Morning Discussions (contd.)

0:00 So it's all finitary. 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 I presented this, Tierney proposed to me later that we should have a packed sort of Allenburg in the Plainhead, never to reveal which part it was, but it was kind of difficult because I already had given several lectures on it. That's what I told you so far. Maybe that's where I met you. No, no, no, no. We met in 64. We didn't meet. Oh, no, no, we did meet afterwards, but it wasn't as old as I thought. It wasn't there at that meeting. Well, Mostofsky was there. Yeah, Mostofsky was there, but I wasn't there. Anyway, so I presented this idea of Boolean value modeling in a way that wasn't completely axiomatized. He was correct, I think. 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 steps, the earlier work that you've done? Well, there's a certain, you know... No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. Because 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.

2:30 Taking the position that category of categories is going to be the element foundation, I still thought, well, maybe for elementary students I should do it. It sounds crazy now. I found out laboriously that, my God, if I'm going to present this all my good stuff and then develop that kind of stuff here 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. And I was planning to give them a reasonable foundation in the first semester, but in the second semester I was going to talk about stuff like completions of metric spaces, and finally additive measures on Boolean algorithms, which I did, but I could only do that because I presented that rigorous foundation based on the actual operations that you use, rather than on some alleged underlying ontologies. 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, Grudendieck has studied abugan sheaves and these axiomatized viewing categories with Grudendieck 85 axioms and all these things, and well, I mean… There are, I knew from Godemont, 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 Bamboo six months later, and 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, and there DA started. So, of course, the idea of variable sets, that was clearly one case of the cohesiveness of variable sets that we were trying to acclimatize, just explicitly assuming that function spaces exist.

5:00 I got it up to this point, I even knew that topology was an endomorphism of the truth value of it, but I had to think, you know, because there's this covering of covering condition in the definition of the sheikh, and if you sort of translate that directly, if you translate that kind of naively, it turns into a higher order statement, you see. 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, now and all over again, implied, you know, they're equivalent to this, just the impotence, I guess, of that operator is equivalent to the… But since it all came, as you said, in the construction of the Associated Sheet, so in other words, in some sense, the truth of the matter object The topology can emerge in some way as the domain, if you like, of the operator that represents a topology. Since you were concerned with coverage initially and then it turned out, of course, that a topology can then be represented as a modality or whatever you like to call it. Well, I mean, Oligo emerged in some ways, in some ways, the support of that. It certainly plays a big role. Yeah, so the other things we achieved during this seminar, sort of more slowly, was because the first idea I had about axioms were not just exponentiation, but the pi-opera, that is the relative exponentiation to life. No, you can see that's the thing you're constantly using in practice, and also the partial math classifier, 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 that the partial math classifier for a single point, that's what truth did, was the partial math classifier for a single point.

7:30 And the global exponentiation, taken together in the quantum quantifiers and sheets, this contradiction then develops into this much richer one, namely the original one, the pi and tilde, 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 having any... Any undue negation or choosing, it works. So that's what the power set functor sort of does that Rodendy didn't have. Mainly the possibility of just constructing all these things in a sort of value set theoretic way. And then verifying, of course, if it actually does what you want it to do. But it reduces the complexity of the typical definition quite a lot. Was Grubnick ever really, sort of, with this work? I mean, he was, but 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, and that's where Mahomet took the theory. Now his followers, Jasinski and Maltinakis and so forth, they have this thing they call the Levere element. There's nothing else but the truth I've watched. No, really. I didn't know that. They've taken that fragment and used it. And he was very, obviously he liked it. Too bad I didn't think of that. The one thing I didn't think of was it sort of served. It sort of simplifies everything because without it you can make all these very complicated constructions, but many of them, of course, but many are really just sort of elementary constructions, but done in the...

10:00 Well, I mean, I remember when I, I mean, 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 representative of a topology, which is kind of a fairly complicated definition, you know, in a small category. And then what is it? It's just a modality, very simple, just a modality of a... It's very, and that reduction, to me, 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, Newton's expolvers are not even using that. They just use it in this odd rule as a kind of generic interval for connecting up things. And that was in the case where the truth was... They are lost. I mean, no, this is... Not to use it. That's 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. And this is some kind of generic notion of homotopy. So they're using the truth value object all the time. They're even talking about sheaves all the time, but they don't use this fact that the sheaves are just, that the topology is this. It's a slow fight. I think it's just because Grotendieck didn't do it in pursuing stacks. He was interested in that fragment. He didn't do the other, so they don't do it either. Sorry to say. This means what you do in category. You have the summary points and what's more important. You're not playing, you're trying out, but what are you going to solve?

12:30 When I say finite, the point is you put yourself at such a vantage point that the things become finite, which are actually infinite. No, it's not that. That's not a measure. It's not a matter of strict finiteness. You put yourself below the universe, and then it's based in small. I mean, that is the expression, isn't it? Yeah, right, the expression is the expression. That's fine. Now, you don't have to be so... But then Monterey, who got the nickname to resort to an outside sector where there's all these complicated, all the resources, the infinite source and so on. He was happy to do that. He was happy to do that. On the ground of that, it was not for him a hindrance. It was a void. Certainly not. So that's why I mean to say he was not interested in it. He wasn't interested. No, and you see, okay. One of Rotendieck'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 is very alien, so they talk about utopos. What's u? Well, u is for them, what you say, in external set theory. But in fact, it could be any topos, even Rotendieck topos. The base topos, which is not the discrete sets, is 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 a 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 only in another sense it's internal, so you don't really use, as you say, an external set theory.

15:00 So essentially, in order to extend the phrase utopos to its proper generality, even from the Goethe point of view, we needed something like an elementary theory of topos. And there is, like I mentioned, there's this joy he seems to have at using the set theory he got from Henry de Saint-Saëns in the Jehovah's paper, and that, yeah, I mean, it's not an important point, and he did like your ideas on that. Well, maybe it was also, to some extent, ultimately, surely, in the end, in solving problems. I mean, there was a strong, surely, surely there was a long term, for example, to prove the big injectors. You know, a much more than, I mean, in other words, perhaps more focused on more specific sorts of problems and in the, you know, but the creation of the general machinery was something that... But he wrote 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 have the problem solve itself naturally. To have the solution to this problem suggests many connections with other areas of mathematics, which, of course, is the solution to this problem. Yes, but still, there is the problem that is the kind of 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 insightful way, you might as well not have done it. Dialectically, how do you know what the problems are? And this is specific, he makes this attitude, maybe it's too personal in a way to focus on this one thing, but the periodic proof of the first Bacon Juncture, 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 wrong words there, too. Colin, I'm trying to say, my mind is blank, I'm trying to say, what's a good concrete example of a base of those other sets, I mean, where there's an algebraic or a couplogical? G-sets certainly could be used as a base. If that's what you want, I mean, Christiane, so does it over topological space.

17:30 Okay, there's a good example. That's the Weierstadt operation 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 any complex variables. And this is useful because there's so much machinery that you can apply in your problem. You can apply the machinery, everything you know about one variable, directly to one variable. Of course, you have to check that it gives you the right result because the logic is not Boolean and so on. But, yeah, it's a good example. And so on and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, Well, Christian Rousseau did that and one or two other things and then left category theory, as many people have, unfortunately. But I have, for example, the project still unresolved of deriving the direct image here for Coleran-Cheves on analytic, proper maps in analytic space. From the Cartan's territory, there are a lot of compact analytic spaces, by exactly the same method, but the idea is that if you prove that over one point theorem in the proper way, you could just transfer this to the topos achieved on the base space, which wouldn't even have to be analytic, by the way, because even this is known in principle, because you could actually derive this as a result.

20:00 This would require making a dictionary which is both mathematically meaningful as well as compatible with the non-classical logic because, for example, one obvious idea is that an internal compact space viewed externally is really a proper map. So this is true. It's a natural idea, but moreover, to show that this is compatible with this strange logic of a topos is another problem, and indeed, that part is true. A good definition of compact really means proper math, if you look at it this way. But then there are complex numbers themselves, there are many complex number objects, they all amount to function spaces in one form or another, But anyway, if we can suppose that, that's solved. But then there comes the very problem of defining coherent itself, which has to do with finiteness. One glaring property of all the topos theory that's been done in the last 30 years is that there is a developed theory of finiteness, which has nothing to do with this mathematical meaning. It's just an imitation of the finiteness that comes from set theory. And so it's an open problem, too. To explain finiteness internally to a topos, which actually means something in terms of coherence or meritarianism or something like that, it must be something rather different from the usual description by dualizing and so on and so on. I'm not saying that the word is useless in any way, because it has several applications, but these, no, I mean, the elementary thing I said before, take a finite dimensional vector space in a topos. That should mean something like a vector bundle if you externalize it. But the finiteness, no, the definition of finiteness that all my friends use, it doesn't mean that. I don't know what it means, but it's not compatible with that intuition, but it should be. And you ought to propose a definition of finiteness.

22:30 Yeah, I propose it. But what about... We are already, we are already rigidified. What you said reminds me of the transfer of principles of Robinson. The importance of the distinction between what is internal and external, whatever the presentation may be, is very important. There is an internal notion of finiteness, which is not the same as the external notion, and the internal notion of finiteness is quite close to the... What? Well, it sparks the logic in... That's right. No, no, no. I know, I know, I know. Easily described, I mean, by, you know, in things like the models of... You know, the fact that the second-order properties aren't preserved in the transfer from a ground model to a Robinson extension, to an enlargement, but first-order are, yes. No, in the topology, it's a lot more complicated because it's true that if you look at, for example, synthetic differential geometry, all the effort to define compact, you know, it's very messy. The usual term definitions of finance don't really work very well in that setting. You have to go through all kinds of rather artificial presentations in order to do fairly simple, you know, things involving compactness. And I think that also leads to this problem of trying to do, for example, measure theory in other areas which have not actually been handled terribly well so far in topos theory, which are very natural, you know, measure theory and all this stuff has been a beautiful theory. It's awkward, you know, still in the general topos setting. And I think it is connected with the definitions of... Finances may be countability too, I mean these cardinal theoretic notions, in other words, at least in first, first-hand, and second-hand. Well, my question about standard analysis is just an analogy, but I'm very well aware that the transfer principle of Higginson's view of all the logics, whether internal or external, involves Boolean logic.

25:00 But what I'm saying is that could we take inspiration from this, let's say, and now this? We spoke about Laubitz. Of course, Laubitz, if you don't take, well, if you take the Luxembourg assignment, you stay halfway and then more or less Laubitz goes, but then you have a non-classical, you have a critical, ideological, and so on, and so, but what can we, can we imagine, I mean, the method of knowledge and analysis and to extend it, I mean, to take inspiration, to transfer it into... Notification that I'm not. Well, there has been non-standard analysis constructed in Mordyke too. What? Pondren, Mordyke. I've done Mordyke to try to do non-standard analysis constructed. In other words, non-standard analysis really did seem to depend on classical principles. Trying to find a construction non-standard analysis that's compatible with constructive reasoning is a program that's actually gone on now. Well, I mean, the transfer, so that means, one that kind of answers your question, is that you should be able to transfer only positive formulas, and this is true, at least for an example, because coming back to the, whenever you, say, algebraic geometry, you can see infinity, or analytic, and many things in between, where you have a tangent bundle, and the tangent bundle functions... This object actually has a property in all those cases that the, it has an identity. So there is a, there is a, I call it a fractional exponent, which if you call it t, there is a representing object, and then the tangent bundle of x is x to the power t, the function space. Well, now, if you look at functionals on this function space with values in any other space, say, W, so X to the T into W, those are actually uniquely corresponding to maps from X itself into what I call W to the power of 1 over T, just to have a notation for it.

27:30 So there's a right adjoint to the tangent bundle, to that operation. The very existence of this right adjoint tells you that it has to do with the geometric morphism, you see, and therefore geometric morphisms preserve arbitrary logic, and so it means if you have, if you consider any kind of structure, grains, ordinary grains, whatever, which can be classified, classifying topos, take the classifying map and compose it with this... This endo, and what that amounts to really is just taking the tangent bundle of the structure, like if you have a single set of structures, you take the tangent bundle of R, and it has the same properties as R, but same in the positive way, without logical negation, without universal quadriplegic hypothesis, probably more than that, but at least, so I call this the positive fragment of Robinson's principle is still true. For this other construction, for this exponentiation by an infinitesimal object, it's not at all like the filter power, but it sort of plays a similar role in the sense of representing differentiation in a good way. I just mentioned that in one of my recent papers, I don't know if it's widely known, that you actually have cancer in that sense. That's, you know, that's less than first order logic because, positive, on the other hand it's more because we have our infinitary disjunctions. So you're trying to speak of the tangent bundle, I mean that, something like thinking of the tangent bundle as being like an enlargement of the original structure.

30:00 It goes to the zero section and the projection, so you have to focus on one or the other depending on the various purposes. RRT is actually the inverse image of geometric morphism whose forward image is this fractional exponentiation. So if you compose that with the classifying map of any structure, as they call it, or some other topos, then the meaning of that is that if you take your generic structure here, then the inverse image here is giving you that structure that I've classified. Another structure of type T, so whatever T, T might have had some very strong accent, but that's all I understood. I say that all the rest of it, but it raises another simple problem that's never been solved. This of course is an essential morphism, because you're going to see the left adjoint multiplying by T. So the models that you're getting there are going to be essential if the original one was. Nobody's quite figured out what it means for a model to be essential. I mean, to have its classifying math to be essential in some kind of general way.

32:30 If you're talking about abelian groups and essential abelian groups, I think it's something like a pure, a connective purity and that sort of thing. I think, I think. But then, to give an account of that in general, you notice that these examples of rings are actually function rings. Then they tend to be the classifying workers and tends to be an essential workers, but I don't have any systematic explanation for that. So it's surely the case that this transport principle is stronger than I stated it in exactly what way or how. In other words, because if the original R were essential in that sense, then so would the R to the T. Of course, these tiny objects are not necessarily infinitesimal. It depends on the pre-sheaf topos. The tiny ones are actually representable and quadruple as such, quadruple of the variables, you know, for which you have a small category of an object which has a product with every other object. They're being called quadruple. Quadruple? Quadruple. That's what it means to show that not-representable puncture is tiny in this sense. It's exponentiation and appreciation of topos has a further adjuvant. Necessary and sufficient. It makes sense that you can take a larger topos and what looked tiny before but wasn't tiny becomes tiny. But if you have a very strong, you start with a site that has products, okay, you've got lots of these kind of objects, but if you impose a strong topology, look down at the sheaves, then only very few of them will survive.

35:00 If you're looking at it as a microscope now, only some of them are still small. And so those are typically in these models of algebraic. In the differential geometry, just really just the literal infinitesimal ones in some sense, in some sense that's due to the, well, roughly speaking is you apply this structure, applying to the 1 over t to a sheaf, does it remain a sheaf? I think it's my question. Enough of me. Thank you. There's something which I call SN on CNN because you have an axon over here, like I'm using the n-factor t.

37:30 Yeah, yeah, yeah. Oh, it takes part. Jet of water, n of... Simultaneous bundles. Yeah, yeah, yeah, sure. The combination of your theory of space in different... Yeah, right. Acceleration intervals in different regions. You see that when you go from Tx to Tx, you double, you double the dimension. So you are 2n, then you double again, 4n. That's what would be 3n. But the operations are not vectors. Right. The velocity to an acceleration to form another accelerates. So the accelerations are not vectors but they form an affine space over the vector.

40:00 But in simple terms, I mean, an acceleration, or rather, the nth potential product and the symmetric part and so on. And so the previous formula I gave, the exponential of cmx 0 and n, which is exactly in simple terms what you do. Take epsilon which is the sum of the epsilon n. And raise exponential epsilon, I mean, develop exponential epsilon by the series 1 plus epsilon plus epsilon squared over 2, etc. And then now neglect all the, I mean, the second term is epsilon squared over 2, to get a multiple term. Epsilon squared is the sum of the epsilon squared plus twice the sum of the double product. Now, if you consider that each epsilon is a first order infinitesimal, so the square doesn't count, and what remains, what you have is that the sum of epsilon 2 squared is twice the sum of the double part, even part, something of degree 2 can make sense of x squared over 2, equal to 2, and that means it's over the integer.

42:30 If the exponential of the sum is a product of 1 plus epsilon, if you can detect the square of each epsilon, but you are exactly in an x-ray or whatever, except that before you are epsilon, epsilon prime is epsilon prime, epsilon, but you can repeat this with the signs. And for Einstein, it's one way to present the topology. It's effective geometry. The differential form is omega, the determinant, etc. And so to each model, if you want to define various categories of supermanifolds, you keep the object, as I explained, you keep the object, but you change the model. And then you come into connection with the point of view about the grade of Manifold, which is that the grade of Manifold is a manifold. And the shift, the shift is locally, well, the model is at the shift, I mean, so local models are just slightly different.

45:00 But you add p functions on the n variable, tensors with the epsilon, the extraparameter epsilon. But if you think of that as a tensor product of a reasonable, smooth function on an open set, which is super understood, tensors by an algebra are an unusual feature to have a mod to it. It has a familiar, I mean, in an exterior algebra, the generator of the algebra you take. This is a maximal idea. So it's not exactly a local thing. It's a local thing with a maximal idea consisting of nilpotent elements. But so we are familiar with natural biogeometry. This is the idea of a sheaf which has nilpotent. Then when you cure this, when you kill this nilpotent element, you have the corresponding video scheme. There could be some type of way, a scheme, to understand that it may have nilpotent subspecies.

47:30 Then you get, we consider as, and the idea, and that means that the attraction is not, but if you deal, if you deal with, if you deal with, I mean, if you deal, well, if you have a bundle of algebra, it's not the same to consider homophilus with respect to, you know, from a bundle of algebra to another bundle of algebra, or to the, for the Christ, for the sheep of sex, and so on. And so, but you have an intermediate step, so to speak. And so you have two categories, really two categories of, I mean, when you deal with the foundations of a supermanifold or graded manifold, the graded manifold of, that's exactly the difference between the point of view of constant and another point of view. In constant, you have the zero-section. And so you have this ordinary manifold with a certain cloud around it. Well, it should be more bubble. I can see that now. Bubble, bubble, bubble.

50:00 And I'm thinking, maybe I got this wrong, does it have some retractions, but just not a preferred one? No, it does, it does. It does always. But it doesn't come back. But there's no preferred one. And in some situations, you select one. And it gives a freedom. I mean, it gives a freedom. In the general case, you don't have a preferred retraction, but in some situations. We did not mention geometry. There are a few papers. There is a general theory that includes... But I did not see one. It's not pursued enough. One of my suggestions to the mechanics is to apply this idea of mechanics. And as I mentioned, next month I'll be giving a set of lectures on super mechanics, supersymmetric mechanics. I came across this idea and said, I do know it. I thought again about it. Interesting kind. It didn't work enough. That's in the back burner. I do agree. That, of course, is also representable by a tiny object.

52:30 However, That's a question. See, because if we just take what you said, that on the function space, you want to take the part, which is the equation. So, if we start with this object, T, which is the generic first order. So, we take T squared. But then divided by two factorial, where that means the group operates. So, on the other hand, from the point of view of, let's say, purely algebraic point of view, we might think of the second-order dual-potent, the spectrum of the algebra, which is generated by one element t whose cube is zero. Yes, and it is the same. Well, not exactly. Because you want to, we're back again to the symmetric part, so there was a symmetry on this second order algebra, and the tensor product of the two one-dimensional, the first order ones, maps in there, namely you just take the sum of the two generic pair elements of square zero, that's cube zero. And, of course, it's a symmetric element of this algebra. But now, in order that this, you see, geometrically, we want this other object to come out to be t squared mod t two factorial, but modding depends on the topology. It's a direct limit, so it depends on the, which certain topology we are dealing with. And so... There's a concrete calculation, namely, there exists a linear retraction. I mean, an instance of the following situation. We have one ring A contained in another ring B. And we want to say that this is really covering, and we dualize, on the basis of the fact that there exists an A-linear retraction.

55:00 Not a ring homomorphism. Which I call, geometrically I call it a stochastic section, because it's not really a pointwise section, but it assigns to each point in the base a probability measure on the fiber, well, probability which might add negative value, but it's normalized anyway. So, from points to probability measures on the fibers. If there exists, and it should be linear with respect to the whole ring of debates, not just with respect to constants, then we should say that this is a covering. So does this occur, is this a recognized root and deep topology in itself? Because I have seen this phenomenon mentioned in exercises, but I've never seen any systematic relating it to... Do you mean Rotendieck? Rotendieck or... Rotendieck mentioned it. No, no, no. Some other books, not even by Rotendieck. Notice that this doesn't have a recognized name. But then, the fact that it is a topology, in other words, it satisfies me. The basic thought is that it's a coverage. The notion of the fact that coverage should be stable by pullback. You want to take this family of covering as a base for proclivity? That's right. And it's stable by pullback, you see, for a reason which is special to the purely algebraic situation. Namely, the fact that tensor product is, after all, a functor in the linear world as well. So it represents the pullback. But it applies also to this section, which doesn't exist geometrically, it's only at the module level, but it applies and gives you the stochastic section on the pullback, you see. So it is stable by pullback. But I would love to make the same definition in the C-infinity world. I mean, all the notion of distributions, I mean, if I could assign to every point in the base a distribution in a way that projects back down to the identity.

57:30 This would be a very convenient notion of hovering, but I don't know that it's stable under pullback, because of the behavior, I don't know well enough the behavior of the space of distributions, you know, when I vary, the distributions on a pullback, I don't know, it's a comparison map, of course, because there is no analog of the tensor product, no analog even of modules. To take account of the full C-infinity ring structure, at least not in a simple way, which would, at any rate, in the coming back to the case of infinitesimal, it means that if we accept this topology, then in the internal logic, every second-order nil-potent, every second-order infinitesimal, rather, is the sum of two first-order ones. The existence is local, and existence means you have a covering and so on. More or less the same question. Because you were thinking of manifolds as some other special kind of space, and when you confront it with all possible sheaves, a larger domain, then it's a question of exactly what the equivalent is. You know, people say, well, in an integration field, we don't have both of them, but then you can be familiar with this. I think it's exactly the same question as you raised about what is non-commutative geometry, it's more basic. What is a morphism? There are many answers, but this definition, I mean, by taking the image of a measure which is very important, it's a reference or whatever, I mean, the distribution, probability, solution of the variable, whatever, it's very important, but it has one disadvantage, if you fix, if the measure is...

1:00:00 In 1962, which is still unpublished, but I have a doctoral thesis, I'm still a doctoral student, I developed quite a bit of courage with this idea. In the fifth chapter, they wanted to do the formula and so on. I suggested at the time, it was printed just before.

1:02:30 I don't have much real influence on that, but I remember I suggested that I should say that there was not an outsider, and that's why it's still an outsider's algorithm. But the point is that there is a restriction on go-back integration, where you first have to go for integration if you don't have atrocious topology, which is a mixture of different factors which have nothing to do with the whole range of categories described axiomatically, because in fact there are several different categories. I said so about some problem in statistics, I mean, which is, I mean, the statistical estimation of parameters and so on. It fits very well with this kind of idea. So, again, we have a natural, we have some situations where the notion of a certain category of space, apart from one space with the other one, is not uniquely defined. To answer, I mean to answer what you asked about the non-committal nature with that, and I just wanted to mention that in probability or in statistics you have similar problems, but it's, but it's interesting that from a historical point of view, this point of view, Sartre and so on, when he came to really interesting facts like the integration theory, he would deny the relevance of them. Very, very unreservation, I mean I remember. In a further discussion about manifolds, I mean, André Wey was quite reluctant to say,

1:05:00 well, I don't really know whether this point of view of a set with a structure fits a manifold theory. Because he takes his mind out of the idea of it without changing the atlas, but you change the morphism. Change the morphism. But then, both are constrained. There is a natural of the scale of time and there is another one which is a category point. It's going to assume that the two notions of isomorphism are the same. Which means that something about, say something about the phantom, the forgetful phantom taking the category 2. But we know it's a phantom. And from inside the conversation, I mean, I have zero doubt, I mean, they and so on, and then came the fashion of structuralists in social sciences, but it's maybe not more than that. About the statistical estimation of parameters and so forth, there's the question of simply the idea of completing a diagram, which, you see, if one of these random maps, the one where we can follow it, come to...

1:07:30 Space of parameters to the experimental space, on the one hand. On the other hand, another space of decisions, and another such map could also be probabilistic, saying if you actually knew the parameter, you'd know what to do. So the question is to complete this triangle, which makes sense, except that usually you can't, and so you want to make it as convenient as possible. And so I noticed that there is, in fact, an intrinsic metric on the HOMs of this category, which is, I think it's a construction of Poincaré or somebody, because it's a convex set, and every convex set has a standard, has an intrinsic measure. So, in fact, this process of, what should I say, yes, I mean, we have the category of... Convex spaces, there are many flavors, many flavors, but it contains as a full subcategory these random maps because they are really mapped between three convex sets, between simplicies in some conventional sense. From this category of convex sets, there is a function to the category of metric spaces. Which takes this, it's kind of, it's sort of information theoretic as well. You see a theme on certain logarithms, possible mixtures, things on the browning and so forth. But my student proved a very interesting thing. This function preserves tensor products. So there's a natural tensor product of convex objects. There is a natural tensor product of metric spaces. And this function preserves it. It's therefore a closed, monoidal closed function. Therefore, any category enriched over the first is transformed into a category enriched over the second. Now, a typical category enriched over the first is what we were, I mean, various categories of Markov processes, statistic processes, decision procedures, all these type of things are actually categories enriched over convex sets. So they all have actually also an intrinsic metric on their haunts.

1:10:00 This idea of optimization doesn't, I mean, there are many, when we consider this problem of completing the triangle as a basic case, there are many methods for introducing the idea of optimization. Assuming some extra metric or some extra measure or whatever, all of which may be useful, but I always thought it was striking that there is in particular this intrinsic one which is well known. I think so. There are definitions of entropy in particular contexts in analysis. There are various notions of entropy, the one by Shannon, but it's a little primitive. It doesn't need to be defined by Kuhlman. Which is relative K2, but that is relative K. It's important. The interpretation of this metric on a convex set, suppose it is probability measures, so then it's a convex thing, right? If you're given two points, a pair of points, then you consider the possibility of expressing the second one as a convex mixture of the first one and something on the boundary. I say on the boundary, I mean you take a hint going toward the boundary. You could just evaluate it. And then you take the logarithm of this of this ratio lambda and then you take, like I said, you can take it in FIMO, but you can also just evaluate it on the boundary if you know where the boundary is. It's a kind of a question of camera. Exactly. Oh, oh, oh, oh, how many parts? That kind of question of camera. This, of course, is a very non-symmetric metric, so, and it's quite consistent with my... My idea that Frechet was wrong to put in the axiom of symmetry because so many examples that are rising and rising... But again, in entropy is a non-symmetrical situation. Yes, very much so, yes. Which is natural because there is a natural, in entropy there is a natural notion of time, you have a certain, you start with some dynamical situation and it evolves in a random way and then you have a new situation and it's exactly the second principle of thermodynamics that the entropy increases because it becomes more and more homogeneous or more and more endocrine, which is very interesting.

1:12:30 So in entropy, in studies, there is a natural sense that it's not symmetrical. The second principle of thermodynamics says that the time has a definite hour and so on. And it's clear that it cannot be symmetrical. So you are thinking of metrics which are not symmetrical. Yes, I mean, for example, the Hausdorff metric... Thank you very much for your attention. And asymmetry, which is not naturally there, because you can take the ordinary difference, just the probability of the ordinary difference, and then it's a very, very non-sense. So you say you lose, almost that one does lose information about something, but people look at it. People study the non-symmetric thing, or they should, but they don't. It's quite strange. There's still this stigma. The word generalize, I didn't use the word generalize, but you know, computer scientists, and they come and say, oh I read your paper on generalized metric spaces. I said, I didn't say generalize, I said metric spaces. It's an important psychological difference. These are the ordinary normal things, and of course among them there are the symmetric ones. You just have three sets and you can't do anything else, you know. To say that the probability, that the distance is zero, says that one set is almost everywhere included in the other, up to measures zero, actually included, so the inclusion, the order relation on the sets is part of that metric space data, or at least it's symmetrized.

1:15:00 Right, exactly, because now and again this kind of thing has come up in classification of types. Models will also find appeals and stuff, and people just rush to judgment in any way. Would this be a good place just to take a break of 10 minutes, maybe 15, and then sort of... And then I'm saying to go just for another hour, and then just... Just go out, yep. Oh, and get rid of it. I'm going for the... Of course, I'll say it. I'll say it. 90 minutes. Just accept it. And what that statement... There's nothing that y can go to. If you try taking y to any polynomial in x, then linearity says x times y has to go to x times that polynomial. But x times y equals 1, it has to go to 1, and 1 isn't divisible by x down here. Yeah, because we get it down to linearity in x. Yeah, well, yeah, yeah, well, I mean, in a simple saying, you know it was when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote a class, you know, when you wrote Right, this is no longer, I mean this is now an extension, but this satisfies some monic polynomial, but that's the case I want, yeah, so I've got some, but this is just y to the n, you can put the constant term in there, now my goal is I need to take y to some element, and I got this right, I mean over r, this is just a free module, right, so I can take y to anything.

1:17:30 So, there's always a stochastic section. Well, so there's always some stochastic section of any of them. In the interview extension, the interview extension is very special. Yeah, yeah. So these are all coverings. Yeah. If we want to accept this code of anthropology. Yeah. You're going to come to a contradiction now, let's see. No, no, I'm just trying to maybe see if I understand this. Yeah, there's so many coverings that you're not sure of. No, actually that isn't many. It's a pretty interesting class. In scheme terms, isn't that telling me that whenever I have a scheme affine over a base, then there's a stochastic section? I mean, it's literally trying it in some global sense. Affine. It's the number of... Oh, you're saying you take the sum of all the affine... Well, no, I'm still trying to form it. No, but I mean, I think what you have is a reasonable conjecture. All I was thinking of was a reasonable conjecture. So the thing is, I might have some scheme situation where, on a cover, it's affine and...

1:20:00 But this has a different, this has different generators that way than this one does, and so I can't, I mean, that's just, I don't know if that even makes sense. I mean, my question is, when does the scheme now happen? Well, see, this very example, right, this very example is, well, Shannon will point it out to me, you can just rotate it a bit, and you get this, and if you take it, if you take it in over a line, then the doctor has a section. Well, in fact, it'll have actuals in it. Then you've actually got two formulas in this section. I mean, this map is just sort of like x plus 1 over x, right? You've got a sort of thing. You're replacing x by x plus y. y squared minus xy minus y. Right. The formula for the math itself is simply that it's such a 1 over x thing that... Yeah. In other words, this is actually the ring of functions.

1:22:30 The group of units of the ring, therefore the group of units is the substrates and that's the function sign of the logistics and the rules of elements. This is that. Because this... That doesn't look like an empty morphism. Oh, but of course it is, it's an unit. Yeah, that's right, it's an empty morphism, so it's equal to the inclusion. Equal to the inclusion. That's one thing about... So the netter normalization picture is, I was looking at the ring of functions on this space, and I thought of it as an extension of this, and that was the wrong thing to think of as an extension of this. I should have thought of it as an extension of that. No, but I don't think, just because it looks like a line, doesn't mean it's like an algebraic equation. No, it does, there has to be. In this case, there's an actual, there's traces of this, half of it. Yeah. And in fact, it does transform into the equation that we had in high school for this hyperbola, but I don't remember that hyperbola. Is that x squared, well, is x squared minus y squared equals one? Probably. Turns it one way or the other. Yeah.

1:25:00 Oh, that's right. It's like A plus B times A minus B times A. Those are the two coordinates instead of the terms. Now that, so that we can think of as the cognitive research, or why, one of those would be in a good way. The imaginary part. And this has stochastic sections, but not... You mean in the complex domain or in the case? Right, because you don't even know that there exists any points. Yeah. In terms of field values, everything has a square root, so it can't be any field, but that's true. So the idea is to evaluate a function by taking its value from the C-points and taking the sum of it to the fixed linear combination, and that's the function on the scale. That's the expected value of that function. And there's no way to pick one consistently the whole way. What about the qubit? I use the qubit as an example of Brouwer's Principles. You can't tell if something is bigger than something you can't define. Except you can because you get overlapping regions which are clearly overlapping. And on each of those you have the perfect non-section. So I wonder what happens to something like that. This, by rights, really should happen, because it's got, even in the real world, it's got, you know, there's always a point there. ...exactions... ...exactions in that case.

1:27:30 I can always think of few polynomials if I just take a while. Yeah, and I'm gonna rotate it to think exactly that's how I think, but... So that's, um, oh, okay, that's okay, that's x plus 1 times x squared. Well, there's another little high school out here. Oh, there's mine. It's not a fifth-degree equation. Hey, that was great. Think about this cube. This is plus 1 times y. Because if you thought that this is rotated, that's a cubic because the roots are 1, 0, and 1, 2. Let's see, so it's y times y squared minus y, minus y. The question is, again, we're going to have to take y to some polynomial in x, in such a way that that gives us a linear map of what's possible. So let's think about this. So we've got some field with x. We're extending it.

1:30:00 I want to take y back to some value in x that will give me a kx linear over kx, but preserve the multiplication of x because I want it to be linear. Not to x, but to some function. I'm looking at these as kx modules, and I want to split this as a map of a kx module. So it should be added to it in the obvious sense, but moreover, x times y should go to x times f of x. But also it should be a section, which means that anything that doesn't contain y should just go back to itself. And I need to know, here's what I need, here's what it is, this, I think, this cubed minus this minus x has to be zero. I mean, just the, I mean, here you would be, what is K? K is that supposed to be? Oh, I see. Oh no, we've said it badly. We've said this badly. We need to know what y goes to. We also need to know what y squared goes to because we're not observing multiplication. We're not responsible for y cubed because it's equal to a combination of y and x already.