but... No emphasis. It's explicit. It's explicit, but it's explicit. You said that it isn't flat. Oh, maybe, maybe, yeah. You said that Gernemont was the first to completely clarify the distinction between a pre-sheaf and a sheaf. Was he actually the first to use the terminology pre-sheaf? Me, I can tell you a few historical points.

1:02:30 When Sayer published his paper in 1950, I tried my thesis using sheaf for algebraic variety over non-algebraic microscopy. And at the time, at the time, Sayer had given his expositions on algebraic chemistry. And I had to repeat in my thesis one or two chapters when I painfully repeated the things for more than a day by Crosby, and there are some difficult points. So you get a very good expose in Florence on this. Indeed, the notion of abstract algebraic reality obtained by doing was defined in the book. The idea that you glue a five pieces was more or less obvious, but there is one subtle point in the definition of they, which is some subtle axiom which was interpreted by In Chauvin's exposition and which comes back again in Sayre. So Sayre realized that there were two notions. There was a notion following the axiom, more or less equivalent to the notion of André Weil, which requires an extra axiom. He called them varieties. And when he relaxed this axiom to have a slightly more general category of spaces, he called them pre-variate. And in the first edition of Botanic, he made a distinction between pre-schemes and schemes. And I think pre-sheaf and sheaf follow the same pattern. I think that was part of the same linguistics.

1:05:00 So we knew that in various situations... We had a notion which was more restricted and with stricter conditions, but to construct as an intermediate step, to construct a sheaf we needed a pre-sheaf, to construct a variety we needed a pre-variety, to construct a sheaf we needed a pre-variety. Until we understood that pre-schemes were real things, that the most general notion was the right one, and that schemes probably became separated schemes. And also the point is that the notion of separated scheme is a certain point. The notion of separated scheme is a relative notion. The notion of a scheme is an absolute notion. But when you come to the axiom of separation, which says that the diagonal is closed, I mean, it's there we remarked, it's there we remarked in this paper that the axiom of they, which was very complicated, even after the reformulation by Ernst. Pashubhadi, which was a little simpler, but then realized that it simply says the diagonal is crossed. But then diagonal means, in the category of imps, diagonal is a relative thing. If you have X over S, then you have the diagonal which is in X over SX, not in X times absolute XX. So it's a relative notion. And saying that X, the diagonal, is closed is a relative notion. That's what convinced Gotending that you should distinguish between the categories of schemes without any reservation. And then, when you deal with valuation, which is again the idea that... The notion of separation is not a notion of a scheme, but of a map, if it's related. A certain map is separated. And I think all this discussion about pre, etc., I think it was modeled around the same linguistic principles. And I suppose that's Gordemont who invented pre. Gordemont, because, as I repeat, Gordemont was commissioned by Baumbach to write an exposition. And Abomacare's position was supposedly at most precise, at most precise. And, I mean, scrutinizing all the concepts and making the necessary distinctions. That was the aim of the Bohemian.

1:07:30 So, Gordimer was a faithful follower of that. I mean, he took it seriously. He took his shift and said, OK, now I have this construction. People were very ambiguous about that. And sometimes in L'Oréal, I mean, in L'Oréal and even in Bohé, the distinction is not clearly made. And then, when Godemont wanted to prove the equivalence between the so-called Lazar definition by Stokes and the definition of the factor over the open set, then he was. And so, pre-sheaf, pre-sheaf, pre-scheme, scheme, prevariety, so I think it's more or less the same. While also, yes, for an incomplete Hilbert space, Bobacki says pre-Hilbert, and I think it was the same idea, pre-Hilbert. It's an intermediate. In my mind, pre- means that it's a preliminary construction. To find that order, it would be more restricted, but the first step is part of the construction. This is much more general in a way, being concerned with pedagogy. It's very often the case that the natural domain for certain constructions is much larger than the natural domain for certain theorems, and therefore it makes it life harder for students if you start with the definition that you need for the theorem, and of course mathematicians don't want to get to the theorem, but the natural domain of the constructions is often larger, and that's I think is behind this particularly. And I remember, I mean, the discussion for functional analysis, I was there when we wrote the chapters for the first edition, which was very short, and the second edition, and insisted that we should have something a little more elaborate, I mean, you know, serious treaties, I mean, you cannot have only ten pages on Hilbert spaces, of course, and there was not even a mention of trace class operators, things like that, so in the second edition, which I took on my shoulders, and...

1:10:00 I insisted that we should, and I had some expertise from physics and mathematical physics, I knew exactly what it was. But Bohm and Key are the first to make a clear distinction between Hilbert spaces properly, which means complete, and incomplete. And people did not consider seriously incomplete Hilbert spaces at the time. They would say, let's see, we have a Hilbert space and within that we have an unbounded operator, so the domain is a dense subspace and so on. But the domain was not considered per se. And I think we should be returning to this topic. The reason why the definition of sheath as a pre-sheath with a gluing condition was accepted as the general one is because this business about global things with supports only works in the smooth case, in a relative case, and so this is good, but as a result, I think... The idea of sheaves as adfunctor on closed sets has been neglected quite a bit, so there is an exercise in Maclean-Murdike about this, and doing this exercise and thinking about it, I realize that there's a vast variety of toposes that are very closely connected or related to the sheaves on the space, but actually things that are defined up to negligible sets. The way that things depend on the closed sets can be different from the way it depends on the open sets, but there's a whole graph that takes some sort of elementary chief theory, which has been neglected because of this realization that the one definition is the best one for the general case, but under some hypothesis. You know this exercise? No, it's definitely roughly in there. Do you have this book? Yes, but I don't think I can put my hands on it right now. No, it's okay, but if we can find it, I'll point it out. No, no, certainly not. No, but I would just finish with some general comments out there. I mean, the Bobacki spirit, in general, is that in any part of mathematics there is an optimum set of definitions and concepts.

1:12:30 And it's part of the work of the mathematicians to discover this optimal set of axioms and concepts. It's a very important principle, but like every principle, it has to be evaluated sometimes. It applies to a certain context and you change the context. And my own idea is that, my one emphasis is about the flexibility or plasticity of mathematical concepts. And if you believe too strictly in the axiomatic exposition, you believe that there is an optimal set of definition, optimal set of truth, but you don't take into account the plasticity of mathematics. Mathematics is a living body and plasticity is important in life, which means that you have to react to slightly different and outside conditions by modifying your behavior. And never forget, I mean, it's good to have the best exposure at some point, but never forget that there are small variations which may resonate someday and become even more important someday. And a good example, I mean, just not far from our subject. Take the history of topology in a few sentences, a history of topology. It started with, let's say, Lefchet's norm in Poincaré in the beginning of the 20th century, which was geometrical, intuitive, and combinatorial, and not completely both, as we know. So you use set theory and you can manipulate quite awful monsters and then came the development of topology as set topology and look at what was done in Hungary and Poland. In the 20s, I mean continuous, I mean the various characterizations of continuous. In purely what you discussed yesterday, the topologies always.

1:15:00 And so, but then we had, then what we had, it was not combinatorial topology, but algebraic topology. Chegg and Hobbes, combinatorial topology. Combinatorics of algebraic variety. We take an algebraic variety, we split it into pieces, we rearrange the pieces, we calculate the cohomology of the pieces, and so on. Also, we express everything in combinatorial terms, but by using the modern technology of combinatorics. And, you know, combinatorics, combinatorics, which was considered... The 40s until the 40s and 50s combinatorics. Look at the judgment of Giordani about, I mean, his classification of the mathematics combinatorics and logic at the bottom, at the bottom, at the bottom. He didn't think much of functional analysis either. Yes, that's true. Really at the bottom. And then, mostly due to Rota, I mean, we understood that really. The borderline between algebra and combinatorics is the first one and it may be a return to the British tradition of algebra in the 19th century. I was reading recently Bull, the book of differential equations. Oh, what a book! What a book! I knew, of course, law of thought, but I did not know his book underhand. It was the best algebra, British algebra of the time. And it was the first one, I mean, people speak of G-module today, that is already very clearly expressed, in an elementary way, but very, for interesting example, clearly expressed in Boole's. So, combinatorics was, well, people did not understand that combinatorics and algebra are more or less, I mean, two phases of the same thing.

1:17:30 But now, of course, combinatorial topology, combinatorial group theory are prominent. And there is a back and forth movement. And if you are too dogmatic and you say, well, I mean, now set theory has conquered everything, or algebra, as we know, has conquered everything. Well, if you are too a little too dogmatic, then you will prevent further progress. Well, I don't have to make this bridge here. I mean, I thought that in other places I would do it in this mode, but not here. I think that's probably towards, just to draw an end to the kind of historical phase. Just one quick question. The recognition of sets as sheaves on a one-point space, is that, as you were saying, originally sheaves were? But they would have wondered why you had asked. What would be the point of... Well, of course, the answer was there, but people would question the question. But this was not a familiar idea. But interestingly enough, this idea of classified living has another interpretation.

1:20:00 But if you work not in spaces and chiefs and so on, but in a non-commutative algebra in the spirit of Kahn, then you end up with another definition. So a port is really an algebra which is Morita trivial. An algebra is Morita trivial too. And then you came with a solution that... What represents really BG is the convolutional algebra of the integral function of the group with the convolution. That this algebra is really the non-commutative space corresponding to B. I think one of the main problems at present is the interconnection between the ideas of Korn about non-commutative spaces and the botanics. To make the real connection. Many people have tried it and, at my advice, Tapia did some work on it. This idea will never be in case. So, again, the idea of space is... I mean, if you are too dogmatic about what space is, I mean, you are too. And to me, I insist very much on the plasticity. If you believe that mathematics is a living body, developing according to some biological, I will not enter into that kind of discussion. But then you have unique plasticity, and I just gave an example where an idea which was considered as dead, resurrects some years later or some centuries later in a new perspective.

1:22:30 By infinitesimals, for example, the frequent case. Well, all in all, I mean, that was something good to do, but the success was not there, was not there. I mean, when we pretended to be the writer in the way of Robinson and his followers, the idea was good, but the idea to be pedagogical and producing a system which would be more natural and more intuitive, that doesn't fit. Well, there again, there was a kind of absolute idea that elementary morphisms are the things. Morphisms that preserve first order definable concepts with all alternations of... I mean, that lies behind his interpretation. No standard analysis is one. I mean, he bought something, he bought something, but it did not bring the revolution that some people... No, no, no, that's true. But another one that I have been among my colleagues who work in topos theory... They learn from Rotendieck that toposes can be considered as generalized spaces. So now this is an absolute principle. All toposes are generalized spaces, whereas, if you look even at the work of Rotendieck, he makes it quite explicit that this is not the case, that there are some... What is really a generalized space in some kind of controlled sense within the huge world of topos is, people are very reluctant to study this for some reason because of this slogan that the general case is what there is, but I mean you mentioned the group acting on a point. This is a clear case of a generalized space in a very concrete way because of the origin of the cohomology of groups. ... covering spaces, fundamental groups, and so forth, so that the spaces... So if we look at it from Toko's point of view, sheaves on the space and the actions of the group, these two apparently very different things, are in fact working in the same category, and they must work in the same category.

1:25:00 On the other hand, how far beyond that should we go and still consider it as spaces? There are many, many aspects of this. It's very hard to consider. I think I finally got Colin to consider that. But I would still say on Groton, yes, sometimes he wants to say, okay, there's this petit gros distinction which has to do with... But other times he will say that the notion of topos is the new definition of space. This of course is correct. Both are correct, obviously. These are not mutually exclusive, they're just contradictory, which is even the better thing to say. Well, I mean, it reminds us of an old tale of the blind people going to the elephant. That's come up so often around Johnstone's book, but the one I want to add, especially when I talk to philosophers, is this other blind man goes up towards the elephant, misses the elephant, goes to a drape, feels it for a little while, and says, an elephant is made of cloth. Because another thing that can happen is that you're just plain wrong. It's not that everything is an aspect of truth. Some people are just wrong. Yes, well, when you have a notion like that on top of these, it's really a kind of protean thing. You have, you know, many different ways, right, of viewing it, and no one of them, I mean, you could call that a judge of space, but all that does, I mean, the term space is an extremely protean notion. I mean, it's exceedingly general. You can't, it assumes a kind of different form, right, in each context in which you find it. But again, the idea that the classical general notion of topological space is somehow the default version of cohesion, for both, you were speaking about the combinatorial and so forth, the five-dimensional spaces, but of course, the central idea of topology is to unite that with the cohesion of infinite dimensional spaces, function spaces. But the idea that the classical definition of topological space is the default notion is clearly wrong, because already with Frechet, the idea was that the function of space should be well-behaved.

1:27:30 It's over 50 years now that Horowitz, late 40s, defined this notion of K-space, basically rejecting the absolute character of the... So-called standard definition, just in order to achieve the simple adjointness, as we would say now, of the function space construction. And yet people go on and on and on and say, well, topological space is this, whereas the algebraic topologists, of course, they don't care too much because they only care in the homotopy. But they always presume, well, it's one or the other of those Cartesian close categories that we're working on. Whereas the analysts, you see, the analysts, as far as I know, still have not grasped this point, that the general notion of topology is a very bad one from the point of view of precisely a function space. Not to mention the riddle of super-semitic manifolds, super-manifolds. While I was working, while I was giving a set of lectures, it was very baffling because, I mean, if you insist that a super-manifold is a collection of ports, it doesn't work. Super-manifolds have two meanings. First of all, you say, well, there are some rules that change the signs and they are quite automatic, I mean. But when you are used to a quadratic form and then all of a sudden it's an anti-symmetric form. And when you write x squared and you say, well, x squared. The noun x square doesn't exist, or it exists, but it's rather subtle, I mean, if you begin with the two foundations of mechanics, I mean, to give an expression, supersymmetric mechanics and so on, I mean, all the things you are accustomed to, it's completely upside down, but there is also a conceptual problem, which is superspace, it is the collection of point or not, it's definitely not. To make the term, of course, you have to compromise between them.

1:30:00 Something workable for the mathematical physicists do not want to understand all the conceptual points, and something which is reasonably rigorous. So again, again, the idea of an optimal set of axioms, definitions, theorems, is not the best thing. So what is the shape of your definition, Zubranoff, that you're going to get? You may change, you may define, if you have a category with certain objects, you can define a new category with the same objects. One of the basic ideas of categories. And so, for me, it's quite manageable. I will test that in my lectures. For me, a super manifold is a vector bundle over an ordinary manifold. Let's say, you have an analytic complex manifold. Suppose that you just know the analytic complex manifold. You don't know the topological species, you don't know the C-infinity manifold. But then you can still speak of the complex analytic manifold with different maps between them. The maps which are complex analytic, the maps which are real analytic, the maps which are smooth, the maps which are continuous. So with the same object you can build a lot of categories. And so my idea is that in order to build supermanifolds, you start with familiar objects, vector model over C-infinity manifold, but you define new maps. That's an idea.

1:32:30 So, the point is that I think we have been influenced by the point of view of structure. We want to define, I mean, if you follow the guidelines of Womacki, you have to define a category by having the object and a phase full factor in the set, so that the isomorphism has exactly one transport action. And of course, that forces you, for instance, in manifolds, I will just take ordinary manifolds, with my definition, a manifold is something covered by charts. I don't identify two different sets of charts. So my object is a scaffold, I would say. And then I define properly the maps. So my manifold is defined with a set of charts. But then, between two different segments of the Atlas, I define various notions of maps. Now, the one which respects the Atlas is the one which does not, and I don't have to change the Atlas. I mean, it comes automatically at the end. Because, and I say, of course, now if you want, I mean, there are some objects to scaffold which are isomorphic in the category, well, you may consider that they define the same manifold, but you don't have really to define it. You don't have to give a formal definition of what a manifold is. You have a category of pre-object, whatever kind of pre-object, and for preshifts and shifts it's the same. You well know that if you work with preshifts and you... You go modular sub-category, you all know that. Inverting some arrows, you can define the shift in terms of the pre-shift. You keep the same object but you change the maps. And it's exactly the same strategy in many situations. You have some object, you have various categories of class over the same object, and depending on the various categories, you find various notions of isomorphism. And if you want, you can identify it and you can say, okay, two scaffolds, two pre-manifolds are analytically, complex analytically equivalent.

1:35:00 They are oviomorphic, they are diffeomorphic. And then if you want, you can speak of the equivalent class. Why to select one specific object? And for supermanifolds, that's the one I found. For people who are already familiar with different structures, people doing mechanics are familiar. I don't have to explain what an ordinary manifold is. But then I start with this definition. Two sentences apply to my definition. Because I knew I was coming here, and I was repeating my legs, I think, what? This fits very well with the examples. Of course, what is behind this is some result of Voronov, which is one of the existing theorems. This is equivalent to the definition by Costant, an ordinary space with a sheaf, a gated committee. Gated over Z2? Yeah, gated over Z2. So the notion, I mean there is a, I mean to substantiate my definition I have to take into account the reason of Voronov which is that this is equivalent to the definition of a costa. All examples, all examples, it's good to have this fear. I mean all examples fit under my definition. So it means, in other terms it means that this definition is low, that you can, I mean in my category of... But the point is that the gluing, I mean, if you start with vector, but not the ordinary mathematical gluing, you don't have to repeat. The point is that you could start again from local models gluing. It takes time. And I don't want to spend two weeks, I mean, six preliminary lectures speaking about charts, super charts and so on. And the gluing of local data is already achieved in this ordinary manifold.

1:37:30 Well, it's not very far from the definition you find in Manin. It's like paying something. Someday you should invite Manin. We are intending to do that, actually, yes. I think he would be happy to come. If he can manage his schedule. Yes, that was one of the things I was going to discuss with you. Yes, of course. Manin may discuss it with me. Fortunate situation to be at the meeting point of two traditions. The one of the Gelfand School. It's the best of both. It took the best of both. And the combination was. Unfortunately, I mean, the people who claimed themselves to be the people who came to take over, both in a technical sense and a little narrow mind. Excellent workers, excellent engineers. Some of you are not so fortunate with me, why don't you fight with a German, why don't you do all that kind of thing, why don't you fight with my German, etc. And at some point you would say, ok, this student, this student, he is for you because he will do nothing. Of course. I say, my friend, I will tell you one thing, when I marry you, The grandfather of my wife came to me and saw and explained to me that when he was young, his job was in the cavalry, he was in the cavalry, his job was to train horses, and horses in the beginning of the 20th century, he was very proud of that, and so he would explain to me, so when I was in the cavalry, I was training horses, and then they brought me a new horse, if the horse was, he said in his thing, flagada, flagada, you will walk. I would say to one of my friends, take it. But when the horse was excited, the horse was for me. Remember so. You will be a teacher.

1:40:00 My job was to take white horses and explain them without taking the steam out of them. Remember that. I think it's a good principle. I think all educators should have that in mind. Our role is to discipline young people. And when I was the columnist, they would send me the people who had too much steam. I like the people with steam, but of course they would not fit exactly the patterns of... They would resist the discipline. Yes, they would resist the discipline. I mean, it's such a place you would favor. That's a place for white students. So that is all. So here, I mean, well, you know the European program IPDE, which is supported by the Max Planck, the IHS, and so on. It is a poster program. And there are 45 fellowships there every year. One year three of my students got there. And a wider student! I said, okay! My wider student, three in the same year! And now I'm the one with, not so wide, but good.

1:42:30 So I said to myself, after all, I may be a good old trader. To my grandfather, the grandfather of my wife, but also to Homer, the prophet, so-and-so breaker of forces, he's actually in the head funeral gate, the very last adjectives. Thus did they bury Hepto in the tank. Categories of space. Are you saying that? You're proposing... Well, I was proposing we might start the discussion now, but if you think there's a better topic, then, yeah. We still have an hour, so it's not much. At least an hour, not much. If you just want to take a ten minute break, that's fine. Yeah, I think so. Let's do that. Yes, let's do that. Yeah, sure, let's do that. We don't want to waste tape. The new star in U.S. mathematics is Jacob Neuer. He spent a good deal of his time, he's finally got into homotopy theory and myoclonic stuff and beyond, but he spent much of his probably very brilliant adolescence doing Samuel Ambrose.

1:45:00 For many years, not just, you know, for a long time. I mean, it's clear to me once you go out to Harvard and you get in a moment, I don't think it's going to be a shame. It's a very confusing concept. I should have given you a little bit more time. So there are a few of these people, and the lesser figures have told me similar things. Okay, so the theme of categories of spaces. Cantor negated the idea of Mangan introducing... We learned how to base mathematics on the idea of structures modeled in the category which was as deprived of structure as possible. So in a way, these principles about choice and continuum hypothesis, even the law of exudate memory, They can view them in their attempts to express this lack of structure as an opposite, in order that the background in which you interpret the structures does not influence too much the results, so somehow this is the idea. So the features that Mengan was supposed to have, which he saw in mathematics, mathematical practice, were shoved into the idea that the cohesion Cohesion of objects is itself a kind of structure. So topological space, chronological space, and so forth. This is a specific kind of structure.

1:47:30 But then there are lots of categories of cohesion. And so, you know, you have smooth, continuous, measurable, and so forth. But cohesion is the same as zusammenhaut? Zusammenhaut, I think. Zusammenhaut, yes. Not Zuzan and Haung as it became a topology. Zuzan and Haung as it became a topology. Well, in fact, it's just the zeroth measurement of Zuzan and Haung, in my view. Connectedness. Connectedness. Because the other ingredient, the need for, say, general theory of cohesion, was, as I know, first recognized by Frechet because He wanted to recognize that the cohesion of function space and the cohesion of ordinary space has something in common. There is a Hadamard. By the way, I was going to ask you, when did Hadamard die actually? Hadamard? Fifties. He was very old when he died. I met him when I was a student. He was still coming to the library once a week. But it was very overwhelming. And I remember one story. So it was connected with the first Schwartz seminar, so it might be. One of the first Schwartz seminars was about... And Adama came to the library and spotted the proceedings of this seminar, which was just out of print. And he came to the librarian begging for a copy. And the librarian knew very well. We knew also that Schwartz was his nephew, and don't forget, I mean, say, oh, Mr. Schwartz did not give me the list, the list of the rest of the experts, so you cannot take it.

1:50:00 So just, but just to situate the things, so this might be 55th or 56th, and shortly before this, late 50s, very old he was. He had stopped being an actor. Oh yes, he was, no, he was not, but he kept an interest, yes, and I remember my first, you know, it was submitted by him. That was the old-fashioned guy to change the rules, of course. So, of course, he was at the very first international congress in 1997. The Zurich one. The Zurich one. He was not actually present, but he sent a note to the effect that maybe we can actually use this set theory stuff to understand the region of function space. I'm turning it into my... And there was some discussion on the Italian side. Well, actually Voltaire has already done this, and Vasconcelli has already done this, and Arcelin was starting already on this, and so it was only a couple of years later that Audemars... So in particular, Frechet is conscious that this is two apparently different forms of, again, I don't know a better word than cohesion. You could use continuity, but that's already got a specific meaning. That these two forms, the forms that occur in the function spaces and occur in ordinary spaces are, again, each more principled. These are similar, so there must be a general theory that includes both. Of course, not metric, but at that time. So Voltaire, Adhemar, and Frechet are concerned with this question.

1:52:30 The determination of, as I say, the default determination that became the idea of open set or closed set, in other words, the category we know as general topological spaces, which in fact has a glaring defect with regard to not only does it contain pathologies like piano curves and all that, which is another discussion, But it doesn't permit the natural manipulation with functions. Precisely, it should have been serving this function, deal with function space. So it was, as far as I can tell, it was Horowitz who first really pointed this out, and to R. H. Fox, who wrote in 1945 the paper about sequential convergence of the... The category of spaces which have sequential conversions, that is, sequences suffice to determine them. But this category is, in fact, what we call Cartesian closed now. And of course that was an old, in the concrete practice, that was an old idea of analysis. The convergence of a sequence of functions f sub n means that given any convergent sequence x sub m of arguments, you take the diagonal f sub n, x sub n, this should also converge to f of x and so on. That's exactly what comes out of the usual formula for Cartesian closure when you just apply it. So this much was... It was more a simplicity quotient. When Gauchy refers to a corner, implicitly the definition he uses is this one. So, Fox refers, in fact, to Horowitz in 1945, and then in the late 40s, unfortunately never published, but in lectures at Princeton, Horowitz gave the definition of k-space. Essentially where you replace the idea of a convergent sequence by arbitrary compact spaces as models, but nonetheless you achieve this Cartesian closure and there have been many variants.

1:55:00 These things are compact open department. On the function space, we'll again give... André Weil played with similar ideas in the inner work of Bobacki. There are some drafts by André Weil, what he called compactology, which is more or less the same idea. Wasn't he doing that in connection with uniform spaces? I mean, that was a thing that he... No, no, no, that's later on. I mean, that's later on when Bobacki was discussing the integration theory. At some point, I mean, very rightly, in my opinion, he pointed out that... I mean, due to Lee-Spark of UOM integration over compact spaces, and that's the definition that Bobakim takes as a measure, I mean, by duality of these continuous factors, but he says that for more general spaces, the natural idea would have to exhaust the space by compact subspaces, and so you have, I mean, to glue together, I mean, many of the other various compact pieces, and he calls that compactology. And he tried to define, I mean, so a set would be measurable if it cuts every compact subset as a more measurable or more immeasurable set and so on. And that idea is resuscitated in a rather obscure form in the final chapter of the integration of Boeck. And when Boeck, he tried to extend his integration and what took over as far as... After my insistence that you should include a Wiener measure in the exposition. And for the Wiener measure, I mean, that's enough. It's one of the things that Wiener recognize. It's very important. The Wiener measure sits up to epsilon one.

1:57:30 So this idea was played very small in the function of the map. And by the way, there was a paper by Weiber and DeGeneres on a proper map. The intuitive idea is that fibers are compact. From the point of view of the category of topological spaces, you must impose also closeness, but in this category of compact regenerated spaces, closeness is determined by intersecting with arbitrary compacts, and therefore if the fibers are compact, the map is closed, and so you have a simplification on the definition of this basic notion designed for this other purpose. It's still a good idea, I mean, to exhaust the space by income spaces. And as I said, in the integration theorem, there is a general theorem, a general theorem which is an extension of the Wiener measure, which says that on a rather general space, I mean, the measure is exalted. So if you have a measure which is a final total mass, then you can, up to epsilon, fill your space by the core of that. It's still a very, very... This idea was resuscitated. This idea was not forgotten, been resuscitated. Okay, compact spaces, I mean, all reasonable spaces, but it's not usually taken into account. Is this the same as compactly generated? And what is the precise definition? There are variants, you see, whether you assume that the bottles are housed or not, but essentially...

2:00:00 But these K-spaces are often mistakenly called Kelly spaces because J.L. Kelly in his very nice book on topology talked about these K-spaces that people assumed that he was doing a Bonhoeffer question. But he never made it in any way, this claim. But there it's just defined in terms of a set of points and a class of subsets. The point is that a set is closed if and only if it's intersection with every compact set. That's the thing. So in other words, but slightly more functorially, it means that you consider the category of compact spaces, and then a given space is determined by a functor. I mean, consider all maps of all the compact spaces. Both the contemporary and the functor of the compact. That sort of thing. But there are many variants, you see. Fox's original construction in 1945, it means essentially you start with, let's say, with compact countable spaces, or even compact countable spaces of finite type. That's your model category. And the general space is just a pre-sheaf on that. Again, satisfying various side conditions for that. The underlying structure is that you use these more finite, more manageable, more decidable or whatever spaces as models, and the general one is a pre-sheaf in effect on that, a sheaf, you should be a sheaf actually with respect to some topology or whatever, but that's the conceptual move to this. Wojewodzki has developed a theory of motifs. I mean, it's more or less the same idea. You start with quite controllable algebraic varieties and sheaths and all that kind of thing. But in that context, almost always, in all cases I know, there's no reason to first introduce varieties. You may as well take pre-sheaths on cumulative rings or suitable categories of cumulative rings.

2:02:30 Yes, of course. It's a small way. See, because of the universal teaching of the usual notion of open set as basic, there's a sort of psychological reaction that if you have to deal with the problem that involves cohesion in some way, you automatically, that's why I call it default, you see, you automatically say, well, it must be a topological space, first of all, plus something else. And this, in the case of schemes, was a very bad mistake, as Grotendieck. In the early 70s he went around preaching, please everybody forget the definition I gave before, the real definition was this. And there's a sound reason for it. I think you may have been one of the first to notice it. If you think that a scheme, or even say an affine scheme, if you think it's a topological space, it's a sheaf of rings, and you forget momentarily that it's a sheaf of rings... This space in itself is a very, very bad invariant because even the simple Cartesian product of two schemes does not correspond to the Cartesian product of the spaces, and so as you said, an algebraic group is not a topological group, it should be, but it's not, and so on, so those fundamental conceptual problems are in a way tied up with this. Relief that on the one hand we need categories of cohesion, but on the other hand there's a default way to do it. They are using topological space as a step zero. That's why at some point I insisted that a scheme should be taken as a factor over the rings. Exactly. So, not taking the first step of gluing rings, but as a factor over the rings. Function over rings. With satisfaction. Appreciate on rings, right, right. Which immediately... And I think, although I'm no expert, I think this... I think this even extends into Galois theory, that the idea that you deal with this Galois theory of all finite field extensions by passing to some imaginary infinite field extension and looking at a huge group and then compensating for this overextension by imposing again on it a profinite topology.

2:05:00 In my view, ultimately, that's all spurious. There's an environment in which all properties are clearly defined, which is just sheaves on the category of finite field extensions. In some sense, the underlying set of the scheme is not a single abstract set, but it's an object. In another topos, but now just using fields. You need only finite field extensions. I think every installation, in other words, what you achieve with this continuity with respect to the pro-finite topology could more easily be achieved directly just by the concept of naturality. Yes, you're certainly right. I mean, I mentioned yesterday this thing that we did in the model theory of Galileo. We couldn't have done it unless we'd taken the second point. There's nothing there. It's just too much set in the end there for you to do one thing with them. If you just buy it down to natural, you get something useful. But just a technical point I want to check. The profiling topology isn't a further imposition, right? The group only has one profiling. You don't have to only look at... In nature, complex numbers. We have the algebraic number within the complex. You can consider that you have a gas content. These are more than one. Most of the fields of interest in algebra, geometry, and mathematics are subfield of them. So, you have a natural algebra. On the other hand, when you go to the PI's, one goes to the so-called field CP of date,

2:07:30 which is the algebraic closure, the completion of the algebraic closure. But there is one thing which is all open. Consider the finite field. What is it? I don't know. I really don't know. And it's a concrete problem. Let's say people who try to do computer algebra, when they deal with finite fields, of course, it's a difficult notion. How do you encode the finite fields and the operation of the finite fields? You can write down even the finest of small degree, you can write it down using the Frobenius, it's only really explicitly, so you write down something that's not a factor, but how it factors depends on... You can say okay, you take... And it's a real problem to encode this calculation on the computer and this is a real technological problem, it's not just philosophical fantasy. This is a problem of computational complexity. Yes, and so you have the awkward situation where half of the natural examples have natural clauses. The other half doesn't, doesn't. And it's, well you can also say, well I take in the field of all algebraic numbers, if you need an extension, I take a prime ideal, but well a prime ideal is defined only when you're going to a limit, so it's again the same thing. But of course, if you take an abstract field, let's say even a function field in more than one variable, how do you define algebraic functions? What do you mean by algebraic functions? That's what it means. And it's quite surprising that this is usually overlooked. We are so happy to have this general existence here. But you know, if you look at the textbook by Van der Waalden, he is very...

2:10:00 Yeah, he's very correct. He tries to be constructive. Yes, he tries to be constructive. And I heard him even late, I mean. So it's not a... Well, the general algebraic existence and algebraic closure is unprovable again, you know, without the mechanics and the terms. You have no real idea what these things look like. I have a question for a logician, I mean, about what is the status of the existence of algebraic closure of the field with p-element. In what sense does it exist? Does it not exist? This is doable. This is really possible in arithmetic. Is there a recursive model? Yes, yes, yes. It's not that accountable. There are delicate issues sometimes. In fact, considerably delicate issues. If someone gives you an explicitly presented field, then the issue of... In the case of the finite fields, I mean, there is an explicit... Low-level, ultimately low-level recursive constructs, but it's useful, it's not useful. It exists, there's no question about it. But in the recursive tapas, presumably, if you have a finite... I mean, that would be the way, you know, sort of invariant way of presenting it. I don't know. I mean, I'm not sure. I'm not sure whether that procedure has actually been carried out in the, in the, in the, uh, in the recursion. It's quite confusing. It's quite confusing. From the standpoint of just classical logic, that is okay. There's an explicit arithmetical. Let's keep it on the tiny fragment of a rhythm, not so tiny, but the tape thing is a good example, and that's an essentially invisible entity, but the closure there is pretty much.

2:12:30 It would be interesting to know whether the algebraic closure of finite fields is actually provable in the recursive topics. You can call it the action of choice, but in another way it's just continuous variation of parameters. In most toposes, the thing definitely does not exist because the thing seems natural because you work in this Cantorian category with abstracts, but once the things are varying then you don't get an algebraic problem. I think this was first pointed out by Wraith, as I recall. Well, the complex numbers are not algebraically closed in some, you know, in some topos. You know, models, you know, were found by Highlander. I mean, whatever it was, quite early on, yeah, there are, yeah, that's right, of course, but I mean, for much of the usual reasonable definition of what complex numbers are, you can't generally show even that's algebraically closed. Which of course brings us back to the general point of Bill's exposition, how this whole issue of algebra and equations fits within this wider vision of the factors of space. Yeah, so the idea about cohesion. In order to understand the form aspect, cohesion, it might be cohesion to analyze it, it has at least two aspects, form and substance. So form is something like homotopy, you see, which doesn't depend so much on whether you have smooth spaces, continuous spaces, combinatorial spaces, you still have the same category of forms. And the way these forms emerge, more specifically, precisely with help of the function space, once you recognize that your category of spaces has this Cartesian close, has this exponentiation operation, plus just the zeroth level of suzamenta, which is suzamenta, in other words, connectedness. And so on and on and on and on and on and on and on and on and on and on and on and on

2:15:00 So, roughly speaking, is the other aspect where the substance is important. When you solve partial differential equations and this sort of thing, of course the homotopical result is of great interest, but you are also interested in whether things are smooth or continuous makes a huge difference there, you see. Something else. But anyway, this is all very much related to Grotendieck because in SGA IV he has this, I call it the chocolate exercise. At the end he wants to be awarded a chocolate medal for having worked out this exercise. There are about ten parts. I asked him whether he was a Michelangelo. You know, various people wrote this, their DAs. He himself was the one who wrote this. And he was still proud of it. He admitted that he was actually proud about having done it. So this is about, now he takes from Giraud this terminology, Gros and Petit. For what sites? Sites, and that's topos. I mean it's a common, I think it's a colloquial usage in this particular context in algebraic geometry because the gros topo, gros site, or the PT one in this context, the gros etal, the PT etal, for example. My position has been for a long time that there's a qualitative difference between these two things. It's not just that there's a method of construction starting with the data that he gives and produces this family of petit topos, which are sort of like

2:17:30 cheese on particular spaces, and on the other hand, one big topos, which is like the category of all spaces it contains, has a subcategory, full subcategory category of all spaces, but indeed... It's not just that they're all topos, as they are, it's very important, but they have qualitatively distinct properties. But this is hard to extract specifically from this exercise because it just is a mode for constructing, starting with certain data, namely a site which is moreover equipped with a special class of maps. And then he constructs these things and the exercise says to prove various things. And from the, again, I think people have neglected this problem because instead of not being so interested in functional analysis where there would be a huge difference, but rather in cohomology, there's a lemma that says that if you take the Grotopos over a fixed space x, And that's the one that's called the Grotopo. On the other hand, you take the key one, which is to put the sheaves on, well, they have the same cohomology, take a coefficient sheaf in either one and transport it. All the derived functions give the same results. So, in fact, this was Giroux's description that, well, the Grot one is something more convenient, pragmatic. To calculate the same thing. Because after all the Petit one, going back to this question of the stocks, typically has these horrible spaces that you can't understand really, except in the complex case where you have formal power theories. But in general this is very bad, whereas in the Grow one, well you can have ordinary Lie groups. Everything is working there. So you can use those directly and then calculate the homology. So it's like a tool. For calculating the cohomology, but it's not something of interest in its own right, but from point of view of functional analysis, it should be, because it links, it combines both the finite spaces and maps between them, not just sheaves on them, with a suitable, with a uniquely, with a unique notion of function spaces, distribution spaces, and so on. So...

2:20:00 What's the conclusion? What terms would you have recommended for the... I tried various things. One thing I tried was simply to keep the same letters and call it general and particular. Giving this distinction of peculiarly particular meaning to general. It's that sort of thing. But again, the point is categories of space. There are many particular generals, that is, there's not just smooth, continuous, algebraic, measurable, those are some main ones, but actually there's a whole infinity of combinatorial, of course, simplicial sets is a good example of a grotto pose. It's not a category of spaces in any sense. What Chanuel and I strive to do is we develop the mode of thinking topologically, but in an arbitrary category of a certain kind, not a totally arbitrary one, but a category that has fiber products and disjoint, some that are disjoint, universal and disjoint coproducts. That actually one can develop the sort of topological intuition, what Rodenby called the general idea of shapes and all that, in any such category, so that it's not necessary to first say, well, let's use the default category and then let's try to correct for the errors that that introduced and so on. But the progress is slow. However, I did find some axioms that are true of all the examples, at least most of the examples, where you have a general category of space as a topos as opposed to the particular ones. Namely, this can be explained very easily that you, there's this pi zero functor. Which doesn't necessarily go to abstract sets. It should go often to the Galois topos, because not only does the points functor to abstract sets, not preserved products, but neither does the pi-zero functor.

2:22:30 But both of them do if you go only down to the Galois topos, as they quote underlying stuff. So it's a huge... But anyway, so one axiom is that the pi-zero should preserve products. Now, as you think of sheaves on a topological space, a locally connected space, there is a uniquely defined pi-zero of Hunter, but it essentially never preserves products. It only preserves products, I think, for irreducible spaces. There are a lot of those, but it's just very special. The other axiom that I found, one that Rodendeek also uses in his big work on homotopy theory, is essentially that every space, that is, every object in the category that's called the general one, can be embedded in a contractible one, where contractible means, well, of course, it's connected, but indeed, it's a terminal object in the homotopy category, so that if you take Pi zero of x to p, where p is an arbitrary object, you get one always. So that means x is protractable. Not only pi zero of x is one, but pi zero of x to the p is one, or all p. Which is equivalent to pi zero x to the x. That's because the connected monoid with zero sort of acts like a unit interval, so you can improve contractability once you've got this monoid acting, because that monoid of endomaps acts on any function space, and so all these function spaces work. Anyway, so the fact that any object can be, any space can be monomorphically mapped into a contractible object is to be expected for a general category of spaces, but is definitely false for the particular categories of spaces. So, for example, there's no topological space whose categories of sheaves satisfy both these axes.

2:25:00 Now, the statement that everything can be embedded into a contractible is actually equivalent, using the internal logic of the totalist, to a very, very simple thing, namely that it should be a connected space with two distinct points, because there's an opposite situation where, in fact, the component structure and the point structure are isomorphic. Which doesn't really reduce to a point at all, but it has a unique point in it. See, that's another completely opposite sort of thing. So, if there exists a connected object with two distinct... This is an argument in Grothendieck's manuscript. If there exists a connected object with two distinct points, then that object can be mapped into the truth value object by taking the characteristic function of one of the points. But the other point will then be mapped into false, because of the distinctness of the thing. And so, once you've got it into the truth value object, which is a monoid under and, a monoid with zero, it propagates into any function space, but any function space is really an arbitrary power set, and of course anything is embedded into the power set by the singleton map. Of course, you don't have to use the power set. It suffices to use the partial map classifier, which is a very small part of the power set. How much of that can be written down in the internal language? I mean, there isn't any way, I guess, just using the internal language. Of course there is a distinguishing group of general for particular right so these are so of course there wouldn't be no there is there is there is very definitely that's you see in the internal it's not that all topos are just sort of not at all they're qualitative in the internal okay yeah can't what is cantor's original move amounted to saying that within

2:27:30 Mathematics as it exists, there should be these discrete infinite sets, you see. So, in fact, it's completely internal elementary language. If you have a general topos, I mean, a topos, elementary topos, which is moreover supposed to be the general sort, then you can define, there's a Galois connection. Defined by the following relation, you say that a map alpha is orthogonal to a space S if S to the alpha is the identity. In other words, that maps from the domain of alpha into S are all constant, basically. You can't move. So this will define a category of all the S's, which are discrete, cantorian, with respect to... So all you need is the parameter alpha and the thing that you, well, there are many examples, many, many examples. This works in simplicial sets and so on. A very nice example is you just take the first order infinitesimals. Take the map from a single bare point into the point that has the first order infinitesimal neighborhood. Just that map alpha alone. And then you take the orthogonal, the category of all these S's. So this will be then another topos automatically, which is reflective, while this reflection is the pi-zero functor. So the needed, I mean, in some sense the whole contrast between cohesive and non-cohesive, which is, everything is based on, is inside the same topos. So, yes, very much so. You can see the pi-zero functor just as an operation inside the topos and this distinction between the categories of space where pi-zero preserves products but not equalizers, again that's crucial because a typical example of non-connected spaces is gotten by equalizing maps between contractible spaces.

2:30:00 Take the line and the line, wiggle, wiggle, wiggle, wiggle, wiggle, wiggle, and you have an equalizer as a disconnected space. So if pi-zero preserves equalizers, then you have none of the interesting combinatorial phases. And yet, there are topologies where that's true, where pi-zero not only preserves limits, but in fact has a further adjoint. These arise in the analysis of the general categories. The two qualities, extensive quality or form, like homotopy, and intensive quality or substance, like Tom's catastrophes, exist in certain spaces. If you throw away the global cohesion of the spaces that retain the catastrophes, you still have something. But that's going to be, again, a topos of this qualitatively different sort where pi zero is so damped, continuous, that it's actually isomorphic to points, even though... So yes, very good point. It is internal. The sets are not external. You don't even have... You don't even have... We're talking about u topos, okay? You don't have to start with u. You extract the u from... The suitable version, and that's a nice thing about it too, if you start with algebraic geometry, then the suitable version of, if you're over a non-algebraically closed field, then the suitable version of discrete is this Galois topos. It's still not, you know, a group snack thing. Just, these are just the things, I mean, specifically, you have this map from bare point into the first order infinitesimals. What are the things that it's orthogonal to, the S's? Well, taking the function spaces, you see, with that, or tangent bundles, and you're saying, you know, that all maps on tangent bundles are trivial, that means that, say you're dealing with a, imagine that S is the spectrum of some ring, well you, this proves that this ring is separable. This is a separable extension.

2:32:30 And therefore, it's one of the, it's part of the site for the Galois topology. The site consists of separable objects, if you like, I mean, just products of fields, you see. And the Graubendieck topology is the bar topology, we call it, where everything is a covering. So it becomes actually boolean, atomic boolean, as Barr showed. But it's definitely not the category of abstract sets. So the thing that you abstract really, in some sense, from the thing, is really the appropriate base for it. And yet, but isn't the category of sets, right? No, it's an atomic Boolean topology, the actual number object and all these other properties. Yeah, it has all those properties. But it's not generated by one, you know, unless we're algebraic with prose to begin with, in which case it is the abstract set. If you were algebraically closed to begin with, it is the abstract sense. Well, that's an interesting, yeah, that's interesting. Because the difference in that case, that you, in both cases, I mean in the case of algebraically closed and non-algebraically closed, you have a common basis, right, or a common construction for, you know, for obtaining a sort of a basis. But you always get a Boolean. You always get a term which has some sort of classical features, which doesn't necessarily coincide with complete discreteness, at least in the case of the non-algebraically closed case. That's very interesting. So what does it mean for the old minimalist? I don't know. I'm trying to think about it. So you construct this, but you construct it from... The category of all algebraic spaces. You start from that. It's like Hegel. Hegel said that we start with being, then we analyze it, rather than starting with a category of sets and then building it up somehow.

2:35:00 But we do that too. We start with being, then we go all the way down, and then we're in a position to climb up. We have a vague... This is an ever sharper idea of what this environment is within which it's possible to climb up. Quite opposite to the constructivist, constructing the void. So I think this is the first page on Hegel. I think I understood the first page. I don't claim to understand anything about that. You know practically nothing about it. What you recognize is you have to start with it. He argues this at great length, that you could have started with being, you could have started with becoming, no, you could have started with nothing, no. You mean in the science of logic? In the science of logic, yeah. This is what, you know, in a certain sense this is what Galois did. Reconstructing history, which I know historians hate to do, but, I mean, reconstructing it conceptually. He's already working in the world of algebraic geometry, and he says, well, let's look at this special case where we have these spaces that are almost one point. There's an incredible stuff there, which is a very useful tool in analyzing the more general spaces. Yes, this is really the opposite. If it hadn't been for the fact that there were surfaces and so forth in the world that he knew, you might not have thought of that either. I mean, this is really the opposite of what you might call foundationalism. Well, we're going to come to that, I guess, when we come to discuss Cantor and the whole... I mean, whether you can think of Cantor as a foundationalist. I'm inclined not to think it's all. I mean, I think it's, well, I presume that's what's going to be the distortion, or at least I'm at, yeah, I presume, yeah. I'm a part of it. You're part of it, anyway, yeah. The fact that he says he starts with man, you see, the fact that he never mentions them again,

2:37:30 Yeah. as a matter of fact, people, but of course there was quite a bit of development. Yeah, yeah, yeah, no, no, no, I, I, absolutely, I, I, I guess that's something that will come up. You know, at length, when we get to that session, it should be very interesting. The hope is that Penning is sufficiently a description of these general toposes. See, the girl in Petit is very misleading. She says they're all the same size. Yeah, yeah, yeah, sure, sure. What they meant was something not really sentient. I mean, they're trying to capture something. They didn't mean that. That's right. In any case, the serious applications in functional analysis have been lurking there. No post-theorists have not looked at them, except in a few specific, you can cite two or three or four particular papers where the partial differential equations, the wave equation, the attack, but again, just in their initial fashion. And on the other hand, the fact that, well, there is functional analysis in everything. So, for example, within algebraic geometry, there's also a functional analysis, a space of polynomial functions, a space of polynomial distributions and functionalities, which, again, has not been looked at very much. I think there's something called the Cremona group, for example. It's real, it's not finite dimensional, but it is somehow an algebraic group. It's an actual group object in its topos which partakes of the algebraic structure because there are polynomial maps into it. But it's sort of an isolated thing that people study. When the terminology of GROW MPT was originally introduced, was it in the discussions just clearly related to the sites or was there...

2:40:00 It was a specific site, it was a specific one, just topological spaces. I did not participate to this discussion. It's in McLean and Murdai, they introduced a sufficiently big category of topological spaces, which is nonetheless small. Again, it's just an exercise. And then Grotendieck formalized part of, I think part of it only, I don't believe it necessarily, his exercise against the most general construction of this type of thing, but it certainly is very general. But the original meaning was merely that with each topological space associated these two topos, it's either the ordinary sheaves on it or all... In fact, Mordeck and McLean give this example of a Grotopis. One of their examples is a Grotopis in which maps are continuous. It's rather nice, actually, because they show how, in quite a specific case, you could realize this Grotopis in a rather natural way. And you can sort of see it's actually quite easy to the internal wheel. Yeah, yeah, yeah.