Afternoon Discussions

0:00 The Kuhner theorem is true, but homology is really a co-algebra. And the co-algebra is from a Cartesian category, and even Cartesian clothes, in fact. One is, so it's got 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 change the maps, but usually they don't put in more virtual spaces, even though it might be very natural to... Well, a spectre. A spectre, yeah, that's right. A spectre. You're right about the spectre. So this idea about the... Homology is really a functor that preserves Cartesian products, from one Cartesian category to another. In fact, Cartesian closed categories to another category. With the special property that the homotopic category also has, namely that the notion of point and the notion of component have become identified, There is a trivial, there's some kind of trivial sub-category. In other words, with co-algebras, using the co-algebras, there is an underlying, graded ones, there's an 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- There are a number of different ways in which we can use these terms, and I think one of the ways is to divide them out by the higher degree part, which is the left adjoint, or simply taking that part, which is the right adjoint. So the right and left adjoints of the trivial inclusion agree, something which is never true for ordinary spaces. There is a very non-spatial feature of the components and points structured coalescing, but at the same time preserve much of it.

2:30 So this leads to one question, which is, you see, if homology is defined by some particular procedure, there are many such theories. 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 it? If a homology is defined to be such a Cartesian closed category with left and right adjoints agreeing and all this, 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. You know, the one way out of Euclideanism, why do you consider the pie what it's well known, already mentioned in the first capsule seminar, that the more fundamental object is a finite group of fixed and points? And in the automotive by Pi-1. If Pi-1 was always being obedient, then there's such, again, we would have known this is a very strong, because if Pi-1 is obedient the same as H-1, and to define H-1, you don't have to, I guess.

5:00 But when you go to higher, you know, Pi-1, in some sense, and I think, touch or not, the same as with Pi-1, we all know that there is... Speakers can behave rather nicely when the base space is simply connected, and otherwise that means you have to deal with the action of pi-1. But the action of pi-1, what does the action of pi-1 say? It says that there are various homotopical or homological constructions that are not good, You put in abstract terms the category and the design. I don't know. I have no answer. Through focus, my question is better than any. Assume everything is simply connected. Because the focus is usually that the stable homotopy is some kind of limit of the actual spaces. Yet 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. I mean, if it were a non-pointed version, it wouldn't be additive. And the construction that I'm asking for would not be additive, it would actually be non-linear even. This may be even more trivial, but no, it's not exactly the action of Pi 1 that's causing the conceptual consternation. If it simply connected, I mean, this is maybe too coarse a remark, but if it simply connected, the points give.

7:30 There's nothing to know about changing from one point to the next. This is true, but when you take the suspension and you make this kind of very tall limit, and so the linearity, you start with a nonlinear category and at the end it's linear. It's not linear. It's not very easy. Well, it would be like the model is the co-algebra. In other words... It's the same sort of thing as the usual homology, but it's 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, the tensor product, into one with ordinary categorical product, in fact Cartesian closed. So nonlinear 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, maybe like the construction sets, it's not clear that you have to construct and get a category from it, you see, and there's this incredibly complicated scaffolding to, yes, does the end product really require that construction or maybe some other construction, so there's many areas, sort of like that, where homology is co-opted, you see. It's more than a whole atrium of cohomology. It has the various, one various operation, it is a still operation, and it's the idea of multiplic cohomology is that the receptacle of multiplic cohomology should not be bare. Bare vector space is also rational, let's say, but it's a representation of the multiplic algorithm.

10:00 And already, I mean, the existence of still... Existence or nonexistence of some space. I mean, how algebra may lead as a receptacle for someone to do something even more sophisticated. So, maybe another Cartesian code category. Exactly, a richer Cartesian code category. Yeah, a richer category. But there is another question connected with this, namely, in my conception of things, Co-homology 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 coherent functions given on a category and then you take an object X, But first of all, typically, if you just took natural transformations between these functors, you would just get constants or something like that. But if you take an object x and pass to the objects over x, then there is the completely trivial forgetting, forget the structural math. But you apply the homology or whatever these functors are to the total spaces, but now naturalities with respect to commutative triangles. For each commutative triangle, I want to have a naturality, because for each arrow, not for each object. So in that way, the cohomology appears as a cohomology operations. Between two homology functors, you get all that. Cohomology, and the cup product is merely the composition. The fact of being... You know, one being natural module. This is just a way of twisting around the projection formula or the convenience reciprocity formula, the change of integral formula, whatever you call it, for the interaction between the two, but if you twist it in this particular way, you can see that the cohomology itself just arises as the natural mass between different cohomology functions, natural in that sense.

12:30 And then, of course, again, it's trivially contravariant because if you change the space by a map, the category of objects over this one becomes transformed objects over this one, again, in the most trivial way, just by composing or forgetting, not by taking pullbacks 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 it. But as such, you see, the homology need not have any structure other than the cohomology, than the co-op, 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. Confess. Good point. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I saw it. I was making besides what is what stabilization would be if you didn't have a baseball. At least that was the best thing to be one to approach.

15:00 Where are we exactly? Yeah, I've just been, the story thus far, quick summary thus far, oh god, no, no, quick, quick. Be an expert. No, no, you will. Something is here. Homology implies homology is not destined to be a group, it's a co-algebra. Right, right, no, that's progress, right? Homology is a form of Cartesian closed categories, so homology is really an interpretation of one higher order theory than the other, i.e. a functor 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. It might not be the only way to arrive at a Cartesian closed category. 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, then we stable homotopically. 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 think nonlinear is similar to linear. 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. Right, right. Because you can follow one operation by another and that is the product. Yes, that was another puzzle. Of course, the term homology was used. Cohomology is obviously a derived term from cohomology. And yet, right, it was homological algebra. No, I mean, cohomology was the term. And then, because of the particular technical possibilities offered by looking, you know, using opposites, but having the maps go the other way, you ended with cohomology. And then that became the fundamental term. Well, it wasn't originally just...

17:30 You have to introduce the multiplication of things by you. I don't know if I know. But then it was just easier. No, I know. Just if you look at the derivation. And the multiplication. Obviously, you then search for the origin of the term. Hope almost doesn't want to come off. Algebras today don't use the term co-homomorphism. They don't use co-homomorphism. But nevertheless, it shows some genuine moving at the term... No, it's a real puzzle to me. I mean, what had been, say, Lefchef's motives for moving into cohomology were no longer the motives in, say, the Sénégal-Cartan for doing cohomology. Something else became the reason why it was easier to do co in the Sénégal-Cartan. And in fact, it seems like for Lefschetz, you've got a multiplication of classes. Well, multiplication is precisely the problem with the second half. And Cartan-Eilenberg says, we'll do that in the next volume. And Grover Deeks says in Hohoku, in a successive version of the paper, I'll do that to them all. I failed to understand the meaning of monolingual category at that time. And so I have a draft which is about producting. 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 then finally, many people discovered that once you have the X, most of the multiplicating structures exist.

20:00 And so, since the X was later on subsumed into Dirac category, and so the only project was, I mean, in the Dirac category, we have just composition of maps in the Dirac category. It is known that the HP of a space with x, with let's say 2, 2, 2, is simply x-ed, and later on came the idea of a monologue, the tensor quality associated with the tensor quality and so on, but if I take the point of view that on the tensor quality, then I take them automatically in a map, it's a phobia, you know, tensor quality for your...

22:30 Then, a linear map is a map from, let's say, x to y to z, and then the composition, of course, the composition is that if you have, well, it's exactly what it's, well, known in logic, I mean, if you have a function f x and y, you can substitute to x a function of both of the labels, to y a function of both of the labels. You can take a substitute to x, a function of g, u, v, and to y, a function of h, w, t, and then you have this composition, but this composition increases the number of variables at each time, so if you have a function in p variable, to each variable there is x1 to xp, substitute to x1 a function in m1 variable, substitute to x2 a function in m2 variable, and mp. So all together you have a function of m1 plus m2 plus mp, so full substitution. But if you look at the practice, your substitution and the same occurs when you are in formal language to make substitution of a predicate for a variable or whatever. Exactly the same. But this composition can be decomposed into elementary terms by making one substitution at a time. Which means that you change the focus from, 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 variables.

25:00 In diagrammatic terms you have a composition is depicted by that guy having input output, I mean, and then you take one of the inputs and you replace, you consider the output of some other process. In terms it just comes to sewing, if you have two diagram trees, whatever, I mean you take one branch of the other and you sew the shoe, or graft, I mean people use grafting or sewing, usually it's grafting, but now it's four trees, grafting, four trees, I prefer to say sewing to this, my dear daughter. Sewing, sewing. It's very difficult because sewing and shoeing, there is no shoeing, but yes, it's pronounced, this word in English is pronounced so, which is unfortunate, but that's spelled S-O-W. Thank you very much. Thank you for listening because I have no business correcting your English. I mean, this notion of function of one very good.

27:30 Michel Nazar, I mean, called, I mean, created this theory of analyzers. It's again, I mean, the dramatization of what you mean by substituting one very, one function into one. I mean, substituting the composition of function of many variables. And it's only, and... I remember a discussion with Sami and where we reformulated everything in terms of the monotone category in one object and all other objects being so you have one basic object x and then you have x times x and x times x and so on so you have one category one monotone category with one generator of the generative object and well the booklet There is also a lot of talk about aggressive function is very close to that, very, very close to that, and we are making this case. And so, okay, and something more primitive which is just replacing one variable by a function of other variables is so important. And then you can repeat. And so, there is one issue in opérate about what you call cyclic opérate. But in many situations, there is no need to distinguish between input and output, and the cyclic conditions where all the strings are the same and love is more similar. And so, the starting point, I remember, I was commissioned by Gutenberg, I mean, to write an account of Kerr's project in general theory of homology.

30:00 And, as I said, when I wrote the draft, I was never happy with his draft and... There are three special instances of the Yoneda 4X, the second is that we are monoid of catalysts, and the third is that we are operatic, and all these address how to make composition of functions. This is indeed this idea of multi-category, I mean, well, category of multilinear maps, because, so, I mean, if you consider all of five linear maps from x times z into z, x times y into z, the tensile model just says that as a factor is z, it's representable. Yes, representable, yeah. And then the y is y, no? Representable, in other words, yes. Yes, but there are quite natural situations where... So, these multi-maps, of course, it's a known concept, I mean, but...

32:30 It's often put in terms of pro-structure, that is, where you have such a structure not necessarily representable in either variable. That you can still, let's say you consider punctures from such a thing into abelian groups or so, then there's a naturally defined actual tensor and actual column, and so on. So this is a known... So plane we still need out of it. Yes. And the con extension along the data. Exactly, yes. It's due to Brian Day, the formalization, the Australian... 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 it not representable, not using the fact that it is representable. Yes. 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 N-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 pro-computers, or in other words, bi-modules, a tensor product of bi-modules, you see, is an example. But what this is, from one point of view, is merely making representable the co-limit-preserving functors from categories of ordinary models, which is strict, you see. So it's somehow that you start from something strict, you try to make it more representable, this will introduce some choice which is however coherent in this whole story. But this is the main example that's, you know, 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, there's something behind it that can be ironed out. And then you can say maybe this can be representable, and then you find the bimodules to do that and so on, but that's related to Alan Cron's notion of morphism, by the way.

35:00 But another point, though, is you see 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 functional precisely because the HOM is the right adjoint, 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 totalities, but some actual functions, 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 is a balance, what you call a give and take involved in introducing this representability. But certainly one wants to do it in order to achieve knowledge, because otherwise you'd be reduced to speaking of the functionals in some sort of external way, which is not to be. Not to be, not to be. But first of all, I mean, I was very reluctant about the coherence condition of my name for a long time, and so, like many people, I stayed in the air until we came with this. There is a double-quiz, I hope, for algebras and break groups and the inventors of knots and so on. And there, I mean, more generally, I mean, the study of modular spaces. And there, I mean, you cannot escape. And it's not only the one-dimensional or two-dimensional curriculum that you need, but you need the whole construction. That means not only ordinary category, but you have to have. Of course, I know the work of Kenneth Brown about... And so on. I've never been really convinced about that. But nevertheless, I mean, such situations exist, and they are not artificial, they come naturally in some geometrical problem. And to deal recently again with a similar problem. Because, well, when you deal with analytic continuation of function in any complex variable, and then you have, so you have certain differential equations, you have asymptotic behavior, and you have some neighborhood, some boundary point, 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...

37:30 But it's exactly the algebra of what is, I would say, is just a stash of polyhedra. And I think, well, the natural outcome, the natural outcome of the... And, of course, again, when I was touching, when I heard about him, I remember that John Moore gave a talk at the Cato seminar You combine and then it becomes associative, but it's associative up to a motor. And Eilenberg and Moore, I mean, devised a method to bypass this difficulty, and so already at a level of pass, everything was associative. I remember when Stashev published his first accounts, I was very elated. I said, wow, by suitably modifying the notion, we could have already associated with your higher level before Benchmark. I discovered that in my study of generalization of zeta values and so-called multi or quadric zeta values, I mean, really to an extent we have some different, we spoke about these specializing differential equations, we have asymptotically we have some solutions and so on, and the Stashev Polyhedron is really there to make visible the combinatorics, not only the combinatorics, but the... So, I came to terms with the Stashe Polar and also the Stashe Polar in the trap are beautiful.

40:00 I've been happy to this criticalization, but there was a problem whether they could be as explicit as convex polyhedra, non-polyhedra. Of course, what Stashe does, it represents them as polyhedra, but as not real polyhedra because they have to subdivide one phase into sub-phase. Let's say the first example is S1, I mean in two languages. And you want to make it a pentagon, so you split this. It's not a true polygon, it's not a true polygon, because you simply divide faces. And that has been a long-standing cause, a natural reality. I've been very a solution cook, and finally I've found, not the key for it, but I've found a view by means of explicit inequality. So, you have your convex polygon, which is given by explicit, or the cycloid, or whatever. But the fact is that acclaim-coherence conditions just deal with the one-dimensional or the two-dimensional correlative of such objects. But these objects exist. Whether you see the fact that homotopy, associativity, or nothing exists, as Bonhoeffer filled a beautiful geometrical object, which are useful in many situations. I would say that when you are, well, I'm also of the opposite vision about any category, isn't it? 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-met, bi-categories with the wobbles and so forth. This is, you know, this is just a discussion really within the category theory. Yeah, yeah. Where I say, well, of course these are beautiful objects, I know that. But, and also in many other, many basic situations, you need, you don't need that. It's something different, that's all. Now, my second remark is a follow-up. It's about cohomology. And during this discussion, I was told by one very mundane remark, that homology is a cohomology rather than cohomology is an algebra.

42:30 Why? Why? When Chek shifted from homology to cohomology, I think it was basically because they wanted to control it. And of course we need to do our own theorem and we need a lot of natural products for the differential form and we want to explore that. But of course, at the time, the idea of an algebra was well in the street. Even if not completely formalized, everyone was, I mean, Casman algebra existed, polynomial algebra existed, matrix algebra existed, and so on. We were very familiar with the idea of associative algebra. And for the record, I think I was the first one to use the word quantum by print in some seminars. So, I remember when I spoke of quantum and quantum mechanics, so now I have to speak to the co-cartists. 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. If that's going to become the fundamental motion, once you drop the code, that's how I meant it. It was meant to be a fundamental motion. That's right. It's an ancient cohomology. The weirdness is that this is some kind of universal imposition to say that that is the basic... It has, but it has become it, hasn't it? No, but that is what I'm saying. That is what I'm saying. Algebras and co-algebras sit side by side. And especially if you have the notion of monoidal category, an algebra is about from x into x, x times x, with respect to monoidal category, or I would say x times x, x times x, x into x. By many years, I think I remember in the late 40s and early 50s, the idea to reduce the binomial math to linear math was not an obvious thing. And I remember the obsession of Carton in every meeting of the Bobacki group, you should say.

45:00 On A, on B, C, is on A, so B, C. I mean, that's what, I mean, every time Gatton would show up in the discussion, we would send such to him. And Godin, Samy, would send such to him. So, the very, I think the first edition of the multilinear algebra was a real place. I mean... No, no, I know. I tried to read it. I mean, it's true that Whitney has similar ideas, but more or less, more or less, but less explicit. And Boba Fett was the first to mention, and Boba Fett is quite cheap. I suppose we did not quote the theory of tensor algebra, but tensor algebra... I don't know, I mean, tensor calculus. I don't know, I mean, tensor calculus. I don't know, I mean, tensor calculus. I don't know, I mean, tensor calculus. I don't know, I mean, tensor calculus. By the number from A to B into C, a representable factor, a term of which we did not state in that way, but it was exactly the same. And by this example, which posed, one of these examples which posed Bobati to introduce the so-called universal program, which we consider now to be a representable factor or a joint factor. But it's interesting for the development of category theory that, I think Bobati actually To a large extent, to be credited for the creation of category theory. Because, furthermore, there was this question... I knew we were going to make this question. Good! No, I don't want to make propaganda. But the point is that there were two major issues with Bobacki during the writing of the treatise.

47:30 First of all, in general topology, in general topology, the Cartesian product of two spaces in topological systems, we don't make any difficult comments. But Bobacki, led by this desire of generality, wanted to define infinite products. And, of course, this has been defined before by Kikota. But he wanted to have a general theory of infinite dimensions. And I know why also, because they had in mind... The ideas were not yet explicitly there but they were implicitly there. During discussions between Chauvalet and Aveli about class, field theory and things like that, the idea of ideas was explicitly stated by Chauvalet in 1938 or 1939. And so the idea of inverse system of groups or infinite product of topological groups was... They are there because of the need of casting theory in the presentation of Chauvelin and Venn. So, Bobacki wanted to have this general notion of Tikhonov topology for an infinite product and Tikhonov theory and that if you have other terms, people say bi-compact spaces, but now compact spaces, the other infinite product of compact spaces is compacted Tikhonov. But then, it's an interesting feature, which is in the fight of Obaki, 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 Mandelbrot, we know. No, not the Benoit we know, but his author, Sholem, Sholem, Sholem Martin Ward, Sholem Martin Ward, who as an expert was in the harmonic analysis of the motors, tuning of the motors, and by the way, I mean, there was a... Nasty competition between Andre Weil and Solène Mandelbault to be nominated to the professor at Collège de France, and I think the Collège de France missed the right decision. I mean, there is a little Mandelbault who is a very, very good mathematician, but I think his name will not survive. I mean, well, no, no. Well, maybe Brecht was nominated, but never mind. No, sorry. I think, so far, so far, the influence upon the MBA has been much bigger than I thought.

50:00 So, sorry about that. But, okay, but I was not in the committee to select a new professor. Mr. France, I was my, I applied myself. What? Sorry about it. The first competition, and at least they all let me go back. Someone was not so well known. I mean, was not the first mathematician, but a very imaginative person. And who was the founding father of computer science in France after that. But there was another kind of fierce competition. André Rech told the wife of me. So I thought this would be the wonderful man 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 liked immediately. But there was no clash at the moment. There were clashes between the two of them, but not at that time. ...to understand what is an infinite product of the topology. And then Mulder was compelled to write a filter. And for the topology in general? Yes, for topology. And Mulder wrote an error. An error. No, it was very... ...to produce infinite product of topology. And he took the wrong topology. He said... I think at the basis of the open set... The product of... Correct. He took the boxed product. It's true. Well, okay. It's an exercise in chemistry. And in the basis of the open set is, you take for each alpha, you open the set, you are buying the x alpha and the product is neighborhood. What is the neighborhood? That's the box of the product. Box of the product. Okay. And in our discussion, go back in, that was called the bundle product.

52:30 It's just that it didn't have the correct universal properties. Exactly, exactly. It's right to the point. It's right to the point. And why was it not the right topology? Because that was the first instance where the Bobacki people discovered that there was a universal problem and that your topology has to satisfy a certain uniformity. It would make the product of discrete spaces discrete. Very bad. True to the X. Never mind, you don't get a movement in space that way, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, that's right, And the first hint of universal product, or if you want, the categorical characterization of an infinite product in any category. So that was me. And then there was a second instance was about the tensile product, which was the first real instance of a representable factor. And, of course, Bobacki did not choose to present the role of Foncteur Archon in the next part of the talk, but we understood very well. And also, it was discovered during the discussion, that was just before I joined the Bobacki group, in the first edition of Amatini Archibaptia, just a few months before I joined the Bobacki group. And then, they recognized that all the basic properties of the tensor product, what we would call the Foncteurian properties of the tensor product, flow automatically outside. So, I mean, what today we would say these are the fundamental properties of adjoint factors, and 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 will preserve some kind of limit or coordinate and so on. So that was the second piece. In the third piece, the algebra of the algebra of the algebra.

55:00 It's a very simple, rather forgetful factor from algebra to the algebra you take is the algebra. But then, when we started to develop the basic property of the algebra, we discovered that all the basic property would flow out immediately out of the social property. That's the count given in algebra and the league. That's the way I learned it. Yes, yes, the theory of Lie algebras. Not that one. You never saw that one, did you? I read it for you. It's a book you published when you was in Columbia. Yeah, yeah, I read this book before I did my thesis. In the theory of Lie groups. So we printed it in the political paper of Chauvin. I just gave it to him right here. The political paper of Chauvin, for example, was for various reasons. By other means, there was only one choice. Another system would not cause this. And you know it's not the kind of thing you do for a price. No, no, no. You do for a price. And I... ...to recheck water polo, many things, and to sketch and so on. And all the other and a half different things. The tag is in my t-shirt. I mean, the widow of Chauvalet was a very difficult woman, and his daughter is as difficult as a model. Well, that will not end. I'm still waiting for a phone call from Springer or Ferdinand. It's true that Chauvalet published... And there are two versions, a slim book which is called Some Constructions of Universal Life, and it's reprinted in the Chauvalet, in the collected paper of Chauvalet, in the book devoted to Cleopatra and Spinoza. I reprinted this book on Spinoza and then I joined this, which is a natural phenomenon.

57:30 But Chouvelry was so that he wrote a book on spinos, which is, to an algebra is one of the best places to learn spinos, he wrote also a preliminary volume which has been included in the final edition about, it was written in the 50s, you look at the index, Dirac is not saying, Dirac, and there is a lot of things to hold the Dirac in mind, and he worked for it, you know, he wasn't a physicist, just... I mean... Of course, when I edited this, or re-edited this, I had an appendix on what had happened to the second wave and spin of individual geometry. I remember trying to read, well, I did read it, Chevalier's Fundamental Concepts of Algebra. Yeah, that's the one we read. Well, of course, he had this thing, this is an exercise in austerity and don't expect any relief. That's right. He said something like that. Well, there it is. It's just a complete purism. Yes, yes. It's such a beautiful thing. It's interesting also that if you look at the first edition of Multilinear Algebra by Bobakty there is of course an account of exterior algebra. And determinants and so on. But the algebra, the external algebra is not considered. For each p one considers the lambda p of the margin. 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. It's not really stated that if you have lambda p of a module times lambda q, the direct sum is an algebra, and interestingly enough, when we wrote the preliminary volume 1, the first volume 1, the algebra, of course, when you have the algebra, it was already clear in the 50s that universal and broken algebra should play a prominent role, and we have already fundamental paper by both Chevalier and Alexandre.

1:00:00 And of course, I mean, we could not ignore that. So when we started writing an account of the algebra, very soon we introduced a non-robbing algebra. And we have a footnote there. If the algebra is an obedient algebra, and so we take it, then we have a special concept. That was the first appearance of the symmetry diagram. So in the first edition, we have just a footnote there. And then they always say it's ridiculous. So, when we revised on the multilinear algebra, we included both the exterior algebra and the symmetric algebra, and then the textual algebra, according to the pattern devised by Schubert. So, these were single instances of representative oratron practice. But then, so it grew progressively, and also, as I mentioned yesterday, the work of Durrnish, Schwarz, and Botanyk. Topological vector spaces led, gradually, to the importance of inverse systems and direct systems, duality between them. And so, all these examples, I mean, all inter-towards a presentable factor and a joint factor, and composition of factors. And, I mean, I agree with Sami, and Sami is what Sami participated from. Starting from 1950 to 1965 approximately to the Bobacky discussion. And so, all these things, well, category theory existed already at the time, but John Fonto was not so prominent. And so, but then, it's incredible, I mean, so, step by step, Bobacky 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 guide by their categorical properties.

1:02:30 But then, Bobacki, so Bobacki, as the practice he made deductively to the development of category and various aspects of category. But given the energy and given the ambition and focus of Bourbaki in the 1930s to do 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 look like? What would have been the equivalent of telling his own song, you know, and all of these, you know, these foundational things, and how would it have been organized in actuality? Well, I know, but you thought about that and you're still trying to do it. I know that. I know that. I don't have it written yet. That's because I don't have a whole team. I know, exactly. That's a very interesting question. It's a collective effort. Of course it is. Of course. Burbanki is a part of the time. And, of course, it had to be later. What could it do? Unravel it all and do it all again? Come on. But then, of course, by that time, in a way, except for, you know, Bill, Lord, and King, Oh, well, that collective, that collective thing, you know, cognitive impulse, of course, naturally, was, well, was dissipated after 40 years. Part of the answer to this is that I don't believe that it was an attempt to rewrite books. Of course, of course, of course. I mean, that's, that's clear. In Terry and his ensemble, they give this theory of structure and the structure of mass, which they don't use. So John, the point is you don't have anything to replace the theory that is in there. So you don't have to replace it with anything. But I can pick up very fast to a point where reactive gravity really did need category theory. Well, you know what I mean is that it's obvious in the organization of the problem. No, but you exaggerate because there is a rigid framework even if it's partly implicit. Namely, for example, as I hate saying, that the default concept of cohesion is topological. Just to take one example, but there's an implicit... Pardon the expression, dogmatism, or whatever you call it. Yeah, no, it's just, this was the framework. It was, and not unreasonably given, you know, come on, that's a great... Well, you're going to say there was also some kind of domain, and so on. I can't tell you, I can't hold it. It's a fascinating thing.

1:05:00 I mean, when finally, what, it's true that in the first publication of the work, there is a description of what? It's really time-serving. You know, I mean, they describe that theory. Well, it's a theory, it's a definition of a morphism between them, what is their word, that's the problem. No, I mean, in the first account, which was printed in 39, I mean, they describe something which never is type theory. Okay, so you start from a set X and you create a scale, I mean, X times X and then a power set, etc., and then you create them. It's more or less a time frame. 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 part. In the first version. 39. And they don't mention at all that you could. They mention the fact that you have isomorphism. And they were very much influenced by the... It was isomorphism rather than any notion of morphism. But that framework doesn't work nicely with morphisms. Because the steps are both covariant and antivariant. To define a notion of morphism, you basically have to make a choice. I mean, that's not the most general definition of morphism, but typically even the simplest ones involve morphisms. It's interesting, a different ideology called quantum. Bohr-Bakke was then much influenced by the Erlangen problem. And also the work of Amy Noether. Amy Noether, the main focus was on isomorphism, because isomorphism, when was Amy Noether? When I Noether, the thing about isomorphism, when I Noether, and so on. Now, my emphasis was on isomorphism. First of all, on automorphism, which is a symmetry group, which is a slightly more general on isomorphism.

1:07:30 There was this remark that if you, I mean, isomorphic propagate along the scale of time, then they say, okay, so they, their ideological commitment at the end of the, we call 9th century now, of course, their account of 9th century at the end, there is a section where they account for that. They commit themselves ideologically. They don't give a very precise definition, but it was later on. I mean, I think that Erosman wanted to give us ideas about that. And I think what you could say is that you want to bridge factors from the category of set to itself with respect to isomorphs. 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 3. Covariant or contravariant. Because you have to mix covariant and contravariant, you can only deal with either one. So, it's really, and Erosman was, I mean, tried in his, in the theory of sketches, I mean, tried to make sense of this idea. So, what we have is that Bobatti committed ideologically to himself, and in this last section, by combining a normal program with structure. Well, they were already committed to it in the later ages. This is not before. After that, Tariq was retired. In the first chapter, they committed themselves to a kind of Hilbertian framework. Well, to some extent, to some extent, to some extent, okay. So, but for a while I repeated whether this original booklet of 40, slimmed book of 40 pages was published just before the war. The intent was that, well, they were quarreling amongst themselves about the foundation, they could not agree on what was the right foundation for set theory, which kind of axiomatic setting and so on, so people were not...

1:10:00 I was completely convinced that self-theory should be based on an axiomatic form of axiomatic theory and the program was more or less accomplished later on by Alborz with his naive self-theory. In effect, what they become is a naive self-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, o is a slash, and so on. Okay, so a set of standards. I think it's a set of standards. Notation and standards. And this is part of the practitioner. But they have already implicitly committed themselves with this last section of Matisse's title, and I think it was mostly under the influence of Erich Kandel and Chauvin, I tell you. I think people who are more practically minded than Gheorghe, they did not want it. Oh, they didn't want it at all? Okay, then I have published some speculations that were published. But he wanted category theory less. We did actually despise foundations, but I mean despised logic. Foundations. That's right. Both Deudonniac very well. Exactly. Exactly. Okay. That's it. So what else 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 over-morphics. Over-morphics are what is out of the mashing, of the general ideological mashing. But of course, everyone knows that over-morphic mathematics. And then, because of new mathematics, they introduce continuous math and composition of math, and they soon discover that all the basic, what we call now, fundamental properties of topological space, are immediate consequences of the... I made it from the categorical part of this and I gave a categorical definition of infinity. Topology, subspace, and so on. But wasn't the topology, general, the first year written before the fascicule du recital? I want to say that.

1:12:30 I actually, well, I have two. I don't have the third year, but I have one from 1947. No, no, no, no, no, no, no, no, no, no, no. Just before the war, they published first the account of set theory, fascicule du recital, 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... Well, continuous function may come early, very early in this account, of course. In a sense, it was the warm, hey, hey, hey, warm, the food. And then when they developed algebra, of course, there were very much interested in the... manoiselle, deuterium, and isomorphism. That's A divided by, A divided by C divided by B over by C, well, that makes sense. But that may have influenced their algebra more than the notion of topology, where the notion of a map and continuous function was a very actual thing. That's it. In the first, in the first, I mean, in the beginning, in the beginning, they consider, I mean, in the first account of group theory, The first and second isomorphism, and also the, when they come to consider both of those, When 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 it. Particular instances of general principles of category. The language of category did not exist. But then, they decided a complete account of the set theory, including the Rothschild Foundation.

1:15:00 I'm not so happy with that. No, no, it's very interesting, nevertheless. No, it's interesting. The only use of the Hilbert epsilon calculus is mainstream mathematics. Yes, yes. And they make it mainstream mathematics. No, no, no, I know, it's just an observation, but it's not important. But, they wanted to write an account, but it has to be admitted that very few members of the Bobacki group took it seriously. Professional ethics. Because professional ethics, they fail to publish a series. Because, I mean, the aim was to publish a complete textbook, then, of course, they could not be satisfied with just a slim account of the paper. So they had to expand on the foundations. And there are a lot of presentations of a formal language... No, but it's still very interesting. Nevertheless, it's fascinating when your mind gets down to it, even if it doesn't take it all that seriously. It's very interesting. 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 fool about those, what are they called? They know variables of blanks, and they actually start to show that one of the footnotes, you know, the Rolex and so on, is actually a blank. No, no, this is all thought out very well. They weren't terribly interested in producing an efficient system. Well, because you rapidly go to abbreviations, right? Exactly. They're wonderful. No, it's a marvelous concept. And so you see, Matthias made this point, you need one thing, but I don't know, so what? You know, they do have hundreds.

1:17:30 But actually, the idea of a variable is a blank, what do they call it, a place marker. There you have it, old man. It's all there. It's all there. No, it's very pretty. There are a lot of ways to make some of these confusions. There are sentences you could simply strike, but they put them in there to get a confusion in. They're trying to be precise, but they're doing it wrong. It would have been better to not try. So, I just want to finish. I just want to finish. What has happened? Of course, more and more people understood the importance of, well, this was not yet called category, but when the category was invented, the name, I mean, more and more people, and in the late 50s, I would say that almost everyone knew the existence that there was, and that was the best way to And, well, of course, the most were Samy and Catton, and the records and the Haut Commissariat, Haut Commissariat, well, it's a political... The High Commission. High Commission. They behave like a gay chief, in a sense, like a gay chief. You know, I'm curious about, it's nearly true that nobody ever looked to expound that book whole and uses it throughout to tell about Go. I mean, Grubnick was obviously delighted with what he was reading of that book before it appeared. Yeah.

1:20:00 The 19th book of the St.Furian Foundation, while the first volume, the first chapter under what is the formal language and description, no one took it in. The second chapter is a rather nice explanation of the basics of St.Furian. It's chapter 3, disagree. It's a beautiful chapter. Well, that's the earlier account. What they did was to reformulate it in terms of willow and rings. And then there's a theorem in chapter 3, which 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 Pritchard and Supply, which I went to papers on, I call it the Zermelo-Burbacchi Lemma, and that's actually, it's a very beautiful, there's so many, there's so many felicities in that chapter. And then the fourth chapter is a bit of a mistake, I think, the fourth chapter, but not the earlier ones, they're beautiful. And yet it's crucial to the project, isn't it? I know, I know, in that way. Well, I suppose so. Here I can just explain what it is. The fourth chapter is called subtraction. And, of course, it's very strange. On the one hand, Bogacki feels committed to his promises to, more ideological promises to, to explain in the day what is this type theory, this expression of type theory. He does that in the first section. Does that in the first section. And so he explains what is an isomorphism and so on, by just the fact that, I mean, isomorphism, I mean, transit through, through this gate from this type, type construction. Okay, but then, of course, at that time they had realized already the strong importance of universal properties, at all photos and so on, and morphics and so on.

1:22:30 And, of course, they wanted, and at many places they would refer to, when they had to make an elementary zoning which, in effect, was the adjoint of the composite factor as a composite of the atom. It's basically that. 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 adjoin of a composite factor is, I mean, 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 appalled. And so they decided, I tell you, they, because it was published. Just a few more before I try to go. No, no, no, it's unfortunate. No, that's right. Witten, but not yet published. They're individuals. Okay, okay. So, what is a movie? And then I've been sadly tired at some point to convince people that when you climb in this, in this... In this scale of time, then you have sometimes covariance, sometimes contravariance factors, and then you have to extend it by using covariance or covariate or contravariance. And there was a drought. I think that Atiyah, Witten, and Samy explained the idea and tried to make the drought. I did not think that it existed in the past. So they introduce independent, in fact in a purely abstract way, an auxiliary notion of morphism, which in effect describes a certain class of categories. Well, they are in effect describing, not a general category, but a certain restricting class of categories. And they give enough information so that, I mean, every time they have an elementary reasoning about a drawing photo of some kind, they can just refer to a general statement, not repeat it. But then, everyone was convinced that if we had to start the project again, it would have been very interesting to see that.

1:25:00 Yes. It's the oracle accident as well. It's the oracle accident of the week. So, I think the best, the most, main account is by Bauer. In a very frank account, Bohr, Bohr was not much, he was a big dog in the cheek, but many other qualities of Bohr, and so, and so, yeah, Bohr is the icon, and then, I think it's only very good, they very well do contribute to the curriculum, and then, in date, and also in geometry, rigidity, are so stable, and it's clear that he... And also, not to mention the full-length paper about Cartesian classes, Cartesian classes, John Graham, and so on and so on. Until then, he would say what some people call a cohomology.

1:27:30 It is just a stenography. Instead of saying, I mean, I have a function which is a certain partial domain, I say I have a section of a sheet. Well, it reaches exactly the same stenography. It dispenses you to make explicit opinions. And, one thing, always this ideological statement by Gregoire to be very careful. Gregoire is a master of perversity. A master of perversity. To start everything from scratch and to put to the category and to talk and we were convinced that mathematics was a hard rewrite. Yes, but then the practical minded people like Boyle and the final compromise was central after a long discussion with Boyle and in advance we support what's said. What a heroic effort though. It was already a heroic effort. And if we do it, it would have been... ...kind of opposition. ...principles, because over 50 should not be...

1:30:00 That's right, because you should retire. Why? He was open, so he should retire. He was open. Like, people who he put an order on. So, I mean, Andre Bell, no various reasons. Good and bad reasons. But he's tempered, but he has conviction and so on. I have a stronger position, too. I think that because he did not... He did not understand how, well, he was basically, I mean, when he was young, of course, he was among the first to take these new tools, like topological groups and inverse 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. And so, he resisted, I think, partly because, I mean, 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. That he would not commit himself to use these new tools. He had an expertise, he had an expertise, and he was already quite confident and moved for good for many reasons. And he did not want to invest in new foundations. And so, as he saw go back, as he saw go back, there was no new tools. He was also fed up by personal rules and the tempers of Goethe and Dickens. I mean, Grodenbeek says something at Bay that actually makes some sense. I mean, you can imagine Bay looking at particularly Grodenbeek's work and saying this, these proofs are too long. Maybe they'll work, maybe they won't. People won't want to learn them. We need a shorter proof and we're going to have to do that by using less apparatus. He looked at this apparatus and he simply said, this is too much. It can't be the right way to proceed. But you have to combine this with the fact that André Weil wrote his own foundation of algebraic geometry, which I see as a great discovery.

1:32:30 So they're longer than chapter one of ETA.