Morning Discussions, incl. FW Lawvere, P Cartier, C McLarty, A MacIntyre, JL Bell

0:00 7th of June 2005, a discussion between Professor Pierre Cartier, Bill Williams, McIntyre, Colin McCarty, in which John Bell will probably shortly join, on a topic which we have loosely headed, Heimler, MacLean, MacLean, Eilenberg, and if I could just say very briefly before I make myself scared, what we thought might be a useful The framework for discussion would be if Colin perhaps could lead off by making some general remarks about the historical background in thermology theory in the 40s, out of which Alan McLean's collaboration began, 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 Grosvenor 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 over-climbed spaces moved from being a very common part of the conceptual step of the homology theory. And so on and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, There's a lot to say about that. The one, the aspect of it that I think is discussed more than commonly is, is Leschetz's role that, that Eilenberger reclaimed in his first paper on it, was written as an appendix to a book by Leschetz, that Leschetz wanted to model, picked these two as the people most apt to do it, that Leschetz had encouraged the whole problem of the sphere, because he thought this might push the subject. He said, here's something that we can't do now, and by figuring out how to do it, we may be able to do it. And the other direction is, talking with Mike earlier, I've been talking a lot more about this here, that there's a real divergence between how MacLean sees group cohomology and how Serre did especially, and that Serre was real important in pulling it in a new direction.

2:30 Yeah. And with what? Kakosini. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yes, yes, yes, yes. Perhaps a little about the general developments in topology, algebra, and the methods and forces that led up to the discovery of algebraic spaces. You know, I wasn't the first to point this out, but this was, in some sense, the key discovery that brought about the recognition of the importance of categories, because the concept of cohomology was a functor. More or less known, even if not formalized, but the concept of representable function was not so well known. 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 we're these spaces. But at the same time, in other words... 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 homotopic category, so that if it hadn't been for this development, one might go on believing that categories have faithful underlying set functions and that sort of thing, which many of the structures considered had had, but here was a glaring example that was definitely not the case, and so...

5:00 It was really a major qualitative change, almost simultaneous, in two or three years. It was very rapid. I would like to make a comment. First of all, I remember I attended the Cato Seminar in Denmark in the 80s. I was very young, I understood everything, but nevertheless. And I was surprised once in the cafe discussion. I mentioned that the K-Pi-N was the thing to represent the cohomology. I don't think it was, I did not study K-Pi-N, but it seems to me that, I mean, there are these papers of, what is it, there are the papers I said on the homology mark two of the K-Pi-N, which end of connection with the schema. It's during these works that Searle, I mean, at least, maybe it was not right in there, but I think the great importance of the fact that the key findings were representing the common defunct rule, I mean, appeared clearly in the work of Searle in the Commentary on Mathematics, Helvetica, of artists in all squares. And then, I remember a discussion with Searle at the time, I mean, he insisted that, well, his emphasis was slightly different. They wanted to consider cohomology operations, that is, natural representation from one cohomology factor to another one, or even endo-factor, I mean endo-representation, let's say. The stenoid squares are operations in the cohomology mod 2. So they should be, and they are factorial, they should be considered as endomorphic for the cohomology factor mod 2. Since the cohomology model is represented by some mathematics and space, calculating the cohomology of either mathematics amounts to describing the cohomology of the nation.

7:30 And I think that this paper in Commentary is one of the strong comments. So I don't know exactly who we analyzed, while the idea of a representable factor was not so clear at the time. But this was really the idea, that the cohomology of the idea of the mathematical spaces corresponded exactly to the cohomology of Penrose. And I think that's one of the strong points of the people. And so I didn't know very well about that. And I think that's another example. I mean, it's about the classified spaces. And already in Boyer's thesis, I mean, when he constructed EG and BG, I mean, it was clear that BG represents the frontal of the principal G-mantle over the principal G-mantle. But, again, in the homotopic category, homotopic classes of X into BG correspond to principal mantles over X, and already in the... In the construction of Borel, of the gas mining space, the homotopic point was very prevalent, because he said, I mean, one definition like that of Borel, take a plot, take any action of the group on the plot, you will not present table of the last set, but in a homotopic angle, because as you know, it's the same as a construction of space, and then you get exactly the Borel construction. Take a point represented by a contractible state, then you let the group G act on this contractible state in a way which is free, which is a new thing, and then I think there were two contractible factors beginning, began in two different ways, yes, yes, and classifying space is more or less a discovery of Habermas or Alejandro.

10:00 And the cohomology of the Iron Bear marketplaces was started by Cerny's papers about cohomology module 2 and then developed by Iron Bear and Cerny and Kant. So, of course, we all know that Cerny and Goethe-Boyle were very, very strong. And when was Goethe-Boyle's thesis? It was written in 1940. It was defended in Paris, I don't remember the year, but it was written in 1940. At that time, Borel managed to visit Paris for one or two years, and he wanted to meet Borel, who gave lectures at Zurich on the ship, and there is one of the first lecture notes, which is lectures on the ship, and then he wanted to go and meet Lorenz, from Austria, to develop ship theory, but after that he came back to Germany to work, and so, but when I remember Borel commenting that when he arrived in Paris, he was a little disappointed.

12:30 It's in my prehistory. I appear on the Mathematical Scene in 1950 when I became a student at École Normale and on the first day of École Normale I attended, that was on a Monday, and I went to the Catton Seminar. We got back to the Catton Seminar. He has given lectures about the axiomatic presentation of the comms. 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, actually, that the concept is, that's a name because of natural, natural structure, natural algebraic structure. On a functor, even which one doesn't look representable, it's a well-defined concept. So you say this is due to ser. I think so. So there's that general concept. And then in the context where you have representability, then you can say to computers the same as computing the cohomology. I think. Or whatever, but it's necessary to have it as a concept in order to make the identification. Exactly. So in some cases, it turns out to be, it isn't representative, it's a direct limit of representation, or something of that sort. But I think the people were really stirred by the discovery of the Stephen. And in his first visit to the United States, Catlin, in 46 or 47,

15:00 when he was enrolled, that was just the beginning of the development of Stephen. And almost immediately, Cato published a note in the French Academy on his own view of the stenot square, not yet a stenot pure relation, and he gave a more or less axiomatic description of it. And the main property is punctuality. I think this is still not widely recognized. You know that the values of some function that you have computed that maybe even has a set, 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. This is, I mean, this is a trivial result. It's one of the results of my thesis, but in some sense people don't normally recognize this. And you have to remember that... But the example of cohomology operations that I learned about from Sammy in Columbia, this was what... And one of the first papers I go in and say was about the impossibility of some vibrations. And that's it. That's exactly it. If you take Newton's space, you cannot have a vibration which is compacted by Wilson, and it's used deeply. I mean, it's used it in a deep way in the Steno definition, or maybe even in the Steno's way. And Morel was among the first people we called that. Well, Steno himself, but that's in my hands. I mean, the... By putting the steno-square of steno-location in general, you have an extra fraction of the cohomology, and that you can, the cohomology group, the naked cohomology group, or the Becky number if you want, I mean, do not distinguish between certain spaces, but if you take into account that you can distinguish between... And you can also prove that some maps cannot exist, some continuous maps cannot exist, because they would not respect the spin-off squares. And all these things, I mean, there is a stroke known by Borel, Borel in one of his first publications.

17:30 They show the impossibilities. And they were very quick to... It seems to me that, so, of course, and the same, I think there was an analogy with the work of Borel on the classified space. Again, the classifying spaces give rise to cohomology operations, the churn classes and so on, slightly different, but nevertheless, I mean, this idea that the cohomology is not a neat object, but it has operations of its own, and that these operations are controlled by the naturality was an idea which was instilled among the work of Eigenberg and Gaetan, Céline Borel at that time. There is also a natural structure. You can imagine, you construct it just as a set, and say, my God, it's a product, and it's a product, because the other part is the modification. But this, I think it was a quite important idea, but historically it seems that people understood that at the time that there was a media, there was a lot of structure on the commerce. But the part that came from that, or after all, they were natural. But the idea that the really, the origin is really the natural was not so well understood. It's said one in his paper about the cohomology of, I don't know if that makes any sense, the cohomology of mathematics in space is the first one really put in. Despite the fact that Sher himself was not so fond of mathematics. Even today, he's not really so fond of category one, like every practice in mathematics. He knows how to do that. That's never been so fond of. Non-important people, I'd say, is a no-no. Playing the concept of sheaf, I mean, Lorraine, what was the sort of specific problem?

20:00 There's a framework 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, probably wrong. This was the problem was to avoid getting the Nazis recognized. That's what he said very often. No, more practical, but more explicitly, I recommend reading the recent paper by Jeremy Shaw, Mathematical Intelligence. Well, it's a historical paper about what happened to Witten in the war, but he has some comments on his mathematics. So, I just got the issue yesterday. And the artist who saw it keeps saying a mathematical tourist. Right, yes. It's called a mathematical tourist. And Peter Michaud went to collaborate with him. Peter Michaud is an Austrian. I'm very happy that he had a... Diplomats, Atiyah had some diplomats and he had support from his government and he was granted some diplomatic favor too so that he could travel in Hungary, Czech and so on, more or less officially, I mean, without visa restrictions and meet mathematicians, organize seminars and, well, Wien is the center of Mitogloub, of course, but he knew it. And it's clear as a tear. So, but just... And so that this explains, he explained that, and the Schauder fixed-point theorem, Loewe's Schauder fixed-point theorem, the emphasis for Loewe, the emphasis at that time was to use fixed-point theorem to prove a fixed-point theorem in analysis.

22:30 So they proved, I mean, Loewe and Schauder proved their fixed-point with the idea of using fixed-point theorem for partial differential. And, but then, in this setup, you are not working with a space but with a map. He was interested in developing invariants not for cohomology, not only for space, but for a map, and the hypothesis of both and even the relative can be seen as an outgrowth, which I realized clearly and why I knew that at that time. And so Leroy was to invent, I mean, his paper is really not about the cohomology of spaces, but about the common, he says, I think that... I mean the spectros, I don't remember the title of the paper, but in an emphasis it's on maps. He wants to bring, and I think the only thing of shape is that when you have a map, in the Leuven spectra, you can consider a map and then you consider all of the bases, you consider the various fibers. Then what you have as a map is that each fiber has its own cost. And the fiber depends on the point in the base. The co-multi of the fiber may change. 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-1. And that was already...

25:00 I think motivated by some example from functional analysis, I wanted to understand what's happening when the fibers do not have all the same, not the same, all the fibers do not have the same cohomology, what is more general, what can happen, and so the shift is a shift, idea occurred by the fact that there is cohomology, the cohomology of the various fibers, each one associated to a form, and you want to organize these things in such a way that you can calculate... The idea of the cohomology of the total space in two steps, first of all calculating the cohomology of the fibers, and then reorganizing to sub-objects, and then calculating the cohomology of this new object. There is much hesitation about associating to each clause that there is a penalty. I also looked at a different idea, the idea that Guillaume explained as gating. He was strongly motivated by the analogy with, well, he wanted to have an extension of the Derham theorem for calculating the cohomology. For example, French mathematicians, they learned it from Ellicott. So, what we have in the Derham theorem is that you have a certain space which is a manifold to the space you associate an algebraic object, which is the differential force of Ellicott. Each differential form has a smaller cross, and so the axiomatization is purely algebraic, which is a smooth, compact manifold, a global differential form.

27:30 At the time, you don't have to distinguish between local and global differential forms, because you have a smooth manifold, and everything is local except for the root. But you have these extra structures that each differential form has its character, a certain cross, which satisfies obvious axiom. 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 lemma, which says that if you take a differential form that is close enough, a differential form would support around a certain point. If the support is small enough, then the differential form, if it's close, it's a common triviality. In modern terms, the aim was to associate to every space, a certain category, a DG in the function of the algebra, not yet a sheet. And that's what in his Harvard lecture. So this is more or less a dual notion of a sheet. In the context of the C.N. Kennedy case, namely every section restriction of something global. And so it suffices to consider the global thing, but then in turn they have the support, which is more or less the complement, I mean. Yes, because what you say, I mean, so the sheaf occurs duly. You take, let's say, you take the differential forms, and you take the local, where you want to have the section, well, the section of a certain sheaf over an open set. Then you take the open set, you take the closed components. And now you take the global difference. The grating is a description of some sheaf which is a portion of a component.

30:00 We all understand exactly that. You consider sheets which are quotient, of course. So, but then, Lohmann was not, I mean, Lohmann did not consider sheets per se. I mean, that was one of his motivations, the generalization of the Duran theorem. I think which is still an active area of research, I mean, in general databases. And especially I've been working for many years on infinite dimensional integrals. So, in infinite dimensional spaces there is still a calculus of variation. If you want to apply this idea to the calculus of variation, it is still an active part of research. And then he redeveloped a true generalization of the... He wanted to understand the cohomology of the knot. And then gradually he realized... The only standard situation when the fibers are all the same cohomology, or let's say when the string is long, the fiber, it's a local attributable vibration, and then it came to exceptional fibers and other situations, which when you have group action is very natural. Group action if you look at the open spaces and stuff. And then it allows that, I think, associating directly to the... So now in the base space you take a closed set and you take the counter image in the collection of fibers above. And then you take the global commons. Easy! And then you have a close asset of the big. And then it took some time. There was some hesitation and I think Cato went and said, I mean, if you look at the various editions of the Cato Seminar, there are some variations in the definitions. And I think it's only the book of Gordemon which is everything straight and took the definitions that we have today.

32:30 If you are motivated by many complex values, Of course, in many complex variables, shapes appear in a slightly different form. What you have, you have the local power series expansion. Exactly. You only have the fibers of the sheet. The fibers are relatively concrete in this case. Yes, the fibers are relatively concrete. The sheet itself is in the back. But the fibers, well, I prefer to see the stalks. In French, we do not distinguish between the stalk and fiber. That's it, the stalk and fiber. The stalks of the sheet. But then, in many complex variables, the stokes are there, before you have it. And so, as a definition, that a shift would be a collection of local stokes put together with the topology of the union, which is a little space. Which is a gain of representability, if you like. That means that there are other parts of space, which is a total space. It's another representability. Yes, there's both a left and a right end on it. But then, in the literature, it was called the Lazar, the name of Michel Lazar, as an aside, Michel Lazar was a Yasmin, or a close friend of my youth, and we collaborated on various things, and he had a very sad life, I mean, his family, I mean, he was rich in France, he had a Jewish family. And I think he lost some of his payments in the death camps. And more than that, he had an elder brother who was a guerrilla and was killed in the fight of the guerrilla law, murdered by the German, by the German, etc. And since his brother had joined the communist guerrillas in France, it was a very tense situation because we had two different guerrillas.

35:00 And the FTPs were the communist gang. You don't distinguish Jewish girl groups. And there is also the Moai, the Moai group, the Moai group, which was part of the communist gang. The Jewish groups, the youth groups were, and there was a lot of, I suppose there was a lot of mission as I was in the Moai. Moai was a Jewish group who joined the communist gang. When I'm not so well trained, I think they were, they were, they were some tyrants. They had some problems, but these are not where they are. So that's the way 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 18th German Army, the US and British Army in the south, 10 kilometers north of us, and in between the communist guerrillas and the non-communist guerrillas for two months of no-man's land. And as I mentioned, my father also. And so on and so forth. He was one of the most promising people at the time, and he participated in the Young-White-Young-White Gatto Seminar, and he is the one who is credited with the exact definition of a sheep as a collection of stocks, putting the topology, what he discovered is that putting the topology of the union of souls, and it's only later than, I think, it's God among us.

37:30 It's in God alone to really, I mean, put the standard in this book. It's very legitimate for me. Yeah, quite legitimate. So, what do you... ...was not an outside... ...was not really an outside... ...dimensional presentation, functional analysis. And for him, she wasn't a site. 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, and as I mentioned during the last year of the Carthaginian University's 100th anniversary, and I delivered one of them, and my talk was called What Became of Say University, and then there was a small I was the first time visiting France, and during the speeches, at some point, he got on me to speech, in his familiar tone, and I'm surprised, I mean, I've had the people organize this very happy day, to call on my former students, and say, even most, we are all your students, if you go for us, I'm not taking your side, I mean, I'm just saying.

40:00 In order to understand, at least in Paris, the development of considers that Carter was the leader, and that most of the discussion which began in the Nobel seminar would continue with the close meetings of Burbank and Carter, and the book of, according to the rule of Burbank, I took first draft, and since Gordon was not an expert, I mean, we, I mean, in the discussion, in the book, inside of the discussion, we said, okay, we need someone who is not an expert. So, we did serious things. And finally, for various reasons, Bobaki did not publish an account. And then, at some point, we said to Goodman, why don't you publish it under your own name? And many things occurred like that. I mean, the Bobo people, non-expert people, had to draft about new subjects. And eventually, I mean, the plans shifted, the plans of Bobaki shifted, and these things would not be part of the Bobaki 3D. That's an important remark, to realize that the neighborhood is very close. In French mathematics at the time was organized, I mean, the central figure in Paris was Carter. Later on came Schwarz, with a slightly different emphasis. And when we discuss both and against functionalities, Schwarz came with a different emphasis.

42:30 I mean, more analysis than topology and geometry. But then there were the two cohorts of French mathematics, and in the late 50s, I mean, they were dominating. At that time, you would have to be a student of Cato, or a student of, or a student of Boss, which was my case. I was a student of one student. But I mean, I participated to various seminars, to the Schwartz seminar. I gave lectures about both identities at the Schwartz seminar and I gave lectures in the Cato seminar and also in the Beaumont seminar. Which means that a king is a strong person, but he has an enlightened mind and he's surrounded by... An enlightened despot. Yes, an enlightened despot. Enlightened despot. Enlightened despot. Enlightened despot. Enlightened despot. Enlightened despot. And the French Mathematica at that time was under this rule, the dual rule of Cato and Schwartz. And it was a time where you did not have many, I mean, many discussions about whom to recruit and whom to appoint. Kantor and Schwarz would retreat two days in some place in the summer and decided the same, because everyone was a student of both of them, at least one of them, and they would decide, okay, well, this one we'll go with, that one we'll go with, we'll go with, etc. Everything was fine. And we were, I mean, we were happy that we were enlightened people. Everyone made a mistake, but very few, very few. And they were deeply honest and made few mistakes more or less. Exceptional of the historical sequence. Okay, but so, just to understand that so. And to come back to Michel Lazare, unfortunately, Michel Lazare, I mean, still under the impression of the Nazi time, he lost confidence in the Congress, like many of us, my youth, I don't have to.

45:00 I think it's in 1956 when the Russians invaded Budapest and said, no, no, no, no, no, that's too much. But many of us, until 1956, I mean, I think 1956 was a turning point. When they realized that the Red Army was invading there, it was more or less, I mean, his tax, how do you say that, I mean, his life. All of these are very important. Because, I mean, he was a communist because his father, his brother, his elder brother, was, I don't know, a Nazi. And then he lost confidence in communism. He was in great despair. The end of his life was miserable. He turned to a drug addict. And he committed suicide. Very, very, as he was, talented. He was a mathematician. One of the most talented mathematicians. The very important now is Pierre de Corbusier. I think as long as he was in contact with CERN, that was good, because CERN was really, I think, a mover. CERN was a mover. CERN was very quick to understand you and he was something you should care. And he was really our, not our father. CERN was a head of us. He played that role for me. The head of us, the white boss. Okay, but CERN was more fond of his approach.

47:30 And he was more or less fooled by this to introduce it. And it's only later, after, I think in 1946, after he came back to France, that he really developed a cohomology, 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 Iron Bear who shaped the final definition of ontology, as you see it in the cataclysm. And you see clearly that in the seminar of 1551, the first part of the seminar is an exposition by Eilenberg 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 don't know how I was 19. Not even 19. I was not even 19. I came to his first lecture. We define an action, we define an homology theory to be blah, blah, blah, blah, blah, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo, whoo Through the, and now with several complex variables, it led to the idea of general sheaf. In other words, was that the direction? I mean, sheaf left, sheaf don't cook for many years. Yeah, and then the notion of sheaf, well, what they called a sheaf in the, I can't remember in Gordon Hall's account, it's a long time since I looked at it. I actually think the notion of general sheaf is already in this category. The book was published in 1969 or something like that. Much later. But the notion of general sheaf was then really gradually... It's alliterating itself, if you like, from the notion of a sheaf of groups. Yes, yes. For many years, a sheaf was a sheaf of groups.

50:00 Whatever definition you took of a sheaf, and there were more or less equivalent definitions. I remember the suspicion of the people when one began to discuss the sheaf of non-groups and I would say exactly the same, the sheaf of non-groups and I think it's really good. Which really does require the notion of general sheaf. A sheaf is a espacetale and the equivalence between that and the idea of a pre-sheaf, of a sheaf defined in terms of pre-sheaf is a recovering condition. Well when was that? That theorem, that result really does require the notion of sheaf, of non-global. Well when did that emerge? I think it was Godemont. That was Godemont. I think it's Gronemann who clarified completely the difference between the pre-sheet and the sheet, to the best of my knowledge. I don't know exactly who, I mean, in the discussion, but it's clearly, I mean, the little exposition is the one by Gronemann, which predominated up in the Womackia. Early 50s. Right, right. I mean, Gronemann's book is 56. 56. 56. It came out shortly before the Tohoku paper, but it refers to the Tohoku. That's right. And he has an appendix where he takes into account it. But of course, it turns also into the cohomons, I understand. Yeah, and the cohomons. That has been another practice. Not only was it really the first one, I mean, written by... I take, I mean, Cato, Heidelberg, Homology called algebra, go to a seminar where Heidelberg and Cato and Sayer want to develop a cohomology for sheaves. But the common theory, as exposed to the common theory, there are many restrictions and many adult constructions, and I, and Voltaire was the first to really understand that there was something common to all.

52:30 If we take a sufficiently abstract construction of a model called algebra, the shape will come down in one stroke. Well, you're trying to create more constructions. Yes, yes, and so that's... but because... Certainly, Cato and Heidenberg had in mind the analogy between the commodity of goods and the commodity of heat, but they could not get rid of some restrictions because they did not have, I think, the idea of injective sheets, and that was the reason why it was brought back. And in the book of Gordimer, it's just in the appendix set, which was written after Gordimer. It had not yet published, but it is still published. It's very odd how remote they were from the idea of injective sheaves. Correspondents, they even talk about injectives meaning something more general than what we now call an injective because they don't think that what we now call an injective, or at least Serre doesn't, will work for sheaves. Yeah, exactly. It means effaceable. Injective loosely means somehow effaceable. He told me that as soon as he saw a room full 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 sheets. But it wasn't even asked in the seminar. No, no, no. In the Catholic Seminar, this question was in the seminar. And the idea that they choose... But you have to understand that I think it's a fact of personal duality. Historically, the CZG is the second-wide resolution by three modules, and while the existence of the three modules is given for a projective out of three, it's not a diva, the existence of a projective resolution is an injective resolution.

55:00 I mean, people consider only projective resolutions, and if you look at Gerhard Eibenberg's homological algebra, you will see that the projective resolutions are more or less prominent. Even if they put forward, I mean, they say, well, if you are the protective, you are the injective, because we have the duality, general principle of duality, and that was really in the mind of Heidenberg, I mean, the contribution of Heidenberg. Buxbaum, what people call Buxbaum. Injecting was considered as rather awkward, and it's only after Buxbaum and Seyd and Rosenberg and others on the thermo-logical method in the Comité Thibard-Gervain-Roubault-Gravins. But one can't really understand why they are important. I mean, there's a parallel existing among logic, of course, because the languages and so forth, these are really projective resolutions. Exactly, exactly. 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 get, say, Schindler. Because the objectives are quite parallel. And I suppose that's also because, I think, the cross-sections of... The Bayer construction uses the Zorn's lemma. And people who are still relevant about Zorn, they might not want to hear it, and see Pogutlis and Pogutlis. So, I mean, projective resolutions are very constructive. I mean, you can explicitly mix and explicitly injective resolutions who are not so constructive sometimes. And so that's why Voltendieck defined an injective shift, well, the definition is not important, but proved that there exist injectives. But by mimicking the various constructions using the strong remark. And if you look at the exposition by Godenow, he makes the point to make things explicit without the strong remark. He has an explicit, very explicit construction. Well, for the case of ideals inside the ring, that's the starting, the conductive step. Yeah, yeah, I mean, now the point, yeah, but... It does depend on the argument. Yeah, yeah, you can. I don't think you can prove it without the answer to the choice. No, no, no, you can, you can. I mean, if you look at it, well, that's a, there's a subtle point.

57:30 I mean, you first, well, the way it's explained in Goldemont's book is the following. You start with a multiple over the ring of intelligences. Yes. Then Q is injective and Q over Z is injective. Oh, yeah, yeah, yeah. Out of that... Dualizing into Q minus Z. Dualizing into Q minus Z. I mean, there's a perfect duality. So, you have an explicit construction of injective. Explicit construction. Explicit construction of an injective for a module, in that case, by dualizing these two over Z. You dualize twice over Z. And then, yes, and then, so, you have an... There are only two things to construct it and to prove it is injective. I'm suspecting that the latter still requires... Maybe, maybe, maybe. The construction does nothing quite. No, that's great. You're right. But maybe to prove that it isn't. Yes, because a lot of those injective things are actually very close to being equivalent to the actual truth. Indivisible abelian groups, all that stuff has been shown to be... Then what you have. There were two things discovered by Godemont. First of all, by the change of ring of scalars, you can go from one ring to another one, by the duality function, or by the adjoint function, the base adjoint function between different categories of models. So if you have a construction of a ship, you have a construction of an alien, and then there's another step which is similar to that, you go from reins and goods to ships, which uses the idea of Goldemar, his basic idea of Goldemar, that if you consider a ship as a collection or group associated to the open ship, then you build the stores by going to the right. Then you have the collection of... Then now you put a new sheaf, an open set, you take the function which takes at each bonding volume in the stone, but without any consistency relation, any continuity relation, a very, very, very hectic, shall we say, a very hectic sheaf.

1:00:00 So the product, so the section of an open set is a product for the open set of the base. That's a new sheaf. And that's, and the whole sheaf injects in that. And then you do that, which is, well, not yet, it would be relatively injective, I mean, with respect to some adjunct factor it's injective, relatively injective with respect to some factor, 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 breeding injective in various categories using adjunct factor in the categories. And so, the idea is that you have a certain model of categories, a Z-module, where everything can be made, and then by using various functions, I don't factor between various categories, you shift that. And that's explained in the book of Goldemar. And Goldemar was not happy with the various construction, which is repeated more abstractly in the Torben paper, and so he made a poem to me. I mean, that was an apocryphal. The definition of objective is that there exists an extension, and this is a non-precise existence in the real life. The thing is going to work, but it must involve the exclusion. I agree with you, because you have non-uniqueness. The point is that we don't move, I mean, when we want to move. It's a good situation to select a base and to define a map, just to define a map of the bases. So implicitly, yes, I suppose that the sheeps were, I think, the standard, and it's the one place where all these concepts are completely clarified and put.

1:02:30 Except that the point of view that a sheep is a factor is... It's not explicit in the bottom of that book, of course. It is there, but... No, it's explicit. It's explicit. It's explicit. He said it. It's explicit. You said that Sir Gelliver was the first to completely clarify the difference between a pre-sheep and a sheep. Was he actually the first to use the terminology pre-sheep? He considered, well, he defined a general notion of algebraic variety over algebraic variables, which was quite a restriction and cost me a lot when I had to write my thesis, because I had to write my thesis using sheep for algebraic variety over non-algebraic variables, and that's a time that I would say I had given this exposure, and I had to repeat in my thesis one or two chapters when I painfully repeated the things over and over algebraic variables, and there are some weak, some difficult ones. If you get a regular expose in Florence on this, you don't have to repeat it. No, no, no. So, and, uh, so, but... Yes, okay. But the point is that self and... I mean, the notion of abstract algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic Fine pieces are fine pieces, it was more or less obvious. But there is one subtle point in the definition of they, which is some subtle axiom which was being interpreted in the Chauvinist exposition and which comes back again in Sayre.

1:05:00 Say we realize that there were two notions. There was a notion following the axiom, more or less equivalent to the notion of Hebert, which requires an extra axiom, he called them varieties. And when he relaxed this axiom to a slightly more general category of spaces, he called them pre-variated. And in the first edition of Botany, he made the 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. So we knew that in various situations we have restricted conditions, but to construct as an intermediate step, to construct the sheaf we needed the pre-sheaf, to construct the variable, we needed the pre-variable, to construct the sheaf we needed the pre-variable. The pre-schemes were the real things, that the remorse-neural notion was the right one, and that the schemes probably became separated schemes. And also the point is that the notion of a separated scheme is a relic. The notion of a separated scheme is a relic. The notion of a scheme is an absolute notion. But when you come to the item of separation, it says that the diagonal is closed. It's sale will be marked in this paper that the axiom of weight, which was very complicated, even after the de-formulation by Schumann, which was a little simpler, but sale realized that it simply says the diagonal is cropped, the amputated line, and says, but then diagonal means the catheosy, means diagonal is a relative thing, because the diagonal is not in x times absolute x. So it's a very deep notion. And saying that X, the diagonal, is close is a very deep notion.

1:07:30 You should distinguish between zero and the category of scheme without any reservation. And then, well, when you deal with variation, which is again the idea that the notion of separation is not a notion of a scheme but of a map. If it's relative, it means a certain map is separated. More than all, the same linguistic principles. And I suppose that Godemont was meant to preach. Godemont, because, as I repeat, I repeat myself, Godemont was commissioned by Robacki to write an exposition. And a Robacki exposition was supposed, at least, by most priests, I mean, scrutinizing all the concepts and making the necessities dense. That was the aim of the Robacki. So, Godemont, for all of that, I mean, took his script and said, okay, now... People were very ambiguous about them. In Loewy and even in Borel, distinction is not clearly made. The difference between the so-called Lazar definition by Stokes and the definition of the factor over the open set, then it wasn't. And so, pre-Schiff, pre-Schiff, pre-scheme, scheme, pre-Varieté, so I think it's more of the same. I seem to recall this using, I'm sure, I mean this linguistically. Yes. Incomplete, Hilbert space. Exactly, yes. So this kind of thing. And I think it was the same idea. It's an interesting, in a way, pre means that it's a preliminary construction here. The final object is a... This is, I mean, 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 life harder for students if you start with the definition that you need for the theorem.

1:10:00 Of course, mathematicians don't want to get to the theorem, but the natural domain of the constructions is often larger. This, I think, is, you know, somewhere behind this, particularly, is the pre. And I remember, in the discussion for functional analysis in 2019, that I was there when we wrote the chapters on the Hilbert space, the chapter on... For the first edition, which was a very short one, and the second edition, which I insisted that we should have something a little more elaborated or, you know, a serious treatise. I mean, you cannot have only ten pages of Hilbert space. And there was not even a mention of Trescaso, Perito, things like that. So, in the second edition, which I took on my shoulders, and I insisted that we should. And I have some expertise in mathematical physics. I knew exactly what was needed. But Wobacki was the first to make a clear distinction between Hilbert space's proper image, incomplete and incomplete. And people did not consider seriously incomplete Hilbert spaces at the time. They would say, let's see, we have Hilbert space and within that we have our unbounded operators, so the domain is a dance of sex and so on. But the domain was not considered pure sex. And I think we should be returning to this. Coming back to this, I mean, 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 worked in the smooth case, in a relative case, and so this is good, but as a result, and I think... The idea of sheaves as puncture-on-close sets has been neglected quite a bit, so there is an exercise by Glenn Merdyke 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 space, but actually...

1:12:30 You know, things that are defined up to negligible sets, and 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 raft of things, in some sense an elementary sheet theory, which have been neglected because of this realization that the one definition is the best one for the general case, but under some hypotheses. You know this exercise? No, this has to be replicated 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. No, but I will just finish the question, please, to not comment on that. Ah, I mean, the Bobaki spirit, in general, is that in any part of mathematics there is an optimal set of definitions and concepts. 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 violated. It applies to a certain context. If you change the context, then... And my own idea is that, my own emphasis is about the flexibility or plasticity of mathematics. I mean, if you believe too strictly in the axiomatic exposition, you believe that there is an optimal set of, on definition, an optimal set of truths, you don't take into account the plasticity of mathematics. Because mathematics is a living body, and plasticity is important in life, which means that you have to react to slightly different 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 some day and become even more important some day. And a good example, I will just show our subject. Take the history of topology. It started with, let's say, Lefchet's norm, precarious, geometrical, intuitive, and combinatorial.

1:15:00 And not completely in Gaucho's field. Then, people at that time, I mean then there was Megan Lane, I mean the author of Megan Lane. So this idea that by using set theory you could give good foundations. And so the instance of pressure. Pressure. And then came the development of topology, asset topology, and look at what was done in Hungary and Poland in the 20s and the 20s. I mean, continuous, various characterization of continuous and so on. You discussed yesterday topology, so I suppose it's part of the discussion. And so, but then we had a, then big, what we had, it was not combinatorial topology but algebraic topology, which was a new thing in the end of Cheikh and Hawking's algebraic topology. Algebraic topology, what has been related? Combinatorial topology. If I just look at my recent work, recent work on modularized spaces. Combinatories of algebraic variety. We take an algebraic variety, we split it in two subs, we cut it into pieces, we rearrange it, and so on. We express everything in combinatorial terms by using the modern technology of combinatorics and non-combinative polynomials and all these things. Combinatorics, which was considered until the 40s and 50s. Look at the judgment of Theodony about his classification of the mathematics. Combinatorics is really at the bottom. And logic, at the bottom, at the bottom. He didn't take much of functional analysis either. Yes, that's true. Really at the bottom. And then, mostly due to Horta. We understood that the borderline between algebraic and combinatorics is not the first one, and that it may be a return to the British tradition of Franchiola, 19th century.

1:17:30 I was reading recently the book of differential equations. The first algebra, the British algebra of the time, and he was the first one. I mean, people speak of G-bar algebra today, in an elementary way, but for interesting examples, clearly expressed in Boolean. So, combinatorics was, well, people did not understand that combinatorics and algebra are more or less, I mean, two phases. But now, of course, combinatorial topology, combinatorial law theory are prominent. If you are too dramatic and you say, well, I mean, now Sestri has conquered everything, or Algebra has, you know, has conquered everything. If you are too dramatic, then you will prevent further progress. Well, I don't have to make this speech here. I mean, that's not for me to write here.

1:20:00 Well, go ahead. And interestingly enough, this idea of classifying the beginning and the action of a group on a point has another interpretation. And if you work not in spaces and shapes and so on, but in non-commutative algebra, in the spirit of common, then you end up with another definition. So a point is really an algebra which is more iterative, more iterative or something commutative. And then you ask what is a group acting on that, and then you came with a solution that what represents the BG is the convolutional algebra of L1, let's say, L1, L2, L3, L4, L5, L6, L7, L8, L9, L10, L11, L11, L12, L12, L13, L14, L14, L15, L15, L16, L17, L18, L19, L19, L20, L21, L21, L22, L23, L23, L24, L24, L25, L26, L27, L27, L28, L28, L29, L30, L31, L31, L32, L33, L33, L34, L34, L34, L35, L36, L36, L37, L37, L37, L38, L38, L39, L39, L40, L40, L41, L41 But this algebra is really the non-commutative space corresponding to beach. So, and what ought to be developed, I think one of the main problems at present is to understand the conic language, the ideas of con about non-commutative spaces and to go to the inside, to make the real conic. And I might advise some work on beauty. But this idea is one of the things that is wonderful.

1:22:30 So again, the idea of space is, if you are too dogmatic about what space is, I mean, you are not asking the right question. And to me, I insist very much on the plasticity of mathematical concepts. If you believe that mathematics is a living, born, developing, and according to some biology, more or less biology, but also sociological, biological, I will not enter into that kind of discussion, but then you need plasticity, and I just gave an example where an idea which is indeed, which was considered as dead. I've been a follower of Einstein for many years, but that was something good to do, but the success was not there. There was a kind of absolute idea that elementary morphisms are the same, morphisms that preserve first order definable concepts with all alternations of quantifiers. I mean that's the lies behind this interpretation. Well, of course, I mean, it was something, it did not bring the revolution. That's true. But another one that I have been struggling with for the last 30 years, among my colleagues who work in topos theory, is this.

1:25:00 They learned from Grotendieck 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 Grotendieck, he makes it quite explicit that this is not the case, that there are some... So the detailed study of what is really a generalized space in some kind of controlled sense within the huge world of toposis, 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... 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 Polko's point of view, sheaves on the space and actions of a group, these two apparently very different things are in fact working in the same category, and they must work in the same category. On the other hand, how far beyond that should we go and still consider it as spaces? There are many, many aspects of this, and it's very hard to consider it. I think I finally got the column to... But I'll still say, on growth, we sometimes want to say, okay, there's this fatigue-growth distinction that has to do with it, but other times we will say that the notion of topos is the new definition of space. Well, this, of course, is correct. These are not mutually exclusive, they're just contradictory, which is even better. Well, I mean, it reminds us of the old day of the blind people, I mean, to the old days. That's called solvent 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 absolute truth. Some people are just wrong.

1:27:30 Yes, well, when you have a notion on top of a really kind of protean thing, you can have many different ways of viewing it and know one of them. I mean, you can call it space, but the term space is an extremely protean notion. I mean, in general, you can't. It assumes a kind of different form 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, but of course, the essential idea of topology is to unite that with the cohesion of infinite dimensional spaces, function spaces, spaces of this. 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 parking space should be well-behaved. It's over 50 years now that Kurevich's late forties defined this notion as K-space. Basically rejecting the absolute character of the so-called standard definition just in order to achieve the simple adjointness, as we can say now, of the function space construction. And if people go on and on and on, it's a little topological space, whereas the algebraic topologists, of course they... They don't care too much because they only care in the homotopies, but they always presume that it was one or the other of those Cartesian closed categories that were working. Whereas the analysts, as far as I know, still have not grasped this point. That the general notion of topology is a very bad one, precisely with function space. Not to mention the redo of Symmetry of Magnetism. There are a lot of misunderstandings about supermanifolds and I'm going to prepare my lectures and it's very baffling because I mean if you insist that a supermanifold is a collection of ports, it doesn't work, it doesn't work and it doesn't work and there's a lot of misunderstanding about supermanifolds.

1:30:00 Supermanifolds have two major sides and they are quite automatic, I mean you change the coherence condition and so on, the commutativity condition. But when you are used to a form and then all of a sudden it's an anticipated form and when you add x squared and you say, well, x squared. No, x squared doesn't exist. Oh, it exists! It's quite simple, I mean, if you begin away with the two foundations of mechanics, I mean, I have to give an expression of supersymmetric mechanics and so on, I mean, all the things you are accustomed to in Hamilton-Giacobbi theory, I mean, the action, the reaction, and so on, and everything becomes completely upside down, but there are also conceptual problems. What is a superspace? It is a collection of quantum art. It's definitely not. And it's not so easy to gather good definitions. Of course, it is a compromise between something workable for the mathematical hypothesis, which we do not want to understand in all the conceptual forms.