Discussions — Grothendieck's Program (Part 2)

0:00 But if we expand the admissible arities of operations and tupleties of operations are just as important, of course, because basically you want to have maps between x to the n and x to the m, where n and m are more general than discrete integers, It's certainly very, very special, mainly infinitesimal objects. Then you come to the situation where, for example, the relation between Lie algebras and Lie groups is an instance of this so-called algebraic adjointness. It becomes smoothed so that one can extract the algebra of a Lie group and also go back by an algebraic function, by an adjoint to an algebraic function. What it suggests to me is that, of course, the recent development of opera art which is a new way of looking at algebraic theory. People are consistent with an rarity is an ordinary number, but we could generalize an operand by taking more or more so than you suggest. I have to think about that. I mean it has been generalized vastly already by taking sometimes arbitrary objects. The theory, the cohomology theory of Beck is based on... Maybe too large. Yeah, but the thing is that certain advantages are obtained by going to that level, but... There's some theorem of Lie about vector fields, I forget exactly, but it can be interpreted as a theorem about this category of pseudo-algebras.

2:30 So, I have to go pick up the email myself. Oh, okay. I'll be back shortly. That's okay. Yes, I think this will be as soon as possible. Why don't we write it on the chair now? Yes, yes, it's cool. Is that okay? This was my reason for insisting that there be a lot of time for Lavelle-Chanel, because it's perfectly clear that it's the same game as, particularly in the theory of integration. Yeah, I never heard the basic thing that he explained. The mojang involves this virtual decomposition of the space into these new things that are, depending on the cohomology, but it's almost like Eidelberg-McLean space. Yeah, exactly. That's exactly what I was going to say there. Well, this is what I was going to ask you, because this is why I was rather harping on the theme in the count of times I did say to you. You know, developing the theme of how Eidelberg-McLean space is kind of fit within this more wider vision, as elaborated, obviously, partly by Gerbenbeek. It's also like the stable homotopy and all sorts of things. Pierre was claiming that... I guess you can say that. Now, the standard conjecture is much earlier than the log. Now, the log work is about topos homotomy. We'll have to talk about that also, which should also come in in our files, but somehow it's a natural part of changing. The theory of motors largely exists now, which I mean, rather bad.

5:00 I know that's more optimistic than what I usually hear, but it does so without at all achieving the goals of the long march. There are more maps than there are multiple co-utilizers to form. There are more and more quotients. Yes, exactly. You know, I had this old, old idea about whatever product one could be nice and work as an object. No possibility. It's sort of soft to interpret. I claim it is a correct formal relationship. Sure, because I mean the things in the product are in fact the, somehow or other, the zeta functions of certain models. Yeah, yeah. That's the point of view. I mean, Manning has this paper, and that's about the zeta functions. He pursues this point of view. We can see the zeta functions, you don't have to get them in there, but the zeta functions are straight against the matrix. There are so many things one would like to pursue here, in the sense in which he inspects the Hodge conjecture to be always true. There are some scientific questions that you didn't state. Yeah, that's right. I mean, of course, it has been verified that there are very small numbers of varieties and very special kinds. It is asking a bit more. Things that sort of project out of the cumulative thing come from accommodations and varieties, but they may come from something to be told, I guess, as various notions of it.

7:30 But it's all business, you know, it's all sort of here and there. I mean, well, yeah. No, no, that's very nice. I mean, that's, but this is, this again is the same kind of play. Well, the whole object is just described as soon as possible. Well, yeah, the whole object I'm going to transcribe later, I mean, I'm going to do it as soon as I can, if that's what you're going to do. That's the reason we're recording here, in the first place. Well, no, but I say, I mean, but we're only doing this because a lot of, you know, we're all like, oh, well, it's still everything that's been said and done, it's just very recent. Well, I know, but of course, the end of it is one person recording it, but I should think it's probably, and to be honest, it's realistic. Thank you for watching. Yeah, this is actually a very important example of what we get, you know, from Barsfield and Grassley. No, no, don't worry about me. I'll do copies for you. I'm just wondering if there's any way I could do them even before you leave. Actually, no, it's because you need a machine to transcribe digital onto audio tape. Well, no, I'll tell you what I could do. You don't have math as kind of an audience. You can't have math on a computer or on a telescope, so you're thrown away too quickly from time. Well, I'm sorry, but that's a bit racist. So if you wanted to take the audio tapes, as long as I've got the digital. I would feel like I worry always carrying stuff. Yeah, yeah, okay. Well, but that's the whole point. As a matter of fact, curiously, it's that, well, of course, it's actually a lot easier to transcribe audio tape than it is digital. That's another reason why I always do the backup audio. Because transcribing digital is much more... If you can get copies of the audio made... Oh, yeah, that's pretty easy. I mean, that's how I could do it very easily. I mean, this gives much better sound quality and is absolutely reliable in terms of sound quality. All of this will last forever, and your participants will prove it for you.

10:00 Or in the sense that anything he looks at, he sees another way he could have put it. Well, that's all the more reason to have it transcribed as quickly as possible, so you can look at it and decide if he wants to revise it. I mean, it's suggesting publishing an absolutely raw verbatim transcript. The popularity is not in that much value, because it just doesn't make sense as a verbatim transcript. Well, unless you're the sort of... Something like Russell, that was just... They are able to express themselves and form sentences whenever they spell, but as a basis for transpirings, of course, in the real world. Yeah, I wrote as fast as I could, but because I knew I had two sets of recordings backing it up, I wasn't too worried. Otherwise, I could have seen it as smoothly as well. It would have been better to have a video, too, because it's always easier to transfer some of the video, but I just don't have a video camera. This is potentially fascinating. As I say in the afternoon, we probably ought to try and... I suggest trying to steer back onto the kind of broadly chronological exposition of the great geometry players of the Great Leap, obviously leading to how did they get to where did Bill start to see the logic, well this is a familiar story to us, the point is not going to be a familiar story to everybody who will listen to these, and that's the point, we're not the only audience for this long term. So having that exposition, even though obviously we people in this room have heard it before, is still very worthwhile. And as you say, Bill always sees a different way in which he might express something, so he may very well have something new to say, even on that. But the problem is it never ends, which is good in a way, but it's a problem from the point of view of achieving a version. I know, I completely see. And then go on to the sort of, you know, the great big legacy. I know that's a terribly big picture to go in any direction. It probably needs something a bit more to control it. And will naturally lead into what we're saying.

12:30 Probably coming to the Montenegro, I'd like to try to get onto that Montenegro while Pierre is still here. Not because I'm in Angus, I think he's got enough that he'd like to say about the Samuel Schenel conjecture. It's very well qualified. I'd really like to hear comments. I actually can't even state the Schenel architecture. I don't even know what it is. I've seen it. But I know that when I see it, given my preparation, it doesn't say anything to me. I mean, I'm just not on that stuff, and I'm not going to hear anything I talk about. That's where Angus is very, very well-placed and has very good orientation. Just a light on personal grounds to your so much attention to Shannon Walsh. Shannon Walsh was great, God. That's always been me. Yeah, absolutely. And I was actually, of course, I would love him to have been able to come here, but he just doesn't travel, so he just doesn't. Yeah I don't think he talks fast either. No, he has a very, very passionate commitment to his teaching. Even now that he's an official at some of the times still. But there are times I've asked him a question about something. I mean, he's so... It includes that he's very kind and generous, but it's just... Well, of course, if I could somehow get the funding, I probably wouldn't have to keep going on TVA as a film or whatever, so I just have to keep funding everything myself. But I really want this to be the first of a series, and Bill obviously is the centrepiece of this, but I was hoping that PA himself would have a similar... He obviously likes talking, as is Greta, I like listening, and everybody who's going to talk to me once we start to work, and then it would be great if Steve Tangle could come, and indeed Bill could come as well, but as I say in this case, in the role of one of the panel rather than... Actually the whole bill of business of having, as it were, Bill as the subject, and us as the panel, has really not, has actually worked out just because in fact it's...

15:00 I told her we'd be there about a quarter past one. You just go across the road. But it might make sense to introduce, because we're supposed to do this afternoon. Well, we are still in the afternoon. Well, I think it would be a good idea, obviously, if there are any points to be picked up from this morning's discussion, I know Angus particularly had some questions he wanted to put here about that, but then perhaps, if I just say very quickly, perhaps, you know, if we give ourselves a limited time for that, and then move back on to the chronological development of Grosendieck's ideas through the 60s and particularly leading up to the... And then stand back from all of this to look at what the larger significance of Grothendieck's legacy is for mathematics at this point in the 21st century. You know, as virtual objects, it seems, oh yes, you mentioned the Kunitz theorem, you mentioned the isolated deduction of co-algebra, and so on, so it seems to me that this whole field of, starting from Harry, which is homotopy category, is somehow... In the case of homotopy, it has many formal properties in common with the starting point.

17:30 It's still Cartesian closed and has disjoint sums and so forth. Conjecturally, you see, even homology is not really a linear thing, because if Kuhne's theorem is true, then homology is really a co-algebra, and the co-algebras form a Cartesian category, you see, and even a Cartesian clause, in fact, if you test it properly. So in a certain more abstract sense, it's again a category of spaces, but now sort of qualitative, spaces made qualitative. And the fact that you have, in such a process, you may well want to introduce new objects as well as new, as change the maps. As a homotopic category, change the maps, but usually you don't. Put in more virtual spaces, even though it might be very natural to... The spectra. The spectra, yeah, that's right. The spectra. You know what I mean, the spectra. So this idea about the homology is really a functor that preserves Cartesian products from one Cartesian category to another. In fact, Cartesian clothes category to another typically. The homotopic category also has namely that the notion of point and the notion of component have become identified in the sense that precise sense of adjoint functor that there is a there is a trivial there's some kind of trivial subcategory with co-algebras community of co-algebras there is a underlying Graded ones, there's the underlying category of non-graded ones, with the underlying function being just to pick off the zeroth component, but this process of picking off the zeroth component is equally well dividing out by the higher degree part, which is a left adjoint, or simply taking that part, which is a right adjoint, so the right and left adjoints of the trivial inclusion agree, something which is never true for ordinary spaces.

20:00 So this kind of abstract quality or form that's being derived seems to have this very non-spatial feature of the components and points not coalescing, but at the same time preserve much of it. Right, so this leads to one question which is, you see, if homology is defined by some particular procedure, there are many such theories, whereas the universal homology theory is usually said to be stable homotopy. But the stable homotopy is done from the start with pointed spaces. And that minor technicality actually destroys my whole story in a way, I mean it doesn't destroy my story, but it destroys the idea, well how do you put in, how do you compute, if a homology is defined to be such a Cartesian closed category with left and right adjoints agreeing and all this, how do you find the universal one? Well the idea for the universal one would be Stable homotopy, but that's pointed. So, does there exist such a thing as not-pointed stable homotopy? That would perhaps be a way of computing the universal such invariant. There's not one way out of this difficulty. I mean, when you consider the pi-1 of a space, of course you have to take a baseball. But it's well known, I think it was already mentioned in the first Council seminar, that the more fundamental object is... So homotopic classes of parts, but with fixed endpoints, which reflects the fact that Pi 1 is not critical.

22:30 Because it is well known that in an automobile like Pi 1, if Pi 1 was always being obedient, then they would have got the N-wheel. There is a lot of difficulty choosing a base form because if pi1 is appealing the same as h1 and to define h1 you don't require a base form. But of course when you go to higher or more topical pi n, they are commutative, but they are not trivial actions of pi1. So it destroys the commutativity in some sense. And I think the issue about the base form is mainly about whether the pi1 really operates on the higher form. We all know that the spectral sequences behave rather nicely when the base space is simply connected. But otherwise, you have to deal with the action of pi-1. What does the action of pi-1 say? Various homotopical or homodrical constructions are not good, but to achieve so, but not over, it means to be locally to be over. Actually, Elijah was going through the speech, putting abstract terms in categories. I mean, these ideas at the point, I mean, I don't know. I have no answer at the moment. To focus my question better, maybe assume everything is simply connected. Okay. In the end, it's additive. In other words, the additivity is picked up through the process, but that's only because it's pointed. It's a non-pointed version. It wouldn't be additive. And the construction that I'm asking for would not be additive. It would have to be somehow non-linear rather than linear, even if it's a different... This may be even more trivial, but it's not exactly the action of Pi-1 that's causing the conceptual consternation.

25:00 If it's simply connected, I mean, this is maybe too coarse of a remark, but if it's simply connected to some points, there's nothing to know about changing from one point to another. That's what I want to make sure. That's just true, but when you take the suspension and you make this kind of Freudian tall limit and so forth. You pick up the linearity. You start with a non-linear category and in the end it's linear. So you would look for a universal homology which would be non-linear, which would be not commutative. It's what you want to have. Well it would be like the model is the the co-algebras. In other words, It's the same sort of thing as the usual homology, but construed in a way that takes account of a little bit more of the natural structure, which is the co-algebra structure, which changes the nature of the category from a linear one with tensor product into one with ordinary categorical product, in fact Cartesian closed. So non-linear in that sense, I mean, it is traditionally built up using linear machinery, of course, but in the end, the end product... It doesn't seem to automatically imply that you must be able to construct all sets. It's not clear that you have to construct. I mean, you get a category from it, you see, and there's this incredibly complicated scaffolding to this area, but does the end product really require that construction or maybe some other construction? So the linearity is sort of like the homology as a co-algebra. Homology is more cohomology for that purpose. It's more than, what, co-algebra for homology, algebra for cohomology. But it has the various, one various operation, it is a general operation, characteristic B,

27:30 or the idea of motivic cohomology is that the receptacle of motivic cohomology should not be bare. And as we mentioned already, I mean, the operation is quite relevant for geometrical studies because it gives you the existence or non-existence of some models of transformation. So, to stop halfway, that's a problem. Halfway off. I mean, core algebra we may need as a receptacle for homology or universal homology, something even more sophisticated than core algebra. So, maybe another Cartesian closed category. Exactly, a richer Cartesian closed category. Yeah, a richer Cartesian closed category. But there is another question connected with this. Namely, in my conception of things, actually, cohomology is nothing but homology operations. So that particularly the product operation is merely the composition of homology operations. Natural homology operations. Now the sense of naturality is a little bit different from the usual one because if you suppose some covariant functors given on a category and you take an object X, first of all typically if you just took natural transformations between these functors you would have to get constants or something like that. But if you take an object X and pass it to the objects over X... Then there is the completely trivial, forget the structural math, you apply the homology or whatever these functions are to the total spaces, but now naturalities with respect to commutative triangles, for each commutative triangle I want to have a naturality condition, because for each arrow, not for each object, so in that way the cohomology appears as homology operations. Between two homology functors, you get cohomology. And the cup product is merely the composition, the fact of one being a natural module.

30:00 This is just a way of twisting around the projection formula or the Frobenius reciprocity formula, the change of integral formula, whatever you call it, or the interaction between the two. If you twist it in this particular way, you can see that the cohomology itself just arises as the natural maps between different cohomology countries, natural in that sense. And then, of course, again, it's trivially contrarian because if you change the space by... By a map, the category of objects over this one becomes transformed into objects over this one. You get it in the most trivial way just by composing or forgetting, not by taking co-ex or something. But that causes the natural maps between these things to go in the other direction. So this is another motivation for focusing on this. But as such, you see, the homology need not have any structure other than the cohomology, than the cohomology, because the cohomology will just come up with all of its operations too, as natural operations on those. I fell asleep in the meadow up there, I confess. It's a good place. It's so idyllic. I saw, I saw you there. Gorgeous. I see the passing clouds, do you know? So, as well as technically, it's a matter of defining suspension without using a base point. The idea about this universal homology should be the Cartesian closed formula that we get by further smashing down the homotopy category by stabilizing it. So it's just a matter of... All of these are examples of making precise what stabilization would mean if you didn't have a baseball.

32:30 At least that seems to be one congressional approach. Sorry, where are we exactly now? Yeah, I've just been... The story thus far. Quick summary. The story thus far. Oh, God. No, no, quick, quick! Being explained, he's always going to... No, no, he will, he will. ...omorocci, omorocci. Homology implies homology is not just an abelian group, it's a co-algebra. Right, right, that's progress, right? Co-algebras form a Cartesian closed category, so homology is really an interpretation of one high order theory to another, i.e. a puncture between two Cartesian closed categories that preserves products, at least. But then the particular construction via the usual linear algebra, homology, co-algebra, and so forth. Might not be the only way to arrive at a Cartesian closed category, and one would like to have the universal in any way. In the usual point of view, there is a universal homology through which all homologies factor, namely stable homotopy. But that doesn't work well for my purpose in its usual form because it's pointed, and so you already force the additivity just by taking the limit because of the pointedness. I don't want to have a linear cup. I think non-linear is similar to linear in some respects. And then that cohomology is just operations on homology, natural operations, and that's why you have a product in cohomology but not in homology. Because you can follow one operation by another and that is the cup product. Yes, that was another puzzle. Of course, the term homology was used. Co-homology is obviously a derived term from cohomology, and yet it was homological, after all, homology was the term, and then because of the particular technical possibilities offered by looking, you know, using opposite, by having the maps go the other way, you end up with co-homology.

35:00 And then that became the fundamental term. Well, it wasn't originally just for technical advantages. You had to introduce a multiplication and things by... I know, I know, I know. But then it equaled the whole homology. But then they could become partners. I know, but then they could become partners. It really was just easier... No, I know, it's just a logical thing. It's just if you look at the derivation... And the multiplication... Well, yeah, it's obviously... If you search for the origin of the term, homology, homomorphism, algebras today don't use the term co-homomorphism much. They don't use co-homomorphism, but co-homology, no, it's a minor matter, but nevertheless it shows some genuine movement. No, it's a real puzzle to me. I mean, what had been, say, Lefchef's motives for moving to cohomology were no longer the motives in, say, the Seminare Carton for doing cohomology. Something else became the reason why it was easier to do co in the Seminare Carton. And in fact, it seems like for Lefchef's, you've got a multiplication on classes. Well, multiplication is precisely the problem for the Seminare Carton. And Cartan-Eilenberg says, we'll do that in the next volume. And Grotendieck says, and Tohoku, in a successive version of the paper, I'll get back to him. Oh my God. I was commissioned by Grotendieck. I was commissioned by Grotendieck. Everyone knows this. Tohoku! Tohoku! It's called the cat fryer. Right, right, yes. I still do that. I failed to understand the meaning of monohedral category at that time. And so I have a draft which I wrote when I was in prison in 1958 about a product in homology. And my starting point was that you should generalize category to multi-category, allowing for multilinear ones. Well, it's one option. It's one option. And I developed that. I developed that but I never published it. And in the process I came, more or less, one of the talks I gave at the Wotanik seminar, obviously, your data product at the end.

37:30 So, and then finally, many people discovered that once you are at the end, most of the multiplicative structure is there. And so... Since the experts later on subsumed into derived categories, and so the only project was, I mean, in a derived category, you have just composition of maps in a derived category, and so many people saw that there was no need to develop a separate notion of multiplication because, well, they, let's say, take the cohomology of a space. It's, let's say, a shift cohomology. It is known that h p of a space with x with respect to some, let's say, to the integral, to the integers, to some coefficient, is simply x-ed p of the constant shift z over x with itself. And then the cut product, it appears... ...which is not completely trivial but not difficult to make it appear that the standard core product is just a composition in the X group. So the composition in the X group was later on understood as just a composition in the derived category. And so, well, it may have something to do with what you were saying. So all these attempts, I mean, were abandoned. And later on came the idea of a monoid room. I had the idea of monologues. I wanted to axiomatize the idea of a tensile product of complexes. But then, I think that I was a little upset by the idea of formulating the exact condition of associativity, which is exactly what Maclean did. But at that time I wanted to bypass this. And I misunderstood completely the situation. I wanted to bypass this and I said, okay, if I introduce a tensor product associated with a tensor product, it's a little awkward to introduce transformation and coherence and so on, but if I take the point of view that you have linear maps and multilinear maps, which are the same as linear maps from the tensor product across the space, then all the coherent conditions are taken automatically.

40:00 These terms are automatic by just assuming that in your extended category you have associativity, in your multi-category you have associativity, and formulating associativity is not difficult, I mean for multilinear mathematics it is easy, but then what came out after that was the development of opera arts, another shift in the concept of emphasis. Let's say, point of view of monoidal category would say that you have a suitable notion of tensor product for your abelian category or your obi-derived category, whatsoever. Then, a multilinear map is a map from, let's say, X tensor Y into Z, bilinear map, X tensor Y into Z. And then the composition, of course, the composition is that if you have what is exactly what is well known in... In computer science and logic, if you have a function f , you can substitute to x a function of other variables, to y a function of other variables. You can substitute to x a function g and to y a function h . And then you have this composition. But this composition increases the number of variables at each time. To explain the function in n1 variable, explain the function in n2 variable and np. So all together you have the function of n1 plus n2 plus np. It's a full substitution. But if you look at the practice, I mean, a full substitution and the same occurs when you are in the formal language to make substitution of a predicate for a variable or whatever. But this composition can be decomposed into elementary terms by making one substitution at a time. You first substitute to x1 a certain number, then you do to x2, etc. Which means that you change the focus. So the new focus is that you have a function of a certain number of variables. Then you select one of the variables, and to this variable you substitute the number of functions of both of them. In diagrammatic terms, a composition is depicted by its input and output, and then you take one of the inputs and you consider it as the output of some other process.

42:30 In terms, it just comes to sewing, I mean, sewing, if you have two diagram trees, whatever, I mean, you take one branch of the other one and you sew it, you sew, or graft, I mean, people do grafting or sewing, I mean, usually it's grafting, usually it's grafting, grafting, you graft, graft in four trees, graft in four trees, I prefer to say sewing to this by the end of it. Sewing, sewing. It's very difficult because sowing and suing, you know, there's no suing and not suing. This word in English is pronounced sow, which is un-sowing and reaping, you see, but that's spelled S-O-W. Thank you very much. And that... No, but thank you for listening because I have no business correcting your English. You know... If I was to correct your English... It's a presumption to correct your English. But nevertheless, so... Now I understand why sometimes my audience doesn't think I understand it. What is this one? Ensue. But you can't say it's a... The ensuing problem. You can't say the ensuing problem. Oh, there's another one. There is the sue. Yeah, of course. Usually I would make the drawing you just say after saying that. You're constructing another tree here. No, no, it's just a circle. It's not a place. So, like that. And... I mean, in the 50s and 60s, many people tried to analyze this notion of function of more than one variable. For instance, Michel Lazar, mentioned before, created this theory of analyzers, which is again an axiomatization of what you mean by substituting one variable, one function in two variables.

45:00 I mean, substituting the composition of functions of many variables. And it's on me. And this, well, I remember discussion with Sami and where we reformulated everything in terms of the monodrome category having one object and all other objects being the product. So you have one basic object X and then you have X, X, X and so on. So you have one category, one monodrome category with one generator or generating object. The booklet of Amesami about aggressive function is very close to that, very, very close to that, and we had many discussions at the time. And so, okay, and what's not understood at the time is that you have something more primitive, which is just replacing one variable by a function of other variables. So it's a sewing process, it's a sewing process, and then you can repeat. With the renewal of operat, I mean it was brought to the forefront, and the other then there is one issue in operat about what you call cyclic operat. In an operat, usually operat, the name means that it's a field of operation. In an operation, whether in logic or in computer science, you have an input and an output. By a block, a black box, and we have various strings, and one represents the output and the other one represents the input. But in many situations there is no need to distinguish between input and output. And the cyclic operand is a situation where all the strings are the same and you have more symmetry. There are many geometric situations where you have more symmetry. And so, and then came, so. Just to say that is... All these ideas, I mean, the starting point, I remember, I was commissioned by Gottendieck, I mean, to write an account of the product in general theory of homology. And, as I said, I wrote a draft, I wrote a draft, but I was never happy with his draft, and Gottendieck was not happy, so finally I gave up.

47:30 At least there have been two answers, two serious answers to that. Well, three serious answers. First of all, most of the products in cohomology are special instances of the Yoneda product for X. One answer. Second is that we have monoidal category. And the third is that we have operas. And all these creations address the same problem. I mean, how to make composition of functions of more than one. And I've been surprised to see that I was listening to a talk by Dreen Felsom. Two days, two years ago when I was in Chicago and for some reason, I mean, he had to resuscitate this idea of multi-category, I mean, multi-category of multilinear maps because the point is that, well, if you have the functo, I mean, if you consider all of, I would say, bilinear map from X times Z into Z, X times Y into Z. The tense of order just says that as a factor is z, it's representable. Yes, representable, yes. And then there's a 1, it's 1, no. Representable in the other way, yes. But there are situations where, there are quite natural situations, where this factor bilinear of x times 1 into z is not representable, it's z. And still, still, still you do something. Right. Well, you could enlarge the category by making it compesantive or by introducing new objects, at least four or ten so far, but there's not many thought about it. So, I see at least four answers, I mean, three natural answers and maybe a less natural answer to that. But, just to say that, I mean, we started with a discussion about Cap-Connect and that came out what came out. So, that these multi-maths, of course, are... It's a known concept, but it's often put in terms of pro-structure, that is, where you have such a structure not necessarily representable in either variable. Let's say you consider punctures from such a thing into abelian groups or so. Then there is a naturally defined actual tensor and actual column, both, which are actually different forms. So this is a known… Yes, and the con-extension along the data. Exactly, yes. It's due to Brian Day, the formalization of Australian…

50:00 So, in a way, this need for coherence, the coherence problem comes about by making something representable. Exactly, exactly. And I wanted to bypass this problem. By making not representable, not choosing the fact that it is representable. And it's striking to me, see, probably if I said this in the company of category theorists, they would say I'm terribly reactionary, but I'm not so convinced about the necessity for wobbly in categories, you see, because they always, they have only one example to give. There's always one example which illustrates the alleged non-strictness, which is the so-called profunctors, or in other words, bimodules. A tensor product of bimodules, you see, is an example. But what this is, from one point of view, is merely making representable the co-preserving functions from categories of... This is the main example that's, you know, for technically for bi-categories as opposed to two categories. And yet it seems a little extravagant to introduce all those wobbles just because of this, when there's a philosophical explanation of why this key example has a wobble and there's something behind it that there's not. It can be ironed out. Something behind it that can be presented in some way. And then you can say maybe this can be representable. Yes, yes. In other words, then you find the bimodules that do that and so on. It's related to, I don't know, Connes' notion of morphism, by the way. But another point, though, as you see, is that if you don't have the thing representable, let's say if you have some kind of multi-hom, which is not representable, then you cannot talk internally to the category about function precisely, because the hom is a right adjoint, which means you know, in some sense, everything about maps into it, but then once you have it as a representing object, then the interesting information is no longer all these tautologies, but some actual...

52:30 Functionals. And of course duly with the tensor product you have co-algebra structures which are in some way precise information in a particular case and not some generality. So there's a balance, we call it a give and take, involved in introducing this representability. But certainly one wants to do it in order to achieve that, because otherwise you'd be reduced to speaking of the functionals in some sort of external way. I have two remarks. First of all, I mean, I was very reluctant about the Korean's condition of my brain for a long time, and so like many people, it's splitting the air, until we came with this precise result about crazy, I hope, pharyngobras, and break groups, and the invariant of nodes, and so on. And there, I mean, or more generally, I mean the study of moduli spaces. And then, I mean, you cannot escape the problem. And it's not only, I mean, the one-dimensional or two-dimensional collection that you need, but you need, I mean, you need the whole construction. I mean, not only ordinary category or bi-category, but what you can. Of course, I know the work of Brown, Kenneth Brown, about one angopoid and so on. I've never been really convinced. But nevertheless, such situations exist and they are not artificial. They come naturally in some geometrical problems. I had to deal recently again with similar problems because when you deal with analytic continuation of function in n complex variables, And then you have, so you have certain differential equations, you have asymptotic behavior, and you have some, maybe some boundary points, and then when you want to understand, I mean, so usually you have more than one asymptotic behavior, and when you want to compare all these things, it's a very complicated algebra, but it's exactly the algebra of the, what it's, I would say, it's just a Stashev polyhedron, a Stashev polyhedron. And I think, what?

55:00 The natural outcome, the natural outcome of the McLean addition is as a Stashev product. And, of course, again, when I was studying, when I never was an expert in homotopy, but when I learned about homotopy long ago, I remember that John Moore, at a Cato seminar, bypassing the non-associativity for the homotopy. First of all, you combine paths. And then, when you go to homotopic classes, it becomes associative, but it's associative up to homotopy. And Eilenberg and Moore, I mean, devised a method which, I mean, paths with very even lengths to bypass this difficulty. And so, already at the level of paths, before going to homotopic classes, everything was associative. I remember when Stashev published his first accounts, I mean I was very alert and I said wow, I mean by subtly modifying the notion, we could have already associated with you, or at least associated with you at a higher level before going to one of the classes. But the Stashev polyhedra has proved immensely useful lately, immensely useful. What I've discovered is that in my study of generalization of zeta values, or so-called multi- or poly-zeta values, I mean, really, you have some different, we spoke about this special function differential equation, you have asymptotic behavior of some solution and so on, and the Stashev polyhedron is really there to make visible the combinatorics, not only the combinatorics, but the analysis. So, I came to terms with the Stashev-Polar and also the Stashev-Polar in that they are beautiful geometrical objects and I've been happy to discover one explicit realization. But there was a problem whether they could be as explicit as convex polyhedra. Of course, what Stashev does, he represents them as polyhedra but as not all. Not real polyhedra because you have to subdivide one phase into two sub-phases. Let's say the first example is this one, and you want to make it a pentagon, so you split this into two parts.

57:30 It's not a true polyhedron, I mean, it's not a true polyhedron because you subdivide phases. And that has been a long-standing problem, I mean, to have a natural realization by a tool called convex polyhedron, and there have been various solutions proposed, and finally I found, by change, because I was not looking for it, but I was involved in it, I was involved in it, I found a beautiful, almost trivial realization. All of this is given by means of explicit inequality. So, you have convex polyhedron which is given by explicit inequality, I could speak about that. So, it seems that the Stashev polyhedron or the Scythrohedron or various things of that time are important objects. But the fact is that MacLean-Cohen's condition just did with the one-dimensional or the two-dimensional skeleton of such objects. But these objects exist. Irrespective, I mean, whether you take C and Z, the fact that homotopy, associativity or not, they exist as bona fide beautiful geometrical objects which are useful in many situations. So, I would say that when you... I also may have a reservation about any categories and... Well, it's a matter of the claim that it's the universal principle again, you see? Yeah. That in every case you must use the bi-categories with the wobbles and so forth. This is... No, this is just a discussion really within the category theory, where I say, well, of course these are beautiful objects, I know that, but also in many other, many basic situations, you don't need that, you can do something different, that's all. Now, my second remark is about cohomology, and during this discussion I was told by one very mundane remark. The homology is a pro-algebra, the cohomology is an algebra. Why? I think, well, of course, when Cech shifted from homology to cohomology, I think it was basically because they wanted a coproduct. And, of course, in the Durham theorem, I mean, you have a natural coproduct for the differential form and you want to explore that. But of course, at the time, the idea of an algebra was well understood already, even if not completely formalized, but everyone was, I mean, Gasman algebra existed, polynomial algebra existed, matrix algebra existed, and so on. We were very familiar with the idea of an associative algebra.

1:00:00 I think I was the first one to use the word Koalchema in print and in some seminars so and so of course the notion was implicit and And so I remember when I first time I spoke of Gaudium and Photogothenica. So now I have to speak to the co-catchees. That was part of my point, you know. Algebra is an old term. It means the breaking of bones. Reassembling, you know, is an old term. Co-algebra is obviously a derived term. Oh, that's what I meant. If that's going to become a fundamental notion, one should drop the co and find some... That's all I meant. It was a... It was going to be fundamental. That's right. Hence the cohomology. Again, this is some kind of universal imposition to say that that is the basic... It perhaps, but it has become it, hasn't it? But that's not what I'm saying. Objectively, that is one thing. In categorical term, algebras and quadramers sit side by side. And especially if you have the notion of monoidal category. An algebra in a monoidal category, as we know, is about from x times x into x, x times x, with respect to monoidal category, or I would say x tensor into x. Well, the point is that long before tensor project existed, multiplication existed. Bilinear operations preceded, by many years, the linear hammam-associated tensor product. I think I remember, I mean, in the late 40s and early 50s, the idea to reduce the bilinear hammam to linear hammam was not an obvious thing. And I remember, I mean, the obsession, I remember the obsession of Carton, I mean, in every meeting of the Mobacki group, you should say. Om A, Om B, C is Om A, then so B, C. I mean that's what, I mean, every time Captain would show up in the discussion, we would say thanks to him. And, including some, we would say thanks to him. So, The very idea, and I think the first edition of the multilinear algebra of Boubacke was a real place.

1:02:30 No, no, I know. I tried to read it. I mean, it's true that Whitney has similar ideas, more or less, but less explicit. And Boubacke was the first to mention, and Boubacke in his time... On purpose did not quote the theory of tensor algebra back then, but I mean tensor algebra was a flow to recognize the importance of the... I suppose Bovaki was the first to notice that in a very large framework, the bilinear map from A to B into C are representable factors in terms of things. We did not state it that way, but it was exactly the same. And, by the way, it's one of these examples which forced Bobakke to introduce the so-called universal problem, which we consider now to be representable factors or adjoint factors. But it's interesting for the development of category theory that, first of all, I think Bobakke had to a large extent to be credited for the creation of category theory. Because, first of all, there were two major issues during the writing of the treatise. First of all, in general topology, in general topology, the Cartesian product of two spaces, topological spaces, may not make any difficult problem. But Bobacki, led by his desire of generality, wanted to define infinite product. And of course, this has been defined before by Tycho. But he wanted to have a general theory. I don't know why, also, because they had in mind, I mean, not yet, Adel and Idel were not yet explicitly there, but they were implicitly there, and during discussions between Chauvelet and André Veil about class refueling and things like that, I mean, the idea of Idel was explicitly stated by Chauvelet in 1938 or 39, and so the idea of inverse system of group

1:05:00 Infinite product of topological group was there because of the need of casting theory in the presentation of Chauvelet and Veyne. So, Bobacki wanted to have this general notion of Tikhonov topology for an infinite product and Tikhonov-Furentzat if you have, well, at the time people said buy compact spaces and now compact spaces, your infinite product of compact spaces is compacted Tikhonov-Compact. But then, it's an interesting feature which is in the fight of Bobacki, that the one who was commissioned to write a draft on that, Was a man which was not really fit for that, it was Mandelbrot, not the Mandelbrot we know, but his uncle, Scholem, Scholem Mandelbrot, Scholem Mandelbrot, who as an expert in harmonic analysis, a student of Sigmund, There was a fantastic competition between André Wey and Solène Mandelbault to be nominated a professor at Collège de France, and I think the Collège de France missed the right decision. They said that Mandelbault was a very, he's a good mathematician, but I think his name will not survive. I mean, why not? Maybe Ben Weiss, not André Wey, but never mind. No, sorry. Maybe not. It's over. So far, the influence of Monterey has been much bigger than the influence of Solé-Montalvo. But I was not in the committee to select a new professor. I applied myself and I was put out. But there was a fierce competition and the reason was that Monterey would never go back to group. Also, there was another fierce competition between us, someone who was not so well known, I mean, de Purcell, who was, well, not a first-rate mathematician, but a very imaginative person, and who was certainly one of the founding fathers of computer science, in fact. But there was another kind of fierce competition, I mean, André Recheteau, the wife of de Purcell.

1:07:30 So, as a result, the person let go back. So, one of them let go back and the person let go back. Okay. So, and Lurie, and Lurie was associated with Bobak in the beginning, but he was not, I mean, he participated to a few meetings or preliminary meetings, but was not so happy and left immediately. But there was no clash at the moment. There were clashes between André Bey and Lurie. So, just to explain that Bobakir was a demand to understand what is an infinite product of topological spaces. And then Moldenburg was commissioned to write the first draft. And the topology in general. Yes, the topology in general. It's a very nice account. No, it's a very nice account. But he was the first one. He had to introduce infinite products of topological species. And he took the wrong topology. I think they are the basis of the open set. The product of connection. Or the box topology. As Kelly calls it, right? The box topology. Somebody would choose to call that. Yeah, yeah, it's Kelly's term. It's an exercise of Kelly. Okay, you have a family of species, example. And the basis of the open set is you take for each alvar, you open, you are writing the example. The product of neighborhoods. The product of neighborhoods. That's the box topology. Box topology, okay. So, and I remember doing, you know, a discussion, go back in, that was called the Montelbrot Topology. Ah, the Montelbrot Topology. Okay, so, but then... Well, he gave this definition, but obviously it did not work. He noticed that it didn't have the correct universal properties. Exactly, exactly. That's right, that's right. And why was it not the right topology? Because that was the first instance where the Burbankian people discovered that there was a universal problem, and that your topology has to satisfy something universal. It would make the product of discrete spaces discrete. Very bad. Two to the X. Never mind, you'll get a boon. That's right, two to the X. That's right, two to the X. Come on. Part of the discrete space. It should obviously not be discrete, but it is in the box of politics. Okay, so, I think it's one of the first times that people have known that.

1:10:00 I mean, the first hints. No, bare space. Oh, that's the only reason. What a ridiculous situation. The first human existence, the first inch to a universal property was the act of universal property or what it is to have a continuous mapping to such a coordinate. And the first hint of universal product, or if you want, the categorical characterization of an infinite product in every category. So that was it! And then there was a second instance was about the tensile product, which was a third-year instance of a representable factor. And, of course, Bobacki did not choose a representable functor of adjointness at the time, but we understood very well. And also, it was discovered during the discussion, just before I joined Bobacki, that the first edition of Amatini-Archibald appeared just a few months before I joined Bobacki. And then they recognized that all the basic properties of the tensor coordinate, what you could call the functorial properties of the tensor coordinate, So, I mean, what today we would say these are the fundamental properties of adjoint factors, I mean, so we all know that now how to play with adjoint factors if you have some property for the direct factor, for the adjoint factor you have some property, that adjoint factors preserve some kind of limit or coordinate and so on. So, that was the second. The third is... And now it's very simple, you have the forgetful factor from associative algebra to the algebra, you take the algebra. But then, when we started to develop the basic property of the algebra, we discovered that at that time I was in the moment. And then we began to discover that all the basic property would flow out immediately out of the universal property, or adjoint. That's the kind of algebra you learned. That's where I learned it. And then there is a book by Chevrolet, actually. Yes, yes, the theory of Lie algebras. Not that one. You never saw that one in algebra. I don't know as it's called. It's a book he published when he was in Colombia. Yeah, yeah, I read this book before I did my thesis.

1:12:30 In the theory of Lie groups. I just gave a general theorem including what you just posted. I don't have the paper of Chevalier. I'm the only one in charge. You know it's not the kind of thing you do for pleasure. No, no, no, no. You do for pleasure. To recheck all the prologue many things and to sketch and so on. And all the others are now difficult. The tag is filled with my thesis. I mean, the widows of Giovanni was a very difficult moment. And his daughter is as difficult as a mother. Well, I will not enter into this controversy. I'm still, I was, I'm still waiting for a phone call from Springer because we have to settle this small topic. So, but just an aside, but it's true that Chevalier published, not published, he had two versions, a slim book which is called Some Construction of Universal Life, or some unbounded algebra, and it's reprinted in Chevalier, in the collected paper of Chevalier, in the book devoted to Clifford algebra and spinos, so I reprinted this book on spinos and then I joined this, which is a natural preliminary. But Chauvelin was so that he wrote a book on spinors, which for an algebra is one of the best places to learn spinors. He wrote also a preliminary volume which has been included in the final edition about before algebra. That was written in the 50s. You look at the index, Dirac is not here. And there is just a one-line hint to all the different equations. He's worked for, you know, he wants to be a physicist. Of course, when I re-edited this book and as part of the collected paper, I asked my friend Boguignon to write an appendix on what happened to the gigantic wave and speed of independent geomatics. I remember trying to read, well I did read, you know, Chevrolet's Fundamental Concepts of Algebra.

1:15:00 Yeah, that's the one we read. Well, of course, you get this thing, this is an exercise in austerity and don't expect any relief. That's right. It's complete pure, isn't it? It's beautiful, it's beautiful. It's wonderful. It's like listening to Weber. It's a pure scale that's beautiful, but don't expect any background material. It's just completely pure. Okay. Okay. But then, it's interesting also that if you look at the first... There is, of course, an account of exterior algebra and determinants and so on. But the algebra, the exterior algebra, is not considered. For each P, one considers the lambda P of the module. It's never mentioned that you can assemble them into an algebra. And, of course, the universal property of this algebra is not mentioned. So there is, of course, something for the alternating product of multilinear form, but it's not really stated that if you have lambda p of a module times lambda q, the direct sum is an algebra. When you have the algebra, it was already clear in the 50s that universal and grouping algebra should be a prominent word, and we have already fundamental people by both Chauvelet and Arish Chandra. Arish Chandra was the first to really use this method. And of course, so, I mean, we could not ignore that. So when we started... I think an account of the algebra. Very soon we'd introduce an enveloping algebra. And we have a footnote there. If the algebra is unambiguous, the algebra is also commutative, then we have a special constraint. That was the first appearance of the symmetry of algebra. So in the first edition, we had just a footnote there. And then they told me, they said, it's ridiculous. So when we revise on the multilinear algebra... We included both the exterior algebra and the symmetric algebra, and then the tensor algebra, according to the pattern devised by Schubert.

1:17:30 So, these were similar instances of representative or adjoint factors. So, it grew progressively, and also, as I mentioned yesterday, the work of Jurnonish, Schwarz, and Grotendieck... Topological vector spaces relate gradually to the importance of inverse system and direct system, duality between them. And so all these examples, I mean, all enter towards a representable factor and a joint factor and composition of factors. But it's, so in a sense, I mean, I and I, Samy and so much, Samy participated to, from Starting from 1950 to 1965, approximately, to the Bobacki discussion. And so, all these things, well, category theory existed already at the time. But Adolphe Fontenot was also prominent at the time. And so, but then, it's incredible, I mean, so, step by step, Bobacki realized the importance of universal property. Which means that you have a categorical definition of the product or co-limit or co-limit. We introduce step by step various limits and co-limits and categorize by their categorical properties. But then, Bobacki, so Bobacki participated actively to the development of category and various aspects of category. Exactly, but the really interesting question, given the energy and given the ambition and focus of Bourbaki in the 1930s to do the mathematics, it's very easy to speculate, had category theory been known at that time, what the thought of Bourbaki, given that energy, would have looked like, what would it have been the equivalent of the Théorie des Ensembles and all of these... You know, these foundational things. And how would it have been organized in actuality? Well, I know, because you've thought about that and you still try to do it. I know that. I know that. I don't have a good guess, because I don't have a whole team working on it. Exactly. That's a very interesting question. It's a very important collective effort. Of course it is. Of course. Bravati is a part of the time.

1:20:00 And of course you have a delay. What are you going to do? Unravel it all and do it all again? Come on. But then, of course, by that time, in a way, some of you know Bill Warkin. No, it wasn't Warkin. That collective impulse, of course, actually, was dissipated after 40 years. Part of the answer to this is that they don't use it. I mean, 40 years ago, there was an attempt to rewrite books. Of course, of course. I mean, that's clear. In Théorie des Ensembles, they give this theory of structure and structure of observing maps, which they don't use. It's true! It's not true! It's not. So, John, the point is you don't have anything to replace. So you don't have to replace it with anything. But I can pick up very fast to a point where, you know, the active reality really did need category theory. Because what I mean is that it's obviously the organization of the... No, but, Colin, you exaggerate, because there is a rigid framework, even if it's partly implicit. Namely, for example, as I keep saying, that the default concept of cohesion is topological space, just to take one example, but there's an implicit... Pardon the expression, dogmatism, or whatever you call it. I know what you're talking about. This was the framework. It was, and not unreasonably given. Come on, that's a great... Well, I was going to say there was also... I won't say that. It's a fascinating idea. I mean, when finally, well, it's true that in the third division of the white, there is a description of what, it's really time theory which is described in this white, I mean, they describe time theory, they describe... Well, it's a theory, it's their definition of a morphism between them, what is their word, that's the problem. Now, I mean, in the first account, Which was printed in 1939. They describe something which in effect is type theory. Type theory. So you start from a set X and you bridge the scale, I mean, X times X and then the power sets, etc., and you bridge it step by step. It's more or less a type theory. But then they consider only isomorphism. And the point is that an isomorphism, as long as you climb into this scale of time, an isomorphism propagates. And so, if you have an isomorphism at the bottom, then it propagates. And they say that's our notion of isomorphism. But isomorphism, not morphism. And that in the 30s. In the first version, 39.

1:22:30 And they don't mention at all that you could have morphism. They mention the fact that you are Mahizan Mopi. And they were very much influenced by the, by the, by the, how do you call it? And so on and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, Bobacki was very much influenced by the Erlanger program. And also the work of Emmy Noether. So, in Emmy Noether, the main focus was on isomorphism. The isomorphism theorem of Mademoiselle Noether, of Freulein Noether, the single isomorphism of Freulein Noether, and so on. The main emphasis was on isomorphism. First of all, on automorphism, which is a symmetry group, which is Erland and Brunner, or slightly more general, on isomorphism. There was this remark that if you, I mean, isomorphic propagates along the scale of time theory, okay. Then they say, okay, so they, their ideological commitment at the end of the booklet on set theory, which is what we call knife set theory now, of course, their account of knife set theory at the end, there is a section where they account for that. They commit themselves ideologically. And they don't give a very precise definition, but it was later on. I mean, I think that Erosman gave us ideas about that. And I think that what you could say is that you want to build, you want to build factors from the category of set to itself with respect to isomorphism. And so you have a choice. If you have two such factors, you can take a Cartesian model or you can go from x to p, to 2 to 3x. Covariant or contravariant. Because you have to mix covariant and contravariant, you can only deal with isomorphism. So it's really, and Erosman was, I mean, trying in his theory of sketches, I mean, trying to make sense of this idea.

1:25:00 And in this last section, by combining Erlon and Pogon, we start here, what time? Well, they're already committed to it in the later, right? This is not before, but after that, In the first chapter, they committed themselves to a kind of Hilbertian framework, but that was already consistent with what they were trying to do earlier. To some extent, to some extent, to some extent. But for a while, I repeat that when this original slim book of 40 pages was published just before the war, the intent was that they were quarreling among themselves about the fundamental... They could not agree on what was the right foundation for state theory, which kind of axiomatic setting and so on. Some people were not completely convinced that set theory should be based on an axiomatic, formal axiomatic theorem, and the program was more or less accomplished later on by Olmos with his naive set theory. In effect, what they become is a naive set theory, which is enough for the practitioner. Putting aside the question about foundations, so they give a set of standards. What is a notation? I remember, I mean, it was acting this way, I mean, in my generation. I mean, what, what, notation, p to the x, always a slash and so on. Okay, so a set of standards. I insist, a set of standards. Notation and standards. But this is what was a partition. But they are already implicitly coming to the same with this last section of what is the structure. And I think it was mostly under the influence of the Hess man and Schubert, I think people who were more particularly minded than you to be on very little part of it. Oh, they didn't want it at all? Okay, then I have published some speculations that were probably false. No, I don't think that very little. But he wanted category theory less, right?

1:27:30 Well, you know, he despised foundations. He despised logic. Foundations, right? Both, you know, in many ways. So what has happened? Burbanki committed itself ideologically to Isomorphic. Then when they started writing general topology and algebra, of course, In general topology, you are not satisfied with homomorphism. Homomorphism is what is out of the machine, of the general ideological machine. But of course, if you want to know that, don't be a morphic part of it. You have to take continuous math. And then, because they knew enough mathematics, they introduced continuous math and composition of math, and they soon discovered that all the basic, what we call now, functorial properties of topological space are immediate consequence of the categorical point of roots, and they give a categorical definition of infinity. Topology, subspace, and so on. But wasn't the topology, General, written before the fascicule du Résultat? I want to say that. I actually have two. I don't have the third. I have one from 1947. No, no, no. Just before the war, they published first the account of Sainte-Furée, fascicule du Résultat, and the first volume in general topology. Just before the war. But they introduced the notion of a map of continuous function very quickly. Oh yes. I mean, it's... Continuous functions come early, very early in this account, of course. And in a sense, it was the worm in the food, the worm in the food. And then when they developed algebra, of course, there was very much interest in the... Mademoiselle Nauterre and Isomorphism, Freudian Nauterre and Isomorphism, that's A divided by C divided by B or by C, if I'm quite exact. But that may have interested algebra more than the notion of topology, where the notion of a map and a continuous function was a very actual thing. That's it, that's it. In the first, in the first, I mean in the beginning, in the beginning. In the first account of group theory, they put great emphasis on the so-called Noether isomorphism, first and second isomorphism, and also the Jordan-Euler theorem, which is quite important, no doubt, no doubt.

1:30:00 And when they come to consider groups of groups, you cannot dispense with defining the notion of homomorphism of one group into another one. You have to. And again, it was against the ideology, because the ideology was to consider only isomorphism, not the ideology. Very interesting. And then, once the theory developed, both on the topology level and on the algebraic level, More and more they understood the importance of continuous math, that means the fact that the topological species make a category, that the group make a category, the ring make a category, and more and more they were more and more conscious of this. And then they were more and more conscious that the basic properties, even the so-called Freudian inertial isomorphic theory were just Particular incentives or general principles from category. The language of category did not exist yet. But then... Now, after the war, they decided to publish a complete account of the set theory, including the logical foundation. I'm not so happy with it. No, no, it's very interesting, nevertheless. The only use of the Hilbert epsilon calculus is mainstream mathematics. Yes, yes. And they make it mainstream mathematics. On the other hand, they become a toy. No, no, no. They put it into open paper. They wanted to write an account, but it has to be admitted that very few members of the Most of the people were not interested in the foundation. And it's just because, I think, of professional ethics, because professional ethics, they felt compelled to publish a series, because, I mean, the aim was to publish a complete textbook, then, of course, they couldn't be satisfied with just a slim account of the basic foundations, so they had to expand on the foundations. And there are the presentation of a formal language.

1:32:30 Well, no, but it's still very interesting. It's fascinating when the mind gets down to it, even if it doesn't take it all that seriously. It's very interesting. On the other hand, they are falling into all the philosophical traps between denomination and object. I mean, if you look carefully, I mean... But I like the fact that they use blanks. There was a serious film about that, you know, with all those, what are they called, the VQ guy, you know, they make this, they know variables of blanks, and they use them, you know, they actually start to show it right in the footnote, you know, that boll X and so on is actually a blank linked, no, no, this is all thought out very well, it may be not, they weren't terribly interested in producing an efficient system. Well, because you rapidly go to abbreviations very fast. Abuse of notation. Exactly. Abuse of notation. Sounds wonderful. No, it's a marvelous concept. And so you see, my bias made this point, you need one minute, but I was so lost, you know, that you have how to use it. But actually, they got the idea of a variable as a blank, as a, what do they call it, a place marker. Place marker. They had all that. It's all there. It's all there. I will not hear a bad word said about it. Particularly for one of its members. Just kidding. But the point is, the point is, I'm not philosophical. They make all the confusion between obeying its denomination and denomination of denomination. But some of them want to put their way to make some of these confusions. There are sentences you could simply strike, but they put them in there to get a confusion in. I mean, they're trying to be precise, but they're doing it wrong. It would have been better to not try. Yeah, exactly. I don't agree. I don't agree, actually. I'm just going to finish, though. What has happened is that, of course, more and more people understood By the way of practice, the importance of, well, this was not yet called category, but when the category was invented, the name, I mean, more and more people recognized, and in the late 50s, I would say that almost everyone in the Bobecki group was convinced that category not only existed, but it was important, a new important way, and that was the best way to organize mathematics.

1:35:00 And of course, the most aggressive to that were Samy and Catton, and we call them the hau commissariats, hau commissariats is a political joke, high commission, it's a political joke, but they behave like a cagey, a cagey, a cagey, I mean, they're insincere. The lack of good faith and so on. Well, it doesn't matter. It's not important. But in that moment, there is something I'm curious about. It's nearly true that nobody ever looked at this stuff, but Grotendieck swallowed that book whole and uses it throughout Tohoku. I mean, Grotendieck was obviously delighted with what he was reading of that book before it appeared. Yeah, that's true. So there were two different questions. First of all, writing the book on set-fueling foundations, while the first volume, the first chapter on what is the formal language and description, while it is what it is, it's what it is. No one took it seriously. Can't confess. The second chapter is a razzle. Very beautiful. Of the basic of St. Fulian. In chapter 3, it is the Greek. And the chapters are about numbers and... Well, it's a beautiful formula. It's a beautiful, gentle formula. They didn't see its generality in the applications. No, they don't. Well, no, they use it. No, no, they give a general... Well, that's the earlier account. What they did was to reformulate it in terms of... And then there's a theorem in chapter 3, which I knew it because I saw that you could use it to unify von Neumann-Ordinal's previous construction of the natural numbers and, of course, also the original footage that was applied. I've written some papers on it. I call it the Zermelo-Burbacchi Lemma. I mean, that's actually, that's a very beautiful... There's so many felicities in that chapter. Oh yes, oh yes, that's right. And then the fourth chapter in this fondable book. Well, I didn't... The fourth chapter... It's a bit of a mistake, I think, the fourth chapter. Great. But not the earlier ones. They're beautiful. And it's crucial to the project, isn't it? I know, I know.

1:37:30 In that way, if you're not... Well, I suppose... Okay, okay, okay. So now... Here I can just explain what it is that the fourth chapter... The fourth chapter. The fourth chapter. The fourth chapter. Yeah. And... Of course, it's a very strange. On the one hand, Bobacki feels committed to fulfill his ideological promises to explain in detail what is this type theory, this version of type theory. He does that in the first place. He does that in the first place. And so he explains what is an isomorphism and so on by just the fact that, I mean, isomorphism, I mean... But then, of course, at that time they have realized only the strong importance of universal properties, adjoined photos and so on, and mortgages and so on. And of course they wanted, and at many places they would refer, when there was, we had to make an elementary zoning which in effect was the adjoint of the composite factor is a composite of the other. Basically that kind of statement. And they did not want to repeat all the time the ten sentences which were necessary. So they would refer to an abstract statement. But of course, to make the adjunct of a composite factor is, if you have a proper language category, it's obvious, interesting, and easy to state. But if you don't have a language category, it's awkward. Say, because it was published just a few months before I turned to go to study. No, that was written before I turned to go, but not yet published. And then there is a second notion about what is a movie. And there have been, some have tried at some point to convince people that when you climb in this scale of time, then you have sometimes covariance, sometimes contravariance factor, and then you have to extend it by using covariance or contravariance. And there was a draft of such an intent, I think. Atiyah, Witten suddenly explained the idea and tried to make the draft. I have not seen that, but it exists in the files.

1:40:00 And then they were convinced that there was no, that everything was ad hoc and did not fit with the known example. It did not really fit with the known example. So they introduced in-depth energy. In a purely abstract way, an auxiliary notion of morphism, which in effect describes a certain class of categories, what they are in effect describing is not a general category but a certain recipe class of categories, and they give enough information so that, I mean, every time they have an element that is only about a joint function of that kind, they can just refer to a general statement, not repeat this elementary proof. But then, of course, at a time that was in 1954 or something like that, I mean, everyone was convinced that we would do it. All these categories existed and everyone was convinced that if we had to start the project again, everyone was convinced that if we had to start the project again, we would start this category. It would have been very interesting to see that. Yes. It's a historical accident as well. It's always an accident of their meaning. It's not philosophical enough. It's a historical accident of their meaning. Ah, never purely accidental. Not at all. Yes, not purely accidental. So, I think the best, the most frank account is by Borel. Borel wrote a paper in the mathematical intelligence and he explained why Bovaldini was considered. I think it's a very frank account. Well, Borel was not a man to speak tongue-in-the-cheek. And so, Borel gave the account. But then, I think it's only Weibull who exists, only Weibull exists. But Weibull has never interested in the foundation, whether in the old-fashioned way or in the new way. And Weib... Of course, let's say about homology, collage, there are a few people by whom they very well did contribute to homology, collage, first of all his proof of the Derham theorem and then in his work and then in class field theory, I mean, in all this, what he...

1:42:30 Proof, which was a complement to Artin-Tate, I mean, his complement to Artin-Tate, and so on. He has proof, and also in geometry, rigidity, has some statement about rigidity in geometry, and his clarity, and also, not to mention the full main paper about Cartesian classes, Cartesian classes, I mean, the Chern classes, the Cartesian classes, the so-called Chern-Weil homomorphism, Chern-Weil homomorphism, different Chern-Weil homomorphism. And so André Day had made a number of very deep and lasting contributions to homology, but until then he would say, what some people call a cohomology class. I remember when we were discussing Schiff, he would say, oh, Schiff after all is just a stenographer. To take into account that you have functions which are not defined everywhere but in some open domain but I mean it's it is just a stenography instead of saying I mean I have a function with a certain partial domain I say I have a section of a sheath. Well it is just exactly the same stenography. It dispenses you to make explicit domain. And so but always with ideological statement by way you have to be very careful. Because Weber is the master of perversity. The master of perversity. I'm one of them. He was a very deep friend of mine. To me, he has been a very good friend for many years. But nevertheless, I can say that he was rather perverse. Okay, so that's finally what has happened and maybe it's... But I think basically it's because it was too late. And there were discussions between Eilenberg, Wotendijk, Lange, myself, and so on, to start everything from scratch again, and to put the category at the top. And we were convinced that mathematically that was essential. It's a hard rewrite. Yes. But then the practical-minded people, like Bohel, like Seville, they said,

1:45:00 I mean, we are engaged in finishing this project. What we can do, and I remember the discussion, Borel. I mean, trying to find a compromise was settled after a long discussion with Borel. I mean, we support what's there. What a heroic effort, though. That was already a heroic effort. Boubaki, a heroic. Yes. Oh, yes. And the redo it would have been... There were mainly two kinds of opposition, and very ideological. But André Weil was already out of the group. I mean, while he still played an important role, but at least he was over 50, and according to his own principles, over 50 he should not play. Actually, he should retire. Well, he was over, he should retire, but he still played an important role. Like, well, I mean, we like to joke that in a republic of people... Members, like Bubacki group, there were some people more equal than others, and André Weil was obviously more equal than others. So, I mean, André Weil, for various reasons, good and bad reasons, partly out of his temper, partly out of his ideological conviction and so on, had a strong opposition to that. And I think that because he did not, he did not understand how, well, he was basically interested in this deep, deep, deep research in algebraic geometry and arithmetic. And, I mean, when he was young, of course, he was among the first to take these new tools, like topological groups, universal systems, and so on. But then, later on, he was already 50, and I think that... To really invest in new tools was too much for him, too much for him. And so he resisted, I think, partly because he could not see what he could prove. He could not improve his mathematics by using these new tools. He was completely convinced that these new tools were very important. But he would not commit himself to use these new tools. He had an expertise, and he was already well-considered for many good reasons, and he did not want to invest in new foundations. And so, as he saw Bobakli, there was no need to do that. But then, this opposition partly also fed up by personal opposition between the tempers of Gauteng Dike and Vein. That just played some role.

1:47:30 They were both very strong personalities. Well, but you have to combine this with the fact that André Weil wrote his own foundation of algebraic geometry, which are six and a half. So they're longer than chapter one of E.G.A. but not as long as E.G.A. Of course. And as a student I had to begin to learn. I started my training by taking this book one day late and I gave up after the fourth chapter. No, no, I don't know. I gave up after the fourth chapter. On palatable, I mean, the book on palatable. Almost as on palatable as EGA. Well, but you can say this in the correspondence, too. This is terrible writing. Bayes is just hard to approach. To me, EGA I can read because it makes sense. Bayes foundations I cannot read because they don't make sense to me. But EGA is longer by a long shot. And EGA doesn't get to anything like the theorems. You then have to go to SGA. Yes, I got it. You're not, yes, whereas Weil's approach does get you to the theorems. Well, in fewer pages. In fewer pages. But finally it didn't get you to the theorems in the very conjecture case. No, no, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. No, no. Did he, was one of the reasons that he distrusted the Grand Equestria was he saw it as attempting too much in the way of generalization and central lubrication? Did he believe that... I'm just asking... You thought you could do both in the 1930s when he was younger. Obviously, he must have felt that it was possible to do both.

1:50:00 When he was younger, you could do both. Hilbert, after all... He managed to do this sort of thing more or less in his old age. He then worked on Foundation, relatively old age. You know, it's possible if you have the energy to go on to do these things. I mean, Hilbert had a considerable influence on the Hilbert School. Surely he had a strong influence on them. It wasn't only the Nerdist School. Of course, the Nerdist School is an outgrowth of Hilbert's approach to mathematics. And that must have had an enormous, you know, very strong impact on the, on Weil and on the, and on the early, but of course, Hilbert never thought, there was nothing mathematical that was alien to him. But Weil said it was never through Gauntlet Harding. Yes, yes, sure. It's not everything Hilbert ever did. No, no, no, of course not. But Hilbert did take foundations very seriously and he made an heroic attempt, you know, to produce a foundational theory. It's just that by the time he got there he was not solving, you know, mainstream problems anymore. He used to work sequentially, right? I mean, that was his habit, to move from one area to another. Well, I'm talking about dealing with a whole other thing at once, right? Seamlessly. Well, that's my question. Did he see that? Did he see it as dealing seamlessly with a general foundation? Or did he see bigger foundations as something... That would have been my guess. Let me finish the sentence. Did he see foundations as something which, as it were, crystallized, came into focus piecemeal from within mathematical development? It's clear from many statements by Wey that the model was von der Weyden and the motivation was that what von der Weyden has done for algebra, we must do it for the rest of mathematics. So there's no objection to abstraction or to unity. There is an objection to 3,000 pages before the first theorem. So it's a pedagogical, if you like, expository issue from the conceptual difference of the nature of probation, I think. And I think it was misplaced, because I think those 3,000 pages are easier to read than any 600 pages of pay. But possibly because they get essentially an average of...

1:52:30 But that's not quite... Fair to compare what an enormous machinery would have been developed later. I also attempted to read. I am the first child to read E.G.A. No, no, no, no. What I mean is that... By the time Goethe was getting ready, it was an enormous elaboration of machinery 30 years on from what Weil and the Burbach students were trying to do in the 1930s. It was an enormous expansion of machinery. And so, yes, weights have been found, but I think in the 1930s... No, I, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, it was, But I think one of the motivations, I mean, they stated repeatedly, but weren't they about indeed, who want to do it for the rest of mathematics? What they had in mind, I mean, on the other hand, also was very, I think, stated that often. It's to include the geometry of a capital in mainstream mathematics with the same level of rigor as what Vandana Devadana did for algebra to put it in those foundations. We have to do it for the differential geometry of a league and the topology of a league. And that was the end. And finally, I mean, when Bobacki published his series on legal politics, I think it's in part in fulfillment of this implicit promise. And unfortunately, I mean, there were many more chapters written as that, but they were never published. I mean, we had an account of Riemannian geometry, we had an account of many, many more, many more things. And I hope that the files will be opened someday, because why don't they publish them? It must be great because, you know, that's the thing, look, the Brobakic are beautiful, they're very beautiful, they're all this kind of prolegomena, they're a prolegomena, you know, just some huge thing that's going finally to come, and that was indeed the intention, and it's all very clear, it's all very beautiful, and it's, of course it doesn't matter to me, but you're waiting, you see, for the culmination finally of this enormous project.

1:55:00 This is not a criticism, because the actual foundation is already, you know, it is a foundation. They had to supply a foundation to the foundation of mathematics, which is called the theory of results, but what they're writing, you know, the topology in general, the algebra, this is really regarded as a kind of foundation for actual mathematics. And they, you can see them, the algebra is beautiful, because I've learned a lot. But somehow there was this great, you know, culmination. I agree that the Lie group was, but there were so many other things you realize that they might get to. Riemannian geometry, but differential geometry in particular, I don't, you know, I thought the point is according to the exact instance of writing. I mean, it was so painful to write, to publish, I mean, to publish it until it was publishable that, I mean, a lot of material was in the five of Bobakir's drafts which were not intended to be published. It was just for self-instruction and preparation for something to be published. To transform them into something publishable, according to the exacting standards of writing, would have been the normal start. And, I mean, the historical fact is that it's a question of exhaustion of steam. I mean, the project started with Gates in 2013, and about 1935. And with driving forces like microwave, cathode, and so on, all first-rate mathematicians use a lot of useful energy. Then there was a second generation, a second generation you can include myself and a few others, and then we were a second generation and then of course we took it on our shoulder and after the necessary compromise where to go about foundation and so on, we decided to concentrate on two series, one on so-called commutative algebra which It was agreed that it should be us. I mean, we would not have overlap with both of these projects. So, we had a discussion to draw the boundary, what Bobacki would cover and what Rottenberg would cover.

1:57:30 We had a very big business discussion about where to draw the borderline, and we more or less respected this treaty. Okay, and under the influence especially of Boel, who was one of the driving forces in the second generation, then, I mean, they had an expert in different geometry in the groups, I mean, and a very practical, he often quoted himself, I'm a Swiss, I'm a Swiss, a Swiss national character, my lady, a Swiss national character, my lady, would quote himself often. and with also a lot of energy and so and a lot of knowledge and a good i mean uh well reasonable like us reasonable like us so he forced us he said well we have to to get out of this impasse if we well it makes no sense to rewrite everything starting from the point of view i agree that it would be better but at the moment We can, I mean, Liebhoop is not yet in a satisfactory state. And with all the people there, I mean, at some point, within the banking room, there were six or seven world experts on Liebhoop. And when you have a group with enough energy, with enough discipline, self-discipline, and with six or seven or eight world experts in a field. You write an account in English. And the reason has been that if you had to rewrite today the series on Nicopet, it would be completely changed because what Bobacki brought in an already established summit is a completely novel point of view. And also various innovations, but not only technical innovations, but innovations of other people. And we saw very carefully, and it's clear that the development... So that was basically written in the 60s, and from that... From time on, the development of leaf theory has been very much influenced by that. As it isn't, it's completely different now, but because building on the foundation provided, and the new ideas, not only foundation, but new ideas, new insights provided by Bobacki, that was exactly what was needed to go farther, to explore piano groups, to explore buildings, to explore all this, and then quantum groups and so on. So now, of course,

2:00:00 I think the progress on Lie theory, I mean, there was, I mean, you can surmise some exaggeration, there was Hermann Weyl and Élie Carton in the 30s, then they provided us with the first account of global properties and the connection of differential geometry, both Carton and Hermann Weyl, then the things were a little, I mean, a little silent for what, steady development, but no breakthrough. Until in the 60s, Robert Lee brought a completely novel point of view. And that made the theory to start again, to start again. And of course, over the next 30 years, there was a lot of developments, whether Pierre de Gaulle, Frédéric Gaulle, and so on, and so on, and to a large extent, the dreams of André Weil about bringing together arithmetic, algebraic, geometry, classifier, and so on, I mean, Pierre de Gaulle, Frédéric Gaulle, and so on, and like that program. Many developments in the groups were motivated in the 70s and 80s and 90s, but this account, and I know from my Russian friends that this book has an enormous influence. As a result, I mean, the book is now outmoded because, I mean, it was too successful. And in a sense, Bobaki, that's what happened to Bobaki. The first generation created linear and multilinear algebra, topology, topological vector spaces. And everyone understood. I mean, the best of this was assimilated. Not only the material but also the way of acting, the way of thinking. All these things were assimilated and became common practice. Then there was another step which is both the... And as I said, at some time we had the best world experts in most fields. So we wrote what was at the time, maybe slightly more than that, forgive me for that, but we have written it.

2:02:30 No, forgive me for that. Well, very cool is this. We'll edit that bit out. We'll edit it. But then, if you want to understand the story of what has happened to Bohm, there was a third generation with some well-specified goal, and to a large extent they fulfilled that goal. There was a second generation, to which I belong more or less, Well, most of the border, but the limit was not the rest of the border, and we have another here, and to produce this series of e-books in algebraic, well, the foundation of algebraic geometry, to help with it. And then, the steam was out, the steam was out, the steam was out. And, well, for various reasons, first of all, something which has been successful for 50 years. This team was out, the period was different, and we had been too successful. So, I mean, we couldn't bring something new. I mean, everything new we brought had been already absorbed and understood and reformulated and, okay, so, and then the team, this team was out, and so what has happened is that the Gospels have been written. Yes, Gospels have been written. In the same way, Gospels have been written. And in the 80s and during the 70s, we had a very painful law case with our publisher, so I had to go to, I remember the many times I had to go to the court and I had to report to the court and so on. But mostly I was the one in charge of this lawsuit. I did not gain much sympathy for lawyers after that. But Pierre, I fell back a little bit taking notes. You said that the two goals were the Lie group book and Foundations of Algebraic Geometry. What is called, I mean, the Commutative Algebraic Geometry Foundation for Algebraic Geometry. And you actually negotiated a dividing line. Exactly, exactly. More than, and then we had this lawsuit and I devoted a lot of energy for this lawsuit, not only me, but it was this lawsuit and I don't think I've got much sympathy for lawyers and judges after that.

2:05:00 Now, to put it in one sentence, I think Boris Bobacki and his publisher were the losers and the lawyers were the winners. They usually are, you know. Yes, I can speak to a personal experience of that, but let us not get to that. Can I ask one quick question, Keir? The date of this informal agreement on the partition of labour with the Grokendieck, on our strokes, on the career... And of course, you have to understand that Germany was a Skype for Bob Hackey. Many volumes of Obakir have been in their final form, printed form and related by Giordani, and what we do knows importance of Giordani in the writing of religion. Do you remember at all what principles were used for dividing? I mean you can look at the works and see what they were, but what principles they had in mind? Yes, we had the principle. I would say the local theory. In a sense, I mean, as long as we spoke, I mean, as long as we spoke of commutative ring, local ring, prime ideas, and so on, without gluing, before gluing things, before gluing things, and so, of course, cohomology could not appear because cohomology comes when you have to glue things, I mean, we know that in algebra and geometry they are fine-model, are topological, are cohomologically trivial, which is a serious result. And Führer in A and B of C in this version. I mean, Führer in B and Carter in this version. So, I mean, that was clearly stated. As long as something is understandable in pure algebraic terms, without taking seriously into account the Zeisky topology and the geometrical insight, it's for us. As soon as it really begins geometrically, it's for you. That was more or less anybody. And we respected more or less these. And it was easier because, due to the U.S., the sky one more sucks.

2:07:30 But already, I mean, for Boubacar, he wasn't. The writing of Boubacar at the time was no more in the hands of the U.S. And various people contributed. Various people contributed. Myself, due to some volumes in the final draft, Sayer, Boyle, Dixie, Dixie was a strong collaborator, strong co-creator, and also Samuels. Samuels, the writing team was more diversified at the time, so I would say basically... Dix-Mies, Samuel, Sayer, myself, and to some extent Bowen. We would write a final account, not only preliminary draft, but an intermediate draft. This was for the community of Algebra or for the group? For both. Was Titz involved? Well, Titz was an advisor. I see. I mean, it's clear that we used many documents given to us by Titz. And with his consent, of course, and he participated to some of the discussions. He was never appointed formally a member of the group and he did not want. But he was our advisor and we drew from his seminar, from his notes and so on extensively and we discussed many things with him, discussed many things with him, but it was clear that it was a temporary cooperation. That means, Tietz was interested in the project of the group but not in the rest. So, every time we had a serious discussion about Legop, we would invite him to participate and we would, I mean, put our agenda in such a way that he could participate, we would devise our agenda such that he could participate, but it was clear that, come by common consent, it was clear that he would not participate to the rest of the project, he was not interested and we did not ask him to be interested. So, the basic rule in Bobak is that, in principle, the full, I mean, Member properly should be interested in everything. That was the basic. You could not be a specialist. You had to be interested in one. Life is life, but nevertheless the basic principle that the member who was really member, I mean the people, when we had occasion to collaborate, it was like kids and also when we wrote the chapter on what is really probability in a discipline and why is it like that.

2:10:00 In our video process, we got advice from... Paul-Henri Meillet was one of the best probabilists in France at the time, and he participated again, I mean, under these conditions, you are into a specific project, we seek advice from you, we will manage our agenda so that you can participate to some discussion, you can write a draft and so on, but you don't promise to be interested in the rest, and we don't promise to take you inside the group, and he wasn't, he had his own project. So that has happened. But basically, I mean, the ordinary members, regular members who are, you know, sat there to swear on the Bible, swear on the Bible that they would be interested in that question. Well, life is life, of course, life is life. And people, some people are more interested in something and some people are less interested. But you have a special handshake. Not that I'm the boy scout. No, the nations I was thinking of. I'm sorry? The nations. The free nations. Oh, the nations. I think it's proper to... It was, to some extent, a masonry. Yes, it struck me. It was a masonry. And what, I mean, the positive side, I mean, like with masonry, there is a bright side and a dark side. The bright side is a sense of community, of cooperation in the work, and public service. I mean, that's a positive side. Absolutely, masonry. As I see them. When I was once, I had to join them. I participated to some preliminaries, I was able to open, I mean, session blanche, it means open meetings, meetings which are open to non-members, so I participated to some of them, but then of course, well, I saw the positive side, I mean, the idea of public good and cooperation and so on, but I saw also the dark side, I mean, I mean, the...

2:12:30 And that's the dark side. And so, after a few meetings, I one day made a formal offer to John, I said, no, no, no, I'm not interested. So, but with Bobak, it was always the same, and there were many positive sides, but on the other hand, I mean, French academic would complain, rightly so, rightly so, that there was a time when all, or when not only the Bobacki, remember, who set the curricula for the various exams and so on, but also that when it came to a point of one, you better be a friend, not an enemy, and in a sense, I was a traitor to the globe because I always refused to play that game, and it's interesting that when I stated in writing, Well, it's ten years ago that was the dark side of Bourbaki. I received immediately an email from Serre, how can you pretend that Bourbaki wanted to control the university? We never did! We never did! I mentioned that Serre was professor at the College of France. Schwarz, Articol Polytechnique, Katon, Atasobon, and Articol-Mobar. Our principle was to select the best people to put them in the proper place. I suppose that Chirac, when he had to appoint a new coroner, he decided to put the best people in the proper place. And he took exactly the same people as he put them. The goals may have been a little different. Yes, one would like to think. Yes, of course. Fortunately, fortunately. But then, speaking about Chirac, I noticed that in the previous government there were about ten women, with some importance, some role of importance.

2:15:00 In the new government there is only one prominent woman in the government, who is Minister of Defence. She is here for the job, no doubt, she is here for the job. And all the other ones are just decoration. I mean, when you see the official picture of the government, just for decoration. And there are not so many. At least a socialist, and there are limitations, but at least when Joshua was a prime minister, I mean, there were a number of prominent women in the government. It was a very prominent role in the government. Would this be a good place to break, because we've come up to 7 o'clock and we didn't, I fear, get quite as much ground cover today as yesterday, which is not to say that we didn't have some fantastically fruitful discussion, but we didn't cover quite as much of the crowd we visited. Can I suggest tomorrow... Sorry, John, come on. If I could suggest tomorrow, if we could perhaps try and start a little earlier, and we are now at a very natural breaking point, which is, as I just said, this, as it were, treaty negotiation between the Bilbaki and Grovendieck in 1960, so it does take us very naturally into the whole of Grovendieck's work in algebraic geometry, including... ... and then to the larger picture of his legacy for the overall shape and direction of present mathematics, which in turn leads naturally into what Bill was going to talk about, the law of initialial programs. So there's a lot of ground to cover, but I think we should try and... We'll tackle as much of it as possible tomorrow whilst Pierre is still with us. Can I add something? Please! Shall I leave that on 4pm on Tuesday? Okay, so we'll still have you for most of Tuesday. No, I will be on Tuesday. Which will be good because Leo Corrie will be here also for that day. So we'll be able to share. I called my daughter and she will be at the end order on 7pm in my home. So I'll have to be there. So if we could keep doing that. So we can keep you till four, which would mean that you and Leo would overlap by one day.

2:17:30 I'm sorry, I had to switch. No, no, no. It's just, it's just. In terms of focus. Forget it. I mean, I think all of this has to be done in today's, I mean, we had reached yesterday evening the focus on the cumulative. The general tense of the methodology in cumulative, we had a very illuminating discussion on that today. No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, But Piero's direction has been, for me, extremely fascinating. I mean, I certainly welcome this elaboration we have today of the discussion of the Kuhn and Staffan. I mean, what strikes me so far is that we were hovering around, just before 1960, we've heard how Grotendieck came from the more general functional analysis considerations to do some very... Concrete things in the complex analytic situation. But we haven't really heard anything yet about his work, as well as St. Lucer's, on algebraic geometry at all, nor have we really heard what the emphasis was around the Bay conjectures for the whole Bourbicky group of the French mathematical community, and exactly at what point Grotendieck got... ... got hooked on this particular thing. I know Saar refers to some, to explaining this to, in some footnotes, I know Saar refers to explaining this to Grothenieck, but I'm not sure if it's a date that... Is it, yeah? That's, that's knowable. I mean, I don't know... Why, why, why? They formulated this project in 1954. Oh yeah, no, but I, I know that, but I, I... That's when Grothenieck, that's when Grothenieck was writing to him about Hawking. Oh, as, as early as that. Yeah. Ah, I see. Because, I mean, Saar mentions some of the enthusiasm of which... Thank you very much for your time, and I hope to see you again in the future.

2:20:00 Maybe they can guess now, but I mean, well, let's now say he has less influence now, but at his peak of influence he was a bridge builder. But to talk more about that, I would very much like to ask you why... It's also the general question of, I mean, references to perhaps these Alan Jackson articles, about the very patchy nature of Grotendieck's knowledge on many things. That's right, that's right. But if you read the correspondence between Sayle and Grotendieck, you will see that. Yeah, yeah, yeah. Knowledge on both sides. But it's true that there is, at some point, I mean, the... Well, already late, I mean maybe in 55 or 55, something like that, Gottingen asking there whether there have been infinitely many zeros of the Zeta function, the Riemann Zeta function. So he would not have had any systematic, very systematic, advanced knowledge on algebraic geometry in the middle of the fifth century. No, no, no, not at all. Now I mean, Sayer was, I mean Sayer has been very, I mean to me, to me the miracle, the miracle is a combination of Giordani, Wotanik and Sayer. Totally different personalities. Giordani with his energy, his organization, his team. Thayer with his sharp mind, very logical, very rational. And absorptive, I mean everything. And the capacity to absorb, fantastic capacity to absorb. And, quote-unquote. And also Zaretsky. Zaretsky to a minor extent. Because Zaretsky was generous enough. After the meeting in the States in 1956 and 1957, there was a meeting on the East Coast. Zariski, I mean the confrontation of the new ideas of Selen and Grotelich and with Zariski. And Zariski was genius enough to tell his followers like Mumford and other, and Artin, the young Artin and so on.

2:22:30 It's clear that what you have learned from me is, well, it's a deep geometrical knowledge, but the method I explained to you. You have to learn a new method to solve them. I gave you problems, but I did not give you the right method to do that. I mean, my methods have exhausted their steam, and you have to learn a new method. Well, as I said, he was old and he was a very genius to tell his students to do that. And in a sense, I mean, as Mumford stated it to me once, I mean... The miracle is that we, meaning the school of the ISD, Harvard, we had problems, but no method. Gotendi had method, but no problem. Quite slightly, yes. He had one problem. The very conjectures. It's not clear that it was in the beginning. Well, there's a certain mismotivation, but no problem, no. At that time, most people considered it was an inaccessible. No feasible problem. No feasible problem. Of course, the very conjecture had just been brought and people said, well, we will prove them in 50 years. But no one expected that they would be proven in 10 years. I think we've reached general agreement on the framework for tomorrow's discussion. We're going to concentrate on the algebraic geometry and that naturally leads into the broader vision of the broader significance of the work as a whole and into the things which I think Bill is probably going to develop in the two or three days after that on the position of logic with respect to algebraic geometry. Well, I'm sure there are many other topics you want to touch on as well. Things like multilinear algebra, you don't have diagonal matters. You can talk about bilinear, but you cannot talk about quadratic. Well, there's something very basically lacking. Yes, yes, yes, yes, yes. Well, the idea of co-algebra is that you introduce as a further structure. Exactly. And, in fact, it's a universal property. That is, among the symmetric and nodal categories, there are...

2:25:00 No, no, no, no, no, no, no, no, no, no, no, no, no, no, no. Thank you for your attention.