Le Foncteur Sonne Toujours Deux Fois (contd.) / Grothendieck's Notion of Descent

0:00 Pernod Gustave n'est pas mort, car il vit encore. Feel free to ask questions in English, if it's better for you. Thank you. I think it was isolated. No, no, no, he did, he did, he did, he did, he did a big article, which is the course he did at São Paulo, on the subject of the lecture topology, which was recently recently. And then there are several articles he did there, which are published in the Sumabas I-M6, the book, the mathematics of Brazil. But I don't think that, in the way, there were several steps. It was after 1945 that there were several French. The Dieudonné, for example, had quit. The Dieudonné had been left for two years in Brazil, the Veil, the Nian, and there were also others, because there was Feynman who was at the same time, and Cécile Vitt, who had been told several times. So there was a... and then it was developed with Limpa, by the way. But it was... I don't think that... I don't think that... I don't think that... No, it was a new Etat, it was a new Etat. It was a new Etat. After he was at Harvard, he was there who met him in the United States. Yes, but he had already done his tournance. The tournance date of 1955-1956. Remember that the tournance is first... ...and then he came to Paris. He participated in the seminar Chevalet on the group Algebraic.

2:30 And then he founded his own seminar of Algebraic Homology. And then there was what was done during the year 1957, it is to say the theorem of Riemann-Roch-Hertzmann, the version protonique of the theorem of Riemann-Roch-Hertzmann, where it is already, since 1955-1956, it is already launched at a point, it has already taken the tourment. It is launched at a point in the functions of the variables complex, in the algebra homology, in the variabilities algébriques. Son grand programme, it is 1958, at the international conference of Edelbourg, 1958, it is there that he announced the program of Schemas. But I don't think it's after. As I remember, the tournoi was already done. In fact, he was nourished in the city. Then he was extended. Like he said, Mumford, the miracle is that we were around Zareski, who knew at least the geometry, but whose methods were a little bit at the end. And then there were a lot of techniques that were full of imagination with new methods, but it might be a little bit of geometry. And the extraordinary event, it's that the junction has to be done, but in part because of the generosity of Zarecki. Grâce to the generosity of Zarecki, who has said to his students, there is a little girl there, go to school! it never it never came so oh to write the textbook on categories no no no the textbook on category there were various attempts at the time to write technical category and that was one and chauvalet gave many lectures i remember many lectures on that there were also mimeograph versions of lectures and there are available in our library and also at the but it never came through

5:00 Attends, Serres a été professeur à Nancy de 1952 à 1954, si mes souvenirs sont exact. C'est après le départ de l'Ottawa. C'était déjà... Ah non, ils s'étaient rencontrés, bien entendu. Non, Serres s'étaient rencontrés. Mais je veux dire, il n'y avait pas d'interaction. No, Serres, I mean, in the first seminar carton of 1948-1949, Serres was already there, Serres was already there, and Armand Boren, and Grotendick did their own knowledge. But, as I said, it does not have been approached. And Grotendick, apparemment, it was not really, he had in mind his ideas of integration, of analysis, it was what he had in mind. And when he arrived at Paris, depending on the questions of personal contact, it was not his sonatee, it was not his sonatee, it was his sonatee. And Serre was a professor at Nancy from 1952 to 1954, but Serre habitait Paris, which was what we called a turbo-profile at the time. And, of course, in Nancy there were not many who were coming, who lived in Paris and who lived in Nancy. Today it's even more common. But there was no interaction, no, no, no, no. I don't have to say that because there was no interaction between Serre and Grotonnik at this time. The interaction between Serre and Grotonnik, she started in 1954, 1955, back to Grotonnik-Apar. And I said, well, the first manifestation, it's the collaboration of Botanique at the Seminary Chevalet of 1956-1958. Exactly. Yes, exactly. No, no. No, no. No, no, no. No, no, no. No, no, no. No, no, no. No, no, no. No, no, no. No, no. No, no. No, no. s'est récupéré, je veux dire, il a mis ensemble ce qu'il savait en analyse avec ce que faisaient les gens de fonction d'un complexe, et à partir de là, il a essayé, bon, après, il a fallu qu'il invente ses propres métiers. OK. Avec votre permission, je pose la question en anglais, Pierre. OK, merci. C'est très gentil. The title of our colloquium is the mathematical posterity of

7:30 I have the strong impression, and it may be quite mistaken, and you'll correct me if I'm wrong, that whereas the posterity of his work in algebraic geometries universally recognized, and its impact was almost immediate, the impact of this great work in functional analysis in particular's theory of nuclear spaces has been very less widely recognized, and particularly hasn't had anything like the impact on the theory of founded operators that one might have expected it to have. would you agree and if so how do you counsel it the people who took it very seriously whether people are home get home yeah but not enough not here not in the West I mean his influence has been less much less important except for for the contribution of the Engelsons, yeah. I agree completely. And that's why very often people have a tendency to forget about his first part, the first part of his work. That's why I'm insisting today. Absolutely. which is much more vast and much more important, so that's why I decided to work here. I can't just produce in an industry, I've arrived in a year, and now I'm going to produce in an industry. So you had the impression that it was a program that was passed, different from the previous year, and perhaps there is an other program that is produced at the moment. And it's on New York. Yes, but that corresponds very well to the things that he said at the time. So it's true that... well, Diodonné has written, Grotendick has taught the subject of the It's true that after him, after his thesis, after his thesis, after his thesis, after the course he has done in Brazil, there is no other thing, except the theory, which is the theory of the matrix of the matrix of the matrix of the matrix of the matrix of the matrix which is always true. But it's true. But I can say that the analysis, as certain with a little bit of analysis more,

10:00 that is, the application of the method with the intuition geometry in the dimensions of the dimension of the dimension of the dimension, that is at a little bit. So, for example, to answer your question about protein analysis, the case theory of Cissarajepal is a mixture of ideas and the case theory and also Cissarajepal approach yes I would strongly agree but it seems to be you know perhaps a less a less widely acknowledged heritage okay yes yes that's quite great so thank you very much I'd also like to thank Layla for organizing the conference in August. It started the whole idea of opening conferences. Maybe a fun word about Nice, which is that it turns out that I actually live in the same neighborhood where Budene lives, and know the lady who lives in Budene's apartment. She said that she told the story of Giudine's wife who was looking for somebody to buy their apartment and told it to them because they were in the university in biology and also my neighbor was a biologist and changed in the middle of late in life I guess started to work on epistemology with René Tomp. So occasionally he would come over and explain how he would discuss epistemology with René Tomp. Okay, so I'd like to talk about descent.

12:30 When Leila asked us to come to Piresk to discuss Grothendijk's work, she wanted to organize everything according to the, I think it's 12 themes that Grothendijk identified as main content. And I was just thinking more recently, but in a certain sense, maybe people shouldn't really be asked to do history of mathematics about their own work. Because the notion of descent does not actually occur as one of these themes. Whereas I think, it seems to me, it's really one of the ideas which comes throughout Grothendee's work in algebraic geometry. I'm not sure at what point it came out of the original functional analysis point of view. But it sort of kept something which, but you know, this is obviously not an idea which Grothendee invented out of scratch. But, you know, I think it's really one of the things where he contributed to revolutionizing the way we look at mathematics in some ways, which I'll try to explain. So, well, the idea of descent, well, it's the idea of gluing pieces together, basically. So, you know, you might have two pieces of something or other, and if you have some, you know, isomorphism between the two sub-pieces of each side, then you can sort of glue them together. Glue together. Like that. And, you know, this is really an idea, this is a fundamental idea, which goes at least back to the idea of map making. If you try to cover the Earth with a collection of rectangular maps, you know, then when people discovered several times, thousands of years, at thousands of year intervals, that the Earth was not flat, then a corollary of that is that it's not a totally trivial path to patch your maps together to, you know, get a picture of what the surface of the Earth looks like cut up into local pieces. And, you know, we still use the terminology atlases and chart and so on. But I think, you know, on a philosophical level, the idea that you would have a sort of a global thing,

15:00 which is obtained by connecting, by interconnecting local pieces, sort of a, you know, it's a philosophical thing which, for example, to, you know, give a hyper-modern example, can really be seen as pretty much a good description of what the internet is for example. You know the internet when you type of address into your computer you don't see the fact that there's lots of servers all around that are talking to each other and so on. You just you see it you perceive it as a single global thing. Whereas in fact it's actually made up by local pieces which are all connected together in some way. And I guess the real point this kind of gets us to where Grothendieck's, one of Grothendieck's main contributions here, is that it's really, it's not useful, or let's say it's better, it's better to not think of the local pieces as all And that, so I'd like to start by saying that, you know, this really characterizes, I think, the revolution which Grothendique brought to the idea of dissent and the idea of gluing with respect to maybe, you know, what was the classical situation beforehand. The classical situation beforehand was, you know, you would have your object, you know, the earth or whatever, and you think of the local pieces as being sort of subsets of the, of the, of your object. Then the, so the gluing data comes from the fact that the subsets would intersect. Then you'd have to say, okay, you know, let's figure out what happens on the subsets.

17:30 So this is the whole idea. This is a local piece of or subsets of the global space. And now, what was Grothendieck's idea? I mean, one of the main things which comes out we can maybe sort of schematize this in the following way which is that you've got all these local pieces but they're not in the same place they're all sort of all out doing their own thing basically and you have the gluing identification and so on and via the gluing identification they create a sort of a global object I mean, which doesn't have to really exist in and of itself in a certain sense. So let's just write here topological space. And then, so this point of view is a sort of diagrammatic way of saying that you can look at a growth in the topology, or a site, which is a category, with a growth So, we can have a category, let's call the category X. And the growth in the topology says that for every turning object in X,

20:00 we're given, well, one good way of looking at this is, as SIV. who are given a collection of things which constitute a topology, which are called covering sieve. And a covering sieve is a subcategory of the overcategory. So, just briefly, what's a covering sieve? It's a full subcategory of the category of objects of the, your category over a given object with the property that's sort of upward saturated, but if you have a, if you have an object in the sieve and another object above it, then it's also in the sieve, then satisfying some axioms not venture to try to write down here, which are the axioms are sort of the translation of the axioms of topological space into the situation I think of it is just a collection of objects. Ah, no. No, the civ is just a collection of objects over x. It should really be thought of as a subset of the objects. And so, you know, this was a, I think this is really due to Grittendee himself. This is really a revolution in the way we can apply the idea, you know, the idea of gluing things together as a way of viewing mathematical objects was, of course, slowly developed along the 19th century, at least. And I don't know if it goes back to the 18th century or not.

22:30 But it was always viewed in the original sense that, you know, your space is going to be pieced together by subsets of the space. And you can maybe look, you can maybe know a little bit more about the subsets than you know about the full space. So you can use that to sort of go between, woo together sort of knowledge about the space, basically. Whereas the, so the idea that you could replace the notion of topological space by the notion of a category with a growth in the topology really revolutionized things in the sense that this made the theory, among other things, it made it apply to Galois theory. It applies to Galois theory in the sense that you can look at a field, right, and the coverings of the field are a field extension, right. And looking at the structure of the category, you know, maybe we have to do things like add in disjoint unions of spec of a field or something. Basically, of this site is equivalent to looking at Galois theory for K. And so this was, of course, what lab then sort of mixed together these two points of view, the topological space point of view and the Galois theory point of view, and then, let's say, the adult topology, which was, you know, among, among, one among many of the technical advances, which allowed growth to need to, you know, blow away a large, large number of problems, which until then had been pretty much

25:00 that this kind of philosophy, you know, you can just keep, you can just keep adding more, more layers of this kind of philosophy basically. Before going on, let me just point out here that this, this application to Galois theory was not just sort of a passing remark, that also led to, in a certain sense, that led to a whole categorical interpretation. And so a categorical interpretation of Galois theory and also of the classical fundamental of a space as being a dual to the category of coverings of x. So maybe in a certain sense, we're not looking at coverings of x. We're not immediately looking at it. It's not really exactly a question of looking at a topology on this category. Pretty much anything covers anything, basically, the same as in Cowell Law theory. But the idea of thinking of the category of objects over x, say coverings or field extensions, as being like open coverings of a space. this is this led to a whole sort of interpretation in which and I guess that was kind of already foreseen by they and others, but it was another step in the whole relationship between topology and arithmetic.

27:30 Now, let me just say, okay, so now there's, I think we can, I think we can pretty much isolate two technical tools, which, well, one of, the first of which, in any case, was definitely growth, and the second of which, I guess, is sort of a general kind of thing, which I think was probably understood by a lot of different people in a lot of different Okay, but so the first one is the notion of hybrid category. So to, in order to express a certain number of descent problems, you know, and pretty much starting with really the, some of the first applications, which goes in the Convision. So in order to express some of these problems, you need to have a notion of functor from x to cap. x will be our, is generally speaking our, our site, those are the categories. And, and you need to have some kind of notion of functor from x to category. What's an example? For example, a typical example would be the F of U is the category of line bundles on U, for example. What is cat? So cat is the two categories of categories. Well, we're not going to get into that question. But universes, you know, universes were another thing which, again, not really exactly invented by Grothendieck, because there was Tarski and people before him, but, you know, Grothendieck systematized the use of that.

30:00 Okay. So, when you're trying to construct the Picard scheme, one of the things you want to look at is the functor which to an open set U associates the category of one when the one you, or, you know, replace that with any other kind of structure. And so, I mean, it is a little bit troubling to look at this as strict as a functor in view of the fact that cat is really a two-category. We have objects. We have functors between objects. We have functors, B, and have different functors from between categories. And we also have natural transformation. So, it's not only are there morphisms between our objects here, but we also have an extra level of morphisms. That's the first and basic example of a two category. And so, viewed in this way, then our functor f should really be thought of as a two-puncter. And Grothendieck's original way of looking at this was by looking at it as a fiber category. I think the needle will correct me if I'm wrong, but this is what you use the integral sign for, right? So, this is called the growth and deconstruction, you know, among others. So, instead of looking at a puncture which to each object of X associates a category, we can look at a, we can sort of integrate those all over X, and I'll call this,

32:30 For any object here, we'll have a category here, which is really, you go on to App of X. But, you know, we're still in this sort of historical frame of mind here, So, I mean, it's, I mean, you should think of it, it's not really clear to start off with whether you want to use a two-punctor or a weak two-punctor. In fact, it turns out that a weak two-punctor can always be strictified to be equivalent to a strict two-punctor. So, I mean, you know, at the time that, and that was really, I think, that kind of question was really one of the reasons why, I guess, Grothin introduced this notion of fibrary category, because here there's, you don't ask the question anymore. I mean, there's just a well-defined definition. And the fact, the weakness of the two-functor, in fact, that given a morphism in X, there's lots of different choices of Cartesian morphism in the fiber category, which give a pullback for a puncture. You can do, you can, I think there, aren't there even four different ways, I think there's even four different ways of getting this, Anyway, okay, so that was the first thing. And so one thing that which this allows us to do, let's say somewhat more easily than the, I mean, no, at the time, we didn't really have the language of weak two-punctors. This was provided, maybe in the 1960s, by Benabu. And I guess one thing which I'm really not sure about is, to what extent this came from the Erishman school, and to what extent it came from the Gothenique school. This is unclear, Benabu said that, you know, he was a student of Erishman and spoke to the Gothenique,

35:00 one of them said very much to him in any case, so okay, but okay so the point is that the notion of fiber of category can sort of up to this point replace this this end thing here, until here. But we can also see one pretty important construction, which is that if you have a fiber category, so let's suppose, you can call me a bold F, but around F. If we have a five-word category, then we can look at this category of sections. It's just the full subcategory of, oh, okay. the category of functors from X opposite to F, look at the fiber of that category mapping to functors from X opposite to X opposite. That's to say, look at functors from X opposite to F over dead-ended functor of X opposite, and look at the full subcategory of those which send maps to the Cartesian morphism. And this has the nice property that it allows us to do two things. It allows to strict divide. So if we look at the functor X maps to gamma of X, this is actually, this is a strict functor. So this is what I was saying earlier, even if you started with a weak two-puncture here, you could do the growth and deconstruction, make a fiber category, and then take the,

37:30 the puncture which to an object of X associates the category of sections. This is a strict two-puncture. From this? I don't know if this is. But you know, in the notion of Cartesian category, there's directions of arrows to be chosen. and there's co-Cartesian errors, I mean, I don't know. Yeah, I mean, not in this terminology, there is no, I mean, they should, the category lives over X, not over X. Ah, okay, but I mean, you know. Yeah, you, yeah. Whatever. You could also have a covariant puncture, okay, so, anyway. Yeah, the puncture is covariant. Well, what we want to think of is the fact that the pullbacks go, you know, pullbacks of functors on the opposite category. No, but here we want to get to a functor. I mean, if we're starting with a functor here and doing the growth indigestion, then taking the definition of fiber category, which corresponds to that guy, then taking the sections, we want to get back to a functor here, which in fact should be equivalent to the original function. So that's the first thing. But, okay, so, but here now we can see an obvious thing to do, just an epsilon change here, which is we can now define the set of locally defined sections. Well, I mean, the example is, okay, let's go back to the original case. Let's take a topological space and, you know, look at this, you know, the category of line bundle is unused, for example. You might say, okay, I'm worried about, you know, whether if you pull back a line bundle and you reap, then you pull it back again. Whether they're set theoretically, that's really the same thing as pulling it back twice and so on. So you have a, so you define a fiber category. then you constrictify that by doing this. So what that says is that an element of here is a puncture from...

40:00 Now it will appear something which, to every y over x, associates an element of f of y, in a punctorial way. But these guys restrict in an obvious way. if you have, you have x prime mapping to x, and if you have a y over x prime, you automatically get a y over x. So if you have a section over x, then you restrict it to the sections over y. That's why it's strictly. But, so, okay, so what can we do now if we, once we've defined our notion using the notion of sieve, We can also do that, so this is the set of sections of our category, but locally only over the sieve. So in terms of our atlas, that would mean sections, you know, say line bundles, but only over the open sets. What does it mean to give a section here if we took this category of line bundles? That means that for every... X here is the category of open subsets of our space. For every open subset, we give a line bundle. Of course, that concludes that all of X is an open subset, so we've already included a line bundle on X. But here we can do the same thing, but we say only open subsets which are small enough, we have a line bundle. And that leads to that, so that gives you the If the map, okay, so, so let me just say remark.

42:30 So this is automatically an equivalent of categories. In fact, it's a limit over a category with a final object or initial object or whatever. And so F is a stack if the map, I'll just use this map. If that's an equivalent subcategories for all coverings is. This says that the category of locally defined sections should be same as the category of global section. Now, I guess, you know, I sort of jumped to the step n equals 1 here, having skipped the step n equals 0. So, before discussing what's a stack, you should say what's a sheath, okay? Okay, but now I can just say, okay, a sheath is just going to be the same thing as a stack, but where the categories are discrete categories. Of course, the theory of sheath came before the theory of stack. And the theory of stacks is sort of a, what Bayes-Golan would call a categorification of the theory of sheath. But, you know, you can think of the theory of sheaves in exactly the same way. Maybe just in that context, a good thing to point out is that this is really a limit. When we take the space of sections of our, say, our sheave or stack, or pre-sheave or pre-stack, over a category, this is really a limit. This is a projected limit. This formula holds for stacks or for, it's a fiber category, it also, in that case, it's a two limit in the category of categories. Actually, I got interested, part, one of the reasons I got interested in this kind of subject was in the paper of Duleen Mumford,

45:00 there's a cryptic sentence where it says at some point, such and such is just a two limit. So, go out and look and try to find the references. Wikipedia was not available at the time. We discovered, pretty quickly, that there weren't actually any references that really explained it, at least I didn't find any at the time, explained what a two-minute was, and so on. Okay. Let's see, now I said there was two technical tools, now I'll go to the other one. Let me start. All right, I'll just do this. Or go over here, maybe. Uh, yeah, that's better. Yeah. Oh, you can. Oh, yes, sir. Right, that was the first technical tool. And the second technical tool is the notion of simplicial object. Now, why does the notion of simplicial object come in? It's actually, I think it's really a mystery.

47:30 So, you know, at the time, simplicial objects had been used by the topologists to represent topological spaces. Based on the idea that, you know, take a space, you can sort of cut it up into triangles. That gives you a collection of simplices. And it seems like a good idea to organize these simplices together by looking at which ones touch each other and so on. It's kind of like the same. I guess that's why. But, okay, yeah, well, I guess maybe the, okay, the answer is why, why simplicial, I mean, why do these occur in the same way? If you go back to the case of a topological space, which I guess is great, then you have the check nerve of a covering, and that gives you a simplicial, simplicial, a simplicial set. So, this is kind of the old idea, if you have a covering, you can look at its check nerve, that, it sort of comes down to doing a very centric subdivision of the, I mean, if you think of the covering as sort of fattening out some triangles in your face, then, You can see it or some bushel object kind of getting developed, which in the case of a nice manifold and so on, incongulatable manifold, it's going to give you back a homotopy equivalent space. But in terms of in this categorical version, it just comes up in a very simple way, which is to say that let's look at, so suppose we have a collection of Ui over X covering family. Okay, so I'm not going to get into the definition of the relationship between covering family and sieve, but. Now, in one, often, not always, but it's sometimes a good idea to sort of add these guys to your category. You can often look at this disjoint union, let's just call that u, disjoint union of the vc. And then, then you can create a natural symbolical object again assuming that all these guys exist. And in fact, it's an augmented symbolical object. X, but...

50:00 So you get, so it turns out that if you look at all the different multiple fiber products of U with itself, they naturally get organized into a simplest logic. If you look at all the natural maps that you have in here, the natural maps that you have are exactly a functor from delta op into our category of objects, in fact, over x. And so, if you break this back, assuming product commute with direct sums on the disjoint union, then this guy really is a, this is the disjoint union of ij of ui cross x uj. And remember, if we're really in a topological space, the fiber product of two open sets over the space is just the intersection. And here we've got the triple intersection. So you can see here, the stuff that you use to define check cohomology just kind of comes out naturally in this bushel object. And, well, one technique of descent says that it's suppose we've got a fiber category over x. Then we can look at a, get a hybrid category and pull this back. The collection of guys of the form F of U cross U. I just pull back the hybrid category here, pull it back by this function.

52:30 And then, okay, so what's one way of doing descent, okay? One way of doing descent is suppose some additional conditions that the... The pullback chupuncturps have left adjoint. And the f of x, the f of y, the different values, the fibers of the fiber category, admit limits. And maybe, I'm not sure we need to pull limits or not. plus some other stuff. Then, what should we do? Then let's just take, then if we have a descent datum, a descent datum for the covering, it turns out can be also thought of, so here we did a section over the sieve, it can also be thought of as a section of this fiber category here. I'll call this kind of term. Cartesian section. And then project, so the first thing to do is to project this into a collection of sections by using the adjoint.

55:00 What does that mean? Concretely, that means what? That means take your categories of objects over here, so let's say line bundles over U cross U, and take the direct image down to X. So to glue together our line bundles, what should we do? suppose we've got a line bundle over u, a line bundle over u cross u, and so on. Then, just take the direct image down to x. And then take the limit. So this whole thing is just a fancy way of saying the following pretty simple description of how would you descend. Okay, so suppose, you know, a line bundle is basically a sheath. So suppose we want to descend a sheath. What should we do? So suppose we have sheaths on u, u cross u, and so on. That means we've got sheaths on our open sets. And we have some way of identifying the sheaths on the open covering. And we want to create a sheaf on all of X. So what should the sections of the sheaf be on all of X? It's easy. It's just the sections of our sheaf on all of X just be collections of sections on each of the open pieces which agree. This whole machinery just comes down to saying that, basically. So this is kind of a souped up version of that simple looming construction. And, which gives us the corollary that lots of things are stacked. I knew that I wasn't going to get to say everything on the resume here. Okay, so that's stats. Now, what to, now, then we can sort of fast forward 20 years in advance. Fast forward 20 years in advance. Grothinique, I said in the resume, he came out of isolation.

57:30 I'm not sure if that's really an adequate description, but okay. As I guess was said earlier this morning, you know, maybe growth index now working on the next stage of some theory, which we'll all have to, I guess the next two or three generations of mathematicians we'll have to work on. In any case, in around the 1980s, Grothinique sort of came up with a manuscript which was then distributed thanks to the work, I guess, of a certain number of people. So I first saw this manuscript in the back room of the library at Princeton University. So I'm happy that Princeton University and some rather got a hold of a copy of this manuscript and hid it away in the back room. So, in this manuscript, Gruth and the Hada, sort of projected that this whole kind of theory, so, should have a higher categorical analog. And so, the idea is to, here we saw two functors and two categories, the idea is to replace two by N, and let N get big, even if possible, let N tend toward infinity, or let N be infinity. And sort of, well, the short answer is that if you put n everywhere, then it still works the same way. So instead of categories, consider n categories, n stacks and so on. Where the idea is that, as for the case of a two-category where we had an extra level of morphisms, the idea for an n-category is to have an extra level of morphisms at each n.

1:00:00 It was pretty well known how to do that for the case of strict n-categories. And so both of these, part of both of these, uh, uh, considering stacks is to sort of ask what a condition is weak in the category. So it turns out that certain expectations in degree two doesn't necessarily work so well in higher degrees. Um, and so for n categories, you have to, you have to consider a case where your objects are not necessarily associative, are only associative up to homotopy, then you need to include the coherence of homotopies for those homotopies. Well, you can read the resume for other things I can say on that subject. So maybe one of the first remarks is that to do this, really the, basically, the top, the topologist basically already knew how to do this. Topologist, let's say, almost knew how to do this. In fact, you know, both of these manuscripts, it starts with a collection of letters, including letters which are to or from Quillen Poulin scratched out. Among other things, Poulin at some point says to Grothendeeke something like, you know, oh, why are you doing all this abstract nonsense? Nowadays, the main tool of abstract nonsense necessary to treat this subject is the notion of model category. So, you know, I think that this probably, this In fact, in a certain sense, it did. There's a thesis by a student of Jim Stasheff, which was recently posted on the archive. The student's name is Worth. The thesis was from the 1960s and was called homotopy transition co-cycles.

1:02:30 Some other remarks, for example, if you look, on a single page, you can ask Larry about this, on a single page of Pursuing Stacks, Grosendieck quotes maybe correspondence with Larry Green. discussing the idea of Siegel's gamma spaces, and on the same page, on the next paragraph, Grothinney recalls the notion of the nerve of a category. And in fact, if you combine the notion of the nerve of a category with Siegel's definition, not of gamma spaces, but of just E1 spaces, you immediately get the definition which was worked out by my student in his thesis. And also, so to sort of complete the sort of all my ideas, to complete the discussion, George Montignotis has a preprint recently in which he points out that there is actually in pursuing facts a definition of infinity groupoid. at least n or infinity, okay, n groupoid where n might be equal to infinity. And it's pretty much the same as Bat-Hanin's definition of n.