Categorification & quantum gravity, lecture 4

0:00 Okay, so I thought I would start by talking a little bit about The type of mathematical structures that are coming in these models, just in general. I thought I would talk a little bit about the relationship between topoid and model categories. I'm not going to talk technically about the definition of the topoids. I mean, the main idea is that they have a power set. So, you have the analog of the set of all subsets of the set. From the point of view of geometry, set theory is a little bit bizarre. You know, there are subsets of, you know, you can, for instance, you can take a ball and divide it into subsets and move them with Euclidean rigid motions so that it then becomes two balls. I think you actually can't do it in two dimensions if you do it in three. So, of course, this is utterly unphysical, and it's because if you think of a region in space as a point cell,

2:30 it has subsets that you could never physically construct. They're constructed by the action of choice. Of that kind that sets have. So you can do, and because you can do anything in mathematics even the kind of things that are set there, all the constructions are mathematics. But you imagine that, you know, it's not like purely a theoretic part though, because it's also, it's about measure, right? You have like the same measure of two balls as one ball, right? Well those kind of, the subsets would not be measurable sets. No, no, I think the thing is that, how to say, it doesn't resolve volume, otherwise it wouldn't be that bizarre, right? You have one ball and you get two balls of the same volume as the other ball. It's all about those integration theories. And then it's, you know, it's what kind of, like, Stoltz's integral which you use for that. So, it's about how you have this very bizarre surface. Well, in some sense they're not measurable, and therefore you can't integrate over them. You could not integrate over one of these subsets. It's not a measured space. Yeah, but in that paradox, I think it would be measurable, because Albrecht doesn't make sense. Albrecht, of course, you can just make all numbers, and you know. No, if they were measurable, you'd get a contradiction, because the measure wouldn't add up. So you cannot do the Podokarsky paradox with measurable sets. The whole point of it is that the measured space has to be only on a very distinguished subset of all sets. A set of subsets, but it's not all sets. They're not measurable. If they were measurable, it would be impossible.

5:00 But how then do you talk about volume? Well, the whole thing is measurable. Right, subsets and the right, absolutely right. When you rearrange it, that's a different measure. You're trying to get measurable, but you go through something non-measurable. That's right, that's right. So, that's just to draw your attention to the fact that the set theoretic properties are unphysical, they lead to all kinds of... A topos would mimic that you're so-so in sets. A topos is just like a world. You can do anything in mathematics you like in any topos. You can describe the category of vector spaces or any of these sort. In the model category, there are topological spaces. In the model category, has a homotopic type. So it's possible to speak of connectivity, cohomology, homology as if it were a topological space. All of the standard topological concepts can be thought of as dealing, as applying to objects in a model category.

7:30 And whereas topos theory is generally thought of in mathematics as a branch of logic or foundations. You know, most mathematicians use the word logic the same way Republicans in America use the word liberal. All you have to do is label something as liberal and you can dismiss it. I mean, it's liberal. I think we've both begun with L, I think. Oh, that's logic. But of course, what they're really doing is just saying, You've got to build on this, you don't like what you get, and so you could be putting it in a box, and you're like, logic, do not open on it, you know. But category theory is mainstream mathematics. It may also, because kind of, I think, more or less assumed view that actually topology has kind of these two aspects, right? And the other comes as, say, Peter Johnson describes exactly also kind of, he would say mimicking this category of topology, but he would say kind of generalization of ideas, topological space, right? And so you kind of like this idea that you have in the same, how to say, object, you have kind of both logic and kind of geometry. So, if you compare not just logical aspect of topos and model categories, and that sounds rather obvious, but say, if you take the geometrical aspect of topos and geometrical aspect of model category, how you would make it? Well, all right, but I mean, well, look, I mean, a topos by itself doesn't have topology.

10:00 If you want to add topology, you have to add layers. The whole structure of modern mathematics is based on layers of structure which correspond to different levels of abstraction. So, whereas Euclid would have told you some theories about the plane, and the plane was the plane, you know. It's naively thought it was supposed to be a real physical plane. We would think that there's the underlying set, and then we would say there's a topological space, and then we would say there's a measured space, and we would say there's a linear space, and it's a linear structure, and the different aspects of the structure are different levels which sort of put on top of one another. That was about Giraud's theorem and this kind of, yeah, because it's a rather general result that, say, not any topos, but in a sense almost any topos, right, kind of reduced to nothing deep topos. Deep topos is something which does have topology. Oh, well, all right, you could, the construction, you could, yes, the site, you know, it's a general method. Rather, it's a generalization of the topology. It doesn't have to be a topological space. It can be any category. All right, there is this different point of view that says that you can think of a topological space as a kind of category, and then you can construct a topos over it. By generalizing she's over there. Okay, I'm not, I'm not, that's not where I'm going with all this, that's not where I'm going with this. Um, you notice that there is no, there are no layers. Model category has topology because of its relationships with other objects in the category.

12:30 The structure of the category allows you to think of it as a topological object. Um, but there is no underlying set. There's no way to recover a point set, John. It's not, there are no, it's no layers. It's just what's there. Now, the interesting thing is that this is not taken as, I mean, mainstream mathematicians do not think, do not think of this as... I don't understand what you mean by no layers. Yeah, okay, so if you think about general topology, general topology... A, topological. A, together. We have sets. There's a projectile monitor. So you can think of these as simpler. So if you take a course in general topology, first you study set theory, at least if you have a chance of stating it down. So this is just what you can say about sets. That's the layer of structure. And then you can have this additional level of structure. And it's very important to remember that the same set can have different topologies. So it's another layer of structure. So if you have a metric space in the tricycle, the topology can have different metrics and so on. So we separate out geometric structure into these layers.

15:00 Whereas here, the model category, there is no underlying structure. There's no set. Okay, so a good example of a model category would be In the initial step, this is like the mathematical generalization of the simplicial complex. So what it is, to say it very simply, is that you just have a set of complexes of simplices in different dimensions. And then you tell me, simplex in my list in the lower dimensions is which boundary component of each simplex in the higher dimensions. It's a set of sets together with a rule, so this edge you would say alpha equals, it would tell you where the initial edge of alpha was the point A and the final edge was the point C. So you want just relations? Yeah, that's right. And then each triangle I would tell you which of its three faces. So this one, one equals, you know, alpha, beta, gamma. And then there have to be certain relations between the assignments. Sets is another example of...

17:30 That would be something different, yes. It wouldn't be the same. It's not the same, but it's actually... But also one of them. Pardon? It's also one of them. So you can make cubes of all dimensions in case... That's right. And then we can formalize this by saying that we have sets S1, S2, S3, and we have delta from S1 and delta squared from zero. Now, notice that these things are not point sets. A simplex, a triangle, is just a member of a list and they tell you what its faces are. That's it. So it doesn't have, so a simplicial set in this sense is not a point set. Now, in fact, Milner is responsible for defining a functor from these things to topological spaces. All of these are called Milner realizations, and it had to be something just as brilliant as Milner because it's so painfully simple. So what you do is you assign to each simplex one copy of the standard simplex, which is, as Hoyne said in RN, you identify them along the box, assigns a topological space, you just take copies of the simplex, You put them together and you end up with a triangular topological space. Now this is a functor. It has an adjoint functor, but tricky. So it's called singular. So it's all possible that that is. You take a topological space and assign to it all possible of it. All possible in the sense that you can just make a...

20:00 You can create all continuous maps from the standard simplex. And you identify it. So, you see, you get two functors, you get two functors, from sequential sets back and forth to top. One is Milner realization. It's not precisely jointed. They are jointed. The two are jointed. The specific name of this is Milner. Well, Milner is geometric realization. It's a Milner realization. It's very similar to abstract. Yes, yes, yes. And it's discrete. So when you write down a 3-manifold, say, if you think of it as triangulated, it's given by a finite set of non-information suffices to describe, like, say, a compact manifold. But the thing is that this equivalent, this actually... Now, the interesting thing is not just that these are adjoint functors, but that imposed homotopy equivalents, homotopy types are the same. If all you want to do is represent a homotopy type, you can represent it by a sequential complex just as well as a topological spectrum. So, in some sense, in terms of the standard things we do in topology, superficial complexes are just as good as topological spaces, even though they do not have quantisets.

22:30 Actually, when I wrote the differential geometry, it was very much a point of, actually, it was Panenka, we did speak about categories in this context, but actually that was his somehow main idea, that it's kind of dual weight. Yeah. Well, of course, there is this rather simple idea. I mean, it's abstract. It takes people a while to say it because it's very abstract. But it's a very simple idea. I mean, nobody raises their hand and says, can I see the details as proof? I mean, once I tell you, you believe it. But my question is because what you say, there's general things about ideas that you told me in the Sunderland. I think it's also very much motivation for a bit different things like locale, like stone spaces, you know, and I just try to understand it. So that's a general idea, right? But that model category is kind of more specific way to... To elaborate on this idea, different from, say, from local others, right? So what here is really... Okay, now what's really interesting about this is that the reason, as I say, that this is not considered a branch of category theory, it's not considered a branch of logic, you know, it's considered an important topic in mainstream mathematics. There are more sophisticated model categories. You lose some of the information. You no longer get homotopy equivalence. You get something simpler and you lose some of the information. But, calculations suddenly become tractable. The reason this is considered important is because things at welfare will be calculated any other way. And be calculated very easily by passing from topological spaces or manifolds or whatever to a certain model category that loses some of the information.

25:00 Okay, yes, there's proofs, yes. Unfortunately, I mean, especially considering that you just asked me for a textbook where you could actually learn something, unfortunately, yes, there are proofs for everything. They render the literature completely formidable and unapproachable. They seem to be quite rigorous. Yes? What do you call this model? Topology, so that you can do homeopathy there. So the definition, I mean, I should probably hint at the definition. I think writing down the formal definition, well, it would take a very long time and I would have to go back and do it from memory. This is... Oh, you're kind of hoping to blackboard it, you mean? Yeah, you know, I was hoping to leave it shut because I'm tired of wearing dust. I have asthma. I'm much happier to do this, actually. So a modern category, since we no longer are working in topological spaces, I have to tell you what I mean by homotopy. So let me remind you what a homotopy is from, yes, exactly. It has, you take a product of your space with the interval, and then blah blah blah blah blah, and you get a homotopy. Cylinder objects for simplicial complexes is by taking a simplicial complex, taking its product with the interval. And what place it will link to? Well, the interval, right? Taking the ordinary common part of the interval. Taking the Cartesian product, cutting it up in a natural canonical way into synthesis. The product of the simplex of the interval isn't a simplex.

27:30 But there's a natural way of cutting it up in the synthesis. So you write down that formula, pretend you didn't already know where it came from, and then prove by hand that everything works, that it acts like a similar object. So that's the idea. And again, if you don't want to draw a formal picture. So the writers, all these people doing like pointless topology and stuff, they would just try to mimic Locke, just how Namer would behave, and here they just wouldn't do it. Right. So actually then, this is actually, this proof even is very sophisticated. The axioms are that in the category we distinguish specialized classes of morphisms. Some of them are vibrations, some of them are cofibrations, some of them are homotopy equivalences, and they satisfy certain relations. And then equivalent proof in this long paper called homotopical algebra, which I think I didn't feel as well. That is to say, you can then define homotopy equivalence classes, and homotopy values, and so on. P.M.C. you can give it in because, see, what I never really kind of felt were the natural home of the theory was exactly this notion of interval. So you have this kind of distinguished object, an interval, and then you have all this map, and then you can say why you take just this interval, you can take something else. Probably you're getting, it's a good question, but why is it natural to do something like this?

30:00 Well, I mean, you're thinking about deforming a map continuously into another map. That's right, but you have this interval, which somehow, it's like a union, like kind of an informal thing, right? So it's more physics. In this context, what would it represent, say, physically or whatever? Well, I mean, it's interesting. I mean, they formalize. Instead of saying you can pick its product at the interval, you just have an axiom that says you have a functor that applies to every object, its cylinder object. And they can be very different from, well, in these, so in a sense that's right. It's intervals without intervals. We lose the intervals in object in this abstract picture. But all the examples I know come from sort of sticking it in the back door. So I don't know why that is. The thing about intervals, of course, join two intervals make another interval so that cylinder objects have to have the next composition. So you can just make the axiom that there's a composition and then that's what that's right. It's a very deep question, so it's abstracting some of the properties, and then you can find examples of categories that have an analogous structure, but there's no interval, so that's sort of interesting. Interval is much too valuable to throw away and much too mysterious to keep, so you want to keep it and throw it away at the same time, and that's what you do. So now, the thing is that there are model categories.

32:30 Where you lose some of the information in an interesting way. If you don't mind, I'd like to, for a moment, digress and talk about something related to relativity. If you have a region in general related to some future observer, and you can ask, how much information can pass from this region to the observer? Now, naively, of course, classically, since this is a classical theory so far, You would think there would be an infinite amount of information that could, you know, you could have a continuous function of time as an extended observer. It's also an extended function of space. You would think there would be an infinite dimensional space of information that would pass out of it. Oh, and I think that's not so. Just within general relativity. There's only a finite dimensional Hilbert space of information that can be communicated to the outside world. And basically, this is related to black hole physics. If you try to put in too much information, information has to have stress energy. It can't be embodied without stress energy. Everything about this kind of physics is completely inconsistent with the idea of the mind of God storing information. You can't put information in the mind of God and take it back out again in all sorts of physics. But it has to be stored somewhere. So if you try to cram too much information, cramming too much stress energy into a region, So, the Frankenstein problem is a generalization of the theorems about black hole thermodynamics, the idea being that if you look at a region, and you look at all the possible states of it, most of the states are very close to a big black hole, and the things with much less energy have much fewer degrees of freedom, you know?

35:00 When you speak of information, is it equivalent to the exponential of entropy? Yes, absolutely. It has to do with the dimension of the Hilbert space. It's the logarithmic dimension of the Hilbert space. So you see, so as an argument before, and this is a philosophical question... Just technical, but why do you call space, for instance, like you kind of... Can you try to use the quantum theory embedded in the space now? Try to... Okay, okay, yes, very good. But I mean it's fine, I mean it's on the question. Oh, but I mean, that's not very surprising. It's a spacetime, so there's a measure on it, so you can ask your questions. There's some collapse of it. The region, especially in spacetime we've been using, has too much information in it. That's served very well as a foundation for quantum theory, because quantum theory has to do with results of experiments.

37:30 You cannot. Observers really should, observables really should be idealizations of experiments you can perform, and the observer has to be exterior to the object being observed, and feel retrievable as high school. So, it's only the information that can pass outside of a region which should experience quantum theory. That every time we try to construct a quantitative gravity over the last 80 years or so, we run right smack dab into the problem that every quantity we try to calculate, theory tells us the value is infinity and has so many infinities we can't even cancel. We have no control over them. That in order to construct a quantum theory that will be finite and well-behaved, we need to reconstruct a model of the region that will be corresponding to the information. The information inside is kind of a metaphysical balance. Yeah, that's right. If there's information inside that you can't see, well, it can't come out in any way, shape, or form. It's not just that you can't measure it. It can't affect the result of a measure. So by instance, principle, we should just say it's not there.

40:00 And anyhow, development, which led us to the idea of Javitrius' Poinsett, which we owe to the ancient Greeks, had to do with a very naive extrapolation of things which can be done microscopically, like cutting triangles into two parts and so on. And just assuming that we could go on doing this indefinitely, I'm not exactly sure whether we could do that. Well, it did lead to a nice simple rigorous geometry, which would be very hard to obtain in a good way. Anything we come up with will certainly be much harder than what Euclid did. But it's been around for a very long time. And physically, there's no reason to believe it. I mean, in fact, just the contrary. It's not just that we cannot subdivide it indefinitely, but that there is a definite limit. Physically, you cannot measure anything beyond a certain minimum. So that leads us to believe that physical space-time has a certain granularity. There's a limitation to its subdivision. And I'm just reflecting on this idea, which originally led me to try having general relativity on a superficial complex rather than a point set. You could imagine a superficial complex, and just all the simplices were on a long scale, and you would just say, that's all there is. It's not meaningful to add points to the interior of the simplices. It's not meaningful to subdivide them. Now, unfortunately, that's too naive. There's no way of picking a particular triangulation and saying that corresponds to what I'm going to speak about. So I thought for a long time that the way to really have a foundation would be to have some way that this is the triangulation of gravity,

42:30 because this is the one which corresponds to the actually accessible information. Now, I don't know any way to do that. If you insist on centricial complexes, it's a bit too many, but in very special things, what the model categories are going to propose as settings for quantum gravity is that there's a construction which is called the minimal. Just a naïve question. You say, okay, model categories in some sense is category of topological space, right, kind of. Absolutely. That's so, just a topological space is an object of that category. That's right. Now, when you are trying to, how should I describe, physical space-time... Do you think like space-time has just one particular object? No, I think regions of space-time. Regions are objects. And then you have, although they do not have points, there is a structure of relationships between regions. We can speak of sub-regions. We can speak of joining sub-regions to make a region. And we can also speak of smoothing the forming of one region in the size of the whole category of a region. So what I believe is that there is a category of physical regions, and it behaves somewhat like the category of topological spaces, but it's not as rich.

45:00 There is no structure that we can divide to some kind of calculus. I mean, if I come up with a category and I make some physical arguments that it's possible that physical regions are really objects in this geometrical category, and that's all I can say about them, then in fact my book is done. They have to somehow be modeled by mathematical objects which enable me to It's kind of a computational program. But did you try to compare this notion of rotative topologies, because in a sense it's also the idea of how regions somehow relate to each other? I had ideas about that earlier on, but what I found is that when I actually tried to start thinking about what can you really see from a region, but actually using results from relativity and astrophysics. To think about the information transmits from a region, I found that it much more naturally translated to a branch of model category theory than it did to... I shifted, it was very strange. I was sitting there trying to construct a topos and I learned about something and I said, well, let's learn that math that's connected to that and so on. I mean, I've gone through an earlier phase of studying all the different approaches to replacing point sets in mathematics, which is very broad. I mean, people very seldom talk about them all at once because they come in different areas. But it's interesting to do a survey and look at all the different things and see what they have in common.

47:30 Gravitational lensing. I felt that I couldn't see how it would connect to gravity topology, when on the other hand it connects very naturally to a certain class of quantum categories, so... It's not a big category, right? Because if you have associativity, you only have to... or it doesn't... or at least if you just take the... Well, it's interesting. I haven't seen anybody actually discuss this. But in fact, final categories really are two categories, because you have a lot of homotopies, you have homotopies of that. So a natural way of thinking would be as two categories, and a lot of things are only done up to homotopy, so you can have weak or strict constructions. You can say something is equal on the notes, or you can just say it's equal up to homotopy. So it's a very special example of this general class of things that people study. When they want to study topology, they want to study two categories. So it fits into the program of higher category theories because the two categories all started much better than you would think. Somebody told me at this conference in Glasgow that there actually is a theory of model two categories, which doesn't surprise me. It seems very natural. But I didn't get the reference. Models seemed very good. I was really surprised. Categorification has become a real industry. There are over a hundred people at this kind of solid time.

50:00 I love them. Yeah. Actually, it was very funny, too. Practically nobody knew anything. When I gave my talk explaining how the idea came out of physics, they didn't know it at all. Lieber-Frankel and I invented it, but they didn't know why. So anyhow, so on top, some relationship to the appearances is there.

52:30 Model you mean like for physical space-time? Well, I mean it means choosing a unique or minimal representative of its homotopy. If you again compare with the stop us approach, because one thing's really... It seems to be interesting, topos, as you mentioned, right, that you can internalize things there, so you can kind of say, okay, we have this topos and our, I don't know, logic, reasoning, we have this notion of internal language, right, and somehow what we say here...

55:00 A topos itself is not a topological object making construction in the topos. And then if you take topological spaces in a topos, or manifolds in a topos, or whatever, you certainly don't get a reduction of information, you don't get effective calculation, I don't see any way to connect it to general relativity, and there's certainly not a model of each other, because none of the above. But these are some strong things to wish for, and when I write it out like this, if I just said, well gee, maybe I had to find something that does all these things, you would think this one, you know, it's like you write down a list of wonderful properties and say, gee, if something had all these wonderful properties, it could prove wonderful theorems, but yeah, that's nice, but... But put it this way, see, in Einstein, just classical general relativity, right, there was this important idea of finding total and intrinsic geometry, right? Well, what happened in Einstein was that he found a piece of abstract mathematics and his hints from physics clicked. The story in Einstein's general relativity is quite interesting. He started thinking about a rotating body. So he thought about a wheel rotating. And then he said, let's look at the little segment of the wheel. Well, it's moving in a direction parallel to its length, and therefore it is still contracted. But on the other hand, its radius is perpendicular to it, and therefore it's not contracted. If you were to sum up the lengths of the little segments and find the length of it, it would not be too high next to the radius. So the ratio of the circumference of the diameter is not pi. So then, that's already curious, but you can say, well, it's rotating.

57:30 But Einstein being Einstein, he didn't go so far as to say, rotating respective to walk. So the only meaningful statement is that it's rotating relative to fixed stars. So, it must be that the gravitational field of the fixed stars determines if this wheel is rotating and the other one isn't. So something about the gravitational field must be, you know, you could just as well keep the wheel still and take a bunch of matter and start spinning it out here. And then, since it's in some sense the same experiment, the ratio of the circumference to diameter would still have to change. So you said gravitational field must cause the ratio of circumference to diameter to change. So he needed some new kind of geometry in which C is not 2 pi r. And then he went to ask his friend Grossman, is there any kind of mathematics where C is not 2 pi r? And of course he said, oh, that's the whole modern theory of geometry. That's curvature, you know. All of these are possible, but natural and inevitable and centric to the structure of this piece of mathematics, which he had no reason to think had anything to do with this problem a priori, although Riemann had already suggested that half a century before, in his essentially unread Habilitations address. It's by having a phenomenal piece of luck coming from mathematics, principally unreasonable effectiveness in mathematics. What did he say? Okay, but anyhow, so now the point is that this sounds very nice, but it sounds like too much to hope for.

1:00:00 It sounds like, of course the circumference is 2 pi times the radius, that would be silly, right? But no, no we don't. So it turns out... We know how to call that. So there is a class of model categories, which are very important in homotypic theory, because they simplify things enough that you can do calculations that nobody knows any other way to do, and important ones, things about homotypic theory, just really fundamental things. And mathematics are the ones that can do, it does very simply, calculate at the back of an envelope things that would otherwise be unknowable and very difficult. And this branch of mathematics has at its very core the problem of how to pass from a space, which I've tried to explain, is essentially the same problem as asking what does the region look like. What does a region look like? Because when you ask what a region looks like, you're looking at the space of all null pads through it, but that's of the same topological type as the set of basic closed loops. So it's, you know, it's not exactly the same, but it's home to the equivalent. So there's a branch of model category theory, and if it weren't for this branch of model category theory, model category theory would be libelology. But because of its topology, and in fact its central topology, and actually I met some topologists of this group at this meeting and they were telling me that model category theory is actually coming on like gangbusters in topology. It's becoming more and more important in more and more areas. And you meant logic also then?

1:02:30 No, it has nothing to do with logic. It's not logic. Because in those logic we are connected to topology. I mean, it isn't thought of that way. It's an effective calculational tool in mathematics, and it has a natural relationship to the discoveries of solvents in this class. It's called the minimal solvents model. It's actually in all of these different related... Whether we couldn't construct the model category of regions of space-time, these already existing model categories, I mean, presumably it would have to be more spatial, I mean, it has a particular dimension and so it would be some sort of, you know, you don't think that regions in space-time can have arbitrary, morally they're still sort of four dimensions. Constructing them that way wouldn't then lead us to a way to construct a quantum theory directly. So, this is sort of, um, now, how much longer should we talk? I'm not tired, but I just thought I should respect my hours. Fifteen minutes. Fifteen minutes is good? I reserved for two hours. For two hours I reserved. Yes, but, uh, let's take, uh, minutes. Fifteen more minutes is fine. All right. No, I'm good, I'm good, but I just thought I should ask, I think, forever. I think it's more fun than listening.

1:05:00 So now I should try to explain what the class of pre-study homotopy types is an approach to it, which for lack of a better term, I might call arithmetic, that homotopy types are a bit like algebraic varieties in that you can do them over finite primes. And you can localize a refining prime. There's a notion of localization, or like in algebraic geometry. Localization means ascending. That's exactly so. You can allow some set of primes to be divisible. So you can take a space and you can say, now, let's let some set of primes be divisible. And you can say, what's left of it's how we took it down. So this was considered just interesting mathematically in its own right. But it also simplifies, it also simplifies the theory. And so there's a lot to use when it was considered very interesting, you know, before even people realized what a powerful tool it was, an intriguing sort of thought.

1:07:30 So you saw rational, you just have all rational equations? You can think of it that way. But wouldn't you have all primes inverse or what's the deal with the... Well, the rational number is all primes inverse. Yeah, so it's what it is. It's like going from z to q, from coefficients in z to coefficients in q. Okay, now there is a very general way of saying this, which has to do with what are called Poznikov Towers. Are these things familiar to people? What? Poznikov Towers. Poznikov? The basic idea of homotopy ties is built up out of Euler-McClain spaces. You can take the Euler-McClain space for the homotopies at each, homotopy groups at each dimension, and then you can put them together in a vibration, in a tower of vibrations. Which... ...homotopy groups are twisted over the lower homotopy. This is enough information to recover the homotopy. And these things are called positive compounds. I mean, it's not just that you can take a product, because, you know, the higher homotopy groups are twisted over the lower homotopy.

1:10:00 ...finite, like... Well, yes, you can truncate them and make it something final, but actually, the limits behave rather nicely, so, I mean, a general homotopy type could have an infinite tower, but if you start something that's a finite dimensional complex, then once you reach a finite stage, it's all determined, and there's still some higher homotopy groups who might have it, but... But you mean by the cohomics that... What is a topological space itself? It's an infinite hierarchy of vibrations. A space is the same homotopy type as... These things can be thought of as dual to cellular complexes, you know. You can construct a reasonable space by adding cells in each dimension. So that's telling you sort of which spheres you put in. But of course spheres can have higher homotopy groups. So instead of that, you can construct something that has only one homothetic group in one dimension by killing everything else, and then, if you construct a space out of cells, it's easy to calculate the cohomology, the domatomic group in the series. So instead, you can piece together the Islander and McLean spaces into a Posner graph tower. The homotopy is immediately obvious in the front of the construction of the homotopy. There's these two dual pictures in a space, one where the homotopy is easy and the cohomology is difficult, and one where the homotopy is easy and the cohomology is difficult.

1:12:30 Yes, I said that. But, I mean, there's two, right? And they're dual in one another. It's not the same space. It's not a common field. It's not homeomorphic, no, but it is in the same way. Do you have a difficult history? Hm? Yes. That's right, that's right. There are logical spaces. There's always something horribly difficult. If you describe it in such a way as to make one thing easy, then the other thing becomes horribly difficult, and it's very hard to get them both at once. If you localize, you lose some of the information, of course. You lose all the information that's contained in the torsion. Okay, so no, no, sorry, I mean in this number theory. Localize it all to the prime. Okay, so now I should explain, there's a map, like that, and this passes to the cost. You construct any number, don't be tight, by piecing these things together, and because there's a map here, which follows from the inclusion of C and Q, To actually pass functorially. And it doesn't only be Q, we can also do in great generality.

1:15:00 And you can also localize just at some price. That's right. That's right. You see, when you localize a function in algebraic gravity, then some function... Yeah, yeah, yeah. And now the interesting thing is there are several different model categories for that. So if you're willing to pass from homotopy type to rational homotopy type, then you can substitute for the spaces one of several different ways to do it. An algebraic complex, which sort of reflects the way these possible things are fitting together in an idealized way. And now, these model categories, unlike the model categories for homotopy type, satisfy that list of desiderata. If you do this, you're throwing away some of the information which we knew we wanted to do, and you're getting a calculus. Models become simple enough that you can compute effectively on them. You get a unique minimum. It's no longer the case that either homotopy or cohomology has to be very difficult. If you're only interested in the rational part, we can compute both the cohomology and the homotopy of any reason.

1:17:30 Suddenly it becomes, if you practice just that part of topology, Both cohomology. And we have two different models. One is built out of the homotopy theory. You construct the cohomology from it. One is really built out of the cohomology, and you can construct the homotopy theory from it. And they're connected by adjunct functions. So there are two dual representations, but we've managed to simplify things enough. You can understand all the information in either one. And in fact, there's a deeper theorem that says any finitely generated model can be approximated arbitrarily closely by genuinely finite models. So you get only a finite cohomology and only a finite amount of homonymic theory. And you couldn't make successive approximations of any object. Yes, it's very, I mean, this has many very wonderful properties. And in fact, I will go on to say that there are at least very natural approaches to constructing a quantum theory of gravity in either one. I mean, you just look at the structure of the results. Nobody ever thought of it this way. Now, you haven't investigated the properties or anything like that, but so it's still, you know, it's still work to be done, but you get into a very interesting picture this way. Okay, that's the hour and a half you asked me to stop at this point. Any questions? Yes, I want to say the region of spacetime is adequately described by some finite decomposition.

1:20:00 It wouldn't be quite as simple as a simplicial complex, but it's something analogous to that. It's more like a handle body decomposition. I want to say this is a decomposition which contains All the information that we can see. Yes, but say... Okay, so now what I would want to say is that then some discrete kind of geometry. So, if I have a simplicial complex, the metric is displayed by assigning a length to each edge. And that's close enough. That's a metric. That was the original idea of Herge and Ponzano. So now I've got something that's a little bit more abstract than a superficial complex. So we'll have to assign something like lengths and distances and angles to my handles. I'll have quantum operators corresponding to handles and pairs of handles and so on. Assigning values to those things I will think of as a discretized metric on the complex and what I want to say is if there were lots and lots of handles then I can think of very little and if I'm standing back it will appear like a metric to me. But it will allow for a graininess at a fundamental scale, so I'm thinking that there is no continuum, so there's no continuum metric, but there's some discrete approximation to it. I mean, an object in a topos can be thought of as a variable set. And points can even join and things like that. So I thought that was interesting earlier on. But I really, I've come around to the idea that, well, there are just no points, period.

1:22:30 It's not that there are points depending on the observer. I said there aren't any points. There are subregions. And there are bits of structure. What I think actually is that a space-time region is like a gas of tiny black holes, and the geometric relationships between the black holes exhaust the observable geometry. That's sort of the way I come to think of it. But there's no space in between them that they sit in. They're just glued together as sort of a handle body, and that's what... Okay, but does all of this moreover be directly related to what we can answer at the end of the day if we had a really powerful microscope? If we looked at a teeny-tangy region, it would start to look great. So if you illuminated it, all you would see would be sets of flashes. And all you could infer about was geometry with the relationships between the sets of flashes. So the flashes are names for handles, or for genomes in one of these models, and ultimately, when we ever have such a powerful microscope, the final task of quantum gravity will be to predict probability amplitudes in relationships between them, and to predict that will be by comparing them to some differential geometric model. Except the lamentable inadequacy of our instruments and principles can ordinate science and do experiments and do other things. Any other questions?

1:25:00 Okay. There's a notion of effective calculability in this book by Wu that I told you was a good book. Now what he says there, what he defines there as effective calculability is that if you have some operation on space, Where you take spaces and combine them in some well-described way that can be used then if you have an algebraic model for the two pieces and an algebraic description for how you combine them, you should be able to calculate the new algebraic model from that without going back and referring to the original space so that the operations of building up a big space out of little spaces are adequately reflected in the category of models. And he proves that the models for rational homothetic theory are, in his name, effectively computable. Now, for instance, cohomology isn't. If I tell you the cohomology of this piece and the cohomology of that piece, you can't calculate. So the fact that instead of giving you the cohomology, I give you the minimal model here and the minimal model here, Then I can describe how they fit together, and then I can calculate the new minimal model, and then from that I can calculate the cohomology. So that says that in some sense, there are geometric operations like this for which the model is just as good as expected. That's effective reproducibility as defined by Wolff. So that makes me think that I can construct a calculational physical theory. All I have to do is tell you how to do some simple pieces, tell you how to fit it together when you do simple algebraic operations, and it's fine. So that's another thing that makes me very optimistic, because it's a setting. Okay, should we call it a day? I'll stop. So, what was your problem? Let me stop this.