Pm discussions concluding part

0:00 The risk-representing objects, and then the tangent bundle of x is x to the power t in 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 maths from x itself. So there's a right adjoint to the tangent bundle, to that operation. Well, the very existence of this right adjoint tells you that it has to do with, there's a geometric morphism, you see, and therefore geometric morphisms preserve arbitrary direct limits and finite inverse limits. In other words, they preserve the positive logic. And so it means if you have, if you consider any kind of structure, rings, orbital rings, whatever, which can be classified, so there's a classifying topos, take the classifying map and compose it with this endo map, and what that amounts to really is just taking the tangent bundle of the structure, like if you have a single single structure, R, you take the tangent level of R, and it has the same properties as R, but same in the positive way, without logical negation, without universal quantifier 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 transfer in that sense. Any structure R, R to the T, have the same positive properties. You know, that's less than first order logic because it's positive. I mean, it's more because we have our infinitary disjunctions.

2:30 So you're trying to think of the tangent bundle. I mean, that's something like thinking of the tangent bundle as being like an enlargement of the original structure. Yeah, yeah. Which it is. Well, it is. It is. Right, right. Sure, sure. Of course it is. There's both the zero section and the projection, so you have to focus on one or the other depending on the various purposes. So raising to the power t 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, say call it R, So the meaning of that is if you take your generic structure here, then the inverse image here is giving you that structure, right, that's classifying it. So phi, or upper star, of the generic structure of T. Composed with Planckian T is this. In other words, this composite is the classifying. ...morphism of another structure of type T, whatever T. T might have had some very strong axioms or not, so that's all there is to it. I say that all there is to it, but it raises another simple problem that's never been solved. This, of course, is an essential morphism, because you've got to figure that out by 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.

5:00 Now if you're talking about abelian groups, an essential abelian group, I think it's something like a pure, connected with purity, that sort of thing, I think, I think. Then they tend to be, they're classifying Morphism tends to be an essential Morphism. I won't have any systematic explanation for that. So it's surely the case that this transport principle is stronger than I stated it. Exactly in what way, I don't know. 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 representables, for which you have a small category in an object. Which has a product with every other object. Everybody calls it quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. Quadruple. It sort of makes sense, so you see, if you take a larger topos and what looked tiny before becomes, what wasn't tiny becomes tiny, bird's eye view, 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...

7:30 We'll survive being tiny, because you're looking at it in a microscope now, and only some of them are still small, and so those are typically, in these models of algebraic and differential geometry, just the infinitesimal ones, in some sense. In some sense, that's due to the... Well, roughly speaking, if you apply this function, applying to the 1 over t, to a sheaf, does it remain a sheaf? In a slightly different question, there is something which I call the banker's principle. The banker's principle. To me, it's one of the fundamental rules of differential calculus. It's a following, it's compound interest. Suppose that your account is credited with compound integrals every day with a fluctuating rate. What do you have at the end of the year? Well, the first day you multiply by 1 plus epsilon 1, the second day by 1 plus epsilon 2, etc., etc., etc., until 1 epsilon plus 365, and the result is exponential of epsilon 1 plus epsilon 2. If you think of the Euler method, the targeting method to solve differential equations, it is exactly that. Continuous product integral is more or less the same.

10:00 And then the tangent to the tangent T, Tx. As you well know, representability, for instance, there is an action of the symmetry loop of Sn on Tnx because you have an action by permuting the n-factor T. Now take the fixed, so do ordinary differential geometry, not in the topos but ordinary. So you start with X, T, X, T2, X, Z. Now take the fixed part of the symmetry group acting on T and X. Oh, the fixed part. The fixed part, what it is. Change of order and of... The fixed part within the tangent tumbled. The iterated tangent tumbled. Yeah, yeah, yeah, sure. In mechanics, it's a good principle. It's just a reformulation of your theory of space in differential geometry. If you want to define what it means in differential geometry, what is acceleration, mechanical acceleration in terms of differential geometry? Velocity is easy now. Of course, the concept of velocity is fully interpreted in the term, in the tangent matrix. But what about acceleration? Acceleration, you have to go through the second polar differential part and take the symmetry, there is a symmetry, you take the symmetry part. And if you, you easily see that when you go from Tx to Tx, you double, you double the dimension, so you have to. Thank you for your attention and see you in the next lecture. In the 60s, I mean, this is exactly, this is exactly, so the bundle of acceleration, so the acceleration are, well, it's not in mechanics, acceleration are not vectors, right, are not vectors, but what they are is that you can, if you are, what, yes, but you can add a velocity to an acceleration to form another acceleration. So the accelerations are not vectors but they form an affine space over the Penrose. But in other terms, an acceleration or rather velocity plus acceleration is just a second order jet of your curve.

12:30 But you can repeat it. And if you think in various ways, you are formulating the same idea. So, if you think really about what it means, I mean, the previous formula I gave, epsilon n is exponential of cmx0 n, which is exactly, I mean, in simple terms, what you do. Take epsilon, which is the sum of the epsilon n, and raise exponential, 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. Forget about it. Epsilon squared is the sum of the epsilon squared plus twice the sum of the double products. 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 to the square is twice the sum of the double products. In super manifold, gradient manifold, you can do the same with exterior algebra, and it has been known, for instance, that if you have an exterior algebra and that if you take the even part of it, divided power, you can define, I mean, if you take an exterior algebra and you take, let's say, something of degree 2, you can make sense of, well, take x of degree 2. You can make sense of x squared over 2, even in characteristic 2, and I mean it's over the integer, the principle being exactly that, I mean you want, if you want to define x squared over 2 and x cubed over a factor of 2, what you have to define is the exponential, but using the formula, I mean, if the exponential of the sum is a product of 1 plus epsilon, if you can neglect the square of each epsilon, but you are exactly in...

15:00 Except that before you have epsilon, epsilon prime is epsilon prime is epsilon, but you can repeat this with putting the signs, and for instance it's one way to present the perfume. The Thauffian, I mean, well, the Thauffian is known at, let's say, in seplectic geometry, you have, in seplectic geometry, you have a fundamental differential form of degree two, omega, and the volume element, as introduced by Neuville, is omega power n divided by factorial n, and, but the x, and if you think about it, is the Thauffian, the square root of the determinant, etc., etc. But this makes sense over, let's say, over ktcp or over the integers, it makes sense over the integers. And now what you have is that my view of a supermanifold. My view of a supermanifold is that, as I said, when I start, I start with an ordinary manifold with a vector bundle, and I take the, for each fiber, the exterior algebra of the fiber of the dual, it doesn't matter. And I create a bundle of exterior algebra and if I want I can go to the sheaf of sections. And so to each vector bundle I have associated a certain number of objects, a bundle or a sheaf of algebra. And if you want to define various categories of supermanifolds, you keep the object, as I explained, but you change the morphism. But in this, so what you have in effect is to associate and then you come into connection with the point of view of the constant about grade and manifold, which is that a grade and manifold is a space with a And the sheaf, the sheaf is locally, well, the local model is that the sheaf, I mean, so the local model is like in differential geometry, you have n real variables, but you add p gasman variables, and locally your superfunctions are the same finitive function on the n variable, tensored with the exterior algebra in the epsilon, the extraparameter epsilon.

17:30 But if you think of that, I mean, okay, so it's a tensor algebra, or it's a tensor product of a reasonably smooth function of a no-percent, which is well understood, tensored by an algebra, but this algebra, an unusual feature to have a mod 2KD, but it has a familiar part, it has a familiar property, which is that it's a ropering. I mean, in an exterior algebra, the generator of 0 squared. So it means that if in an exterior algebra you take all the elements with a 0 component equal to 0, this is a maximal idea. So it's not exactly a local ring in the standard set, but almost, almost. Up to the slight difficulty of the greening model, it's a local ring. And it's a local thing with a maximal idea consisting of nilpotent elements, but so we are familiar in algebraic geometry with this idea of having a shift of a space with a shift which has nilpotent elements and then when you kill this nilpotent element you have the corresponding reduced key. And it took me some time anyway at the beginning of the scheme to understand that when you do this thing, I mean, so you have a general scheme and locally you may have an important element, then you kill an important element, you produce a new scheme, which is a new scheme, which is a subscheme, it's a subscheme. I played with this idea for the infinite human group. That's one of the reasons which confused Cottenley and Gabriel also. So we have this group with one element which is infinite.

20:00 If you have a general supermanifold, if you kill an important element, which are exactly the Kastmann generators, then you get an ordinary manifold, which has to be considered as a submanifold. And the idea, the intuitive idea is that a greater manifold is an ordinary manifold surrounded by a cloud. Surrounded by a cloud. It's a good intuition. Surrounded by a cloud. But then, because you said, I mean, the zero section, and what I define is really corresponding to the zero section. So you have a zero section, but in general you don't have a projection. So here is a natural situation where you have a zero section but not a projection. So the class doesn't have a projection. Yes, exactly. And of course it's the rest of the class, and it's exactly the difference between... If you deal with a bundle of Casman's theory of algebra or a sheaf of sections, if you work with a sheaf, I mean, so the transformation that you allow enables you to mix the Casman variable with the ordinary variable, and that means that the retraction is not fixed, there is no fixed retraction. But if you deal with, if you deal with, I mean, if you deal with... If you have a bundle of algebra, it's not the same to consider homomorphism with respect from a bundle of algebra to another bundle of algebra, or to the corresponding sheaf of sex, and so on and so forth. And so, but you have an intermediate step. And so you have two categories, really two categories of, I mean, when you deal with the foundations of supermanifold or gradle manifold, the gradle 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. Well, it should be more bubble, I would say. I see another bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble, bubble,

22:30 And then it gives a freedom. In the general case, you don't have a preferred retraction, but in some situations, you select one. Well, I published that already in a paper with Cecil DeWitt, the manuscript of our book. We just hinted in our book to that, but we did not repeat it. Well, we are all in synthetic differential geometry, but the ideas which are only slightly different come from supergeometry, supermanifolds, and to my knowledge they have not been exploited. That's what I suggested. Yeah, I think there are a few papers. There is a general theory that includes both the super case and I think only one. I think that's one of my suggestions. Of course, since we can't say about mechanics and this applies to this idea of mechanics. And as I mentioned, next month I have to give a set of lectures on super mechanics, supersymmetric mechanics. People were not preparing last week these lectures. I came across this idea and since I do know it, I thought again about it in these terms. So, well, I don't know whether it has anything interesting that can come out of that, but it's just interesting. Yes, it's a good one. It didn't work enough. That's in the back burner, so to speak. It should be brought forward. I do agree. That, of course, is also representable by a tiny object. However, which tiny object? That's the question. See, because if we just take literally what you said, that on the function space you want to take the part which is symmetric. On the other hand, the representing object, therefore, should be a quotient.

25:00 So, if we start with this object T, which is the generic first order infinitesimal. So we take T squared, of course, but then divide it by two factorials, where that means the group operates in the same way. 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 nilpotent, in other words, 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... You just take the sum of the two generic pair of elements of square zero, that's of 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 and modding depends on the topology. It's a direct limit so it depends on the topology we are dealing with. And so there's a concrete calculation, namely there exists a linear retraction. So, in other words, it's 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 a covering to be dualized on the basis of the fact that there exists an A-linear retraction, 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, a probability which might have negative value, but it's normalized anyway.

27:30 So, from points to probability measures on the fibers, if there exists, and it should be linear with respect to the whole ring at the base, not just... Then we should say that this is a cover. So does this occur, is this a recognized topology in itself? Because I have seen this phenomenon mentioned in exercises, but I've never seen any systematic relating it to... You mean Grotendieck? Grotendieck or... Grotendieck mentioned it. No, in some other books, not even by the Grotendieck school, actually. Because, notice, I have several problems with this. First of all, this doesn't have a recognized name. But then, the fact that it is a topology, in other words, it satisfies the basic, well, it's a coverage, the notion of... The fact that covering should be stable by pullback. This family of covering? Yes, this family. Okay, I understand what you mean. You want to take this family of covering as a base for concatenating topology? 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. See, 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, you know, if I could assign to every point in the base a distribution in a way that projects back down to the identity, 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, take it to, the distributions on pullback.

30:00 I don't know, it's a comparison map, of course, because there is no analog of the tensor product, you see, no analog even in modules to take account of the full C infinity ring structure, at least not in a simple way, which would, at any rate, this, in coming back to the case of infinitesimal, it means that if we accept this topology, then in the internal logic, every second order nilpotent, every second order infinitesimal, rather. This is the sum of two first-order ones, right? This is because the existence is local and existence means you have a covering and so on, so... More or less the same question, I'm sorry. Yeah, yeah, because you're thinking of manifolds as some other special kind of space, and when you confront it with all possible sheaves, a larger domain, then the question of exactly what the equilibrium is... When we came to discussion of morphism in general, people say well in integration theory we don't have morphism and it's not true, I mean we have a first class which is if you have let's say a probability, if you have a space with let's say a probability measure, if you have a map. From one space to another, but then you can define the image of the measure. But I think it's exactly the same question as you raised for non-commutative geometry. What is a morphism? There are many answers. Well, this definition, I mean, by taking the image of a measure which is very important in probability, it's a so-called... It's a graph on statistics, it's called marginals or whatever, I mean, the distribution, probability, solution of a variable, whatever, it's very important. But it has one disadvantage. If you have the first two spaces and a measurable map between them, as soon as the measure is given here, there is only one measure on the other side which fits with that.

32:30 But then there is an immediate for that, and it's exactly the same idea as bimodule, which is diffusion. And what is a map if you have two measurable, let's say, two topological numbers? What is really a morphism? It's an operation, it's a diffusion associated to any point in the first space, a probability measure in the second space. So a random fact. Well, it's not a random fact exactly. It's a diffusion. A point explodes into a cloud. But that you can compose. Yes, I have written a paper about this in 1962, which is still unpublished, but I have a doctoral thesis. I'm a side-doctoral student who developed quite a bit further this idea. There were various operations on measures that you can do. You can take the image of a measure by a map, you can take the tensor product of a measure on different spaces, you can do various constructions. They wanted to unify that. And it's done in a very complicated way, very complicated by the notion of the formula of measure and so on. That was printed just before I entered the group, so I did not have much real influence on that, but I remember I've seen the draft and I suggested that I should change it, but since I was not inside the group and I was still an outsider, I did not have any influence. But I suggested such an idea. But the point is that there is a restriction of Go-back integration where you first have to... You cannot speak of integration if you don't have, at first, a topology which is irrelevant. One, irrelevant. In many examples it's important, but I think for foundations it's irrelevant.

35:00 It's just a mixture of different structures which have nothing to do with each other. Adverse, adverse, of course, application is very important, but then... So it's this kind of thing where the foundation should apply to a whole range of categories described axiomatically, because in fact there are several different categories, not only topology. And then so, I mean, I thought about some problem in statistics, I mean, which is, I mean... So again, we have a certain category of spaces where the notion of a map from one space to the other is not uniquely defined. There are at least two candidates. And that's just to answer, I mean to answer what you asked about the non-committal nature of the class. And I just wanted to mention that in probability or statistics you have similar problems. But it's interesting that, from a historical point of view, it's interesting that Bobakid, Why he advocated this point of view of structure and so on, when he came to a really interesting fact like integration theory, he would deny the relevance of the point of view of structure to this situation. He would deny it. But I remember also that André Weil had reservations. I mean, I remember in the first discussion about manifolds, I mean, André Weil was quite reluctant to say, well, I don't really know whether this point of view of a set with a structure fits manifolds theory. Take his mind out of the idea that you should first have to take charts and covering and of course, as we said repeatedly, I mean, if you're from a category point of view, a space with an atlas of charts is a norm, right?

37:30 And you can't keep this, define this as something without changing the atlas, but you change the morphism. But then, Bobacki was constrained by this idea to objectify about the meaning of isomorphism. There is a natural notion of isomorphism like climbing the scale of time, and there is another one which is categorical. I don't want to say something about the fung talk, the forgetful fung talk, taking the category to the category of set, but we know it's unfortunately drastic. But again, just a historical remark, I'm not sure that Boubaki took really seriously this point. And from inside the conversation, I have serious doubts. It's a little strange. I mean, they made great firsts about Stryker and so on, and then came their fashion of structuralism in social sciences, but one has to say that Leo Coric would say more about that, I mean, we can't wait to discuss it again. But it's not clear that Bobacky really believed in Stryker. It was not just a political slogan and no more than a political slogan and I have some doubts. It's maybe not more than a political slogan. 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 or whatever you call it, from the space of parameters to… Experimental space, on the one hand. On the other hand, another space of decisions and another such map could also be probabilistic. If you actually knew the parameter, you'd know what to do. So the question is to complete this to make a cumulative triangle. Which makes sense, except that usually you can't. And so you want to make it as communicative as possible.

40:00 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... That's what I say. Yes, I mean, we have the category of convex spaces, of which, I mean, there are many flavors of convex topological measures, many, many flavors, but it contains as a full subcategory these random maps, because they're really maps between free convex sets, between simplices and something that's in a dimensional sense, so... There is, from this category of convex sets, there is a functor to the category of metric spaces, which takes this, it's kind of, it's sort of information theoretic as well, as you see in Fiemel and certain logarithms and possible mixtures, things on the boundary and so forth. But my student proved a very interesting thing. This functor preserves tensor products. So there's a natural tensor product of convex objects. There's a natural tensor product of metric spaces. And this, this functor preserves it. It's therefore a closed, monoidal closed functor, 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, is what we were, I mean, various categories of Markov processes and statistic process, you know, 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 Haas. So 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, 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. Is it related with the Kullback definition of entropy or not? I think so. It's just a...

42:30 Sorry, can I touch your bill? There are definitions of entropy. Ah, right, yes, entropy. In particular context and analysis. Well, there is a various notion of entropy, the one by Shannon, but it's a little primitive. It has been refined by Kullback. Thank you very much for your time. 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 an nth going toward the boundary. If the boundary is there, you could just evaluate it. And then you take the logarithm of this ratio lambda, and then you take, as I say, you could take it in fimo, but you could also just evaluate it on the boundary if you know where the boundary is. It's a kind of Poisson camera. Exactly. For how many parts? This, of course, is a very non-symmetric metric, so it's quite consistent with my idea that Frechet was wrong to put in the action symmetry, because so many examples that are biased... But again, in entropy, you have a non-symmetrical situation. Yes, of course, very much so. Which is natural because there is a natural motion. In entropy, there is a natural motion of time. You start with some dynamical situation, then 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 random, which is rather the same. Science is not symmetrical. The second principle of thermodynamics says that time has a definite hour and it is clear that it cannot be symmetrical.

45:00 So you are thinking of metrics which are not symmetrical. Yes, I mean for example the Hausdorff metric. Subtitles by the Amara.org community In asymmetry, which is not naturally there, because you can take the ordinary difference, you can take the probability of the ordinary difference, and that is a very, very non-symmetric thing. So you say you lose, obviously one does lose information in that house of things, but... The second look at it was... People studied it as a non-symmetric thing? I keep telling them they should, but... It's quite strange. There's still this stigma. The word generalize. I didn't use the word generalize, but you know, computer scientists like to say this kind of thing. 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, you see. These are the ordinary normal things, but of course among them there are the symmetric ones. To say that the probability, the distance is zero, says that one set is almost everywhere in the other. So the inclusion, the order relation on the sets is part of the metric space data, whereas it's a symmetrized distance. Now and again, this kind of thing has literally come up in the classification of species of plates and models, and I'm starting to find that fiddles and stuff, and people just rush to judgment. I mean, you know, quite interesting. Would this be a good place just to take a break of 10 minutes, maybe 15, and then perhaps aim to go just for another hour, and then... Sounds good. Well, let's take 15 minutes. We can take a little longer if you don't mind going later. It's 20...

47:30 Why don't we say, why don't we say to meet back again at a good show, should we talk about the past, say, yeah, let's see if there's, let's see if there's, okay, Pierre, we'll say to meet back again at the past, say, so, so do I get to keep it, which is, I'll take a long, I'll take a long, I'll take a long rest, and then we'll try to keep up to the report, and this has no stochastic sections, it seems to show there can't be any, any retraction, any, but it seems to me, What I think now is sort of a preferred notion. There are several candidates. Here's T squared over 2,000. So I mean this is worth, I think that's the idea of the second order differential equation. Well I'm going to take it on myself as the facilitator since I've already run the suggestion past Angus and Bill and the colleague. To suggest that we take as a relatively self-contained topic, just for the last hour or so, a theme which naturally comes out of the discussion of how one sees logic and logical notions as fitting within the framework of algebraic geometry which is the unity of algebraic, geometrical and logical conditions in topology that are expressed by what Bill has referred to as the underlying The ability of SUD, which is the subgroup separable and grammatically decidable, the conditions for objects to be, respectively, those things, in a topos, and how it also relates to this issue of localex, of the conditions for topos to be localex, and the distinction between Petit and Gros, topos is now characters of generalised space and characters of space in general, are in fact perfect timing.

50:00 So, which is a subject which Bill was talking about briefly at lunchtime, so I invite him to take that up and to give us an account of this. Well, the SUD simply means, well, they're all three the same. There are three different, it's an unusual, just to remind you that they are the same. But decidable is taken from logic, it means it's an object. The square has a diagonal map which is complemented in the most naïve sense that there's a subspace of x squared and it's co-product with the diagonal is all of x squared and it's intersection with the diagonal and it's a Boolean complement even though we're in a non-Boolean world and it may happen that occasionally we have this what's called decidable or decidable equality. The equality might well be decidable when other predicates are not, and this is a common situation to consider. On the other hand, Shaniro likes to call it separable because it's exactly the same thing. It's just the geometric expression of the idea that in number theory or ring theory there are no repeated roots. Separate the roots because they're close enough to get confused. So the S and the D are just two different logicians, word and world, and they're geographers, which is why they're the same. Unramified is also the same thing, except that the word is typically applied in the category which is expressed as the categories over a certain object. The separable objects over a given object, x, one says that the map is unramified. Basically, if you start going along the curve at the top, then you can't suddenly separate into two parts, because being different and being the same are rigidly opposed.

52:30 And for you, separable is an absolute notion or a relative notion? Separable is an absolute notion and ramified is the corresponding relative one. I wouldn't be fair to say separable is an absolute notion, but if... Maybe in a relative context, of course. I mean, a scheme is a scheme. Maybe it's a scheme over a base. Yeah. A scheme might be over a base. You use only one category and only one method. Yeah. But a category may be over a base, and you're looking at it that way. Right. Right, and so the reason that that came up was because, well, basically, Peter Johnstone had a... Topology is a maker of what you call QD, QD topos. Q means quotient. So this comes about in the following way, in any topo, even in a much more general category, let's say in a topo you can consider those objects which are SUD, that have this complement, they are closed under many operations, sub-objects, Cartesian product, finite number, disjoint suns, and so on, but not under quotients. So they typically don't form a topos on their own. However, one can consider locally SUD objects, which are the recipient of an epimorphism from an object that does have this strong separation property, SUD property, and these always form a subtopos. So there's a subcategory which is a topos, however it's actually a quotient because the inclusion, the obvious inclusion function is actually the inverse image function with geometric morphism. You see immediately by what I said already that it preserves arbitrary co-limits. Of course it has a right adjoint which is just to, given any object, To take the SUD core of it, which is just the joint image of all maps from all possible, so, you know, Peter Johnstone found a characterization of which Brotendijk toposes are QD, and he found that, in terms of sites, so for which sites is the corresponding topos QD.

55:00 And it actually only depends on the category of the site because it passes to any subtopos and therefore the topology doesn't seriously affect the question. So appreciative topos, so say on a small category C, is vocally... I'm sorry. Q.D., he now in his book, in The Elephant, he calls it locally decidable. We call it locally separable. Oh, I got too many words, but the characterization is that the category C consists of epimorphisms in its own right, that is, its maps are not epimorphisms. There's a Yoneta embedding which inserts C into the topos. It's not that the maps are epic taken in the large number. Epic is a cancellation property with respect to a certain class of co-domains. If you just take the co-domains in the category C itself, this cancellation property should be true and conversa. So this certainly includes the case where C is a partially ordered set. So this is a traditional case of sheaves on topological space because the open sets form a coset. An epimorphism even in the extended category. So it's a kind of, in that sense, generalized space, because certainly two kinds of things that should be generalized spaces are sheaves on a post-set and actions of a group. These are subsumed under this condition that was in the site all maps started. Of course, so would be in the previous idea of Grotendieck about etendue, which is essentially equivalent to every map in the site is monic.

57:30 But in a sense, the epic condition is related to consideration. There's a certain way in which it's more general, because one feature, one drawback of the QD notion is that it's not stable under... Slice it, so that if you have a topos E, and E slash X is QD, E itself may not be, even though X covered the terminal object, which it covered, so to sort of locally, with not two senses, you see, locally in the sense of covering the topos, and then locally in the sense of there are enough maps into a given object inside the topos. So locally, locally, SUD. It turns out, I worked this out, this is in several of my papers, sites, this category C has a following and using property, a cancellation property still, but as you have a pair of maps, parallel maps, you can conclude that they're equal, provided there exists both a map before and a map after that has equal, so from those two equations, with some map before and some map after, then it follows. The two maps themselves are equal, so that turns out to be equivalent to this thing of there exists some cover in the topos on which it's QD, but obviously it's also a special case if every map is monic in that signature. If every map is monic, or every map is monic, it's included in this locally. So in some sense, the etadus are also locally QD, even though it's first clans to epi and mono seem to be so. That's, roughly speaking, why they came up there. And now, as I said here, as I said, this notion of generalized space is in fact general enough to capture the main concrete idea of generalized space beyond these trivial things about co-sets and groups, namely the etal. So the Petit etal topos of Brunton is certainly a pattern.

1:00:00 Well, I'm not absolutely certain of the final proof. I mean, I asked for advice and nobody answered. We have to process it some more. I'm making sure I didn't miss some separability condition someplace, which I don't think I did. Yes, you take the Petit-Atal site, you look at just the affine parts, right, that's sufficient, right, because everything has an affine cover, you look at just the connected affine parts, and that's a site equivalent to the Petit-Atal site, which has no idempotence, because of the fundamental property. If you have one of those SUD objects, and also a connected object, and you look at two maps between them, they're either equal or their equalizer is empty, and it's just because a pair of maps is really a map into the square, there you have the dichotomy, equal, not equal, and if you had the wrong thing, it would be dividing the domain into two parts, but that you can't do because it's decidable, so it's a basic lemma. I should have mentioned that it applies here because you're dealing with these objects that you assume they're connected and that's of course enough for the site because the sums come up automatically and they're generated. It's just the fact that they are in the context that it applies to all the maps. It's a kind of unique continuation. I don't know if for a long time there was a reluctance to species in a very natural sense from a logical algebraic, total tokos point of view.

1:02:30 And of course the other thing from that epoch, which is very, very hard to get my friends to look at, is the fact that he had two different kinds of toposes going there, not only the generalized spaces, but also the categories of spaces, especially the analytic one, in his Cartan seminar about techniques of construction of analytic spaces. The technique of construction was that you take the category of analytic space, but you don't need to take a site that big, and you look at sheaves on it, huge topos, and now you say, well, I want to construct an analytic space that should be an analytic space which represents the whole theory of representable functions. I want a space which represents such and such a concept growing out of the... Analytic function theory, analytic space theory. I first show that it's a sheaf. It has to be a sheaf. And then, you know, you can whittle it down to see that it's representable by one of these special objects, analytic spaces, you see. You start by working, you know, in this much larger world of the punctures, contravariant punctures that preserve the notion of cover. Now, you can say that this was generalized space. I mean this is a category that's an extra little step which is obviously coming up all the time. You look at the sheaves on schemes or sheaves on magnetic spaces and so forth but obviously these schemes and scales should also be called spaces. They're very you know they may have you could always prove more properties with special objects but they they represent they represent concepts that arise in the study even if they're very special spaces. And the beautiful virtue is that you can manipulate them geometrically, even in the same sort of way that you manipulate the particular. So I claim that even without using the word, he was constructing a category of spaces in January 1960.

1:05:00 Colin, do you want to say a little about how this connects with your own paper, your own JSL paper? The one which was the more philosophical version of which you published is sex as sets of points and spaces, because you made the point there, of course, that that paper didn't make it. Yeah, I guess I, yeah, I said, let's see, I don't know what that said. It said in two versions, the weakly designed and the solid version. In a topos with Nostralen sats, where non-empty objects have global points, decidable objects are sets, and I showed pretty much, and basically that's what he does, not only are they sets in terms of the motivation, but they are a category of sets. You've got a topos with what we call the Norschtelensatz. Every object in it has a global element. Every non-empty object has a global element. You've got the decidable objects. They are disjoint unions of global elements. Ah, right now, if you hadn't required this to be too valued, well, your global elements wouldn't be too valued. You'd get some Boolean topos. If you didn't require it, you'd get sets. And the real motivation for the paper, I explained in a talk in Philadelphia, this is for if you've got any students who are familiar with synthetic differential geometry and haven't yet learned set theory, you could motivate the category of sets in these terms. Remember Peter Freud sitting in the front row thinking. But the more serious version of that was a historical claim that, for Kantor, Kantor says a set is a collection of well-distinguished objects, and I say, look, this has a precise topos meaning. We've got toposes of spaces where space is not a collection of well-distinguished objects. The ones among them that are collections of well-distinguished objects form a category of sets. And so this was a real point to make in the 19th century.

1:07:30 If we decide that things are completely decidable, then we'll get Cantor's Cardinal's Isle, because there were not intrinsically all decidable spaces. Yes. As Cantor himself did, we extract these things within the world of meaning, cohesive meaning. So the serious point here was to say, yeah, we've got these objects which are not all decidable, the classical spaces, Peirce talks about this and Aristotle talks about it, John is the expert on those sources. And one way to depict what Cantor did was he said, well, let's just ignore the fact that non-decidability, let's treat them as all-decidable. Oh, and that was the other thing I showed. This is, in fact, a sheaf subtotal spread topology. So it's not just that you pick out the discrete objects. You can also map to them by saying, take the sheafification. And I say, that's Cantor's abstraction process. We'll take double negation sheaves. I think in the usual models of SDGs it's essential. So I said, yeah, this is Cantor's abstraction operation. He says, take any object and ignore everything except the cardinality. I said, in Topo's terms that means ignore everything except double negation identity. That is, take double negation sheaves. In these models of SDG, those things will be decidable. They will be sets. They will form a category of sets. But it brings out the geometric meaning of Cantor's abstraction process. Yeah. Yeah, and says here we've got a nice precise model. I'm not claiming I was trying to do SDG, but I'm saying here's a precise model where the kind of reason Kant talked about makes sense. Yeah, well I mean the contrast of course is more general than anything. Specifically, infinitesimal or differential, with any kind of, it's given the Nullstellentots. Well, Nullstellentots and also two-valued if you want to get the two-valued category of sets. Right. When you say category of sets, the point is that it will behave, you know, like the category of sets. It will be completely without cohesion.

1:10:00 In some sense, you see this decidable object condition is like a small step in the direction of discreteness, but the objects that merely have that property can still be very non-discreet, like a tall covering and so forth, but they're sort of discrete in the vertical direction. ...whose tangent bundle is always equal to itself is really quite discreet. Even though you're only talking about infinitesimal paths, they don't get anywhere. They have to, yes. So in other words, s to the power of t equals s along the diagonal path is the condition... Yes, which I was still worried about. Yeah, of course it's... I mean, literally, let's say, in algebraic geometry, it forces the... S to be the spectrum of a separable algebra, even though we started this with the infinitesimal thing, but this is the condition, you know, that all derivations into everything is zero, right? Because the space S to the T, of course, has figures of all kinds of shapes, so it really has to do with maps between two rings and derivations. Typical, but then you restrict that to these simple objects, and you find you get a Boolean tokos, namely the Galois tokos, because the restriction, for example, this, look, we literally think this is the thing we were discussing about stochastic sections. Of course, any finite field extension has stochastic sections. This just means the bigger one is a vector space over the smaller one, and so there do exist many linear maps which can be chosen to be rejected. So that topology, although it's not boolean globally when you restrict it to the cyclical objects, it turns out to be that. So it's the so-called bar topology where everything is a covering, and that gives all the... The Boolean atomic, atomic Boolean topos, which is basically Galois topos.

1:12:30 So, the separability idea combined with the topology gives actually a much stronger idea of discreteness. Still, the Galois group is acting, but otherwise it's discreet. Yeah, well, I mean, if you have the Nellstall and Zatz, of course. I mean, there are versions of the Nellstall and Zatz which don't require the global sections, for the moment. Well, literally, I mean, basically what you have is that these discrete things are both reflective and co-reflective, so a general cohesive space has not only a Cantorian set of, quote, set of points, but it has equally a set of components, and connected components, and these are two functions, which one is left and one is right, adjoining to the same inclusion. So there's a natural map from the points to the components and so the Nullstellensatz should say that that's an empty work for them, roughly speaking every component, because components are always not empty by definition, so any component has a point except that it isn't a global thing because you're not talking about abstract stats, you're talking about whatever this discrete thing has turned out to be. But of course it brings it up again. I mean obviously it's well known to everybody. Again, I think... The stability of the global sections factor? Totality, stability. Isn't it? Well, it just means that you've got a global sections factor from one topic to another. The way it was normally termed, the stability of the global sections factor.

1:15:00 Preserving epimorphisms. Yeah, yeah, that's just the way of saying it. Global sections preserves epimorphisms. Isn't that equivalent to... The internal choice. I thought that... It means that it's exponentiated by... Exponentiated preserves epimorphisms. If the map from A to B is an epimorphism, then from X to the power B back to X to the power A, no, excuse me, going in the same direction, from X to the A, A to the X to the B should still be an epimorphism. So no, in other words, no actual existence statement. That's the property of epimorphism preserved. It's called internal choice. You see, even the Boolean models with groups going tend to satisfy that, actually. But then the business about the global sections, you see, is if you apply the global sections factor, if it preserves epimorphisms. Then you will get a real point out of that, and the real point of that will be a choice-choice function. There are two aspects. There's internal axiom of choice, which is a very strong restriction on topology already, but it does not imply that the epimorphisms actually have sections, it's just that in some internal logics that's what it is. In fact, there's a paper by Alberta on this very subject. Is that right, Colin? Did I get that right? Well, in my recollection of reading Alberto's paper, you probably know it, if I look at his paper, his forms of extensionality and topos theory, he makes, in fact, I don't quite know the bill he just made, but he also makes a remark, which I always wanted to understand better, about the relationship between weak extensionality and relatively uniform separability conditions.

1:17:30 I just don't know if you have this paper. Yeah, I do have it. I do have it and I will dig it up. You don't sound like definitions at most. Okay, I thought that, well... What was it again? Uniform relative, uniform separability, which he... Relative? Uniform separability, which I understood is a version of his Hausdorff separability, but maybe slightly stronger than Hausdorff. What he terms, he has about six different versions of... I noticed that this SUD condition is way stronger than Halsthorpe in one point of view because Halsthorpe just says that the diagonal map is a closed map. He's saying it's a Clopin map. I mean there's no topology as such around, but it's in the spirit of the diagonal being Clopin and being really closed. Is that what he's talking about? But yes, I think it could well be that. Can you say there's no topology around? There's no specified notion of open and closed. We're talking about objects in a category which may or may not be an anthropological character. It may or may not be a category of spaces, and even a category of spaces may not... Well, my feeling is that really these are two different terms. It's unfortunate that the word separable or separated or whatever it is, For Hausdorff, schemes are separated pre-schemes, as the previous terminology and so forth. Now that's been reversed. So the separated, which basically means the diagonal is closed, is one thing, but then to say that we have this logical decidability or separability is another thing, and so the fact that they both have the same name is confusing, because I don't think there's any context in which they become the same. This is not one of those cases where there is a single very general theory of separation that specializes to all sort of one kind of category. I guess that's why I was asking.

1:20:00 I don't know. I can't think of any. I do have the paper and I think although I can't put my hands on it in two minutes, I would be able to. It's just confusing. I think what is confusing is my reflection, I'm sure. Maybe in the classical case of a separable polynomial where you look for the roots. You care whether they're separated or not. The two things might possibly come, but it's just that's sort of too special for the House Orphan Institute. It just seems that the use of the word separated has too widely separated, which cannot really be united, these two things. I suppose that I didn't. I mean, it's just a remark. As soon as I made the remark, it was verified by, guess who? He's very good at these things. I mean, unramified is an old... Oh yes, yes, exactly. It basically means you don't have branches of things. It's a property of a map, it's a property of an object, and the category of the object is over. As such, it's just the same. Were there terms for it, or terms for special cases on it? No, I mean, it's unramified. Oh, we have another, we have another notational trick. Because the French word for unramified is neat. Noam, Noam, Noam, if you want. In, in, in, in, in, in... We're going to use net, net. That's how, sorry, net, excuse me, net. Gabriel, you get Gabriel. Yeah, Gabriel, but I don't think he has been followed very much. Well, okay, anyway. Anyway, so we took this word into English, which is neat, you see.

1:22:30 But we added connected, so neat means separable and connected, so therefore in one word instead of two, this category of connected italics and mathematics is neat because they are taken as both separable and connected. In any category, the categories that need objects in any category automatically has the property all maps are epic, because they have both this property, although the limit is that it's connected, separable, to maps, but if both are both, then you can always have... In other words, because any non-zero thing mapping to it would necessarily map epically, and so the two things are equal. I mean, sorry, they are equal because, on the other hand... On the other hand, Rostow and Svante, the direct image part of the QD reflection of the general object, takes out the part, we call it nettoyage. Nettoyage, because you see we're cleaning up the general object, we're rusting off the stuff that's sticking to it, and just taking the really unique part, the cleaned up part. So you see, out of this we can make a whole story of linguistics. We commonly say that you've been taken to the cleaners, that your money has been taken. But you see, your money is around the superficial aspect of your body. Your body is still there, you see, so any process of removing sort of the secondary part of something. It can be called Netoyaz. You remove the dirt from the clothes, you see. And your clothes, of course, your clothes themselves can be removed. And your clothes themselves, that's right. You remove the clothes. And then you skip them and they're all scrubbed clean. So there are all sorts of layers which are all functors of the same sort of general pattern, because you know exactly what means SUD, that's which category you're in. So if you change the original category, you will therefore change the notion of what the Netoyaz is.

1:25:00 This kind of word sort of helps you to remember what it's all about. That's very useful. I take your point about your paper on the motivator. The condition that's in Colin's paper is showing us how to get the category of sets out of categories of spaces. Where it started can be categories of spaces that either specialized... Right, you see that, okay, that's kind of, I don't know exactly how this works in general, but you take any topos, it's got this QD reflection, which is where the right edge of my portion goes ahead, it's extravagant and that's what happens. On the other hand, you always have the double negation key, which is the subtopos, so you have a composite functor, geometric morphism, going from... The Boolean core of the topos to the nettoyage of it. Now, what you're saying is that under your assumptions this was an isomorphism between a quotient and a sub. But I don't know exactly when this might happen in general. This would help a lot to understand this Greau-Petit business too. If we could freely say that these are the same, for example, under certain kinds of assumptions. It's a little bit confusing, this fact, you see, that you have the forward and backward maps, and sometimes you even have the essentiality and the localness and so forth, so what is a quotient topos, and what is a sub-topos are distinct concepts. Occasionally you might have a retract, and so on and so forth, but especially this one. I have to apologize for the horrible color of that bead, but it was quite revolting. I didn't paint that myself, I didn't mean to get around to repainting this.

1:27:30 The sub-category of topos, which is itself a topos, But the thing is that the sort of intuitive inclusion that you have is really the left adjoint of the restriction, not the right adjoint, in many of these examples. So for example, in this relationship between the infinitesimal, he was talking about the power series and so forth, and the infinitesimal neighborhoods. All of these terms are part of an infinitesimal object, but it's via the left adjoint to the inclusion, to the restriction, that you get actually an infinitesimal object. On the other hand, the right adjoint, which is the standard sheaf inclusion, you see, the sheaf inclusion is always this right adjoint, you're actually getting quite another global space. You're getting one which is very, which is informal in character, rather than tiny as in character. It contains the same information, but it's plugged in in the opposite way into the world, so this is not ever easily pointed out, because these are actually just two different spaces. It's like co-discrete and discrete, but with structure. Well, one of the things we'll certainly come on to in the next couple of days is the exposition of what we loosely call the idea to finish

1:30:00 Yes, it's already nearly quarter past seven. That was a bit of a bang, as I know. And then tomorrow morning, Pierre, we were going to ask you if we can keep you until lunchtime. Well, at 12 noon, I thought you said you... Yes, no, you said at noon. Okay, well, I'm sorry, in English, you're a fuck. Yes, true, true. Literally, literally, I... The English use very... In my childhood, the expression lunchtime and dinner often used to change your meaning. That gives us three hours or so for discussion and I thought it would make it very self-contained to discuss the whole issue of... What's this? I found out something about stochastic sections of a cover of the line where one branch goes to infinity at some point. Let me switch off, please.