Discussions — Grothendieck's Program

0:00 The category of a billion chiefs on that site. They named the category of a billion chiefs on that site. They talk about sheaves of sets on that page. Yes, they do. They treat that category, but they don't name it. They don't name it. Somehow it's, you can name the category of groups, that's not hard to think of, but you must not name the category of sets. The set sheaves of. That which they dare not name. Oh, yes. And it's true. In Tom's book, Introduction to Metacohomology, which I've looked at in some detail this way, you would actually have to add a couple of pages to Chapter 2 and a couple of pages to Chapter 3. To introduce the topos. And make no heavy use of it. Tom's Introduction to Archaeology is a little Springer-Berla graduate text. Yeah, which I think is out of print right now. You have to add a couple pages to each of those two chapters, but you could cut lots of pages later on. Sure, yeah. But yeah, it's just this bizarre idea that you can talk about sheaves of groups, talk about sheaves of sets, name the category of sheaves of groups, deal with the category of sheaves of sets, but not name it, because then you'd be talking about topos. Well, yeah, that's a reflection of this general... The belief that something is difficult leads to the propaganda that's difficult, which leads to it being even more difficult, rather than any attempt to simplify or take advantage of simplification. About spaces versus shapes, I mean, I just want to mention that in the 15, with the Canton seminar, there was a great shape emphasis. You have various cohomology theory. Each cohomology theory is defined by a functor over the whole space. You define the rational cohomology of the whole space. So you have a category of spaces and you select the coefficients and then you have a functor through the group.

2:30 If you want to use that practically to calculate what you have to do, why don't you use various relations between the spaces and your stratification, excision, projection, spectral sequence and so on, but you deal by manipulating the spaces, which if you think that this is completely from one space, much freedom with the coefficients for the possible shift, so... It's really a different empath, a completely different empath, which means that when you have to calculate practically, it's quite different, I mean, quite different. The idea of resolution, let's take the idea of resolution. I mean, in an abelian category, let's say, like the sheaths of groups of space, the notion of resolution is quite easy to understand. And, of course, the game is to which you want to calculate the commodity of a space, you look for what you have first, and first of all, the differential form will give you a certain revolution, but it's too big, so it's difficult to calculate, so you take another one, etc. But the game is to as small as possible. The solution is quite simple. But then if you deal with a fixed coefficient and a category of spaces, you're moving from a... Linear fluid to non-linear fluid. And then you need different tools. And what is the resolution of space? It's not so obvious. Of course, Simplicial Sets provides, as I explained yesterday, about motifs, at least tape motifs, that you can do, you can define what is going to be a resolution in space. It's something, in a sense, which has a much deeper geometrical meaning, and on the other hand, which is much difficult to handle, and I think that's topos, I mean, so, in a sense, the sides, sides versus topos, I mean, the sides is really the geometrical part, I mean, all the spaces you're covering and so on, and then the sheaves, the sheaves are the other aspects, and there are, of course,

5:00 I think if people have reservations to speak of topos, because they see more clearly the geometry, it's easier to visualize the various spaces and cross-structures between spaces than to imagine the whole category of machines. I don't know if you've seen Ulysses, a short article in the American... Mathematical notices called What is a topos? This is to accompany a long article about Grotendieck, which is... Oh yes, yes, yes. The paper by Annie Jackson, just a supplement to the paper, to the two installments. Yeah, the two installments. It's maybe November of last year or so. Well, in the notices there are two installments by Annie Jackson on Grotendieck. It's very good, I think. They mention topos, you see, and so there is this what is column that they have, so appropriately they have one called what is topos, but then, so Uzi writes this, and he gives almost all of it as a bad exposition of what topos theory was 40 years ago, and then at the end he has one paragraph. Which mentions our work, but consigns it entirely to the scrap heap of logic, you see, and so this is a very sewer point with us because we don't believe topos theory is all about logic, we think logic is an important auxiliary that more algebraic geometers should learn about because it helps, but for God's sakes, the actual content is geometry, and so we see the topos as geometry directly. I mean, but again, it's a question of there's all these rumors and propaganda which said, well, it must be logic, you see, it must be logic. Didn't you work on the independence of the continuum hypothesis? Didn't you introduce intuitionistic logic and all these kind of things, as though these were slanders?

7:30 Well, the kids didn't understand that. Autopolis theory is a synthesis of geometry and logic, a big synthesis of all these things. If you emphasize that it's true, with one purpose you might emphasize the logical facet of it, and for others, if you like, the more specifically geometric. But it goes anything that smacks of logic, you know, at all. You see, immediately it gets tarred with that brush that mathematicians, on the whole, have... I know. So I thought I was striking a blow against this prejudice, but instead I've got to counter-curve it with even more prejudice. It's really terrifying. There's no way I can deal with it when I won't. I reported about my piece and now I will talk about this with all the limitations which I admit, I mean, this paper has been often quoted outside of France, but very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, I'm sorry that some of my young collaborators, and I did not put that in their mind, but spontaneously they came to me with the same comment than some of my young collaborators, that, well, they do excellent technical work, excellent technical work, and they know every sentence in SG, every sentence in EG. But Grottenig is right in complaining that. He's biased, but because he's too honest, he can't tell what perversity is and other research. He doesn't see the point. He doesn't see the point. He's too honest. He doesn't see the point.

10:00 He has been raised in a certain subordination, at least what he learned from Goethe. It reminds me of the, in the south of France, a small sect called the Derbys. I suppose it's one of these revival groups. And there is a small group of, and they are called the Derbys. And it's very interesting. The Darby's, the Darby's, they are, they call themselves, les frères et trois. And they don't, the other one, they are the frères et trois. Contentful. We won't go into it now, but there is an exact parallel. Okay, well, maybe not completely different. They're all on the same dominant. Each is trying to say that they're narrower than the other, but that's their claim to the... But getting to the mathematics again, when you talk about this shift, cohomology was a functor from, say, a category of spaces, becomes a functor from a category of sheaves, I don't think the Seminar Quarton ever talks about functoriality in the space or in the group. To work with it, you would have to use it somehow. But Rodendy does bridge that gap. He's always talking about... He'll phrase it as cohomology in the category of thieves, but he'll talk about functoriality in the base. And the bridge is really geometric morphisms. Geometric morphism is full of point-derived factors. That's why we call it the five-lock categories, exactly, for that purpose. The five-lock categories are to take into account the double-functionality.

12:30 For a given space, your functionality about the shape, but your also functionality about the space. And I think, as you say, it's full of point-derived factors. I mean, fiber category is exactly an answer to that. I mean, trying to make a synthesis. But I think still psychologically, I mean, I'll try to explain why topos is so resistant to topos. I think this idea, this global idea, which is perfect, I mean, so it means that you have topos plus geometric morphism. That's okay. So the category of topology is geometric morphism. But the point is that in the mind of many people, these are two different steps. Two different steps. Historically, I mean, in the beginning, I mean, they were two different directions, and they were united by using five-word category, but... Even there, if you're doing the cohomology of a group, you normally don't look at it in the whole category of toposes. You might look at it in the category of groups. Yes. I mean, that's the more useful functoriality for a lot of calculations. Yes, exactly. No, no, but you know, I mean, so... Really, I mean, the point of view of topology unifies two rather different trends. And that's what my point is, that since it unifies two rather different trends, people have experience in one direction or in another direction, or maybe in both, but they don't feel a necessity to do those things together. And it's true that in actual practice in geometry, either you work with resolution of coefficient, meaning shifts over a given space, and that's quite useful. In the actual practice, you rarely do both at the same time, and that's why people don't see why it should be advantageous to unify these things. Well, it's possibly an explanation. But Eilenberg claims we're already trying to bridge just that gap between the group cohomology and the space before group cohomology is even defined. They came up with group cohomology by figuring out what was it we were trying to relate to the topological cohomology. So it's been there, the problem's been there from the start. And then Grotnik unifies it more thoroughly with derived categories. Oh yes. And you said you had something to say about that.

15:00 I will first discuss the first steps, historically. Golden period. It started more or less, in a very modest way, as follows. He has an axiomatic framework which is more or less from a given group, a representation of a given group, so you have a group G and you would consider them a category of G-modules, so modules over a group algebra which is the same G-modules, so of this category, and then the cohomology is not just one factor but a collection of factors, H0, H1, H2, from this category to the category of a meaningful group. Then, there are two constructions. First of all, it's a collection of factors, which means that if you change, for each i, if you change the g-module m to another g-module m prime, you have an induced map from hg, hi of g, with coefficient in m, into hi of g with coefficient in m prime. Somewhere in the Cat Island there you will find something similar. But also you have the exact sequence. You have the exact sequence of cohomologies. That means if you have an exact sequence zero, short exact sequence zero, m, m prime, m double prime, zero, then it induces a long exact sequence in a cohomological. It can be, and a new feature is that you have the coboundary operator which maps h i to h i plus one.

17:30 This can be rephrased as follows. If you know already, it can be as follows. If you start with the category of G-modules, you have two things which exist. Org, from one model to another one, and X1. X1 is easy to define directly. And what you have is that you have the X1. And the X1, there is a partial multiplication of X1 with Ohm, which is X0, and if you express the axiom of the delta factor, you will see that you start with something which is not yet a category, something which has as object the G-modules and two kinds of Ohm, Ohm and X1. And then both have to be represented as homomorphs in the dominant category, with some combination. If you read between the lines what you have, is that, if you want to be a little more... You see, out of the G-modules, you have a somewhat incomplete category, because the product of two OPs is defined, but the product of two X1 is not defined. So, you can't get one way of understanding, to understand the Yoneda problem. We want to combine them and to say that these R and X1 may be generators of a new...

20:00 The end result is that the object will be the same G-module, but the homomorphism between two G-modules will be the direct sum of the XI, X0, X1, X2, etc. with the ordinate universal unit number. So, you are building the first appearance of the life category, and that's why I gave a lecture at the I-55 along these lines. It's recorded, but in the library of the Institute Poincaré. So, that was the first example. So, the idea is that x1... They formulate the definition of the delta factor by first introducing a category defined by generator in relation. The notion of x1 is easy to define, I mean the exact sequence and equivalence and so on. ...category by generator in relation, which means that you allow composition of x1, x1 into x2, x1, x1, x1 into x3, etc. And finally you sum up the definition of the delta factor by meaning it should ordinary factor from this new category into So, but it's not yet the derived category, it's not yet the derived. But then, in this presentation, resolution plays no role. You don't need resolution. But if you want to calculate the resolution, of course you need resolution. If you want to work with a module, it's better to work with projective resolution, as we discussed already concerning injective resolution.

22:30 Instead of building the X, so this is one step further, I mean, I first took the exposition of Ironberg in the Cantor seminar, but restricted to the homology of group, common group, and then, and now I move to, to, then the X, the general X was presented in full generality. And then, the first remark is that you can build, you can build your general XI by pieces, which, by combining various X1. I came with the remarks and so it can't tell you what you have. If you want to define XP of M with L, the factor which is, if it's contravariant, you go to projective resolution. If it's covariant, you go to injective resolution. And if it's a mixed fun tool, you can use simultaneously, well, you have three options. And a balanced functor is a case where the three options coincide. The most complete version is to, since m of m into n is contravariant in m, you replace m by a projective resolution. Since it's covariant in n, at the same time you replace m by an injective resolution. So you have two resolutions at the same time. They form a bicomplex. You take the total homology of the complex. You can simplify things by taking only a resolution, a projective resolution of M and keeping N, or keeping N and taking a dejective resolution of N. And then I remember, I came to a summary with the following. Suppose in O and N, I simultaneously replace M by a project and N by a project.

25:00 And I say, well, if you do that, you are now the composer. Now I have a third, M and P. Each one is replaced by a projective resolution that can be compounded. And that exactly is the alien adapter. Once you have the alien adapter, you can come back to zero. It's totally furious. It's a gated law of nature. As had just been decided two years before. It's a gated law of nature. It's a standard. You cannot do that. And I say, but of course, now, and I remember very well. For model, projective resolutions are easy to define. So, okay, we do that. And we have the X. And as said already, the current product can be used as a product. We were interested in sheaves. Of course, the main work we did was sheaves. And I said, well, at the time we did not have yet injective resolution of sheaves, but we have some substitute fine resolution or whatever. And I said, well, now do the same for sheaves. If you have a space X, of the space X, let's say, with rational proof, it's just of, in the category of the constant shift corresponding to Q with itself, and again, and in more general situations, you can back then, for shift, you have to use injective resolution. But I knew that, I knew that in order to calculate the X, I could at will replace both term by a protective resolution, both term by an injective resolution, without mixing protectively. If you mix projective and injective, you lose some composition. You cannot take three terms.

27:30 So I think that was what it saw. Understood. Grotanik understood. What is interesting is not the cohomology per se, not the cohomology per se, but you go from a resolution and the corresponding cohomology. What are you doing in it? You are working. So you complex the resolution. But in many situations of a category, you kick the object, but you change it. And so, I understood perfectly. Now, of course, in my description, I would say, I want to define the commodity of a sheath. I'm in place, so I would consider only the resolution, I would consider only the resolution, which were, I mean, the resolution of a sheath with all modules, which means that complexes which are, whether I assume or think of something located in one degree. But of course, and also I remember that... I rephrased the lower spectral sequence in such terms. Instead of having three coefficients, now we have three spaces, x, y, z, but then okay, but you can rephrase things in that sense. But then, Guantanamo, he said, we don't have to work with it. We have to deal directly with the complexes, up to some increments, which means that we keep the complexes. And he was, I think he was influenced by the idea of one topic category. In my recollection, I don't care whether he was, he majored in it. The connection is the homotopy. So that's more or less, I think, the origin. They're repeating often. Don't take cohomology, don't take...

30:00 Yeah, exactly. Don't take cohomology was really the instant. I mean, I myself was still interested in the cohomology, as tools to calculate the cohomology and to combine in the operation and so on. But what did they do? They took the next step. Keep the complexes, but... All the complexes, yes. Up to a point. And so that's, no, that has been a very fruitful, of course, and for instance, there is an out, there is a, which is interesting. About the same time, Sayre proved the Tor formula for the multiplicity of intersection. And he says, well, he says that you have two. You have two sub-variety crossing each other into a larger variety. You want to calculate two curves crossing each other and you have the intersection, you know that in some situations if the intersection is not transverse you have to count the number, to count the multiplicity. And there came the formula with alternating sum of tau. But in the derived category I don't know who was the first to remark that. But in the derived category, a sub-variety corresponds to a function of space, and then the sub-variety corresponds to a prime ideal. Of course, it's not really the prime ideal which counts, it's a quotient a over b. So we have a which represents the base and then space, and a over b which represents space. So we have a over b and a over q. And Cez said when we... We take the alternating sum of L. But the point is, a slightly broader point of view is that if you have a complex, that was taken by Mumford later, if you have a complex of modules over a satisfying suitable restriction, then to this complex you can associate a cycle, that means a linear combination of irreducible superheight by going to the cohomology and so on. And then, but now...

32:30 Well, this has been elaborated by Monfort, Knudsen and so on. But then you have the effects of modules or, if you want to be global, coherent sheets of your algebraic variety. Now the idea is that instead of having a calculus with cycles and so on, with multiplicity and intersection, But in this category, you have a tensor product, which is a derived tensor. A tensor product is a derived tensor product. It simply exists. Two suitable equivalence classes are the same as a cycle, an algebraic cycle. Well, you don't have to define n. And there are some difficulties for the associativity and to use a certain spectral sequence for tau. But this is for free. These spectral sequences are just, I mean, after going through the cohomology, the expressions of the tensor product in the derived category is associated. And so finally what we get is the idea that in algebraic geometry where you go and test it in cycle divisor and so on and calculation with them, you replace all them by complexes in some derived category and the tensor product does exactly what you want. One of these, I think, in print that was elaborated by Boblitz. The original idea was understood and, interestingly enough, there was, I mean, I took this idea from some notes, from some draft of self for Bobakir. Well, we were interested in, well, of course, Bobakir was interested in... In chapter 6 or 7, we are interested in, well, we don't come to the point where we define multiplicity of intersections, but we define the so-called Hilbert-Samuel polynomial, and Hilbert-Samuel polynomial.

35:00 And I remember in some of the drafts that José provided, there was this idea that you have a certain category of modules. And you have also an additive function on the Cateo-module, which is a predate in K0. It's a preliminary form of K0. So, you could say, maybe in a sentence, that this resolution provides you with a definition of K0, and then you have a product in K0. Simplify ways. But it's important that Serre wanted to prove the existence of the Hilbert somewhere. As a polynomial, an important asset of this Hilbert-Samuel polynomial is the behavior with respect to exact events. And there, action, more or less, well, he did not put it really in axiomatic format. He considered... These are all functions of additive, what you call additive function of modules. So it means that to each module we'll associate a polynomial, an integer for every short exact sequence of certain values. And that played a very important role in the exposition of Hilbert-Samuel polynomial in Brumbach. But that was about the same time that Say invented his door formula. It was soon recognized that two things were the same. And I think that, if you want to understand the prehistory of the right category, this is more or less a description. So this polynomial serves to help to define the equivalence relation that the complex has? Well, the polynomial is the outcome of the definition of the equation. But of course it immediately had the opposite effect too, because this phenomenon that you have one normal quotient relation, you identify maps, then you also invert some maps. These two steps were taken, I don't know if the first one, but at least by Gabriel and Zisman in their book on calculus of fractions and homotopy theory.

37:30 And it's interesting because for the ordinary construction of homotopy, ordinary spaces, you don't notice this because already the identification is good enough, but they make it very explicit that if you start with the combinatorial category, like some special sets, then you have to do two steps, and these two steps, I've never really fully understood why you need these two steps, except to say that... I mean, there's something similar already in ring theory. You have localization, which is inverting something. You have passing to a quotient modulo, an ideal, which is identifying some things. But if you look, the category of modules is more or less equivalent to the ring. If you look at the category of modules, you find that if you look at flat hunters, you see. That the equality part is more or less subsumed under the inversion part, because two maps are equal under a certain functor if and only if their equalizer becomes an isomorphism, assuming the equalizer is preserving, but the functor is preserving equalizers. But then precisely the construction of homotopy or derived category is not exact, and so you don't have this assumption of one of these two kinds of steps under the other. You need them both. Why do I have to mention the book of Gabriel and Zizma, which was another fortunate outcome of our... So-called European seminar of Falcetta, which was a joint venture of Heidelberg, Zabrick, and Strasbourg at the time when Dorot and Gabriel, myself, and Verdier were in Strasbourg. There were these intensive meetings in the 60s, and it's when Verdier joined us. Verdier was just finishing his work on the Dirac category.

40:00 You have to understand the job situation of the time. We appointed Verdi a full professor in Salvo before he defended his thesis, which brought to some embarrassment afterwards, which brought to some embarrassment and to a complete mistranslation in Eisenbud's commutative algebra. I forget the exact French, but he quotes Grotendieck saying that Verdier is about to write up a presentation of derived categories, but Eisenbach translates this as Verdier has just written up a presentation of derived categories at a time when Grotendieck was complaining very bitterly that he had not written it up. The complete manuscript was found after the death of Verdi. I remember in the 60s, Verdi carrying his manuscript in his hand and showing out glimpses of the manuscript that for various reasons one would not enter into the reasons that he did not publish, but I see the resentment for Goethe and I don't want to enter into the personal issues. And so, but then when he died, I mean, the paper was found in his files and then it was resurrected by Martin Yortys and eventually published. But the published version won't really replace Cartan-Eilenberg, right? I mean, but it was meant to. It was meant to read the whole of that. No, no, I mean, no. The complaint of Grottenig is right. I mean, this is slightly incomplete. There are three chapters, and Malsiniotis took great pain to title him in tech, and I helped him to proofreading. I helped him to proofreading. But Grottenig wanted to have a fourth chapter about... Well, he gives a description of the Dirac category but not of the Dirac factor. This is really incomplete. And it doesn't completely replace Agatha and Ivan. That was part of the misunderstanding. But the point is that Gabriel and Zisbon was an outcome of the years spent by Verdi.

42:30 And we gave the direction of the seminar to Verdi at a time. And then, Brady began to explain his idea about derived category and so on, and I remember I gave a preliminary thought about what it means, inverting how it was in a category, in a localization category. I gave the first talk. And the homotopic theory form, and the homotopic category form. And then, but then there were many lectures. And finally, but this is a slightly different... ...direction because it really is a non-linear direction, there are categories in the non-linearity direction, which is more important for a botany theory. I think I have this right, that you could get the derived category by taking the category of resolutions and just inverting some arrows, but it's not a calculus of fractions. You make it a calculus of fractions by first doing the identification and then inverting arrows there. The class of arrows you're inverting is not as nice if you don't form the equivalence relation first. Yes, yes, exactly, exactly. Okay, so the inverting is not exact, I see. Now, you first go from complexes with ordinary maps to homotopic classes of maps with new complexes, and then you invert. And that way you have a calculus of fractions in the now-standard sense. Is that Gabrielsism and the defining term? Whatever. All you had to do was invert the arrows, but it's not a nice class. Where the fractions, instead of being just binary, instead of being words, are just binary, because you can bring the denominator past the numerator. So therefore you would join maps, there are more maps to be inverted than just the homology equivalences, homology, algebra, physics. And this is what they do also in the nonlinear theory by introducing vibrations, cofibrations, trivial and all this, where again the main step is said to be inverting certain arrows, but you go through all this first.

45:00 Well, I think it's still a good observation, but I think it's still a good observation, but I think it's still a good observation, but I think it's still a good observation, but I think it's still a good observation, but I think it's still a good observation, but I think it's still a good observation, I don't know whether I had this idea of working directly with a complex lecture afterward, but Serre was not convinced. And Serre, he saw that was an unnecessary step. Well, one reason that occurs to me, I mean, the point of the derived category is to avoid taking spectral sequences. You are exactly right. I mean, while I don't remember all the discussion we had at the time, but I could imagine it was more or less that. I mean, I said, why do I have to prove the associativity of this complicated spectral sequence when it could bypass that? Why? Spectral sequences are always there. So, in a sense, that's me explaining mathematics. But it's true that this idea of additive function of modules, I mean, that in the draft, maybe, I don't know whether it survived in the printed version of Boba Key, but it was certainly an important part of the draft.

47:30 That's where I got the idea. The important part of the manuscript was devoted to the construction of the K-group. And this idea that means that the K-group, yes, and that the point is that there are two present. You can say the K, K0 group represent, I mean, so you first define it by generator and valuations. You say to each sheaf or to each module associate a generator and each exact second give linear value. But there is another point to do, which is to deal directly with complexes. The definition of K0 would be that you first define the derived category of complexes and then you take the isomorphic classes of complexes in this derived category. That's the keynote. The K-NOT is immediately obtained, which is an extension of what I said before, from the tensor product, the connection between K-NOT and K-NOT according to Atiyah. Atiyah is the result that even dimensional cohomology with rational coefficient of space can be calculated from the K-NOT, which is one of the best known results of Atiyah.

50:00 Reasoning as when I said in the beginning I start with the delta factor which has two pieces and I want to unify the two pieces. And the interesting fact is that starting from the two pieces of the delta factor you build the neonatal product and it acts by the neonatal definition, the neonatal product. And then without mentioning any resolution. But then after that you prove that you can calculate everything by a rule. And then you come directly to... So, you put first the X and the unit up on that, which is of the arm in the derived category. But only after that, historically that was. It's true that Sam was always reluctant to go directly. Able at manipulating spectral sequences that he did not need to bypass them. On the opposite way, I mean, on the contrary. My first exposure to spectral sequence came from some talk by Sami at the Garton seminar. Well, it's a little confusing, it's a little confusing. Well, it's a variant of the exact couple, but it's a little confusing. And so, for many years I resisted spectral sequences. And I even went as far as writing a few papers where I imagined various methods to bypass spectral sequences. Well, of course, in the one way I found new wisdom, but they could also be obtained by spectral sequences. But I was motivated by the idea of not using spectral sequences because I did not understand them. I'm now with my student Brown doing very complicated calculations on the cohomology of moduli spaces, and well, we are very happy to have spectral sequences, although one would be more cost.

52:30 And of course, spectral sequences, I think for the topologists, if you have one open set and a cross complement, you actually exceed an exact sequence. But you have more than one open set. You can use repeatedly the excision. I don't know the exact sequence, but it's much better to take everything as part of a spectral sequence. I would say exactly that, and so of course you can be more elementary, but it's less. Now the spectral sequences you're using, are these Grotendieck spectral sequences? I mean, a lot of the ones are cohomology. Yeah, I know, I know, I know. No, it's more or less the Lorentz cohomology corresponding to covering or hypercovering. It's slightly different. There are many ways. A certain point is that at some point, sometimes we have simplicial resolution and co-simplicial resolution, and sometimes you have to meet simplicial. And simplicial versus co-simplicial are the same thing! But what we have, when spectral security is not in a good region, spectral security is in this quarter, in this quarter. And you know, it's a little complicated. The arrows keep pointing, the differentials keep pointing into it instead of crossing it. No, no, but the point is that the diagonal is infinite. That's when, yeah, yeah, instead of, you've got these things crossing your square, you've got zeros on both ends, they're not going into your square. Now, I mean, when you climb along the diagonal, there are infinitely many components. Well, get slow, get slow in this calculation, both of you. It was very close to get slow in the same part. You could argue, but it's technically, it's a massive thing. You can control the situation, but it's really more complicated. And this is very...

55:00 No, I meant what I'm... Modularized space system in my application from MAMBL. But we want to interpret them as cohomology classes. Yeah, yeah. Okay, that's fine. But you have to identify the various cohomology classes. It's a painful process. It's a painful process to identify very... Because the situation... The geometrical situation is very complicated. And to identify the proper cohomology classes is a difficult job. But what we hope is that once you are back on this problem, we will find immediately a good formula to prove the transcendence of it. I mean, people so far, I mean, analytically minded though, with classical mathematics, they play with complicated formulas. Of course, they have a feeling for complicated formulas, but so far they have not been able to find the right formula. And so, we hope that by... Giving a canonical interpretation to this formula, we will be able to find the right formula. At the end, we will be able to say, OK, take this formula! But the point is that it's cheating, in a way. But of course, you know, there are practical in mind that people who are outdoors, people that they would say, well, if we speak of multi-convergence, they will not understand. But if we say, here is the formula, here is the concrete. But I liked it. The other day, when we're talking about motives, and we quickly get to defining real numbers by lines possibly without points in the integer plane, motives are not harder than that. I mean, well, sure, particular problems can be, but thinking about motives leads you directly to simple... If I can make a suggestion. We've seen a very big illustration of how this rich cohomology, which was developed and then transferred from the analytic into the algebraic categories, is also now being used to attack very subtle issues in number theory. And this might be a suitable place to break and then after lunch go back. And look at the way that it was also seen, particularly in connection with the relation of global and local calvaries, to have these deep consequences for the understanding of how logic fitted into the geometric literature.

57:30 Going back to the topos and to Verdi's presentation in Bahia, which would allow Bill to take up the story from there. Would that be a good way of proceeding? Okay, in which case, let's go for lunch now, and let me switch those off now, okay, and waste the battery of time for this one too. Oh yes, in fact, that's a very good idea. In fact, I didn't close it last night because of the noise from the kids in the street. You weren't here. No, no, no, it's fine now, but last night there was an awful lot of noise. What your old sparring partner might call a bunch of rowdies in the spirit of a bunch of rowdies. A bunch of rowdies. So, what's next for media? Well, precisely what we said just before we broke. Just wait for Bill to arrive. We thought that the next item to take was precisely the... Perhaps break away from the purely chronological account of the development and look directly at the underlying ideas of topos theory as they're crystallized in the Lorbeer-Tenney axis by seeing the logical structure as naturally falling into place within the framework of the algebraic geometry and in connection with that I was particularly going to suggest that Bill and indeed Colin might want to say a little bit about the relationship, for instance, of the local and global coverings to the multiple structure. You know, this business of localization of coverings and the weakly-discipled solubility conditions.

1:00:00 One of the things that picks out the topos of set sort of vision is set-like toposes, from toposes of spaces in general. We'll talk a little bit about the relationship between the projective and injective limits and co-limits in connection with choice and extensionality principles, but obviously other people will have other... I mean, that's a specific thing, but I think more of... Well, it's one of the things that we had... Before you got here this morning, that was one of the things that we had said very much that we wanted to discuss. What Bill was actually looking for, and that's really what made the difference, of course, in terms of the... You were looking for something, right? Different, or at least in terms of your own motivation. Anyway, that's what I'd like to see emphasised. Well, the first approximation, that's in fact exactly what we had said at the beginning of the morning session that we would try to discuss. Yeah, to a rough approximation, I think that's a fair summary of what we said. We'll try and tackle this afternoon, so yes, yes, fine. I think that'll bring us back to McLean again as well, to some extent. No, I mean who was interested in founding logic and, you know, in some more general framework. And one of the things which perhaps might make an actual lead into that discussion would be for Bill to tell us, as it were, what your first reactions were on the hearing of Bernier in La Jolla with this, as I understand it, what was the first expose of topos theory, two categorists, and how soon you saw it as connecting with your own earlier.

1:02:30 A program for expertizing the category of sex without using membership. Other concerns which have been present in your work. I'm not planning to give autobiographical details here, but I think after all it might be appropriate, because as I just discovered, the only published biography of myself is totally in error in nearly every line. I've just published a book that seems to be confused recollections of an old man and consult his notes. Well, if the biographical details naturally lead to conceptual development, which they do in this case, by all means, yes. So I'll just give a few more. So it was, I guess you could say it was from electrical engineering that I started. Amateur radio and... I went to physics in the university as a cyclotron technician and planned to study a major in physics but decided soon on that the mathematical level of the physics courses in the mid-50s was not up to explaining things clearly and therefore I switched to mathematics. He interviewed me for a scholarship when I was just still in high school and was impressed by the fact that when he asked what is the trajectory of a stone thrown into the air, expecting me to say parabola, I said instead it's an ellipse, one of whose focal points is at the center of the earth. I always thought that when I got positive comments from teachers, I always thought they were exaggerating, that they were really overreacting, because this is really only one little thing, right?

1:05:00 That's a good story. Actually, I was thinking about it again, just recently, because it means that you have some fourth-degree infinitesimal, up to a fourth-degree infinitesimal in books. It does look, so it's more, it's not just up to second degree or something, it's actually up to fourth, because it's quadratic. So anyway, so I started to study under the guidance of Truesdell, but contrary to what MacLean says, Sammy was not in Bloomington. He had long ago left, I mean, three or four years at least, long before I even entered the university. It's not that he taught me categories. I learned the categories eventually from... Actually, his name was Ernst Schnapper. He was an algebraic geologist. Schnapper. Ernst Schnapper. Yeah, so one of his courses, he mentioned homology. Did you already live from Zorn, who was in Bloomington, that surely is still teaching at the time? Well, okay, let me say. Sorry, Ernst. So I already became, in effect, a graduate student after three years. I'm taking only grad, you know. General topology and functional analysis, various courses in functional analysis. And I had a teaching assistantship. I mean, this was in order to support my family. Even though I was an undergraduate, I was given a teaching assistantship. So I had an office, which was right next to Zoran's office. And Zoran was a wonderful gadfly. He would often come over and say, If a function is one everywhere except zero, what is its value at zero? He felt there should be a rule of inference which would yield this result. This kind of thing was always sort of playing around on the edges. It never could be made quite rigorous, but nonetheless it was very, very penetrating. And also in Truesdell's lectures he would sometimes attend and make remarks like this. There was a theory of mixtures, you know, a very elaborate theory of mixtures, different kinds of substances mixed together and what will they do and, you know, so Zorin says, have you ever actually mixed a mixture?

1:07:30 That's sort of funny enough. He's quite a character. I really enjoyed having him there. And also I must say, both he, Jumping Head, and Bear Nice, Bear Nice, they both have... There was really an interest in k-theory, for example. They felt this kind of thing is really the next wave. They weren't in any way limited to some narrow definition of logic or something. So, one of Truesdell's favorite students was Walter Noll, who was in a certain circle, the Society for Natural Philosophy, which I remember. He was for a long time basically the god. Even higher than Grotendieck in the sense that everyone followed his notation terminology without any... He himself would change it from time to time, but it was out of respect for his fundamental contributions. He was claimed to have solved Hilbert's problem, one of his problems was to axiomatize physics. And so physics is a broad thing, there are many different... Novel had certainly made a big step toward axiomatizing the continuum mechanics part, and in such according to Truesdell, that's the main part that there is, and therefore he had axiomatized physics, which is, you know, it's not unreasonable in a way to be discussed. So anyway, I was impressed, very impressed, by the way, which on the one hand, Novel would Describe these physical concepts like a body and a sub-body and force and motion and all these things. Very clear concepts. But then, in order to implement this, one had to invoke Cauchy sequences, sequences of Sobolev spaces, atlases, a whole lot of presentation as well as particular determination of what I came to call the degree of cohesion of the spaces. So I felt that really there should be a way to express this kind of physical idea.

1:10:00 In a way which takes account of cohesion, I use my later term, which takes account of cohesion in general without committing itself to continuous, smooth, analytic, algebraic, combinatorial. So I wanted to provide a language which is powerful enough to express all those things, and even indeed powerful enough to, by its own internal properties, distinguish between them, and yet to go as far as possible. In order that, it's an old idea, right, that through abstraction you can actually come closer to everyday intuition, because these concepts are close to everyday intuition, whereas the particular mathematical machinery under so-called foundations are not. So this was the basic conviction and program that I started with. When I heard about categories, I thought in a rather reckless manner to change my whole career, from having begun a graduate career in one area to completely switch, different advisor, different place, and so on, and so I chose Sammy as a destination because he was the one with the most joint authorships. So there was Carton-Eilenberg, sorry, Carton-Eilenberg, Steinrod, Eilenberg-McLean, Eilenberg-Moore, so there was a whole sequence. So obviously this is the guy who is really the moving force in category theory, which is what I want to learn. Very fortunately and fortuitously, Clifford Truesdell and Sammy Islandward were close friends, not because that their mathematical interests were very close, but because they were common art collectors in that sort of milieu, their second lives, were much more closely connected. And so Truesdell simply phoned up Sammy and said, well, here's this crazy guy, you can take him. And Sammy said, okay. So it was just a very bizarre way to get, you know, no entrance exam and all that stuff. But again, I felt a bit guilty that they're, why should they do this? But anyway, so I did that and I learned quite a bit from Sammy when I finally got there, of course, but it was not so.

1:12:30 As I later discovered, Truesdell himself had taken time off from his graduate studies in mechanics at Brown University in order to study logic, even considering to commit himself to the study of logic by going to Princeton and taking notes for Church, which later became his book. Because of these struggles with foundational questions and category theory, I decided to go to Berkeley and study with Tarski and Vaughn, Dana Scott, William Craig, and so forth. The name I knew was Tarski and Scott. So there's, again, something MacLean says in this completely mixed up book of his. That he came to Columbia during my first year there, and Sammy told him to talk to me because Sammy thought that what I was doing was probably, well, was dubious, and so McLean talked to me for one hour and then he decided that this would never work and told Sammy so. Now, at that point in the book, McLean says LeVere lost his graduate stipend and was forced to move to Berkeley. I don't think that's true. I think my stipend, my meager, meager stipend would have continued all the same if Sammy was not so easily swayed. But the bad thing, the really bad thing that came out of that interview is the slogan, you see, which he claims that Sammy told him. All of this was my program. And in any case, E. McLean is the one who publicized it far and wide as being nonsense. Namely, do sets without elements. Some mysterious way to do sets without elements, you see. I never said that. I said sets without elementhood. But the elementhood relationship was giving, again, far too much spurious structure to the things that we wanted to simplify. So, it took me years, and I probably still haven't succeeded, especially with this new publication, to overcome this one slogan. So many people think that's what it is, it's just...

1:15:00 Nevertheless, I remember that some of you, often in discussions, would say, I mean, let's try to make a proof without them. Yes, yes, I understand this, but I mean, it's... It was quite consistent. I remember Bobacki discussion or seminar discussion. That's impeccable, but without... Without elements in the sense of going down to the nitty-gritty of the individual elements. There were generalized elements always. There were always generalized, so-called generalized elements. I always advocate that the word generalized gets too... But that's not it. Yeah, no, elements. But I don't think that's going to change. Well, maybe. Well, I mean, the very word element, of course, is supposed to connote some kind of smallness, right? Basically, as opposed to something more general. Anyway, so at Berkeley I met with considerable opposition, except for Dana Scott, who reluctantly accepted, step by step, some of his advocating. But also Kreisel. So Kreisel made a point actually talking to me. He drove me out to Stanford and back to Berkeley and we'd come and we'd have these discussions and so later, many years later, it turns out that he'd been on the phone to Gödel telling him, at least according to Dana Scott, telling Gödel about this. So Gödel and Berenice were corresponding actually. About this question, not naming any person, but at the same month that I was writing my thesis, February 1963. Okay, so, let's see, is this enough in the way of... Anyway, so I did an initial thesis, and in the thesis, of course, presented the idea of using category of categories as a framework for all these discussions. It was slightly inelegant, and it had a small error in it. Yes, and it's never, you know, it's a small error. Well, the inelegance is worse than the error. The error was easily corrected, but I haven't republished it because I feel, you know, it should be.

1:17:30 Much better, although I think that's the framework in which categories unconsciously work, even if they claim, you know, in church that they are doing girdle bear knives or something. It's clear that that's not true. So the practical framework is really the category of categories, I think. That's his seminar because he lectured about essentially the Groves-Zariski topos, in other words, the topos of algebraic spaces and how this was reconstructed and how the infinitesimals worked in this category and so forth. So, you know, I saw right away that this was... And so on and so forth. Oh yeah, I mean that book is an outgrowth of the fact that a bunch of that material was presented in the seminar, right? Yes, but there's no two ways about it. André Weil presented to Bobacki a foundation of infinitesimal calculus, differential geometry using a kind of near-potent element as far as 53. And he has just a summary among his collected papers. There were four pages which he presented in the Eosman Seminary in Strasbourg. But I have a much longer draft which was presented to Bobacki rather long time. I draw inspiration from that. And there was much more in the Kahler's long Italian paper, of course. And, I mean, the idea went back at least to Studi around the turn of the century. The so-called dual numbers.

1:20:00 The dual numbers, right. Which I think somehow couldn't be properly exploited without the categorical point of view. Just the idea that domains and co-domains are definite at that point. Yeah, right. If you don't have that, then it all sort of dissolves, I mean. So, right, so... Oh yes, that's right, that was 1965-66, this seminar. On January 1, 1966, can you remember that date? That's the date which Dana Scott wrote in large, in his nice handwriting at the top of the page where he introduced Boolean value logs. Yes, yes, yes, right. There's a well-known book on that. Perfect edition now, right? At any rate, this was really, in some sense, even though I didn't fully understand it, this was clearly the signal, you see, that because, you know, this small, logically small difference in some sense between the Boolean algebra and what you get by reducing this module of an ultrafilter or something of that sort, you know, it meant that the previous treatments by logicians had always been tied up with these. Well, you saw it, but neither... Well, I mean, people would have known because they wouldn't have known about Boolean-valued models on one hand, and Paul Cohen, who wasn't into any of this stuff, he wouldn't have noticed it either, although he had vetted the original construction, as I said. I went to Scott's lectures. He immediately started lecturing on this stuff, and suddenly there was never a lecture. No, no, no, there was never any talk. No, no, no, no, no. And Colin went for about a day and then just walked out and didn't stay. Solovey might have made, but he didn't, but he, Scott says in his preface to my book, which is a bit startling at the time, that people thought, that he admitted that he didn't, the real ideas had come essentially from Bob Solovey, and Solovey, I mean, he was the one who first had the idea of summing up forcing conditions to, you know, to get a boolean algebra from them, and that's what Scott says.

1:22:30 Yes, that's what that's what he actually said it was always called the Scots. Yes, that's it almost called we never wrote anything about it. No, that's right. But it was Sullivan who had the original idea and that was I think the one. Yes, that's my point. Yes, that's the point. He was the one with actually that kind of background in a way. I know the day-to-day is amazing, but You know, after all, he'd come from algebraic geometry. I mean, he'd done his thesis of Riemann-Roch theorem, and he was a reclaimed student, etc. He was a reclaimed student. He was accustomed to Bourbaki mathematics. Exactly, exactly. But what you're saying is that it's not clear that even he saw so clearly that this was going to fill in a fragment of the bigger geometry. Oh, absolutely. No, he didn't. He never did, but he was the one, if any of the logicians could have worked with him, that would have been him. But obviously these issues of credit are... If we can get back to the ordinary exposition of... But Cohen was actually... Cohen was actually on algebra and geometry at Stanford. I know, but Cohen wasn't even... Soloway became a logician. Cohen never was a logician. He went back and did that amazing work and then got out of it. Yeah, it's a little bit important to show a lot of this. Yes, of course. I don't understand the line of... Cohen wasn't interested in logic at all. He was interested in solving problems, and he's such a brilliant... Exposed to quite different... In other words, that year in Berkeley was definitely not wasted. Yeah, no, certainly not. Nor was the trip to Wimbledon. Well, sure. It was incredibly... So then came... McLean always wanted to hire me. He decided after all that I was alright. You know, there's the whole story about my thesis and how he accepted it in San Miguel. That's a well-known story. But anyway, he wanted to hire me at Chicago and so... So I said yes and oh but could I please have the first semester leave of absence because there was a gathering in Zurich of people working on triple theory now called monad theory and I wanted to participate in that so he said okay so fine so but when I went to Chicago it was with a definite program in mind with Marshall Stone because Marshall Stone had developed an interest in these matters.

1:25:00 Oh, that's interesting. Yeah, and so I corresponded with Marshall Stone about the idea that the Scott-Solivay models of set theory were really special kinds of... The topos and that this would be relevant to physics so he you know he you can understand he loved all that and so it was he who was really going to greet me when I got to Chicago except that you know he wasn't there for a time and so as a result I had very few discussions with him but this was this was I went there with this program of combining let's say combining Scott Solovey with With the ultimate aim of simplifying the understanding of physics. So this whole program was present there. McLean again says in his book that he gave a course on mechanics and that's what inspired me. He knew that wasn't true. In fact it was a joint course I had proposed. We were both teaching it. I mean, I'm really, I'm reeling with, I'm surprised to see that I got this manuscript, the degree to which the inaccuracies in this book of McLean. Pierre had a question, I think. No, I mean, it was long after they left for Chicago. Oh, yeah. It was long after Sammy left Indiana. No, but the influence of Bay was still felt in Chicago. Oh, I suppose so. I just didn't recognize it as such. Your arrival in Chicago with this program already crystallized in your mind was 1966, the last day of 1966, December 30th. Of course, in logic, the cohort effect was felt enormously. As Dana says in his preface in my book, that was when the bomb fell. You know, frankly, for logicians. I mean, they'd been thinking about the continuum. It was a specific problem, Hilbert's first problem, I think. Was it the third problem? Yeah. Well, anyway, it was a major problem and nobody had made any progress on it, really. Ah, there have been some by Shepardson. You know, there were various limited results that have been, you know... And then a non-logician comes along and does it in a very... It looked very specific. I mean, you know, it was incredibly ingenious, of course. I mean, he's an extraordinary, he's a brilliant mathematician. It was very difficult to see what the general, to extract what the general significance of it was.

1:27:30 Particularly because, well, intuitionists and logic was apparently. Why? You start with a classical model, you end up with a classical model, and somehow there's the actual methods that you're using. In between, it was intuition. Yeah, yeah. Very puzzling. Or rather, hiding, I would say. Yes, okay. But the rules were… Right. And that was extremely puzzling. Now, Emily, you weren't attracted so much to… I know, I know. But the bridge was brilliant, that was really brilliant. To the general picture. Well, that was incredible because it really revealed that these models were not these sort of super syntactical things. Exactly, exactly. There was some sort of obsession there with finitism. Yes, yes. He went out of his way to make it syntactical, but he need not really have done so. He could have only been less scrupulous if he had done it slightly differently. Cairn is on record since then on several occasions of defending him. Yes, a lot of auditions did this, there was this feeling, he went adventuring to logic, even maybe at that time, he had to be very careful. It was like Abraham Robinson's first Formulation of non-state, you know, when he does the whole, it's belt and suspenders, he uses tight, you remember, and then when Marsha Markover came and others came along and simplified the whole thing and made it more freewheeling, you know, rather than this, you had to be cautious. Logically, you were supposed to be punctilious about the details. Right. And I think, I think... Oh, I forgot to mention... Yeah, sorry, sorry. I was in Jerusalem in 64. Ah, yes. Announced... Formalism 64. That's right. At least the idea was more firmly convinced to formalist philosophy because of his construction of non-standard analysis.

1:30:00 And again, of course, I played with that, you see. And again, you see, much later I realized that there's very little in it that's not in, what's his name, the German guy. You simply take the fresh air filter and take the power with respect to that, you've got all the content there. The fact that the logic is interpreted in that. If the interpretation were not conserved, it would be detected already there. So, following this by the choice of an altar building, making it a point, making it pseudo-constant, is really almost spurious for any independence result. But there's exactly the same feature there as in the independence group. It's really just a Boolean value where things are variable. So I got this whole philosophy. Not just from that, of course, but from the word general sheafs. In fact, can you say a little more about exactly how that crystallized and the different sources of that? Again, the Grotendieck theory, as presented first by Gabriel, and of course I studied Grotendieck as well, all these things, I mean... Bundle as a variable space, parametrized by the base, just everywhere in geometry and analysis and physics and logic you have this phenomenon of structures that are variable and should behave like structures, all the same, so the question is, and you see that with Grotendieck topos is a very general answer to the question of what kind of a universe are they living in while they're varying. In the sense that a vector space interpreted in the category of sheaves over a space is really a bundle over the space, but it is a vector space, and the point of view, if you put yourself inside this topos, is merely a vector space, but if you interpret it externally, you see that it's moving. It's varying continuously, in this sense.

1:32:30 No, it's internal motion. I mean, it's internal variation, not comparing different models. Not among different models. Varying in itself. Right, right. And that's when I used to say variable sets. I had to make that precise because the immediate idea is, well, you're changing from one set to another, whereas one set is varying in itself. This is sort of a typical. But later I realized that this is not quite accurate, that, roughly speaking, the real toposes are having a quality which I would call instead cohesion rather than variation, and that the general toposes have both these aspects, cohesion and variation, and that they're sort of pure cases of both, and so on, so that... Well, throughout the 70s, I talked about variation. You did, you did. Very effective. Which is a very important thing, but, you know, so in other words, for example, remember Anders Koch, under this influence, he was saying, well, the gross or risky topos consist of variable sets because the ring of definition varies. Well, that's true, it's varying, but the content that one is expressing that way is really the algebraic style of cohesion more than... You know, unless you pass to the parameter space, say you're doing it over the integers over a field, what varying the ring is doing is really just tying things together to give a model of cohesion more than just a variation in some more precise or narrow sense. So developing that whole dialectic is still underway. Crudely, it's still a case, crudely speaking, it's all variation, but it's controlled, smooth variation. Except for the Boolean-Calvalian case, which is random variation. Well, the axiom of choice, then, holds internally, which is random. I mean, you could just... The fact that it's Boolean means that you can specify something here and something on the complement without worrying about any boundary conditions and you've still got a good thing. I mean, just in that sense, random. You can always take something and it automatically has a complement that will fit with it whatever the shape.

1:35:00 Subjects are a product of the sub-objects of the sub-objects of the sub-objects of the sub-objects. But if every sub-object of one has that property, then you have a Boolean situation, rather than you have no boundaries. No facts. No facts. Sorry, John, you wanted to possibly say it another way, did you? Well, no, I do see it as a kind of fitting. In other words, in the Boolean case, you know, there's certain natural joints, actually almost none in the case of where, well, you know, where you have a top, or coming from a topological space in which you have no Klopin sets at all. In general, you are always going to have these rather large boundaries. In other words, it's like jigsaw. You actually have two things. There's one there and one there. There's no way of fitting them together in general. And the very few cases where you can fit them, and there are other interesting intermediate cases where the Klopin sets, there are some Klopin sets or something, the complemented sets, which are non-trivial complemented sets, and those are exactly the parts of the jigsaw that you can sort of fit together, but in general you can't. Anyway, that's the sort of vision I have. Actually, this talk of boundaries is one that actually makes precise, because I was amazed last week, my colleagues who mainly work on topos theory, and most one of them had realized The lattice of subtoposes of a topos is co-hiding, co-hiding. In other words, the internal logic of the topos, there are two totally different questions because given any topos, then the sub-objects of an object form a hiding album in the sense of the infinite unions distributed over finite intersections. So there's a kind of implication operated on sub-objects. Whereas if you look at the lattice of sub-toposes, it's the opposite. So that A and not A intersect in the boundary.

1:37:30 There's a definite notion of the boundary of a sub-topos, even though there's no notion of the boundary of a sub-object within a topos. Because they do these elementary calculations. Tierney and I worked out this, Tierney and Tierney worked out the aspect that's extremely simple, that the topologies, the notions of covering, just boil down to one simple endomap of the truth value of it, satisfying three simple axes, and one easily calculates that this forms a hiding algebra, as you see. The growth operator. Yes, yes, yes. The covering operator, local operator, whatever it's called. But then, you see, and these parameterize the sub-objects. So it's a striking situation that even though these are large categories, there's only a small number of sub-categories of the type in question. And even they're internally parameterized. Then they stop and say, well, okay, the subtopos is from a hiding house. It's nonsense because it's a reversed correspondence. The more coverings, the fewer sheaves. The more sheaves, the less coverings. So it's actually a co-hiding. So that means that there is a boundary for any subtopos, which is another subtopos. Totally unexploited, as you said. No, not really. There's an immense amount of structure there which has not been delved into at all. It should have direct significance for model theory because, as Grotendieck again pointed out, essentially these toposes are classifying toposes for positive theories. So if you considered model classes for positive theories... Then, there should be a boundary model class. Distributive lattices among something else? Well, maybe it's the boundary of another variety. So this kind of thing, there could be very simple examples where this comes up, but no one is carrying it out. And perhaps leave a nucleus of a logic of contradiction. Well, certainly in the beginning. Not a paraconsistent or anything.

1:40:00 I don't know, I don't know. But it's a curious, anyway. But we should discuss that later. It's an important fragment. I mean, why is it that Gopin said, you know, I mean the they used to call them what was it called? The cohorting algebras were called barurian algebras. Yes, although, actually that's... Well, I suppose Brouwer thought that Heidemann was getting things from the wrong hand. Yeah, okay, but he never... It has nothing to do with Brouwer. No, it's just that... It's just that Dana Scott published the so-called Brouwerian lattice. Dana Scott explained it this way. That Tarski had this fixed idea that algebraic varieties are described by saying f of x equals zero. And so he turned everything upside down formally. He was talking about hiding algebras, you see. Formally, it looks like co-hiding. But he always thought of it as that. Because f of x, whereas, of course, in logic, you care about f of x equals 1. It would be more natural to take f of x as true. Yeah, that's quite a notational tangle there. I think Brouwer had nothing to do with it. Well, it's just his name was floating around. Yeah, yeah. Tarski knew that formalizing... He never used the term. Well, that's an alpha subject. I didn't know that. Yeah, he never used that term. No, no, no. Tarski worked, of course. Tarski and McKinsey. Yeah, but all that stuff by Tarski and McKinsey, they didn't use. They did very fundamental work in representation of these things. You know, and lattices, represent lattices of open sets and all that stuff. And Garrett Murkoff calls them barolettes. He calls them barolettes, that's right. But those are... I think so, maybe... In his lattice theory, in the lattice theory. I thought they were called residuators. That term was used, but I think... I wouldn't swear to this because I did a reading course as an undergraduate on these things. And suddenly it was really about to go by. I think I probably was reading other papers, certainly, where they were called Brauer letters, maybe by Montaio or somebody like that, I can't remember, but they certainly were called Brauer letters as long before the book was published. Well, that's true. That's true. But those letters, they were written in the 40s. I'll take that as an issue.

1:42:30 There was an issue in the 30s. 30s, yeah. The first one was in the 30s. It went through Sandoval. Yeah. It went through Sandoval. There was a serious byproduct of this confusion, which was... That those people who were perfectly in position to do so never even noticed there was this boundary operator. In particular, not the boundary operator that satisfies the Leibniz product rule, this powerful stuff which is really just, if you wish, hiding logic turned upside down, but then it has this completely different significance. That's what some of your friends down in Australia don't understand. Now, well, that's it. That increases my first Ph.D. Then if you, you know, they think if you just... Well, I wasn't responsible for mathematics. No, actually, the topos theorists have the same problem. Well, since the topology is form of hiding, I was ready to serve the sub-topos. You know, there's the question if you, okay, if you have this abstract lattice, fine, you can turn it upside down with the other kind of lattice, but if it's embedded in a larger category as being, you know, the sub-objects, well then there's one way of looking at it as material, so to speak, and the other is purely formal. The one that's compatible with actual sub-objects inclusions is quite, you know, it's distinguished. Yes. And so in those cases where it's colliding, Then you have an intrinsic notion of boundaries, not some additional structure, and it does satisfy this Leibniz rule, but nobody ever noticed that until we around, what was it, only a few years ago, noticed that it was, well, see you later, that you have this Leibniz rule, and it's just amazing, I mean, it's a very powerful rule, you can start calculating stuff, all of which could be translated back into intuitionism if you wanted, I don't know. What it meant, because again, the thing is that the closed sets in the space are somehow much more tangible than the open ones, and so that's quite true. They have their boundaries, intersect them and so forth. They're walled. There's a definite end to them. But, of course, in the case of the internal language, one striking thing I like as a little exercise, you know, in sort of local set theory or internal language of the toppers is that, you know, omega... Can't be, you know, a co-hiding algebra, unless it's Boolean. I mean, you really are very, you're stuck in that.

1:45:00 Exactly. It's a nice little exercise, actually. It's a very beautiful... But if something is not true, it doesn't mean... No, no, the thing is, also for sub-objects, because there are many toposes. For example, any pre-sheaf topos over a Boolean set theory. Where the individual sub-object lattices are actually co-hiding, they're naturally hiding, but they're individually co-hiding. Yes, it's a good point. I got a lot of insight out of the fact that even though they're not totally natural, they are preserved by some morphemes. But then in that case one needs a more general framework so that you get to hiding the magnets rule. For example, relative to Cartesian product, for that it's sufficient to know that the projection maps, which are very special maps, you see, the projection maps substitute into the... But then in that case, I always felt one, yeah, but I understand it, that one needs a more, if you like, more general, flexible, formal framework, you know, which sort of transcends the internal language, of course, in order to do justice, to make all those calculations to be, you know, at the same level, without having to talk about something external. See, so in the same sense that the Hencken models are functors that preserve products but not exponentiation, nonetheless they must therefore compare the exponentiation, so you can have homomorphisms of, say, co-hiding algebras that preserve union and intersection, don't preserve the difference operation, but of course there's an induced comparison. Hankin really discovered something there, I don't know what philosophical basis he had for it, but it's a constantly recurring property that you have these, once you get into the two-categorical world, functors can preserve certain things, but then if those things themselves have adjoints, those adjoints won't be preserved, but there'll be a comparison all the same, so you don't lose control completely. What you need is some kind of co-local set, something that combines, something where you make the same kind of calculation, which is similar to the type of calculations that one makes in the local, in the internal language.

1:47:30 Where they look the same. After all, boundary calculations don't look any different, of course, from closed sets. Yeah, yeah, who cares? And you sort of do it in set theory, but set theory is, because you're essentially, what you're essentially is Boolean, and so you're just using Boolean properties of complements. But formally, the calculating with open sets and calculating with closed sets is the same. Look, in the set theory case, it's because you're... As long as they stay inside one space, but if you can move the space... Yeah, neither part of it makes it look like one space. Well, it will never look like one. No, no, no. No, but I mean, there is such a thing as possibly a formal theory of the two categories of topos at a given base. We want a formal theory of topos, but Bain and Boo actually made some. I mean, that's where the logic of contradiction would live. That's right, that's right. Because you have inside each one, you have the hiding logic between them. You see, notice you have, by the way, you have existential quantification. Given in geometric morphism, it can be factored into a comonadic and a full inclusion. This basically gives you a notion of image, which will behave, which satisfies the rule of inference through existential quantification of, you see, thinking that the logic is in the glattis of subtopos, in the narrow sense, and then you can substitute, of course, again, not preserving difference, but preserving union intersection. And there's an adjoint to that, taking the image. So you have existential, but apparently not universal, quantification. So that's relatively interesting. Even this tin can problem, it's in Nekomo 90, where there's a long article called something about the future of topos theory, where I mentioned this Berdier's talk on the beach in La Jolla in 1965.

1:50:00 Engler and I were in the booing section. He said, oh, this can't be found. He started off by saying this is about foundations. He goes, oh no, this is not foundations. This is based on set theory, external set theory. It's not expertized. So in perfectly acceptable French fashion, he threw down the chalk and abandoned the lecture because he'd been insulted. This was incredibly stupid on our part because... Very narrow, very narrow... I'm sorry about that actually. I mean, there's nothing more to the story in that sense. Later we found out. Very soon thereafter. Anyway, this is about the Tin Can Problem. I'll try to describe it in the short, as a short article in the Komen Heidei, called The Leibniz Rule for the Kohaiting Boundary and Certain Preciate Topologies. So, every appreciative topos has this co-hiding structure, but, and within a given object, the boundary formula, the Leibniz rule about the boundary is automatically true, it's true in any co-hiding algebra, it's just a property, it's a consequence of the rules of entrance of the co-hiding algebra. But the... The tin can, this is when I teach calculus, you see, I talk about the tin required to make, to contain the beans, you see, so the boundary of a cylinder, which is itself a product, Cartesian product, object, is a union of two parts, namely the, one of the parts times the boundary of the other, which is the two ends, and on the other hand, the opposite thing, you take the circle times the... So it's a Leibniz rule, but it has Cartesian product in it rather than intersection. And we're actually talking about sub-objects of a bigger product.

1:52:30 See, I'm thinking of taking the disc out of an interval out of a line. We're inside a bigger product. So products are just intersections. In fact, it's hard to be called a cylindric algebra. The idea is that you're substituting along projections, you get these special sets called cylinders, but then their intersections are the sub-Cartesian products. So, as I said, these are the results of substituting along the projections, and so in order to get the tin-can formula, the Leibniz rule for Cartesian product, in a topos that already has both kinds of them, you just have to know that substituting along a projection All of this preserves the co-hiding knot, because the boundary is just A and not A, right? So then, I actually found a condition on the small category C, so that for pre-sheets on C, this formula is true, for which the Leibniz formula is true for the co-hiding boundary. And it's kind of amusing because it's just that a map can always be factored, but not in the image way, like the graph of a map, factoring it through the Cartesian product. More exactly, if you have two opposed maps, Say, A to B and B times A. Then in A times B, you have the graph of one map and the graph of the other map, so the graph always retracts onto one factor, so really you've got two impotents on this third space, A times B, such as you split them, but then you compose the two corners of the splitting the wrong way, so to speak, because you take the projection from one and the injection from the other one, then you get your map, you see. So any map can be expressed that way. Clearly, if you have products, because I just told you how to do it, use graphs, you see, but equally well if you use co-graphs, if you have sums, you use co-graphs, you get the same result, but there are a lot of small categories that have neither sums nor products, but where this kind of factorization is possible.

1:55:00 You get these, this third object is necessarily a bit bigger than the other, so you get an infinite, even starting from the smallest thing, you get an infinite sequence of... Objects equipped with idempotence, and it turned out that the simplicial sets have, sorry, delta, the simplicial category has this property, and so the tin can property is. So after simplicials, that really is about topologies, just like they've always been. This boundary formula is a crucial intuitive component. ... about quantifiers and sheaves, and they've been in 1970 because, well, I mean, it was a sort of general thing, but I was going to mention the, you know, on one hand, the, you know, the contribution from, you know, from Booley and Valiant and from all submitting with Gabrielle, but on the other hand, the other input from your emphasis, of course, on adjunction. I mean, the notion of a junction, of course, was, well, you didn't discover it, but you exploited the notion more than anybody else in areas that nobody, you know, where you went boldly where nobody had gone before, and in particular, of course, for logicians, and the point that I'm still astonishing first-year logic students with, well, you know, these operations are junctions. Now, of course, this was your idea. And one coming, I guess, from work, early work, just in your thesis, of course, but there was some kind of these two notions in the idea of quantifier, your talk, quantifiers and sheaves, it was some kind of a fusion, if you like, of these notions in discovering, of course, the axioms, developing the axioms for an elementary tapas. I thought you'd like to say, well, I think that's what helped you get, I think that's what helped you get, yeah, to get to that, yes. Right, so, okay, right, so we have Dana Scott, 1966, Gabrielle, 1965, went to Chicago to talk to, with Stone, actually Stone had a student.

1:57:30 The idea was the student was going to develop the million-valued model of algebra, the topos, was a student of Marshall Stone, and in fact you don't know him because he, you know, seconded, in fact he, when I moved to, when I moved to, when I moved, when I moved to Stanford University in New York, this student came along with me, and then one day, Most of the books from my library... Oh, he really described it. Oh dear. And he was missing too. I could probably conjure up his name, but he didn't do what Marshall Stone hoped he would. David, what was his second name? I don't know. My wife knows. She knows all these things. Yeah, so anyway, so again, as I said, contrary to Saunders' story that I was inspired by his course on mechanics, which was actually a joint course, it was not a bad course. The thing was that these ideas about applying what I'd learned from Gabriel basically to this project of having a simplified framework in which continuum of physical problems could be discussed, it just sort of jelled, you know, and I was giving a series of lectures, so in May of 67 I gave three lectures where I expanded. It's essentially the paper that was eventually published 12 years later. Yes, this was, but again this paper is essentially what I said. I mean, Ken McClain for some reason thought this was terribly wrong, but I put in comments, later comments, but they're clearly marked in the paper, so if you just carefully excise the things that have three stars, then you'll have essentially the original. He did give me his notes that he'd taken in Denmark to compare this very thing on and I just couldn't see what he was concerned about. No, I couldn't either. Eduardo Dubuc was there and he had his exact notes. So during this controversy he reproduced them and mailed them all around and said,

2:00:00 So part of it was the adjoints, that's right. In other words, just take specifically the notion of function space, which is absolutely crucial to all this stuff, as Horavitz recognized, so you see in the 40s at least, Horavitz incited Fox in that he gave his own development of... So what he wanted was exactly the property of the Cartesian closed category, except in the concrete context, so he didn't put it that way. But then when you come to Kahn's paper, Kahn was, and still is, an active researcher in combinatorial topology and the simplicial sets and so forth. So in his paper on adjoint functors, he points out that for simplicial sets there is this internal harm. Again, he spells it up. It's a concrete example. You can give the definition of this, you know, in terms of elements, and then verify that it's adjoint. But I noted, well, adjoints are unique. That's a general theorem. So therefore... You can use it as an axiom. In other words, if you set things up properly, everything you want to do follows just from adjoiners, or really from interlocking systems of adjoiners. I'd already pointed this out about the quantifiers long before that, but... What's the term, I just, I'm sure that, just remind me, can you use the term adjoin? He did, and that would suggest that according to McLean's paper it goes back to presumably the idea of a jointness in functional analysis, right? Well, at least that was what McLean says. But why did he use it? I joined operators. McLean had to give some speech at an anniversary of Marshall Stone, 60th, 70th, 80th, and probably 70th birthday.

2:02:30 And so he asked me what he should say about Marshall. How does Marshall Stone fit into the development of the categorical point of view? So I said, well, there's the adjointness of function algebras and the Sohn-Chak compactification, all that sort of thing, the Sohn duality, which by no means limited to Banach, Tingle, Buhlmann algebras, what the logicians seem to think. It was actually about continuous functions and so forth. So I said, well, that's clearly, and that was one of the topics that I'd been teaching in Truthbell's course that helped me to discover the idea of adjointness without naming exactly that. So I told him this, you see. So Stone inspired the notion of adjointness. Explained that way, McLean carried this into the lecture hall and turned out, well, adjoint operators, you see, you write it as... Yeah, yeah, you know, with inner product. Yeah, yeah, sure. And inner product notation, aha, wasn't as fairly common. So, but I think that's completely misguided. I haven't practiced, you know. It's incited various students to try to concoct some way that adjoint operators or adjoint structures don't... none of them have ever found anything. So it was actually kind of misleading. So, it's not a term, but then in that case, the term adjoint was simply some general con- But again, the notational similarity might have been what he was talking about. There's no, it's not necessarily wrong. It's just a very inadequate explanation of the role of stone, you see. The role of stone is much more profound, I think, than just transferring some notation. One thing which it strikes me might be helpful in providing orientation, including as it were to the wider audience, would be if you could just run over again what you were saying to Angus briefly during the lunch interval about the QD objects in the topos in relation to the connectedness of the sub-object classifier and the... This relationship between the decidables, in other words, what you refer to in the Kamo lecture as the kind of unity of S-U-D objects, and in relation to this misperception, this propagating, was just as it were the applications of topos-theoretical ideas to logic, whereas in fact, as you made clear this morning, the whole point was to see the logic as already being there in the...

2:05:00 Yeah, well, since we knew that quantifiers are adjoints, and the crucial point in this thesis, you see, I wanted to axiomatize the category of categories. I knew that the functor category is a crucial operation, in fact, practically everything you think of is part of a functor category, if not a whole one. But then, to treat that axiomatically... Adjoint has presented itself. I think that's probably, that may be the first time that adjoint was used as the defining property of the X-Humanoid system. So, with this program, which I call categorical dynamics, and already there, of course, there's the observation, well, if you have a suitable category of cohesive spaces, then you can also construct categories of dynamical systems over that. The change of time, you know, if you go from linear time to circular time, this was a construction that, say, Smale used in a special case, without ever pointing out that it's just a con-extension along, it's a change of, you know, internal category, or going from discrete time to continuous time is also an adjuvant. So all these kind of different things were... You know, obviously in each case there's a tremendous amount of particular information, but at least there's a uniform conceptual framework, which anybody can understand, quote-unquote, for doing that, you see, because you want the universal smooth time dynamics for a given discrete time. But all that's sort of clear, it's just a matter of making it. So I recognize that it would be necessary to more fully expertise the notion of topos as a framework for the categorical dynamics.

2:07:30 In the article, in the talks, I mentioned the topos as a verdié, modendié, I forget which names I mentioned, but at the same time, in order to have this sort of conceptual simplicity, I realized one would have to... So, again, I made the next step when I was living in Zurich at Hickman's Institute in 1969. Namely, that in fact there is a sub-object in every Groton-Diktokos, and hence a power set. Oh my god, a power set! Kantor should rise up from his grave! There's a power set in every Groton-Diktokos, and at the same time, the characterization of it is elementary. It surely was the input of logic, you know, that led you to see that, or you and Tierney, or you to see that. What I mean is it didn't come from, you know, we really weren't interested in, any of a subjective object classifier was something that may have been implicit in what Groton was doing. Let's back up a bit. That was the purification of it, so to speak. No, the original desire was that, well, okay, some kind of general logical background that I wanted to have. So the fundamental construction of the associated sheaf, well first of all the definition of sheaf, and well for the definition of a Groton-Deke topology with respect to which you can define a sheaf, so that whole sequence of requirements, but it starts from the associated sheaf. How do you construct the associated sheaf? Isn't that a huge direct limit over a class of coverings and then you adjust it twice and isn't it all this infinitary stuff? Well, no. You were doing all that to reflect into the sheaf. What's a sheaf? Isn't that something that satisfies a universally quantified,

2:10:00 And so basically the first idea was partial sections, that any object has a space of partial elements, we might call it now, but if you're going to get in a sheaf topos, where the pre-sheaf topos will be partial sections, so the idea of a sheaf is one for which the... The partial sections whose domain happens to be a covering, in fact, are equal to the object itself, in other words, the elements of the object itself. So there's the partial classifier, which has a morphism to the sub-object classifier. So it's the inverse image of the topology along that that you want to restrict to in order to say that the object is equal. So that's how the truth of object value comes out as the partial map classifier for the one point space and the role of the partial map classifier is to facilitate in finitary terms the construction of the associated sheet. So of course, it gradually became clear to me that this was also logic, but that's where it started from. Tierney proposed to me later that, you know, we should have a pack sort of like Eilenberg and MacLean had, namely, never to reveal which part we were talking about. But it was kind of difficult, because I already had given several lectures on what I've told you so far. In fact, maybe that's where I met you, at Oberwolfach? No, no, no, no, we only, we met in 65. We didn't meet, oh no, no, no, we did meet afterwards, but it wasn't at Oberwolfach, no. I wasn't there at that meeting. Anyway, so I presented this idea of Boolean-valued models in a way that wasn't completely axiomatized, but at least it was correct, I think, that using infinite products over a given Boolean algebra and so forth, that was before, that would have been...

2:12:30 So when was that linked up in some way with the earlier work, which may have always been in the back of your mind, but you know, work on the elementary theory of the category of sets, the earlier work that you'd done? Well, there's a certain, you know, connection there, which I'm sorry, but I mean, when was it made more explicit? Okay, okay, okay. So, elementary theory of the category of sets was actually kind of implicit in my thesis. I mean, I talked about category of categories there, isn't sets a category of categories? But it was because of teaching this course at Reed. Reed, I'm going to be... It was a college for exceptional students, so it was considered the right thing to do to give the freshman in calculus some kind of novel foundation. And as I explained in the commentary on this paper, which has recently been reprinted electronically, I spent the whole summer trying to figure out how to do this with ZF, because even though I had already made the... Taking the position that the category of categories is going to be the element foundation, I still thought, well, maybe for elementary students I should do it. That sounds crazy now. I found out laboriously that, my God, if I'm going to present this element of stuff and then develop that time of set theory and then translate it into the type of operations we need to do this and then translate that into calculus. That'll take more than a semester. I was planning to give them a reasonable foundation in the first semester because in the second semester I was going to talk about stuff like completions of metric spaces and finally additive measures on Boolean algebras, which I did, but I could only do that because I presented a rigorous foundation based on the actual operations that you use, rather than on some alleged underlying ontology. Yeah, I'm moving back and forth here, but certainly it was in 1964, before I went to Europe at all, that I of course started thinking along the lines, well look, Rotendeek has studied abelian sheaves and he's axiomized abelian categories with Rotendeek 85 axioms and all these things.

2:15:00 There are, I knew from Godema, there are sheaves of sets and so they must form a category which is similar to Grotendieck's categories, only different, and so I mean I sketched out just this idea before I even... I met Benelou six months later. He told me that there existed this thing called topos theory, but I had no idea what it was directly until six months after that when Verdi started off on the beach there. So of course the idea of the variable sets and the sort of starting set theory, that was clearly one case of the cohesive and variable sets that we were trying to acclimatize. Just explicitly assuming that function spaces exist. And then Czerny's contribution. So I got it up to this point. I even knew that topology was an endomorphism of the truth value. But I had the thing, you know, because there's this covering of covering condition in the definition of Schiff, right? Yeah. And if you sort of translate that directly, naively, it turns into a higher order statement, you see. So it was not apparent to me, but Tierney worked it out. We had a joint seminar, which he directed really, in Halifax for one year. And during the course of that, he realized that all these very simple actions, now well known, implied, you know, the equivalent to this, just the impotence, I guess, of that operator is equivalent to the... But since it all came, as you said, you know, in the construction of the associated chief, so in other words, in some sense, the truth value object... Topology can emerge in some way as the domain, if you like, of what turned out to be the operator that represents a topology.

2:17:30 The whole thing kind of, since you were conserving coverings initially, and then it turned out, of course, that a topology can then be represented as a modality, whatever you like to call it, on omega. Well, we know we've emerged in some ways, in some ways, the support of that, of that. Oh yeah, it's certainly played a big role. Yeah, but yeah, so the other things we achieved during this seminar, sort of more slowly, was because the first idea I had about the axioms for the whole thing were not just exponentiation, but the pi operator. That is the relative exponentiation, if you like, because You can see that that's the thing you're constantly using in practice. And also the partial map classifier you see as something that you're constantly using in practice. And so it was kind of, neither one of those implies the other or anything, but somehow together... The mere assumption of the partial map classifier for a single point, i.e. See, that's what Shue said, it's the partial map classifier for a single point. And the global explanation. Take it together, in the quantifiers and sheets, this contradiction develops into this much richer one, namely the original one of pi and tilde. It just comes out. Which of course is, how do we know this? This is because set theorists have implicitly done this already 50 years ago, we just have to be careful to make sure we're not doing any undue negation or choosing that it works. It's just a naive, so that's what the power set puncture sort of does that Grotendieck didn't have. Mainly the possibility of just constructing all these things in sort of a naive set theoretic way and then verifying, of course, that it actually does what you want it to do, but it reduces the complexity of the typical definition by a lot.

2:20:00 Did he have any real interest in it? I talked to him about it. He used the truth value object in a very crucial way in his work on homotopy theory. Now his followers, Chesinsky and Maltinatis and so forth, they have this thing they call the Levere element. It's nothing else but the truth value object. No, really. I didn't know that idea. They've taken that fragment and used it. And he was very, obviously he liked it. Too bad I didn't think of that, but the one thing I didn't think of, but it sort of simplifies everything, because without it you can make all these very complicated constructions, but many of them, of course, not all, but many are really just sort of elementary constructions, but done in the Tocco school. Well, I mean, I remember what I meant. One of the most impressive things, I mean, as far as I was concerned, was the fact that you can recognize, I worked it out on my own, you know, was the representation of a topology, which is a fairly complicated definition, you know, in a small category. And then what is it? It's just a modality, a very simple, just a modality. It's very, and that reduction, to me... It was so impressive. I mean, because it just makes, you know, something very complicated written in the external, relatively complicated, and then all it is, you know, mu squared equals mu, it's just a closure operator, it's so pretty. But see, those scholars are not even using that. They just use it in this odd rule as a kind of generic interval for connecting up things by a, you know, in other words, in the case where they are lost. I mean, no, this is not to use it. This is my idea also, right? When the truth value object is connected. And it gives you a notion of homotopy because you can say, well, if I can parametrize the difference between two maps by that, zero and one, then it's a homotopy notion. And it's some kind of generic notion of homotopy. So they're using the truth value object all the time, or even talking about sheaves all the time, but they don't use this fact that the sheaves are just, that the topology is just, well.

2:22:30 It's a slow fight. Seems strange. I think it's just because Grotendieck didn't do it in Pursuing Stacks. He was interested in that fragment, so he didn't do the other and so they don't do it either. Sorry to say. I was not interested in a finitary argument because I think one of his main contributions was to use this largely infinitary construction in categories. From the very beginning, I mean, we were, I think, Samuel Fulton was more interested in McLean, more interested in everything finite, but we were both accused of infinite sums or infinite coordinates and things like that. Well, when I say finite, the point is you put yourself at such a vantage point that the things become finite, that they're actually infinite. No, it's not. No, no, no, no. That's not a matter of fixed. It's just, you put yourself to the level of the universe, that every space seems small. Well, you know, I mean, the expression is finite. Yeah, the expression is finite. To what end do you mean to resort to an outside set theory with all the resources there from infinite jobs and so on? He was happy with that. He was happy and he was proud of that. It was not for him a big thing. Certainly not. So that's why I mean to say he was not interested. He was interested. One of Grotendieck's very fundamental teachings is that you should always look at the category slash an arbitrary object. You see, so on the other hand, he talks about, he and Verdi, they talk about utopos. What's u? Well, u is for them what you say, namely an external set theory. But in fact, it could be any topos, even Grotendieck topos.

2:25:00 Precisely working over the base topos, which is not the discrete sets. It's very fundamental, I mean, Giroux used it also after I persuaded him about this. But in order to do that, then one has to have the proper algebraic conception of what the base topos could be, what sort of a thing is the base topos. So it's still, so to speak, external to the topos you're looking at, but in another sense it's internal, so that you don't really use, as you say, an external set theory. So essentially, in order to extend the phrase utopos to its proper generality, even from the Godendieck point of view, we needed something like an elementary theory of toposes. No, certainly you don't have my set of objections, but I think that was not the aim of the lecture. And there is, like I was mentioning, there's this joy he seems to have at using the set theory he got from Thierry de Lassonde in the Tohoku paper, and that, yeah, I mean, it's not an important point, and he did like your ideas in principle, but it's, there's a style point there. Well, maybe it was also, to some extent, ultimately, interesting in the end in solving problems. I mean, there was this strong, surely there was a long term, for example, to prove the vague conjectures. You know, much more than, I mean, in other words, perhaps more focused on more specific sorts of problems and then, you know, that the creation of the general machinery was something that... But he writes a lot about this, and what he says when he writes about it is that solving the problem is the least of the points of the project. The point of the project is to... So, to have the problem solve itself naturally, to have the solutions to this problem suggest many connections with other areas of mathematics, which of course is the solution. Because our solutions will flow out. Yes, but still, there is the problem that is the ultimate goal in some way, even though you want to find the easiest way of solving it. No, no, because if it's solved in an unexciteful way, you might as well not have done it. Dialectically, how do you know what your problems are?

2:27:30 And he takes this attitude, maybe it's too personal in a way to focus on this one thing, but the pietic proof of the first vague conjecture, he's happy to say, well, it's the wrong proof. It's not worth having because it doesn't make connections. Well, of course, there's some personal rivalry there, too. Colin, I'm trying to say, my mind is blank. I'm trying to think, what's a good concrete example of a base topos other than sets? I mean, where does it, algebraic or topological? I would say G-sets. G-sets certainly could be used as a base. If that's what you want, I mean, because John Rousseau does it over a topological space to make the Weierstroth cooperation theorem. Right. Okay, right. There's a good example. That's the Weierstroth cooperation theorem, which is about, let's say, n plus 1 complex variables, is actually a result about one complex variable. In the quote set theory where everything depends on in complex variables and this this is useful because there's so much machinery that you can apply uniformly into these cases. You can apply the machinery everything you know about one variable directly to input one variables. Of course you have to check that it gives the right result because the logic is not Boolean and so on but yeah that's a good example. Yes, in the work of Tony, also as a minimum and complete variable. Considering a function in one variable over a function of a minimum variable is a common tool, very common tool. And yesterday I spoke about the Dolbo lemma, it's exactly that. And when Kotanik uses the tensor product of function spaces, it's exactly for the same purpose. You know, Christian Rousseau did that and one or two other things and then left to the category theory, as many people have, unfortunately. But I had, for example, the projects.