Recent research in non-commutative topology

0:00 You know, I will use this for all t0, that is that, okay, for every, every, so there are, there are temporal points. e t prime, where t prime is in some closed interval that I denote by e t zero, in closed interval . I would close, that's very important, in t, not open. When you take open and open and open, maybe you don't get very far. Sometimes you need to close. There are temporal points such that their representatives, such that for all elements in the topology at the moment, t, or t0, it is the union of these points, but their points are somewhere else, so I have to write it correctly as, let's do like this, the union of, and I will write here, the image of the point that comes from where, so you should read this, it's the image, and when it is in the future it's a free image, like I did here. But only, let's use this notation for the moment, it's wrong, but you know what I mean. It's either a past point and then this is correct, or it's a future point and then it says there is here, I think that matched with it. It's really, but okay, for sure, let me say this. So it just means that, enough points means that every open is a union of the points, but of the images of the, of the pre-images of the strange points, okay? So every open is characterized by points, but now the points are in all of the time. That means that there are enough points. And then, then I can construct the space now, computer space. So, I will define in spec a set of d0 by d0, Let's use this in the notation. Well it's just a set of points, points in this sense, the temporal points, the points at d0. But in this sense, you look at where they come from and you take the image of the previous one. So, when there are enough points, I can characterize all opens in a topology by points, at the price of being in another time, but controlled by a closed interval.

2:30 Not at infinity or something. That's why the information doesn't come from too far in the future, in my story. If you are too close, then there's nothing new, but if you're too far, then you don't follow that anymore. So there is a closed interval where everything comes from. Then this is a set, and now I can define a real topology on this. So I call this spectral topology. Okay, how do I do that? Then I introduce so-called accessible elements of T0 accessible case. This is a series of elements in the topologies, a series such that the pi t prime t, xt prime or t prime smaller than t, let's say, is smaller than xt. So they are sort of increasing strings. they are bounded by, you go further, it's bounded by the next one. And, oh maybe let me be a little bit rough here, I can always, we can also look at the exact forms. But, so that, so when you say that a certain point belongs to ux, an open set, which will be associated to such an accessible element, because that's not going to be the only sense. So I have to say when is the point there. So the point is there when, roughly speaking, the phi T prime T prime is smaller than the XT dominated by that same cos T, XT. And then for all things in some interval. So there's some, it has to be satisfied in what I call relative open in here. Relative open means it's the intersection of the closed with an open one. In some relative that's only a technical. This is a technical problem isn't it? Technical problem because I just want to take an open interval where the things are happening and intersect with this one that is given to me because all the points come from there.

5:00 So I only have to look at this intersection. This is I, t, zero, intersected sum, really open interval, in t, which is called this relative open. Relative open in the closed one. It's a trace of an open one. I only work in that close interval if I want to talk about the points at a certain moment because it's given to me. So that's only a technical word here. So I look at all the points satisfying this condition with respect to an accessible element, and that defines a topology. This is a theorem which in fact was surprisingly not completely clear to prove. So these forms of topology, the UK's are of topology. A set theoretical topology, a real topology commutative. People say this commutative. Okay, so now to go a little bit because of what's the time we have to do? It's about quarter past twelve. So you can read, or we can look at it, or you can correspond by email if you have to. But these dynamics also have a few continuity-like assumptions But everything is very, well, very trivial and intuitive. They are not, the most important one I told you, this ambiguity is perhaps. Don't rub off that bottom part. Oh, too late, too late. Damn, I haven't got that. I can tell it again later. Yeah, okay. No, I'm sorry, I just haven't got that last part. Sorry, I can't bring the crew. I like working out. That's easier for some of us. Where did you go? I just got as far as you are, just your last couple of lines. It's accessible, deaccessible elements. Yeah, I got those. T is in your X. That's right. If Pt' is at a certain moment, this happens at the moment, P0. It would be useful to have this. Then you say that Pt' is 0 Pt' is dominated by the Xt' that is given in the series, which is a series of this X. This was a deaccessible element. So you have this kind of, DxS2 means that you have a nice string that grows but controlled and that's the points, that these are temporal points and you have the condition that they are also dominated by these things. And that is then on a certain interval, on a relative open interval, relative open.

7:30 That means IP0 intersected with an open interval, the trace topology if you want on the first like a closed sat variety. But that is nothing. It is okay. You have an open set, you have a closed set, you work in the intersection and that traces what matters. So this tells you that then there was the theorem that the ux for the accessible elements is a topology. And the computative, it's a set with a topology. And of course the whole construction comes kind of behavior of these things allows me to prove this theorem, but it's not, I thought first that this would be a three line proof or something, but it actually uses all the things. But it's completely written in the notes. Well, just to finish the global idea, at least. So now we want pre-sheaves. So I want pre-sheaves, pre-sheave, then I want maps between, so the maps, they should be respecting the pre-sheave. You can imagine what you want, you want an open set, the sections to go to the sections, but the open sets are So this map phi t1t is living over the map from 1 dot t prime, phi t1t to 1 dot t, and here you have your, you know, that pre-chief is just a functor to a category if you want that, your control variant usually, and here also. So there is a connection here that satisfies the pre-chief diagrams and so on, so this system satisfying obvious diagram conditions. Just at the pt prime on an open set, you see what the image of the open set is, and goes to the pt of that open set, and these maps must be also a system and so on. Okay, so then you can say about separated pre-sheaves, that's the same definition, that's not a problem. Some condition about coverings and going to zero, the usual definition of separated.

10:00 But now the sheafification is done on that spectros. You cannot sheafify on the same manner because there's not enough points. But I can define from this dynamic system of pre-sheaves or separated pre-sheaves, I can define first a separated pre-sheave on, at the moment, T, let's say, on this spectrum, on the temporal spectrum, on the moment spectrum, on the moment, of course, on now it's T instead of T0, whatever, T. So I constructed this space with the topology. This system, this dynamical system gives me, if they are separated pre-sheaves, it gives me a separate pre-sheave. If it's not separated, then it gives me a pre-sheave. Now, here, because this is commutative, has enough points, it's a commutative thing, I can chiffify. And this is my chiffification of the pre-sheave at the moment t. So this is the chiffification. This is classical, because now I'm in the classical commutative world. So in a sense, this thing is the commutative, what you observe now, that this topology that is done by these accessible elements, the UX for some accessible elements, they determine that the opens, all of them, so the topology, topology that's called this part, so this is given by the UXs, is determined by the topology on the lambda t, on yp, let's call it yp plus t, is determined by lambda t. It is just like this, you take an open here, it was the union of temporal points, because there were enough points. So I look at the points, the union of these points, that open is a set of points, that set of points is the corresponding open set here. If I give you, of course, a set here, it is a set of points. I take the union of the representatives at the moment t in lambda t. These are the border points, but I take the string until lambda t. I take union, and that gives me the open here. So they determine each other. But there's no direct map. There's this space. There is no direct map from the lambda t to the spec to this. This is the non-permutative thing. there is some kind of thing here in both ways. The objects in the topologies are

12:30 completely determined by each other via the set of points, temporal points, defining the open set and then you take the union of the correct images in that moment t. Conversely, temporal points take the union of the points that gives you the open set here. Okay, you have to check something. So I don't have a, here's a commutative world which you measure. This is space-time with a lot of time. There is some space, R4, if you want the fourth one is the time, let's call it down. If I now move to any other moment, then my spectrum will change completely, so I will have a yt and a yt prime, and I will have a lambda t and a lambda t prime, and these are well connected, and they are so well connected by this topological description. but I don't... YT and YT prime are not connected? No, there is no map here not that I know a priori there is no map here corresponding to this map so this kind of thing is broken you observe reality, I suppose in this sort of measurement space but it's, so you see this space is much better than the commutative shadow because the topologies correspond completely and in the commutative shadow you have many opens corresponding to the same projection. here, no, they determine each other. So this is sort of the best commutative approximation of the non-human space. But it's not so punctorial in some sense. But there's some kind of, I can say that there's a kind of observed truth for a map here, but I cannot describe these maps globally. So again, that observed truth is in the picture. So it's kind of a relation It's the same spirit that I was using, but it's done at a... It's a formal thing here. I didn't start up with the idea to do physics, although, of course, I cannot deny... No, no, I mean, once you put that up there, you're thinking physics in a certain way. Yes, yes, yes. When you think about how things evolve, you know. I want the mathematics that the points behave like particles almost, and that the points are not always available, but you still, it is bad to make an abstract construction if you cannot calculate.

15:00 So I want to calculate in this community, this could be R4, whatever, a nice space. I already described for R4, for example, the deformation among community variety or topology that corresponds to four-dimensional affine space. and with the cell of intersections I can play I can make examples so this is a very general still generic model I think just one point Freddie on that other one over there what you eventually do is you integrate over the size of the cell so we don't have a limit as it were on it like you're putting on here we actually integrate it but weight it in a certain way so well of course which is the fact or the assumption that there are enough points locally in some closed interval. Then the thing works, and then these connections are as strong as it gets, because you cannot have a map here, because, well not in the classical sense, I don't have points. Here I have enough points, I could think of maps here, but they do not really, but it is clear, this is not really determined by points, it's topologically determined by points. so in a sense there is a kind of topologically determined map if you do that so on open sets they determine each other so it is observed truth but I don't have a map that tells me which point goes to which point and the trouble is you could remedy this by playing with an integral there should be a way of describing that condition the breakdown of that condition you see the thing that worries me I don't know who am I to say, but the thing which worries me slightly about this construction and also about that notion of Chris's about the arrow fields is that this way of thinking of mappings in between these structures, well, let's say between these spaces or something, it is very, well, it's not very unfunctorial, but I don't see how you translate it into the functorial way of thinking that. I think you can do some other categorization, right? Why do you think of maps as defined on points if you... Well, you don't necessarily, of course. But if you do it on some topological aspect, you can induce some kind of motion of map here, I suppose. Yes, that would be very interesting. There is some kind of thing. As I said, you can go like this, but in the vertical strips, you only say something about how opens correspond and not points.

17:30 There may be a mixing of the points in open, so perhaps if the open is really the terminal point by some Hausdorff properties in every point or something, maybe you can find maps. This brings to mind very much, I remember, an exchange between Bill O'Beer and Cartier in June. By the way, I don't have a priori a Hausdorff topology, but I guess perhaps if it is, if you have me post this, if you know that your deformation leads to a space here, which is Hausdorff, then I perhaps can define a map. Because I have an open and I take filters defining the point and then I probably find the point, of course, and not probably. Well, the remark that Cartier made to Orvera, I think, I'm trying to remember, this was four or three or four months ago, but it was something on the lines of it until the thing that the most important, we were discussing, you know, what was the most important lesson that Grodin-Deacon taught in algebraic language, and I think the soundbite was that the questions about the components of spaces, and particularly about the components of connected spaces, are always more fundamental questions than questions about points. that you can only, as it were, get to the questions about the points once you've understood about the components. And that, of course, it's itself an expression. Yeah, of course, I read it very generic still, so I didn't put conditions, but as I say, I assume that once you say that these spaces are known to you or given, and that the other things are deformations of that, and different deformations are connected in time, then I can make a matter of that. But that's an extra assumption for the moment I didn't make. Because in the algebraic geometry, you only have T1, not T2. You don't have Hausdorff, you have semi-Hausdorff, or what's it called? T1, the Zalinski topology is not Hausdorff. So I want to keep it open for the Zalinski topology, but if you take a sort of real topology or whatever on this space, and suppose that it's a nice space that you know, then I'm sure that, you know, that opens the terminal point by the limits, and you have a map, whether it's a good map, who knows, but it's a map. But I didn't put this in because, as I said, to me this is just as well the Hausdorff or some real topology or some Zariski topology of some type. So I didn't want to put in things that were not necessary. And maybe, yes, so maybe there is a map, but it's a different map than this one. So still I think we have to go to the non-committative world to see that.

20:00 In fact, to determine this map we have to go there. Yeah, these identifications of the topologies. So it's strange. I've never seen a map between topological space where you just say, okay, the topological opens, determine each other, and that's it. I don't know more. Although I have a construction by strings. So I know I have a procedure to construct it. But I never saw that before, that such kind of map between brackets. But it could explain why the measured things are really living in different worlds. And what was pleasant for me was it solves the sheevification. Ah, by the way, the very interesting thing is that the stalk, otherwise it would be stupid, but the stalk at a certain point in another time is defined and is in fact the stalk of the pre-sheev, which is always defined at that point. So, this is very interesting. It's really the sheathification you want, because what do you want? You want to construct something which is a classical point-wise set, and the stalks of this sheathification, this is a theorem. They actually come from the stalks in another time now, this is at moment t, but this pt prime, I mean it's a temporal point. It exists at moment t as an image or as a preimage, the stock of that temporal point, so I should really like 5 t prime. Then I get the stock at the real point of that real space at the moment t prime. So in this string construction I have kept the connection with the non-commutative spaces at least on the level of stalks, sections also. So this thing reappears here as the right stalk in the shape. So it is really a chifification, it's not just a chif, it's really a chifification of this kind of string-wise defined dynamic system. So that's also a very important point here. It's also a non-trivial theory, I had to work to prove it. But you know, it's a force from the structure. Well

22:30 I call this a space continuum without timing it. Space continuum. That's in chapter 4, isn't it? Yeah, that's chapter 4. That's the thing I didn't ask. No, but this was not done the last year. It took me longer than I expected. That's very interesting. So I don't know... This would be a nice way to tie with the increase in entropy. and I'm pretty sure. Yeah, the way that the thing that Omazawa and you were always trying to get from the 3D Omazawa That was very limited as you can get an entropy operator but I was never convinced by what he was doing and yet somehow that idea of the not having in my case not having a unitary I mean I was just talking about unitary and non-unitary transformations and trying to control the class of non-unitary transformations that one needs and this seems much more fundamental It's much nicer in the sense. It is... I don't know. I didn't want to use anything except what I needed. No, no, no, no, no. I didn't have a preconceived solution for this. I started to think about... The stuff for me was the sheepification, for the non-community to make a sheepification on some space. And by accident, when Arthur and my first doctor, I think he came with that probability spaces. By the way, so if you put in our probability spaces, the thing is to check that on the limit, on the limit you get the probability space also. I think. So these maps should be nice maps between the probability spaces. You can put extra conditions. So that's maybe the advantage of trying to be generic. The things I assume, I need. And if you don't like them, then you have to sacrifice some property. Yeah, yeah, yeah. So like the ambiguity, the non-ambiguity interval. But to me it was kind of intuitive because if you, at every moment there's infinitely many, or whatever, I find it, but many possibilities to go back, then it's too chaotic. So I needed at least some small time interval, small as you want, but some time interval where every element has a, if it has a pre-image, if it has one, it's unique in that interval. And that outside can be, and it's very important for this future thing that outside this small interval things become unpredictable. But in the very near future, which in my opinion is sort of logical, in the very near future you can almost see what's going to happen.

25:00 And if you see the car coming in and I'm going here, then you can say, well, it will hit me. When you're in the self-same moment, then things, as it were, are fixed. So this is what the effect of that is. This is absolutely a fovenal thinking to the Planck idea. Because most of these people have been trying from chaos to get order. And this is different. You've got order, but you haven't got order. order but you haven't got global order no that's the point well that's exactly the point yes and that is observed true i have some statements that i cannot really prove but i can prove that i cannot contradict them in in a certain interval so i need more time to contradict them and i don't have that time perhaps and this is the same problem where you have it's all coming down do some uncertainty principles and mathematicalize now. Very interesting. Yes, because what you're doing is you're reflecting this uncertainty principle in the structure of the mathematics itself. Yeah, it's not in the mathematics. You're plastering it all. Yes, that's what I like. Yes, exactly. That's what I thought. You see, that's what happens when we're doing quantum mechanics. We plaster this on. We plaster non-locality on. Yeah, definitely. And, you know, you've got a theory which is based on absolute locality, and then suddenly local. I mean, this is crazy. And just say, oh, it's a non-local, that's quantum mechanics. That's not really getting to the heart of why non-locality is a necessary feature of that structure. And it's the same here. This is beginning to show why maybe, I'm talking as a physicist once again, that the entropy, the increase in entropy is more to do with this, the fact that these global relations are not possible. I was happy with this observed truth. That's an extremely important point. I think you're absolutely, conceptually, that is absolutely true. It's a new idea, at least for me, I think it's observed truth. And I cannot contradict it, but it's not a proof of the truth of it. It's also important that finite, but of course, all things in physics and mathematics that you really do, that you really do. They're all fine. They're all based on finite. We don't have a continuum. All this finiteness is there, and then you cannot contradict it, so it holds in small enough intervals, and then you try to connect them, but you can never get outside an open interval. And therefore you don't know how far you can get with your...

27:30 And there, here's a... So how is this different from chaos theory? Because you put a lot of structure in chaos that... Well, chaos theory, I cannot make the connection directly. I'm sorry you just got to process I don't know it well enough maybe there is in a normal space maybe there is some notion of closing points or something I'm not sure I think there's still deeper conceptual issues I think to be uncovered about the way that this tremendous deepening at the C&CM structuralization of the notion of point is connecting with the other features, I mean particularly this the fact that you don't have a notion of direct map it's not essential at all, it is whatever definition you put, you make a construction that has enough of them by, of course, the main action that you assume that in the beginning there were enough points because you've got a sufficiently rich and structuralized notion of point or surrogate for the notion of point But this business here that you had in this construction down here, there's a little diagram right in the middle of the board, that, you know, you don't have any direct map from the lambda t to... Yeah, but the objects in the topology completely determine each other just by the structure on the opens. That's very interesting, as you say. I've never seen a notion of a map between topological spaces. It's a strong topological condition. wondering, sorry, this is King Charles' head, if it comes back to some of the things that Carter and Lovere were arguing about this summer, which I've got to go through those tapes and do transcriptions of the key passages. I mean, this may be going up completely at a tangent, but there was this big debate as we spent most of the day on as to whether the whole of the mainstream of 20th century topology had been going in the wrong direction, which, of course, Lord V.A. wanted to claim that he had been. I can almost agree, but... But because... Well, no, but, okay, the soundbite... The soundbite is because they have taken the notion of open sense. Continuous map. Continuous map and open sense. That's fundamental. What they should be looking for is something which takes account

30:00 of the relationship between the covariant and the contravariant and the functoriality of maps, and they should be looking at something more like the notion of figure inner space, of the kind that, well, we'll take it right back to you. These maps of the points, they never looked at. Yes. Because they look at the points, and then they want to express an inverse image and so on. And there was a very good, rich discussion on that, and it puts me very much in mind of what you're doing here. And I wish, I'd love you to get together with Carter. I'm really sorry you can't come to this meeting. But you can get together. We'll get you together sometime when we get back from China. Someday, maybe? No, no, not someday, maybe. It has any value. Yes, we must make plans to get you together. And maybe you guys, at the meeting in London first, sort of made me believe that I should perhaps formalize it further. Because at that time it was just, in my coverage, it was just a hobby, which I included in the last chapter of that. But I certainly never seen a notion of math like this. It is clearly an important topological condition, and it's telling us something about the inadequacy of the modern version of the topology. I know the technology of things as well. It seems to be determined by, I have a lot of projects that I didn't solve, because I'm lazy. myself, I should look at it, but there's a lot of homology and so on you can do that. Yes, and it's not going to tell us something... It's a new meaning because you have these correspondence between opens, so I have a feeling that there's a map on homology. That kind of stuff. That would certainly connect with that whole kind of of translating everything into terms of conditions on homology. So I have a feeling that when you start to do this because the opens it's not determined and there is, and perhaps That's what I put in the project. If you want to do some chronological maps or interpretations, you have to do the non-commutative chronology, which I started to do with Willard in a very concrete case for non-commutative algebras, calculating the sheave chronology over some non-commutative rings by using check chronology. But I think this is in here. That would be really, that really would be interesting. But I put it as a project. There are 24 projects here. If somebody wants to do something, I gave a lecture about it. I really think that would be a very, very, very worthwhile project. There are many projects I run from. That particular one is because, as I say, it seems to me to connect up in such a striking way with this really useless idea. And also, as I say, this is also telling us something about that there's something not quite right, or there's something too classical about the way that we think of categorical isomorphisms in general.

32:30 And as I say, the more I look at... And I'm more and more struck that this idea of Chris Isham's about the arrowfields may well be on to something really important. But the trouble is he's expressing it in a language which is too dependent on physicist notions of configuration space to appeal to the mathematicians. But I think he is on to something important. And this is another aspect of it. We have to go. Yeah, we've got to go. I think in aspect of both of you, you know, you're just, you're already happy. You have the homological maps and so on. You don't want these maps on the spaces themselves sometimes. Yeah. So there is some, perhaps to find some root for that kind of idea. Well, I'm assuming that this is what Karkir was getting at with that remark about components always being more important than points. So... Yeah, I'll put both of this together. Wait a minute, let's get ourselves organized. No, no, no, no, this is the committee.