Categories & sets in a contradictory world

0:00 Thank you. And I said, you're going to take the next step. Oh, I'm going to start the quantum theory. But now, I have to talk to the quantum theory very well. And my answer to this, is the quantum theory that contains what can be measured in the geometry of the space and the range? So I'm trying to keep one of those people Thank you. And then I actually But the disease When they actually And there's going to be a few more years But that's Separation between space and time And I've always thought I used to be But one thing they never so find this in the ground the process is the process the individual are like find the diagrams I know, I'm small than things, though. So, I think that you have to be very, very careful. You have to notice why your intuition is coming into play.

2:30 Whenever you notice your intuition coming into play, you have to rudely suppress it, because your intuition is the devil. Unfortunately, when people put, okay, okay, you have to deal with it. I mean, it's the devil. You can't base it in an electrician. So he has an intuition that says space and time is separate. Because he deserves a very easy, space and time is separate. And so that's the feeling. When people go to graduate school in physics, and there were problems, and they develop a mindset. So there's a great tension between that and the other one. you have doing mechanics and relativity, which tells you the types of many things. Well, if you lose it, I mean, how do you know that? No, no, it does. It does. So, so, so, so, this is the only context of the barrier. You just chose it. No, no, no, no. Please, did I? Okay. If you're asking which things should you start with, and so you're starting with space-time and space-lust, if you set the edge, the loop-gravity, you start with 0.8% of your own, sorry, I'm going to argue with a whole set of data, and then you start with a piece of sense, and then you start with an average of a whole set of ways. Okay. The loop-gravity will begin with ash-decker variables. and then it talks about the symmetry group is broken down Colleen? Colleen? Colleen?

5:00 Thank you. Thank you. Thank you. Thank you. Thank you.

7:30 so should the projector be off I'm satisfied with living in France because a country with three or six of my friends is a cheese. Thank you. So, you end up with the idea that it doesn't change. So that gets very seriously weird. I mean, I tried thinking about that for a while, and then I just decided I did not want to think any more about an unchanging state of a new universe, and that's why I got into the Hamiltonian picture. The idea is that the past, present, and future, and universe were all contained in a single universal state. And then we divide it in parts, it's not all of the nations, it's right. We would create one part of the universe and the rest of the universe. So the universe is in one single absolute unchanging state, and the only reason we see things changing is we're part of the universe. When we're outside of it, we would see us as present and future altogether in one unchanging state. So at that point in the Senate, I wanted to do something else. I struggled about that for a year, and I could not turn it into a maturation.

10:00 Thank you. Thank you. I was a Marxist when I was younger. It's a little bit more complicated because I am a person who does science, sort of. It makes my life a little bit difficult. As I said, probably we should try and steer clear of some other issues in the next 48 hours. Okay, we... Well, no, in fact, because he's here now. Ah, right. No, he's here, don't worry. He's just making himself come. Thank you. Professor Cartier is a Professor of the Institute of the J.O.T.A. Institute of Science in Paris, since 1971.

12:30 He is known for the introduction of Cartier operators in algebraic geometry and Cartier divisors. In 1976 he was awarded the Ampere Prize in the French Academy of Science. He is also a great inspirational teacher. more recently he wrote two books one published in January 2007 Cambridge University Press which has us in the room we have as a guest from the University of Texas Professor Cartier is working on one of the more of the history of my friends and some things less known he did an important work of probability theory rather more in 1960 my subject today will be if I write limits in the category theory I intend to show that with the Torku paper of Goldenbeek, which is 50 years old now, there was a turning point in the category, and since we have to study this mathematical representation of space, of course I will accept the point of view of Goldenbeek that the space is presented by a category of ships. And so I will not question this point of view of many, many possibilities, but I will accept this point of view and concentrate on the sheaths themselves. It is known that the first paper or category was written by Eisenberg and MacLean in 1945 and called Natural Transformations.

15:00 Yesterday I spoke about naturalness in mathematics, of course, it has subsided. Natural transformation. So, that was something where, of course, the main notion is the notion of factore and transformation of factore, natural transformation. And it is interesting then in the title, factore, to look at it. By the way, the name factor, I suppose, was invented by logician in the 30s, it appears from time to time, that the idea is a higher level function, not a function at the level of a given set of ports at a higher level, and it has been used in various ways with this connotation. So that was the important notion. But what's the distinguished function of the mind tool is the following. Usually, I mean, geometry is the middle of the 19th century, when people were aware of the importance of geometry transformation, symmetry, transformation, rotation or whatever, and even more sophisticated transformation like the Li transformation associating point to circle and polar reciprocity associating point to light. But this kind of transformation fits very well with the paradigm of mathematics all embedded embedded into a century, because they made it, once you are set, the next thing is to introduce a transformation. Despite the fact that historically it's not exactly this way, but it occurred, I think it's supposedly delicate in system of the transformation of the functions. Of course, function in analysis where it was slightly more restricted in sense by order in the middle of the 18th century. But I think that Dedekin really understood the importance of general transformation, and the notion of one-to-one transformation or bijection is explicitly due to its foundation of numbers. So, this is different. I mean, in fact, it's something different.

17:30 It's not taking one, let's say, one set of ports and bringing it to another set of ports or to a set of solar or light, whatever, transportation. It's a much more ambitious thing because it takes, it associates to any space, for example, any space. not questioned at the moment, which says space in the sense of half of topological space in the sense of over, whereas of itself is right half of a symbol of separation number, and, but it is associated to any such space, another object which may be a space, for example in topology, when one of the transaction is to come, associated to a space next, the loop space, or next space, some point, which has been very efficient to the topology, or some algebraic invariant, which is the very beginning of the motivation of algebraic topology, to be associated to, let's say, to a space, some goals, let's say, the petty goals, So, this construction, of course, is not done by an acting force of the space, it's more abstract, and we have to consider the collection of all possible spaces and associated to any space, or another space. Of course, now, of course, in modern language, people can't even be better. I want to be rather historical, but not to use words which have been... the word put in the mouth of the harsh language. So, that's one thing, and the other important idea, I would never play, The paper was the idea of natural transformation between factors and natural transformation. They were not yet called natural transformation between factors, they were called natural transformation. And it is known that one of the motivations of Heimenberg and MacLean was duality in elementary linear algebra. and how already in 1945 the distinction between the finite dimension of the vector space and its dual was very staffy

20:00 and in differential geometry it means that you have no more indices and apparent indices and transformation And in the beginning there were more or less implicitly skeletal vector spaces, you plead in geometry, so you don't have to distinguish between space and its dual. So, but the question was that as soon as we develop linear algebra, if you associate inter-space, the dual, it's natural to iterate its construction and to take the white wall. The white wall, I think it's Banach, you put emphasis on this. Because in Banach spaces, while inter-space, by definition, the dual is the same as the original space, which is not too difficult for it by matching, by matching. But in the minor spaces, of course, that's a place where duality actually will be taken seriously, and where you could not confuse a space in its dual, one of the first, let's say, for example, it's very known analysis that if you take the NP function, B function with B is power, would it enable the dualism Q function with NP, you have a certain relation, and it's only in the case where B is equal to 2 squared, N will function with Q, or 2, and that's an offset duality. So, it was very natural. I mean, it came with the development of a large space, et cetera, not out of the poll, because you remember that Gottlieb started his mathematical life in functional analysis, and there has been much influence by the idea in this later development. It's one of my poll that, contrary to the impression you can get from the fresh leaf of Gottlieb, of whom I wasn't ready to, I wasn't ready to, there is almost no mention of his work in functional analysis in this, especially if it was my gaitry, well, okay, it's okay, let's say, but in a matter of spaces it's important, and first of all, in finite dimensional spaces it's clear that when it's not difficult to show that the jewel of the jewel is the original space, just the counting of damage, the same dimension but the ultimate space so when you repeat that you get a space which has the same dimensions so we expect them to be actionable in the banner spaces it's very far from me so

22:30 uh if you take the jewel of l1 it's infinity and then the jewel of l50 is the monster so we were used we were used to this idea of functional that we should consider seamlessly the dual of the space, distinguish the dual from the space itself. And then ask the natural question, is the space isomorphic to the dual of its dual? But before we can ask this question, the important thing is that, as usual, when you speak of isomorphic objects, there are two levels. First of all, you can have a weak definition of isomorphic. There exists some isomorphic that you don't construct explicitly. Or some natural, some natural map is given the isomorphic, the thoughtful isomorphic. And, of course, in balanced basis, it's easy to, for manipulation, what I call efficient use tautology efficient use of tautology that means efficient use of tautology is when you speak of f of x x is a broad so true x of x that means you have to consider f of x as depending not only on x from given fraction So, let's go back to the financial basis, once we have discovered what is an actual from V, the space, to V, star, star, star, the star, the star of V. It remains the point that is nice of all. But what makes it natural? That was a question addressed by, I mean, David McLean, and a very thin, and I think they were inspired, I think we made the discusses again, inspired by the, by the Erranden program, Erranden program. And at first you would say that if any automorphism or re-ohanizer-morphism or re-with an auto-space transport, first

25:00 through the dual, then through the second dual, then you want to have some compatibility between all this construction. But I think one of the breakthroughs like that, MacLean, and why they went farther than the environment program. And what distinguishes mathematics as expanded within the framework of structure for the combat design and the point of view of category is that we replace automorphism or one-to-one map by multiple maps. And so the question of the naturality of the identification of space with a double dual has two levels. First of all, automorphism. Let's say, if you have an automobile space, an automorphism gives an action on the dual of the end of the second dual. And it should coincide with the original one. But this was a familiar idea, of course. It means that people were very familiar from geometry in the 19th century with so-called intrinsic construction. I mean, one started with the idea that when you use the coordinate system to do some construction at the end, you take the scaffolding out and you will want to verify that your construction was really independent of the coordinate system. In physics, it's exactly the same, physical law should be at the end, independent of the observation system of the frame. So that was exactly the same thing, that's the important thing. But the point is now that we begin to think about what is a fact. And I think the next step was taken by Iron Bear in 1950, in the Carton Seminar, the Carton Seminar. I remember, I mean, being a freshman at a corner bar and leading with the announcement that Iron Bear would speak. I didn't know who Iron Bear's was, but I was curious of everything, so I went and said, I just did not skip those. But after a few weeks, I was a friend of Erdogan, but you know I look better, frankly, and supportive of young people.

27:30 My claim also was such a man. So, let's see if it's an important point, which is, in this cartoon seminar series of lectures, the cartoon seminar, maybe six or seven lectures, about the foundations of homology of groups. Because homology of wood was discovered by Heidelberg and Mark Lane in the topological study, well, there were preliminaries steps also, but they took it seriously and that was one of the starting points of homology collage. But then, I remember, in the exposition of Izo-Belkosinkito-Strange, we are supposed that we associate a group G by what is that means we have another group, or other sequence of group, that we call HIG, the homology group of G, coefficient in blood. And then, out of that, we have some constructions, and we have some axioms, but the main part, which will shock you, if you think of the general philosophy of mathematics at the time, expounded by Hilbert and the Parvodaki, that everything should rest on an explicit set construction. Let's say, if I want to define what is a group law, a group law of the set, well, of course, a group law, it's a map from G times G to G, or if you want a terrenary relation, a subset of G times G times G, but you start from the natural construction of the product, that is a product of two sets, then a subset is a member of the power set, so we have two basic constructions. And then we built out of that a successive extension, and the book law function in an element in some sets in disque, and what was supposed to be the rule is that you should You can have a representation of any mathematical, any mathematical precautions according to this pattern. That was the first time, I mean, I don't know, was not explicit about the way we do.

30:00 But it's a similar step which was taken when, in the 19th century, people began to introduce arbitrary functions. At the time of order of function had to be defined by some explicit way. A formula was not clear, we would say, to divide an algorithm like an algorithm. The fact is something that you can define an algorithm, always a definition of a algorithm. One is not completely explicit about what's the kind of algorithm. For him, an infinite series is part of the algorithm. It answers the question. But the same thing, the step was taken at the end of the 18th century, beginning of the 19th century, from an explicit description of the fact of school and algorithm, which is a general function. The function would just say, have a rule to associate to x. But that's true. I'm not excited. And I don't know what's taking the next step in this development. Assuming that we have already mastered the construction of set theory and application So a good model of all the existing mathematics, of the geometrical algebraic, topology, etc. within Centrion. Of course, in your explicit construction of factors, if I tell you how to define the jewel of a vector of space, it's an explicitly constructed factor according to the criteria of Centrionism. But to consider it, now it's the next step. to start from an explicitly discontractor to general, and that was taken by Herney Heiberberg. So that had what was important also, if you look at the book of Heiberberg and MacLean in homological algebra, they are explicit definitions of a diagram to Mardut without thought, which is important. And I remember, it was almost a joke with Eigenbeck, how do you do the Pood without thought? Every time you produce the Pood, give me another version of the Pood, which is without thought. It was almost a joke. I don't have to tell you I'm not joking, but then came the point and it's interesting. They defined the direction of a module as follows.

32:30 Let's say you have two modules, M and M. You want to define the source of S between M and M. And this should be given by a certain, so S should be a certain, the same ring. And what characterizes S as a knife? An arrow, a certain arrow of math. In this way. So, you have S which morphs into M, and M which morphs into S, and M which morphs into S, and M which morphs into S. You have to satisfy something normal. You should But the point is that we don't give an explicit construction of S in terms of that one, the natural construction is S is a set of pairs of moles, one in N, one in N, etc. But here we have a question which is purely in terms of mass, linear mass. And the important fact is that this kind of evolution doesn't really define S, difference really up to an algebra, a unique algebra. At that time, you are already used to that principle that you have some constructions which are more or less ad hoc and artificial, but you are very happy when you can give characteristic property, characteristic property, which ensures that the object is unique, up to a and better to a unique isomorphism. If you have two solutions of the problem, there is a unique isomorphism to be satisfied natural condition. So, it's important to know that this definition is purely categorical. but of course it has more good property except that it doesn't extend to an infinite number of models. The point is that embedded in this there are really two notions, quite an A-component, and some of the A-O's character has a product, some of the A-O character has a component. And what is a special feature in this element, I guess, is that the product and the co-product coincide.

35:00 But if you take a collection of modules, you have a natural notion of product, a natural notion of co-product, but they don't coincide with you. And that's exactly where I'm in the thinking of Golden Week came. Ironberg partly partly partly because they were financial and for people who are a little knowledgeable in logic everything can be expressed in first-order logic by the evil object and so on but relation and so but it can be expressed within first-order as presented in by DASC, the so-called dark distraction. And here, you don't have a point in that we don't have an implicit notion of sense. Well, we have a collection of two engegs or three engegs in the category, but it doesn't mean that we know what is a set of engegs in the category, what is a set of engegs in the category. Well, as long as the point, that, of course, when you define it as in a way, you say, you have some collection of options, and they don't form a set, just a collection of them, that can be expressed easily in terms of your thoughts or not. But we speak of the set of all the four fields, of y, of red, with the other one. And I think it's a branch to some extent. It's a branch, and if you think of Yolanda's lemma, Yolanda's lemma tells you how to present, to associate to an object, a phanto, one-to-one-one, between object and representable factors. But, when you say that you represent your human category as a category of something into the size, and you assume that the connection of homomorphism from one object from one is a sensor.

37:30 And of course, you could bypass this. You could be both axiomatic and say, are two different notions, the notion of a map and the notion of an object, and they are related by that, and you don't put the morphism of one of the different ones from a set. They are just a variable, a variable add from M to M, and you can quantify, but it doesn't require you to make them as a set, as a collection, which is important. Of course, all my talk is about the comparison of getting rid of themselves, as I've said in a minute, in a time. So, there was another development, quite simultaneously. It started in the book of André Weil about topological groups. In the book of André Weil about topological groups, it created systems of locally compact books because it is there that the r measure functions as well i spent 15 years of my life working on the local functional litigation which was referred to before and uh i've tried with to make sense of general ideas of fireman about what is to integrate in infinite dimensional of space, of course, it's quite developed, probability theory, I mean, it provides integration theory by our space and so on, that's a different thing that we want, and so, what I want to I would say that we would like to have, yes, functional analysis. So, functional analysis indication because it refers me back to locally compact books. And in this book, of course, there is mention of the progress made at that time. The book was appeared in 1940. At that time, about the fifth Iberfis problem, which was already retranslated, was by von

40:00 Neumann, I guess, as looking for the precise structure of all local incompatibles. And that was the way it was solved later on by Montaglietziby. In the book of Andre Vey, he refers to development in the 30s, especially to von Neumann. You know, Neumann was able to show that he studies the structure of compact good and, in our language, to show that any compact topological good is an inverse limit of compact limits. That's the way, quite more or less the way it was formulated by von Neumann, and as a matter of fact, it's a very easy consequence of the Petrovirium, but I don't think Petrovirium took us this, because it belonged to von Neumann to make this happen. So, if we have a natural, we want to discuss the class of all compact topological components, continuously. And then we have a compact input, and compact input, according also to the apply, to be face-to-face, and he can't not be face-to-face represented as matrix. which are quite explicit object, and we take an inverse and we get to post-general compact code . And then we get the force to extend this, and in the development, and this by the way itself, and similarly to the community, and in this case, There are direct limit of inversely means, direct limit of inversely means. And in the Book of Unleaven, there is, in small things, a general description of what is the direct and inversely means of course. Unrevelled in my own application to number of urine. of course, already Koula introduced the topology on the Galois-Goultre, in an automatic sense of non-Hinai degree, and then, again, it can be reformulated by saying that Galois-Goultre

42:30 is inversely with a pro-objective, a particular finite. So, Andreje was putting great emphasis on this, and he, well, he has signed the discovery of the inverse by the lead to Airborne, and Airborne, I think, was motivated by this work in logic, especially the compactness film. The compactness film of Airborne is a film about some inverse image. But he says that Hermann used only this in the case of the seasons. Hermann defined the notion of a direct image of the seasons of the book, and also maybe some authors, but not a generally text. And I read, I think it's one of the contributions of our daily soldiers of Doctrine and Remar. I was reading, because I read the book of Andrei Weil, around 49, until 49. And two years ago I was already acquainted with Andrei Weil and I was husband. He was never my Jesus advisor, he was always a Godfather, Godfather. And then I ask people to advise at some point, and I say, why not work on the fist problem? Garbage. Garbage. It's not interesting. What is interesting has already been a shift. And the rest is what I think it was like, because the outcome of the fist problem is very reduced. Except that Gromov at some point uses the solution of Munger-Pincipi, but it's one of the rare occasions when the work of Munger-Pincipi now gives way too further development. So, but then this idea of inverse and direcum, it wasn't very, very much in the mind of Andrebeck, which looks a little strange because, I mean, it's not the kind of mathematics that was to accept gamma. Okay, but, so, and I know the point, I know the point where the inverse and direct limit were used, there are two different, there were two lines of development leading to direct limit, not direct limit.

45:00 I suppose that, if I remember what was happening at the time, direct limit were considered as a natural object. We had to study that. Interest limit were more or less obtained by a formal analogy. Inveraging the arrows in the category was already a familiar idea. I even better push very much of that. And so, once you have the definition of it, it's immediately universal, you have the definition of universal. But it's true that in the beginning, there were many more examples of it, except that's what I mentioned, compact. But, there were two instances where, first of all, a functional analysis study. And there is a famous paper by Schwarz and Eudony and Schwarz, which is about the so-called F and LF spaces. F means fresh space. And Voltenig very soon discovers that the fresh spaces are inverse limit of monarch spaces. If you read his thesis, it's one of the things that he wrote. So that was a first natural occurrence, a second natural occurrence of inverse limit. About the planet limit, in his work on distribution, Laurent Schwarz introduced perhaps a new idea in functional analysis. First of all, the idea of duality was not a new idea. Banach already exploited it, and von Neumann and Giordani had extended the motion of dual from Banach spaces to a larger class of so-called locally convex spaces. And so, the idea of the reality was that in order to define the distribution, you have to take the dual from perhaps an unusual space. The space of smooth function is compact support. And in the beginning, Schwarz did not consider the space of functional incompatible as a topological space, but rather as an in-object space, in-object. And of course at the time people were not happy about his will, now with a better perspective

47:30 because of the definition. He has a direct system, you take the fraction, both in a given compact set, and you inflate the compact set, you get a direct system of spaces, of topological vector spaces. And then, by duality, the dual-other in-object is a poor object, which is not true. But, but, the Jordanian, especially, was unhappy with something, and in the first painting of the book of Gofferts, of the distribution, he speaks of the pseudo-topology, one is slightly unhappy about the pseudo-topology, and did our form for pseudo-topological space. And then at the end it was, I suppose it was Giordani who put, at the time, Schwarz said, Giordani were quite easy, that's it. And then Giordani said, it's not, it's not passion to do this way, we should do things in the Borax way. And Eudoné introduced the idea of a direct limit of topologies, which was brand new all the time, brand new. And while maybe in the most general case, but at least in the case of linear topology called Spaces, which was the case of Perkins. And it shows that it has all the right property, but was the first instance of an existence of a direct unit in a not so obvious situation, in very such a white situation, the notion was obvious. But, in this case, it was not obvious, and that's nice, one ought to be careful, I mean, the proper category is not a category of topology for the extra space, but of locally under the extra space. And so only we must know that the notion of directivity or inversely could depend on the category we consider. The same object can be considered as object of different categories, and the notion of It depends, not only on the class of objects, on the maps. So that does it. And if you read the thesis of Gottel Dick, in the light of his further development, you will see that it is full of category variety. One of the basic ideas is that any space is an inverse,

50:00 And when you do analyze, the first limit becomes time-limit and so as vice versa. So the product was already very familiar, and especially if you read his last paper in instructional analysis, which was published in Brazil, Sumapaziliensis, in 1953, if I the idea is that you start with Banner species and they can be sent by the vessel directly to anything, getting away from their thinking. And he ends up at the end of the paper, he said, now I've defined 14 different kinds of textuals. These are just one which are more natural than the other, but I could define two of them. And he says that this is under 14 months of untangled escape to consider. And then, he states the main theorem of his being. Factor being is a theorem of the factoring. It's completely critical. The definition is factored, which is an outlet. In a sense, it's derived factored. It's a derived factor, left and right derived factors, so a kind of derived factor. And, of course, the idea of projective and objective is already very visible in this visual setting of functional analysis, but projective and objective. And one of the great discovery of, well, of the great research in the thesis, is this notion of nuclear space. And nuclear spaces are spaces which, in the realm of constant analogies, are very close to final-establishing spaces. And then for this category, you can play an inverse and there are many exactness research about the tens of water. Such a category where you can define the tens of water, which is both land and height. So, and of course this idea of Nougat's Basics was taken with straight advantage by the school of Gertrude, the school of Gertrude back in the West, the school of Wiedemann, I don't know, at this time. So, the heritage of children, the ecstasis, is more of a certain form than one.

52:30 But, let's go back to that point. And, so, Gottlieb was familiar with his idea of diversity and diagogy. At the time, Wobacky was revising, there have been many revising, many times revised the first volume of topology. follow various editions and more and more when you see when you look at the various edition of uh you will see more and more existence on the inverse and directly and then when after some revisions the algebra books first series of not committed but about that then each chapter was concluded by a special section inverse and direct unit. Why? Because in between we were familiar with the idea of sheaves and in a sheave there was a long debate in the beginning of the Council Seminar. There was a long debate and there were two completing definitions for a sheave, but we But then it was realized by Carter, due to his work on Cousin problem, and when he worked with complex analytic function, it's delicate to define what you mean by complex analytic function. On a closed set, on an open set, it has a power series, it's much more important, but on a closed set, it's a little more complicated. I mean, anyone wants to teach elementary complex In one able knows this model. Of course, you have to consider it, but what is the function of the normal figure to close this one, but it's constricted. So, I was very familiar from this world cycle, complex and able that the true notion, the real true notion has to be associated to every open sense of something. But then, at the time, the difference between sheaf and brief sheaf was clear, and as a reason, there was some confusion in the book, in the catcher scene, some confusion in the talks, in the catcher scene, and then Michel Lazard, who was the example of dumb cheaters.

55:00 He came with a very precise and very clever mind, and he came with a definition which you, not a section of the object, but a stone. So in organizing, it was his idea to organize, but the idea of a stone was already there, but to organize a collection of stone as a space, another space, which is eternal, And for a while the two definitions competed, and Gator was not completely committed to one or the other definitions. But when Ser wrote his paper about algebraic brain and cheese, there was no more question. There was no more question because it's clear that, well, hey, Stoke exists, local thing exists, but it was clear that the best thing that you could do is to represent the general world's algebraic variety as governed by a five varieties. And such, his idea was already produced in the book on the foundation of Hathya-Pakshamati, but with the development of, say, to the elite. Do you mean the so-called Gaga paper? No, no, fact, fact. Okay. So, there was no doubt that Shakespeare was important. And then, in the direct limit occurs repeatedly as strokes of chips. In analysis people are familiar with the idea of the germ of a country, germ of a society. I think it was everyone who put this idea forward. So all these ideas were, the idea of direct limit was very familiar and thought it was important. it may be nice but still. If you want to be really coyote from the point of your category, you want to consider more such a thing. So, well, if you follow the various editions of Gorgaki, you will see that there is no remote existence on these direct and inversely.

57:30 But, at the same time, I mentioned that I learned there, in his first lectures in the Carter Seminar, November 1950, developed, axiomatically, the notion of homologico-perfugur, And he took it when this was the factoring speech. And Carton had, well, first Muray, then Carton had developed the notion of common people ships, but the original idea came from it. It was pursued by Carton, and Carton was the one to, you know, first to realize that his His work on many complex variables, by him being a roca, was much better understood in the free world of ships, and that's why Carton began to put so much emphasis on ships. But on the other hand, you need a commodity of things. And unfortunately, if you follow the exposition on the Garton seminar, it's rather appropriate. You have to put restrictions on your surveys by complexity or whatever. And your tools, they adopt tools and it's rather complicated. and we began to think again about this, so that we should have natural definition of of sheep. And of course, the model was there, the description, I'm so much in description given by I would have told the common homology or common rule. And Moten, he began to think about that. And I think it was his main motivation for the doorway. In between, I mean, most of people, which I remember, middle of the shooting companies, and Moten did, which was quite When he visited Kansas in 56, there is a technical note, he had an AFLOR contact, I suppose he did not understand the act of the Air Force.

1:00:00 So, at the end of his day, in Kansas, he was supposed to write a report about his research and he wrote a so-called technical report, which is about a hundred pages long, rather sure, right? On the other page. And where he developed this idea of a fiber space, not with a structure of group, but And it's an important development also in this direction. What do you consider the following? If you want to define the type of principle, of course, standard definition is via the cosycle, I mean, you want colouring, and you want the matching functions, and the satisfies of the law, you call it now the one cosycle, So what you have is that you have a space covered by open sets, once you have ui, uj, and then you have this is uij, the intersection, and what you have is that on uij, you have a certain function phiij, which value from Uij to G, a certain group, has so much interest, some blue incondition. The point is that you have, in order to describe your fiber space, your principal fiber space, you need not only open covering of space-based, but also you need this and this factor. But when I write this down, it will be witchcraft. Witchcraft. And there were familiar examples where you can consider that I think continuous, or real-analytic, or differentiable, or complex analytic. And the work of Spencer at the time, I mean, Codin and Spencer at the time, put great emphasis on comparisons. why do you have co-cycle which comes in, you have to find a function, etc. So, this, already in the work of the ancestor of Kodaida and Spencer, it was obvious that we should have to consider, in order to define a principle of fiber space, you should be able to see in which category you have these maps.

1:02:30 And so the right notion is not a notion of a structural group, or a principle but a structural shift. And you should consider, and you should consider the common legal, as that's the way he invents the non-commitative common logic. And each one which was perceived by other people. Unfortunately, after many years, we don't know yet how to define, in general, a third order non-committative camaraderie book, second order, to some extent, third order, and it's just out of reach. So, but this was the turning point of Wotenly because it was the inspiration. But my point is that it was not really a job. But what Wotenly did at that time was an exact combination of what he had done before and what was the main emphasis on the carton. So, the problem was, we needed, of course, we needed chromology theory for shift, many instances. And this, before this, the Cousin problem, the Cousin problem, as you say, was a complex number additive or multiplicative. The Cousin problem, and it was very early recognized, both by Cantor and by André Vey, that it is a problem in chromology. There is also an un-published paper of Andre Vey, his lecture notes in Chicago in 1948 about a pilot space in algebraic geometry. The pieces that have not been published in the library of, or I suppose in Chicago, but in the library of Institute for Advanced Study, you can see them. But the pieces have not yet been published. means they don't know okay so and so so you see all the time all the threat all the threat as contrary to what most people expect andreway should be counted as one of the founders founding

1:05:00 father of categorical and common if you will despite the party but what he liked those and When you mention a category in front of him, bah, bah, bah, but it's part of his personality. People who knew him well knew that he was very friendly, but he made a point to be nasty, at least in the public group. Okay, so let's, so all these things get together, and I want to try to disentangle if it's possible with this thing, but what is the contribution of God. The contribution of God's name used is about a technique, generalized, generalized, generalized, and we will have a discussion about that. generalize. And then we say, well, it's a pity to achieve former category very similar to the category of modules, achieve for groups. And chromology axiom are very similar to the chromology of an axiom or chromology of a group. It should be a unified exposition. But now, what is the way to define chromology groups, let's say, The idea of resolution was well known, but of course, the natural idea of resolution which comes is projective resolution, because a free module is always available from the very beginning. A projective module is just a slight extension of a free module. In fact, the iron factor of a free module, and you can do everything with free modules, instead of projective model, but I don't know what's very new, such as characters in property of free module, what makes them projective, the category of definition of projective model. And of course, in my work of the world, I don't know, I don't know what's very systematically, I mean, use duality very systematically, he said if I projective model, it's an injective model. But at first, we were considering this of this thing. I mean, in order to prove that any module can be measured in the injecting module. It is tricky. It is tricky. Injecting module at the beginning

1:07:30 were considered more as ad hoc and just for the symmetry purpose but it would not be interesting. Except that the alternative lies that in the case of shield, no projective resolution that you have You should have injection resolution. And it was even more pressing after the paper by Serre, actually by sheaths, going on sheaths. Because at two occasions, Serre had to screw the definition of chronology to the most extreme. In his thesis, in his thesis where he extends Lorraine's sequence of mapping of a physical space to another physical space, to the case of loop spaces and path spaces. And then he had to, a major part of his thesis is purely done and technical, he had to take all the results of Lorraine to each present in a slightly different form or different kind of work. topological space, a different kind of homology, and that's why he invented the Kubikov-Philomelium. And then later, the cells were consequential of path, which was, of course, the main tool in this thesis devoted to what other people would say. So, once, only we had to say that to transgress natural border of homology in the field. But in this paper on the antibiotic shift, we had to do it again. Because it was the first time. Zeisky topology was a natural thing to do. And that Zeisky topology came into being with a lot of resistance. There is a story, I don't guarantee, that sometime in 1942 or 43, Zajnski gave a talk about the dissingularization of surfaces, and in his paper is published, and look at the collected bit of scarcity, and at one point there is an important compactness argument. Compactness are given in exactly the same sense as airborne with compactness, while compactness

1:10:00 are given in airborne and logic by airborne, just so that if you are finally approved of something, it will use only finitely many axioms or finitely many instances of the axiom. and that's a little more or less a spirit and something similar something similar occurs in other situations so there So, they wanted to take, so we had to use this area center, that is Zeitzky topology. So, and Zeitzky topology can be used as an angle of compactness. We know that, in some sense, Zeitzky topology is compact. You know, as an angle of sense, because it's not out of space. And so the story was that, to finish this group, Zeiss King used a compactness argument. And then he said, well, now I have to go back to the complex field. So far, my reasoning was about an arbitrary field of galaxies. You know, at this time, I had to use a group of physical arguments. And it seemed that Chauvalier was included. And that he said, well, why not choose, in the preliminary way, that's a compact topology. And that he defined, on the spot, he defined the Zeiss-Keyt topology. Well, I would not check it here, I think it would be a lot of difficult. Where would this lecture have been? Where? In Princeton. In Princeton. At the time, Chevalier was a visitor in Princeton. First in the Institute of Arts and then in the University. Then he moved to Columbia. I suppose it was in Princeton. Okay. And, okay, but then, the idea is that this was Aesky topology, but there was a Aesky topology was invented so this is true uh to have compactness up and in a little developed by zayat itself in good case emphasis on the compactness but it's nice it was not exactly the one you but it's similar as compactness so part of the society what it was as a record and i think

1:12:30 If the book by O'Grebe of the foundation of algebraic geometry is so complicated and so boring, it's because it really has the idea of anxiety, but it doesn't want to introduce it explicitly. So, he defines a manifold as we do in differential geometry by covering my charts and doing charts, etc. the fact he doesn't want to introduce the topology it may have been influenced by the way the method of analytic number three of course as usual if you find a new method it's good but if you are obsessed and he was always, always very obsessed by pure, very pure method. Okay, so, the scientific topology was, for the first time, he introduced, I think, used in this modern sense by Andre Weil and his 14 people, But there was some resistance, of course there too, really five years later took it and developed that. But I remember that Schumann was very relevant, and I participated to the Schumann seminar in 1956, and now the proceedings I've been published by myself recently about algebraic books. But then you see that there are many places where you could use this advantage, but it doesn't. Fortunately, he is also one of the authors of this literature note, Amin has no reservation. And Bohel was half and half, half and half. Though I knew very well, he knew the theory of Liberti, compact Liberti, in a fantastic way, and so he knew all the topological arguments. And if he could mimic a non-topological argument, he would do it. He would do it. But, it was just a trick. And, uh, but Gaudenig was the first to take it seriously. But, the price to be paid is that this, as I said,

1:15:00 it was so exciting that everything that we have been able to build about Sheafield and Cobalt is completely out. And Gaudenig solved the problem going to the top of the roof. And, so, I said, okay, obviously the properties of the category of Schieff are such that we can expect to be injective and not projective Schieff, who can forget about projective Schieff and use injective resolution, which in the case of Bardi, we are considering some reservation because projective resolution is easier to deal with. Okay, so he deal with the injective resolution. a sheath category, if you have a complicated category, how do we prove that there exist injective sheaths? Not always. More lady before module is quite tricky. She said, what? I will not give an explicit construction. I will set a collection of axiom out of which, for general category, you can prove existence of injective sheaths. And that's this famous EP5, or EP5 star axiom. And then, when you have a sheet, you know, you can put an exam, now she is a high injective and you can calculate your comology using the injective approach of sheet. And that's all, that's finished, you have nothing more to say. And the theory of homology of sheet per sheet is that you just use a general label of the Yes, and now, of course, that was a time where Goldman published his first textbook on chip. And if you read, it's clear that most of the book was written by Goldman before he used this paper by Gordon Dick. And at the end, one tried to make the comparison between the two and then, he was the one who produced an explicit elementary construction of injective sheets. And some people tend to remember only the construction of Goldman. Well, in the book of Thomas there are many other things and many interesting things, including an appendix where for the first argument the exact rules about the monads and I explained. It's the first people, but I might not know it's one of the first people to look at the monads.

1:17:30 Ok, so, but, let's go back to the table, right, the topical table, and the topical table, so Grottenich took these two lines of development, for topological space, for global, local, compact, or topological vector space, we have a motion of this interactive, in that way, they are very arbitrarily settled. of arbitrary cardinality. Okay, it's this method. And that's what it does, and so we will consider infinitarity, and so on, limits, and so on. And, of course, once you've used the same things about it, all these constructions Now, if you have two categories, C and E, you can have the categories of C or C or whatever. That means the second of all factors, from one category to another one, are the object new category whose morphism are the natural transformations, so we are back to the original paper by Adam Perry MacLean. But the point is that what is the fact of, and I started my discussion about it, it should assign to every object, let's say to any topological space, a compact perm. And, of course, even if it's easy to describe within the framework of self-fueling the paradigm of rebirth, what is a group, as said, group G, if a subset of G is a subset of G, that is a type G, which express a multiplication, satisfies maximum, it's much more complicated to define a cumulogical term, not a given rule, but a notion of, on purpose I don't

1:20:00 speak a category, a notion of, or like you call the species of, the species of triage. Because it's not, you cannot phrase it in two elementary terms, I mean, for each set g you can't speak of the totality of the gopros on g but the point is that how to divide by from a practical problem you have no doubt that you write the axiom of the action of the group and everyone understands that if you have a little more sophisticated philosophical problem what are you doing at which level of upside are you working here it's really clear that you are working not at the level of unfunctional set but at a higher level in the 40 50 means that you are leaving mathematics towards meta mathematics and that's why at last go back in the 50s try to give a general description of what it means by structure to be miserable. First of all, already at the time, in a sense, psych-cell theory is just an expansion of the analogous program, with emphasis on isomorphism. Automorphism, isomorphism. So, natural extensional group, they are always good. Pants So there was There was a notion of movement. That means that what is crucial is a notion of light of movement. What we want to do to have that. And I think it's Erosman who really described carefully what it is. When you define the notion of movement, while you are in effect considering what you want to use,

1:22:30 group of structures took in the application to combinatories it's a factor from the category of set with isomorphies to another category or maybe to the same category to any set the associated class of all possible co-structure meet and to any isomorphic set into another one to any by injection by injection the associated transport if you have g maps one to one to h which is amazing. So, when you want to define carefully what is the structure, not the structure on one specific set, but the general notion of the group, the general notion of the house of age, etc., you really deal with the factor. And this factor, this factor, as I said, belong to a higher level of abstraction, that's my point from the beginning, a higher level of abstraction than the notion of groupings, the groups themselves versus individual groups. So and that's why Borbaki tried to make a formal description, but since he wasn't very happy about it. So first of all, he knew, Borbaki knew that you had, not only isomorphic, but also morphine, genomorphic. And second, he knew that we had to go to a higher level of abstraction, but he was very because he had to formulate it in terms of meta-mathematical terms which were completely at board. And I remember some discussion with the Bobaki whether we should include a fifth chapter on the strike on the theory of sets to discard category. And we wanted to fit with the logical standard of the work, and I was assigned the duty to by the preliminary draft. Later on, it was taken by Samuel and by Chauvinet, but by Joseph who tried to comply with this framework, and so I discovered in mathematical terms of what the category should be for the conflict abroad. Of course, it could not go through. Of course, history is at this point that Bovaki, I mean, and Bovaki wanted to develop categories because, of course, all the founding fathers of Bovaki were members of Bovaki at the time. I remember there was one, Maclean was, to some example, not so chic, I would say, and Cato was there,

1:25:00 Sam was there, Courtney was there, and all the people who developed homology theory and category theory were. I'm a very active member of Bovaki. So everyone knew that Bovaki getting the wealth, but because we took ourselves in a very strict ideology of stride, publishing stride, if we had not done that we would have to start with all over the project again. We gave up, it's called, we gave up. So, now, are modern day, of course, and no reservation introducing these methods, these infinitely methods in category. But as I say, if you want to be consistent, you have to discuss it at a meta-mathematical level, or you have to admit that mathematics is part of mathematics, but just a higher level of abstraction, and of course you can think of two categories, But we had to solve the level of abstraction. But as soon as we did that, we made it with the standard security paradoxes. You had 14 set of forces, category of four categories. Here we begin to consider the category of category, et cetera, et cetera. And if you want to define the set of fundamentals, you have to go to this higher level. And the point is that it's well known, it's well known to Russell in this type theory, that the reflex of the principle makes logical difficulty. Because if you, if you, if you, if you describe, if you describe logic as a certain level of abstraction, by referring to, to, to, to, to, to, so-called non-predicativity logic, which is a way of controlling ideology. So, that was, yeah, I think, I'm not a, a fanatic, not in chronological terms. So he went, he went through, he went through, and developed this method. But, of course, if you assume this covers it, to speak of the category of the country is difficult. Well, of course, you can concentrate, see real-time, see in the two zones, see more categories, if it makes sense, but of course, it's a realistic

1:27:30 category. You can say you're not good, you can say you're not good. This is more like a view from point of view. So, and you all know the pitfalls, for example, some people repeatedly approved that every factor of the joint by going to so infinite limits and so on. So, as Vedamu pointed out, that there are more subtle contradiction of paradoxes connected with the so-called Richard paradoxes, which is a set of definable number, a definable object, that's what we're known formally in the narration in the regression field. So, Ben-Abu pointed out that when you say, well, I define KZO, the KZO loop of the space, as the set of equivalence classes of sheaves or vector-binder or some kind of class. It doesn't really make sense. You can easily refer to a very beautiful paper by Ben-Abu in the Journal of Symbolic Logic, Well, it turns out that because of the exact image we shall paradox, I mean, if you use freely this notion of isomorphic gas, of course, there are model-like problems everywhere, as we know in geometry, but you have to be careful. So that brings me to my last point, and this is a tentative conclusion. What is the present sequence? If we are just interested in getting one, if you are using very easy. Well, you can either set a small category and assume that every person in a small category is a large category, lack of faith, or you can do that. But you can also formulate it, as before, in terms of first of the logic. It's very easy. You have primitive notion, primitive notion, primitive relation, composition, etc, etc. But cell theory is easy, it's axiomatized in an accepted way, in a certain way, with or without choice, axiomotricism.

1:30:00 The two are quite reasonable, despite the reservation of the zebra, but when you put them together, it's a plan for the reason I've experienced. Of course, there have been various cues tried. For instance, what did he himself introduce the idea of a universe, a universal sense, which was taken seriously in the book of the Masur et Gabriel, an algebraic book, which is not so pleasant, I would say. They mimic the method of André Weig, who introduced a so-called universal domain in algebraic geometry, a big field, and every field should be a subset of a given, very, very big field. These methods are not very satisfying because of the reflection principle that I want to speak in the philosophical term. And the reflection principle forces you, once you have a universal domain, to consider it as part of a larger universal domain, and to go maybe one, two, So this is, these universes are not very convenient, and also, they are not very pleasant, also, they lead you to very complicated problems, except for the existence of an accessible cartoon, all these problems, these problems are difficult problems, not interesting per se, but totally relevant, etc., totally relevant. So, we are just introducing arbitrary difficulties in the field, which are already many difficult points. So, what can we do? My advice is to think of what people did in the 18th century. In the 18th century, people spoke freely or freely disillusioned. And one never gave a question. Berkeley was happy to point out the contradictions, the goals of the market, the quantities. And if it is a value, sometimes zero, sometimes zero, divide zero, by zero, et cetera. Well, I would speak about a solution offered by . It is just one solution, which I don't recommend pedagogically.

1:32:30 But, so this is infinitesimal. But, if you read, if you read carefully, let's see the textbook of Euler, which is much more important than you read, provided you accept that infinitesimal exists, that they obey a certain rule, and that, once you admit that there are more or less implicit rules not to do something, The main difficulty is what is forbidden. What is forbidden is clear. What is forbidden is less clear. But if you accept that what a man is at a given time is not completely formalized, that a man is at a borderline, borderline, and which cannot at the moment be put in a completely formal system. It's like doing seashore. I've been trimming at this case of the seashore. I know that there are many dangerous spots and that it's better to see the price of someone who lives there and to think where is the dangerous spot, where is the stream and so on. So in mathematics, we have something similar. There are dangerous spots, why not? And provided we are still a little away from that, we are safe. But I really like to use an allegory, which was for him. He was living in Chicago in the 50s. And he said, at that time, Chicago was unsafe. Maybe more, not more than today, I don't know. but at least even the downtown was on sale. And he said, well, if you have to leave your car for the whole night, better choose a well-lead car release, and put your car just below the big polar with the big lights. If you put your car in the dark corner, you get a chance. And Hermann Weyer expressed similar rules, so in mathematics you see axiomatic is this huge pole which puts a good light over a certain area. If you move within the axiomatic realm, this is a part of mathematics which well understood, but there are many difficult things to do. but at least you are in a safe area but if you go to the

1:35:00 border, you take a chance but before, it is what is interesting Thank you very much for such a wonderful talk Any questions? You were saying that He thought that the category of Jews shouldn't be expected to have projectors. Can you tell me more about how he would have thought that at that time? What would he do? At the time there went more of this projectors. I think development started from Czech, I've shown from the Yeah, well, and that you will see in the seminar, oh, I know. It's really common, and the experience of cobalt is nothing, so I project it, but I said we don't do anything, and I'm going to so-called talk, I'm going to do anything, I'm going and so on, and so on, and so on, but it's, it's a thing, a place in the general, if you are common, but there is a problem. one of the things which is also a it seems to me when you saw it over the paper he just kicked himself and said this is bearish proof for modules you just restated it and it should have been obvious to all of us it wasn't but in hindsight he thought this is just bearish proof I know I think you know probably some of that but people have not told it That was the question. I write what I call the plasticity of proofs. There is a general principle of plasticity.

1:37:30 A good proof is plastic. That means it can be modified. And that's why in a given question it's better to have two proofs or three proofs. Because the plasticity of one proof is not the same as the plasticity. is also good and then use plasticity to extend into certain new situations the second group work in different situations now i mean the now the point is that i agree the method is exactly but no one was all in that the good system, the good most general are in a category well in a category where invented not only by Gottlieb but also in fact and which was that's where Well, this is why he hit himself so hard, because he was a little bit to see, but in my sight, he said, I should have, everybody should have seen that this would work in my sight, too. No, but Kotlin was here. And I think Kotlin was, maybe, among the first ones who really considered the category of full function, and you know, and the general notion of the limit with respect to diagram and so on, and to express everything, please, once you have this, so, the beginning of two categories, the two categories, the four categories. and what they need to say. Excuse me. Godemann simplified proof of the existence of objective, not the objective resolution. That's the plan. Is that true? Somehow I'll go back to the case of Modules. Is there an analogous construction to Modules? Yeah, I know, I know. Change of, change of reason. Oh, yes, with the Harvard C and all that. Now, there is one proof for injective margin. You first do it for the injectors. Then you know it's implicitly injecting Q and Q. Q over C, yeah, yeah. And then it's very easy to manufacture an injective resolution. And then you make a basal change. And it's exactly... Goldemar's trick is to compare a given space with a space-based discrete, if you think about it.

1:40:00 So you have the space X and the space X-beta, which is a space-based discrete thing. And you use the change base, the change between the two. So a sheet over X and a sheet over X-beta, which is a collection of stocks. And so you have the sheet, which is glued together, and a collection of stocks. two little different categories with various factors and natural factors between that and that's exactly the meaning of the construction of so and then that reduces the question of objective shift to objective models that's okay and of course it is inside all these I don't know, we're alone in particular, but it requires someone as bold as the woman needs to extend that to the works in the arts. I can't tell you, I can't give you a second. Before coming here, I had a conversation with myself, and I said, while I'm speaking about and talking to people, because I'm so important, I said. Does it deserve you to go to the office? I don't think about it. We knew, we knew already, now, it's clear, the idea of, let's say, to combine them, let's say, for example, to have, before Walter Miggs approach, the science, not only that we have, the general description by injective evolution, but it is applied to the science, And in sales people, they have to give this check-comology, I'm not thinking about check-comology, thank you. I use check-comology very often, and that's a very interesting role, but it's good to know that this also can be a change, the general property can be a change. Actually, you can say, let's speak, Lord, about what you like to do. You have some people that ask about the exactness of what you're doing. We have some exactness of what you're doing for each other. Thank you. Okay. Something that's mystified me for a long time, I mean, being at a distance from this, is Grofnick's attitude to physics. Could you comment on that? I mean, because it seems like there's an enormous resonance there

1:42:30 between the problem of the observer and quantum mechanics and all these ideas about tokoy and so on. Okay. What? That's a complicated thing. First of all, at the end of his stay at IHS 69 or so, I have testimony from my physics colleague I mean, he tried to understand, I mean, he asked them to include basic physics, Gugliel especially. He asked Gugliel about 2020, and Gugliel is, of course, one of the mathematical physicists who is very sure, I mean, he is a very formalistic mathematician, who are these books are beautiful, but very formal, very formal, and he found one, his notion of power, I mean, he was one people to contribute to the understanding of cow, but there was a reason it wasn't for. And Greg told me, recently, that at some point, since John Cicillia was asking him various questions. But, at the same time, he was also asking questions about biology, because that was a time where Tom was his colleague, started to put emphasis on biology, and now, of course, many, many years later, it's one of our main scientific embassies in my institute, by HGS, to do mathematical biology. We are fondly more than six But there was always something in the mind of Rothenberg, an equation, physics equal Hiroshima, which is true, he was in price, in price Hiroshima, and for him, according So to him, physics was the work of the deputy, and he is now, for many years he has been obsessed with the deputy. And he believes in the material benefits of the deputy. So, he has been, so, for him, physics was the work of the deputy, with the outcome of nuclear weapons and so on. for mathematics, the whole Achimedes, Nagus-Megis as well, so his default to work for the defense

1:45:00 of his time, of his time series, and Leonardo Di Vinci was not only a painter, but also a military engineer, not to mention Moritz. So, of course, we don't have to go to a discussion about this point, a political discussion, that to all the people, I trust this kind in the mind of Gottlieb. Gottlieb was observed by the idea that physics is one of the physics. And all the more, because at the time, NATO was putting plenty of food to subsidize, to subsidize the same people, both the EU, and one, I mean, tell you about all our political fights at the time, I said, well, many people participated in this political fight at the time, and both of all, I think I was totally under the, totally put out of myself. At the other hand, I heard, at the second hand, that at some point he was obsessed with, again, he said in some letter, which I have not seen, but he transported to me, But he said, well, look, the deputy is at work everywhere. For example, when I was a schoolboy, they told me that the speed of light is 300,000 km per second. Quite a beautiful number, 300,000. It's the speed of God, etc. You see this label, I could trust him, he could show me, but what about many bytes per year? How many bytes per year, et cetera. And so, to be it was really strange. You said mathematicians don't understand units. And the point is that someone who is so aware of the idea of intelligence principle should not understand what is in the physical description, what is just arbitrariness of practical system of units and what is fundamental.

1:47:30 And for the physicist, it's a dimension of analysis which develops this. But every physicist, trained physicist, knows how to use the dimension of analysis to distinguish what depends on which order you do it. And, as I said, if you say that it's almost all the one-dimension, there is a light bulb that looks at the length, that was a light bulb that looks at the time, and you are exactly a replica of the dimension of a disease. So, but, today it is something, I think it is, I can only come to the conclusion that the mind of the disease is in this respect, I can see more of that in this respect. And he said that, according to one who reported to me, would do my trust, no reason not to trust him, but he doesn't even understand me, that's why he, and so, of course, physics, I mean, I've been working, when I wrote my, I wrote my paper in the politics, I mean, my piece worked, just with the purpose of showing that physics and mathematics had something to do together, but, well, it would be difficult to say. to send you a liquid copy that you would just discard it or send it back to me. Isn't there an anecdote that I don't think it originated from you, but from somebody who was in contact with him about some phone call in which he said, you know, I have solved all the problems of physics or something like that, and I will tell you the solution if you tell me what a meter is. It's a variant. It's probably a variant of this. It's a variant. OK, thanks. Pierre, on this general question of the increasing recognition of the centrality of the structure of the functors from one category to another, how does the Tohoku paper relate to the developments going on at the same time in scheme theory? Was that slightly earlier? That was a lot earlier. So it was known already at that time,

1:50:00 and I think the idea that the category of schemes does not have a full and faithful country into the category of sex. Did you wish to do anything? I think the theory of schemes, the category of schemes. No, no, I mean, the local paper was published in 57 and it turned in 55. Right. And you just after that, you had to see the number of schemes. Oh, so the scheme of development was later? Oh, right, okay, that's what I know. No, it is. So he was not aware at the time that he wrote to Hoper that there is no full of faithful function from the category of schemes to sets that it's... Well, I mean, I myself, myself, given a formulation of schemes, as a faithful function to getting a year of 5,000 to set, well, in my point of view is that a scheme has pointed, which It was at that time when you all talk to people, they had not yet invested in algebraic geometry. The paper was written out of 55, and then they came back to Paris at that time, and there There was a Chevalier Seminar, there were two Chevalier Seminars, one on Gevalier Geometry, which was the foundation of Gevalier Geometry, and where the World's Key was invented. The first notion, the first time the World's Key was used by Chevalier in this seminar, to warm and pours more than the end by a central limit. And there were two seminars by Chevalier in the sequence. The first one is painted with the carton in our series. The second one was never printed and I published in some years. So at that time, it's quite a scheme. About the history, yes, of course. There is a paper which is from Cotton of Chevalier. And some time in 1949 to 1950, after the appearance of the textbook of Andre Vey about the foundation of algebraic geometry, then there is a paper by Chevalier who tried to formulate these things completely in their interest.

1:52:30 With the same restriction as on Rive, at irreducible variety which, over an arbitrary field, which remains irreducible, will extend to their tobacco, absolutely irreducible variety. Because there, people who are still instilled in the ideology of function cleats, cleats, not things, function cleats, and that's the place where it is natural. And the reason is that there was a problem, there was a problem of tensile product of two fields, extension over one field, which I think there was a force to solve in the Chevalier-Carton City, and especially that is it. And then, so, but Chevallet took the definition, the definition, that the idea was how to glue a fine variety was understood, and how to glue a fine variety. And now Chevallet took from Zeiss and his own world the idea that what is crucial in an algebraic variety, first of all the function thing, so restriction to inducible variety, But the collection of all low-perlings associated not only to the bond, but to the sufferings. Which was a very important tool in the dissingularization program of Zeiss. Why? Because the basic motion is blowing up. So you transform a blow-up or point into something of higher damage. So you have to put them side by side. When you do the blow-up, I get the injection. And Zeisky was well aware that really what is the structure of an algebraic variety is more contained in the collection. And so Chevalet translated rather a good definition of Reveille into this setup. He said, an algebraic variety, or rather the scheme of an algebraic variety, the scaffold, the scheme, it's not a variety, the scheme of an algebraic variety, right? What's the mean? The blueprint, if you want, the blueprint scheme is a certain function field and a collection of group elements. And now, to any, to describe a nafyan variety, you just have to take, within the fraction tree, a summary, which is finitely generated, and then, which is quotiently given tree,

1:55:00 for more depth. And then, to look at such a, as I say, generate automatic e-collection of the locally, all the locally obtained by local trees. And then, if you want to If you want to grow them, you consider the two aquariums as contained in the same fields, and then each one generates a collection of new goods. This strategy was well known as a protective case to Zeisky, but Zeisky always considered projective reality. And Réveille is also considered a non-projective reality, but as you know, it took some time to find an explicit example of a very idea project, a way which has no project. And then, so, but then, Chobali expanded this and took the idea from Christ, you see, it's a collection of wood. But it's not Réveille, it's a wood, it's a wood, it's a wood. Yes. And then I took literally food and step-by-step, I mean, in my thesis I introduced a certain way in this definition. You know, you couldn't produce the variant of that, until we came to the general definition of the script. But it's only 57 or 58, since it wasn't. Well, very short. Very short. I've never seen this before. I've never seen this before. If you try to read the history, the fact that the total paper was published on 1957, delayed, or purely editorial, instance. The report is that it was too long, and the editor complained that it was too long, and the paper was intended to be published in 1954, 1975, and purely editorial, this was published in 1957. Thank you very much. Thank you very much. I think we must have discussions here. We'll have two-hour discussions. Thank you very much. Thank you very much.

1:57:30 So is there so much interaction with the Chevrolet, I think? Well, except for... Why did Chevrolet resist to the risk of technology as Pierre told us? Well, I think that there is a fine chronology that she was in astronomy. Chevrolet was... It's been significantly related. But I think it's actually... There is an answer question about seniors. It's because... The first case of reading. It was the first experiment chapter where there was no four in fact which one would use an accurate sense. Okay. So therefore it really was. I shouldn't say that. I'm getting to the back back. John, the geometrical framework. John, the double. The colonizer is very much more. Okay, don't go back. And if you're looking at the pulling back, that is a very broad brush. Yeah, yeah. Yeah, yeah, yeah. Thank you. Thank you very much for the debate. Sure, sure. I wanted to ask him... There's a general point about government management here, because it's a very kind of philosophical. Yes, this is very important. Thank you. It's already done.

2:00:00 Thank you. Thank you. Thank you. Thank you. Thank you.

2:02:30 Thank you.