Categorification & physical spacetime

0:00 Thank you. the schedule, um, is that we go continuously between the two toilets. Oh. Um. Wait, so somebody's going to interrupt and start talking about something else? Yeah, so, in announced time, in announced time, I guess we'll have to, in practice, we'll have to have a five minute break. If there is a zero minute break, we'll have to have a five minute break. So, if, if there's a way of, um, let's put it this way, coming in under the hours, it will be better than coming in over the hours. That's as good as possible. I don't know how the circle goes. Okay, should I start? I'll cock loudly off the mic. Okay. Okay, yes, so we've got a tight schedule this afternoon, so we might go get started. But first off is Louis Crane from Kansas State, talking about categorification and physical Okay, so I'd like to thank the organizers for having me. I was actually completely astonished. I hadn't realized that categorification had turned into this giant sort of industrial effort. I mean, when I wrote the paper with Igor Frankel, where we more or less started it, I can't really say that the reception was particularly positive. People went on for a long time saying that they didn't like the word or that there must be some way to do things without categories and so on. It seemed like it caught on very slowly, and then one day I started realizing, oh, gee, there's all these people working on it. So I have two ambitions for this talk. The first is I was going to explain to you the origin of the subject, that what were

2:30 the thinking that led to the formulation of what we called in our paper, Igor Frankl and I, the categorical ladder, and why we thought it had something to do with construction of diffeomorphous and invariant quantum theories. So, I thought that the origin might be interesting to people and what the original problem was, and I want to emphasize that, for me at least, from the very beginning, the problem of quantization of gravity was what motivated the whole thing. Ideas relating to Einstein's equation were what led to formulating the picture. And I'll try to show you how that fit together and what I was looking at and why that happened. And then the second ambition I have is to tell you my current program in quantum gravity, which is categorification really in a broader sense. And to me, the categorical ladder is one part of a larger picture for which I have physical motivations. So to begin, I'd like to state the broad categorification principle. The classical point set continuum has proven

5:00 quantizing general relativity. In categorical formulations, of space-time Einstein's equation and quantum mechanics axioms quantum higher category theory. So, we're in the classical continuum that seems to be impossible to write down a quantum theory of gravity. In categorical descriptions of space-time, it writes itself down. It's just there. Both sides. So, that's a strong claim. And it was really that observation that led to the categorification. So what I would call, I was thinking of calling petty categorification, but actually after seeing what a big thing it had become, I didn't want to call it that. So let me just call it the categorical ladder. in order to construct If you're more for some invariant quantum theories in 2-D, we need algebras.

7:30 And there's more than one way to do it, and you've got many different classes of algebras. But what naturally flows out of trying to construct it is you need an associative algebra in some special conditions. In 3D, we need ring categories. with all due apologies, uh, to, um, to Jenner, and she used to live off the N. Um, uh, whose are categorifications of the 2D structures. So we already knew when I started working on this, something in 2D and something in 3D. And there was a pattern. But the people who had constructed theories in 3D were physicists and they didn't know what a category was. So making the mathematical generalization, as I'll explain, was a critical point. Now, iterating this gives an approach via two categories of constructing 4D theories. Now, I was never interested in a bigger picture. I know some people are in going to all dimensions. Because for me, the critical thing is to construct things in four dimensions. In physics, of course, it's the most interesting problem because it's where we live.

10:00 But actually, in topology, the most interesting problem is in four dimensions, too. And so there were three different goals here. There was Donaldson floor theory. There's topological quantum field theory, which is the one you end up with is BF theory. And then there is quantum general relativity. So there's a large spectrum of problems here, and they're all related. Now, the one that I first worked on when I was formulated the categorification proposal with Igor Frankel was trying to construct Donaldson floor theory. And that seems to be the problem that attracts the most interest. I mean, I think basically in some sense the suggestion that we made in the original paper that you would do that by categorifying the quantum groups, by going to categories of perverse sheaves over flag varieties, a la lustig, still seems to be, I mean, it's harder than I had any idea it would be, but it still seems to be very much alive and kicking and very central to what's going on here now. interest, as I explained, was really in gravity. And as I gradually realized that this problem is really not so closely related to gravity, I gravitated to the other problems. So I will summarize the results of this program as applied to all three of these problems. And I think it can be said to have had a great deal of success. 4D topological field theories, yes, we really know how to construct them, we have at least a reasonable model, and there's some variance for gravity. And of course, Donaldson floor theory is much harder, as I'm sure everybody realizes. But I didn't realize at the time that Donaldson floor theory is not just naively a topological field theory. So as I came to realize that, I realized that that was one ball of wax, and gravity was another, so I went over to trying to work in gravity. So I'll begin by explaining what led me to this, what I'm talking about in 2 and 3D. And, um, so, um, the slogan is, the categorical ladder is the dimensional ladder.

12:30 So, let me explain what led me to this. So there were two pieces of work that had already been done by physicists. The most interesting was the development of spin networks. So I should explain in 3-D the quantum theory of spin. Now, the whole development of this was by physicists who didn't know what a category was. So they wrote down a mathematical structure that allows you to describe the quantum mechanics of spinning particles. And there's one Hilbert space for each half integer, depending on the total spin. of the particle. And then there are two operations which you have to do in quantum physics. And you have to have a mathematical method for expressing them in order to be able to solve problems in spin. The two operations are a combination. And this is very little. This the spin contribution to multiple molecular collisions. That's how the theoretical theory reserves are developed. You have particles who spin. They collide. They form a new particle. What's the probability of what the spin will be for the new particle? And they had sequences of events where you could get several collisions and then something could come apart. And they wanted to calculate probabilities for that. So there are two operations. One is combination. And the other is superposition. If there are two different roots, and you're careful to design your experiments so you don't measure which one you're going through, then you can get a state that could be a superposition having gone through either of these two processes. Then combination and superposition.

15:00 Now, it's a quantum theory, so there are operators. the components of the spin. It's a quantum version of angular momentum, which looks like a vector. Then there are basic spins. It's labeled by integers or half integers. And then R n equals 2 n plus 1. So there's one in each dimension. r0 is one dimension, r1 half is two dimensions, r1 is three dimensions, and they had a set of rules for what happens when you combine these spins. So, R, I, combined with, I'll give it away, R, J, and it turns out to be the direct sum of R, K, where K goes from the absolute value of I minus J to I plus J, and I plus J plus K is in Z. So you get each intermediate spin value between the maximum and the minimum just once. And they have, moreover, specific ways of writing how the basis here is written out in terms of the basis of valor, which are called clutch coordinate coefficients. And they discovered that in order to try to calculate very complicated probabilities like this, it was really enough to calculate probabilities for closed diagrams. So they wrote diagrams. They were trivalent diagrams. And you ended up closing the thing, so you ended up with a closed trivalent diagram. And on each one of these, you put the appropriate Hilbert space. And here, you put the appropriate tensor, which is the operator, if we could cross up to that. And then you composed them, and when you had a closed thing, you got a

17:30 that if you had any complicated spin that it can be written the closed spin that you could subdivide it into tetrahedra and you could write as a sum over all internal label you have to in order to cut it up into tetrahedra you have to add new edges and it doesn't change the evaluation to do that so it's a sum of all internal labelings and then there's some factor here product over the edges of the end in our end and then it's the product over all decomposition of a number called a 6J symbol. And the 6J symbol looks like, it has an asymptotic form. The thing is, a tetrahedron is a rigid figure. So if you take the spins you're putting on the edges and treat them as lengths, you get a Euclidean tetrahedron. Does everybody How many people are reading on this? Oh, most people don't. Okay. I should be careful with it. This is really, as far as I'm concerned, this is the bottom that all came to G and R. So this looks like if you draw it as Euclidean tetrahedron, it's rigid, so it has dihedral angles, theta I, and it has a volume. And it looks like 1 over the square root of 12 pi V the root, cosine, sum of all terms like j i theta i minus pi over 4. If you know something about quantum mechanics, the pi over 4 comes from the WPV approximation. So this is this funny expression. And you can then take this thing and put it in here for each of those, and you get this funny oscillating expression that when you sum it up, gives you the evaluation of the spin net. And this was discovered

20:00 by three physical chemists in Lithuania in the late 50s. Strange story. But then, you see, the physicists had the idea that spin is a kind of geometry. Spin vectors are like vectors, and when they're adding, classically, they would just add like vectors. Angular momentum just adds like vectors. But here, instead of having vectors, they're described by these Hilbert spaces. It's a quantum theory. So it's a piece of geometry we already know how to quantize. Spin geometry, it's the one kind of geometry we really know the quantum theory for already. So they said, okay, let's take something big and complicated like the geometry of the manifold that we want to quantize. And let's cut it up into bits and put spins on the bits and use the combination diagrams to fit them together and make a quantum geometry out of the little bits of geometry that we already know. So, like, locally it'll be, it'll be, like, flat, but then when they piece together, it turns out they have curvature around the edges. And it was discovered that this expression, this expression, gives a discrete path integral for 3D quantum gravity in the Euclidean signature. This is the work of Brecce and Ponsano, but it was very strongly influenced by the work of Penrose. He already had the idea that you should try to reproduce quantum, I mean, general relativity is a geometric theory. theory. So we reduce it to the simpler bits of geometry and figure out how to quantize those and piece it together. Now, why is it that this gives you general

22:30 relativity? Well, it's because the thing is oscillating depending on the dihedral angle. So if you think of one spin in a triangulation, it's sitting in the middle of a bouquet of dihedral angles, okay? It's got a bunch of dihedral angles, and as you vary that spin and leave everything else constant it's changing by a phase which is the sum of the denhedral angles so it doesn't change at all if they sum up to 2 pi so that means if you think of this as a path integral the classical solutions to it are given by stationary phase so we get stationary phase to 2 pi. The angles head up to 2 pi, and the geometry is flat. And in three dimensions, Einstein's equation reduces to the metric being flat. There's no Ricci tensor, it's a Riemann tensor except for the Ricci tensor. When you set the Ricci tensor equal to 0, the whole Riemann tensor is 0, all the sectional curvatures vanish. The discrete version of all the sectional curvatures vanish is that the angles around each edge head up to 2 pi, So there's Einstein's equation in three dimensions of spin geometry. And one of Penrose's students tried to extend this to four dimensions, but it really just didn't wash. He made something called fabrics, but it really didn't work nicely at all. Now, I looked at this, and I said, I have another name for the quantum theory of spin. The quantum theory of spin is the tensor category of representations of SU. How many people are reading that? Oh, that connection's still. That's a funny thing. I've never seen it in a book. This is a bridge. This is a very important bridge. The Hilbert spaces have to have actions of Jx, Jy, Jz. And Bohr's quantization says that the Poisson bracket has to go over to the commutator. So they are representations of the Lie algebra.

25:00 And in fact, any quantum theory is just a representation of a Lie algebra. You can take the Hermitian operators, multiply them by I, their pseudo-Hermitian, then they form a Lie algebra, and the condition has to be the Hermitian operator spoils on having unitary representation of the Lie algebra. And then if you think that a quantum theory, instead of just being about one system, should combine systems, which people always leave out in the axioms in physics books, because it makes it very complicated, but then they sort of by hand tell you how to combine systems. But the actual rule is that you need the category of all representations of the layout of the election question, which is a tensor category. Okay. And then, investigating this further, I came to the realization that the connection with gravity has nothing to do with spin and everything to do with category theory. ingredients in the connection to general relativity all come from the pentagon. You can start out with the pentagon, which gives you an identity on 6J symbols. 6J symbols are bits and pieces of the associator. Everybody know what the associator of a tensor category is? How many people know what the associator of a tensor category is? Okay, huh? We had it this morning. We had it this morning, okay. Do you remember it? Okay, good. I won't say together. Okay. Okay. So, this observation put me in a position of being able to generalize. Whereas Penrose and his students thought this was something specific about spin. Okay, and I should say, this formula that gives you this oscillatory behavior, you get it

27:30 Pentagon with a very small change, and if you change it twice, you can get back so that the thing is just one spin larger in one direction, and that gives you a recursion relation which, it's a three-term recursion relation which gives you the cosine and the limit. That isn't quite rigorous, but it's good enough. So it really just comes out of the structure of category theory, and this reduction where you can cut the thing up. It just comes out of fundamental axioms in category 32. It just has to be a semi-simple tensor category. So I realized this thing that the physicists had discovered, which seemed to be the best hint I knew toward how to really get a quantum theory of gravity, was really not specific to spin. It was really just inherent in the axiomatic structure of a tensor category. So the only problem is we're in three dimensions and I want to get up there. So I said, okay, if you have something that's very high and you're here and you want to get there but it's sort of steep, you back up and you get a running start. So instead of trying to go from three to four, I went back to two and I tried to run past two to three four. Okay? Now, it turns out that in two dimensions, a topological state sum, this was originally invented by some physicists who called it something else. So, we have a triangulated, There's two ways to do this. I'm just going to do the one with the triangulation. There's another very nice one where you take handle body decompositions instead, but I only have so much time in a horribly ambitious... I mean, the organizers are breathing down my neck after talking to myself. I don't know. So you have a triangulation of a surface, and you want to construct a quantum theory on it. And then you want it to be topological. What does topological mean? change the triangulation. So you need some labels on these things. You need a labeling set. So you want to put states on these things. This is very much in the spirit of people

30:00 who do lattice statistical mechanics or something like that. So you put a label on each thing. And then you have a local contribution to the partition function, right? So on each triangle, then there would be a certain energy, depending on the interaction between the terms you put there. And you form the evaluation, and it's the sum over all labelings, and then there's some normalization factor that we don't worry about, and it's the product over all triangles of C, I, J, K. And I'm oversimplifying a little bit, but if you know better, don't ask any questions. so you need this data to define a statistical mechanical ensemble if you like, or a Euclidean quantum theory, they're pretty much the same so this is a way of doing physics in two dimensions on an irregular lattice if you like, and now I want the condition that says if I change the triangulation I get the same thing so you know that you can get from any triangulation to any other triangulation partner moves. Is that common knowledge? How many people know about partner moves? No, not everybody. Okay, so there's a theorem. It was proved relatively recently. It was proved about six months before we needed it, that basically you can get from any triangulation to any other triangulation by taking a simplex, the next dimension up, and pushing it down, and canceling out one half of its boundary and putting in the other half. So it's sort of get from any triangulation to any other by like a combinatorial cobordism, if you like. So there's two moves in two dimensions. There's this, and then there's this. And those are the two ways. And you should think of these as halves of the boundary of a tetrahedron. And so these impose conditions on these Cs, and when you write them out, you discover, lo and behold, this is something we already knew about for a long time. Because if you think of these Cs, think of C i and j k's as multiplication constants for an algebra.

32:30 So take the vector space generated by the labels and use these to write down a multiplication. then the basic axiom here boils down to associativity this axiom if you just write it out says that these are the coefficients of an associated algebra and this one here says it's semi-simple so when I got as far as learning this I told it to Igor Frankl and he said do you realize that and we've seen some more things that he took out an old algebra textbook and he said do you realize that half the headings algebras are actually names for quantum field theory. There's also the Frobenius algebras and the semi-simple algebras. They all show up in their place in this sort of hierarchy. So that's something very curious. I mean, quantum field theory is really a branch of abstract algebra. I'm surprised. Now, so in 2-D, we need a number on a triangle, and we have an equation on the boundary of a tetrahedra, and it corresponds to associativity. Now, in the 3-D theory, which I did first, because it's really the source from which the whole subject developed, at least from my angle, the basic thing we have is the 6-J symbols, which are really the associator of the tensor category. These are matrix elements for the associative. It's, you know, it's alpha ijk from ri tensor rj tensor rk to ri tensor rj tensor rj. So this little building block, which is giving you the basic geometry, which allows you to find quantum gravity in three dimensions,

35:00 is the associator, which is the basic element in the structure of a tensor category. And as I explained, when you have a tensor category, you really sort of have a quantum theory automatically. The other thing I could tell you about that marriage between categorical theology and quantum mechanics would be finomenology. Nobody ever thought what kind of mathematics is finomen diagrams. I mean, because, you know, that was back in the 50s. were just inventing category theory, the physicists were just inventing Feynman diagrams. The physicists thought category theory was totally crazy and nothing about it. The mathematicians thought Feynman diagrams were totally crazy and nothing about it. So what do you think? Fold over until I need speed. What Feynman really invented was categorical physics, because the diagrams are just, they're defined by being things that respect all the symmetries of the problem. vertices to respect the symmetry. That's the rubric when you go constructing a quantum field here. I'll find them. So what they are is just diagrams that are showing you morphisms in a tensor category. And in fact, the thing that Tuft proved that meant you could actually use gauge theory was just he showed that he had a regularization which preserved this property of them acting like an interpoint. So the category theoretic structure is actually at the core of Clinton field theory as well as the Clinton genetics of spin. So that's a very strong connection. And then here, when you go writing the terms out and expanding, then you end up getting things that oscillates so that they act like, if you think of them as geometry, they act like Einstein's equation. And none of that Spin is just the, um, the category of representations of SU2, and the two physical operations, which I call combination and superposition, anybody want to guess what they are? What's, what's combination? You know. Pensive product. Pensive product, right? What's superposition? Direct sum. So you get a ring category. Spin is a physical ring category. You can see the

37:30 morphisms in a bubble chamber. Those little diagrams are the morphisms, and the operations are physical operations. Okay. And the other things you see in a bubble chamber are just more complicated tensor category we have symmetry but the symmetry is still defining the tensor category so then you see in 2d you've got an algebra and it's got a whole bunch of axioms a plus B equals B plus a a times B and C equals A times B times C ad nauseum. And now every single one of those axioms is an axiom of a tensor category if you put circles around the operations and change the equations into isomorphisms. So where we had operations, now we have founders, because we're in a category. Where we had equations we had isomorphisms. And, of course, the exomorphisms have to satisfy a new set of equations called coherence. And all of this fits together so that the geometry goes up one dimension. The easiest place to see this is in these two examples I just showed you. you see that in the two-dimensional topological state sum this is the basic element this is the basic equation if you fit these two things together you've got a tetrahedron because the diagram of the associator if you draw it as a categorical diagram is the same as the diagram of the associative law you draw it as a diagram. It's still association. So you go, you know, A, B, C, A tensor B, B tensor C, and here you can go around the tetrahedron both ways and get A tensor B tensor you see, and so the tetrahedron has to commute. So that's the same as saying the top face

40:00 is the same as the bottom face, but now when you've categorified it, what it is is there has to be an isomorphism. And then the coherence law, the coherence law for the associator is what's called the stash-f pentagon. Okay, if I draw you a picture of the stash-f pentagon, it looks like a force implex and the five associators in the pentagon are the five faces of the force implex and in three dimensions the partner moves come from taking a force implex and pushing it down there's the categorical ladder when you categorify the equation becomes an isomorphism and from that isomorphism you can extract matrix elements building blocks. And then they satisfy a new equation, which is the simplex in the next dimension. And it doesn't really work with simplices. There's also a version of this with handles. I thought I should only do so much of this. So, we iterate this process and we go up to four dimensions. And we have, by this process, approaches to all three of the problems that were so interesting for us in four dimensions in the intervening years since composing this. And it's worked very nicely. So what do we get when we go up to 4D? I don't have time to do details. So logically, I should really tell you about the first, about the model my student did, as my graduate student. So he took a tensor two category, and he got a state sum model. I like to call these things categorical state sum models. The name spin form somehow stuck. I hate that name. I'm sorry, because it's not about spin, it's about category theory. Spin is just the simplest example. What they really are is categorical state sum models. It's part of the theme that fundamental physics is all going to be built up out of category theory, which is not an enormously popular idea for physicists.

42:30 Okay, anyhow, categorical states of models, a.k.a. skin foam models. So this is McKay. Now, but that wasn't actually the history. David Yetter and I worked out the same thing, But instead of using a general tensor 2 category, we got away with using a modular tensor category. So we were able to write down a four-dimensional theory. We used the braiding in a fundamental way. It's because if you try to write down the connection diagram for a four-simplex, you cannot write it in the plane without a crossing. So you have to use the braiding. But that's a hint of the fact that it's really a four-dimensional structure. And thank God I don't have to tell you this, because John Baez just explained to you that this really is a tensor two category with one object. So I don't have to explain that to you again. So it's sneaky, but we have just enough structure here that we can actually get an interesting four-dimensional theory. And here's a combinatorial formula for the signature. Now then, the Donaldson floor theory, there's my work with Igor. But of course, that's only programmatic. It turned out to be enormously hard, as I'm sure many people are very aware. So, in general relativity, it turned out we were able to modify this. It turns out general relativity is a TQFT plus a constraint which breaks diphomorphism invariance. And by, and And this is joint work with John Barrett.

45:00 So we were able to write down the model. Now what we did was we took the Lorentz group, which is SO31. like the rotation group in special relativity. It's non-compact, so its unitary representations are infinite dimensional. Nevertheless, we were able to discover a regularization, which divided by infinity in an absolutely canonical way, so that we got finite expressions for closed nets in representations of the Lorentz group. And based on that, we were able to write down the state sum model for four-dimensional Lorentzian Signer general relativity. And this is sort of a cheap, because it doesn't have a braided. But there is actually a braided version. There is a Q deformation of the Lorentz group. It's through the Bufanoir and the Locher, also to some Polish authors whose work is hard to do with this. But there is actually also a Q deformation that uses the braided. And at this point, I think it's fair to say that the most active and hopeful direction in quantum gravity goes around various modifications of this kind of categorical state-sum model. and the finite evaluations of closed diagrams in them that we were able to work out, we could write them out as oscillatory integrals in hyperbolic space, and by an incredible set of miracles, they converge, even though nobody thought they would converge. People wrote papers saying, well, that's obviously silly because those are in the dimension. But it did work. And so it's sort of like, from what I've been picking up on the Donaldson-Florid branch of this program, that nobody knows quite why it works, and it's very hard, but it really does seem to work. So you see, the whole motivation of this grew up around quantum gravity. And I don't think this proposal could have ever seen the light of day by any other process.

47:30 It wasn't like people were just waiting to discover this and it was just like a slight little poke. I mean, just the opposite. It was horrible discovering this. there's enormous resistance to it. I'm really amazed that I can walk into a room and say I'm going to categorify something and nobody says anything sarcastic anymore. The nicest thing they said was that it was a, I mean they said, well isn't there some way to get away without the categories? Oh, that was horrible, it was an absolute career disaster. But now it seems like maybe I'm not accepting of it. Okay, so anyhow, so the thing I wanted to get to was that, okay, so now we have the very cranial, and it depends on a particular triangulation, but it's finite. But this, depending on a particular triangulation, is very important, because the constraints keep you from using the theorems about invariance. The totems don't work anymore. So, the answer depends on which triangulation you use. Now, Carlo Rosali suggested something called group field theory, in which you interpret four simplices as vertices and tetrahedra as edges. And you interpret the expression of our model on a particular triangulation as a particular Feynman diagram in some fundamental theory which itself is non-geometrical. So you get the picture that the fundamental vacuum of the theory is point-wise, it's kind of geometry, geometry, but the Feynman diagrams look like bubbles of geometry. This is very attractive because you live in a universe that seems to have popped into existence that is nothing. So this kind of idea has been around as a suggestion to formulating quantum cosmology, so it's nice to have it come out this way in this mathematics, in this very elegant way. But the problem is it's not Feynman, because you're summing over all time in your Okay. So what I believe now is that there must be some third. And the way of finding

50:00 the third thing is to trot out another branch of category for it. Over the last few years, I've really learned that this theme of replacing the point set by categorical structures of various kinds as the foundation for geometry and topology is one of the fundamental themes of modern mathematics. In fact, most mathematicians don't realize what a big field this is, because all kinds of people are doing it in different ways in different fields, and they don't talk to one another. I had a seminar in Kansas State University, and it turns out half of our faculty had some version of pointless that they could teach to the other people, the algebraic geometries, like its meanings, and we had a post-declosure with model categories, and we had David Yedder, who'd done some logic, talked about topos, and, you know, and then there's subversial complexes themselves, and the profit of these other human categories, and then, of course, all the stuff that John was developed. So it's a very, very broad, rich thing. And my feeling is that we haven't really used the deepest part of it. And I would say the deepest and most powerful part of this whole thing, at this point, is not in category theory, sorry, it's model category theory. Model category theory is amazing. It's a very powerful branch of mathematics. And what's so great about it is that it's very computational. the description of a space to the point where you can compute all kinds of things with it. So what's the idea of monocategory? Well, I'm not going to get a chance to explain the constructions. But the idea is closely related to something I learned many years ago in graduate school from MacLean. Saunders-MacLean actually had a very radical picture of the role of category theory in mathematics. He felt that set theory was not really the ultimate foundation. He said, we get the continuum from our interaction with nature. We get set theory from interacting

52:30 with discrete, well, sets, and so from a different aspect of nature in a different way. And we shouldn't really assume that we just reduced one to the other. So he had the idea that it's not sets that are important in all the models we make in geometry and topology. It's the relationships between them. We have regions, we have maps, and we have homotophies between them. And it's really the relationships between the regions which are meaningful and survive rather than the underlying point sets. Now, model category theory is an implementation of this point of view. Topological spaces are just objects in a category that has enough structure that you can do homotopy theory in it. If I say it like that, it takes two minutes. If I tried to write down the definition that Quillen gave, I would have used up the whole hour. It's very technical. But let's not worry about that. Vibrations, co-vibrations, blah, blah, blah, blah, blah. So what I believe is that in order to make a synthesis two points of view, and relativity, we have to use something called the Bekenstein Bound. I didn't tell you about the Bekenstein Bound. Bekenstein Bound tells us if you have a bounded region in general relativity, there's only a finite amount of information that can be communicated from it to the outside. Because if you try to communicate too much information, you put too much energy into the region, and you make a black hole and nothing gets out at all. So a region in space-time, as far as external observers are concerned, only contains a finite amount of information. That was what made me start wanting to use simplicial complexes instead of continuo in the first place. Because I knew, somehow physics is telling us that there is no continuum. That's the shortest distance. You can't measure anything shorter than that shortest distance. In quantum mechanics, if you can't measure it, it isn't there. You should remove it from there. It's not like classical mechanics where you can pretend you can measure anything without

55:00 interacting. That's critical to the transition to quantum mechanics. So I believe we haven't really understood yet the implications of quantum mechanics for mathematics. And what's going to happen is that we're going to find a way to construct a particular model category. It will be constructed out of the tools that Sullivan and Quillen used to get rational homotopy theory. But it won't be the same category. It will be specifically adapted to describing simply connected regions in a space. And the thing is, there isn't just one way to construct this category. There's two, and there's adjunct functions connected. One of them is related to the path space, which is the same home Adobe path as the loop space. And that's a graded Lie algebra. And the other one is a differential graded algebra, which is related to the differential terms of the space time itself. Well, what I believe is there's going to be two ways of constructing a quantum theory in these two models, and the functors are going to connect them, because in quantum mechanics you can't measure everything at once. So if you study a region by asking, what are the multiple images I see of some light source coming through it. That's going to be analogous to ordinary quantum mechanics to momentum space. Those are like beams. And on the other hand, if you ask, what are the relationships between subregions, that's like the position representation. So there's going to be two different representations with two quantum theories on it, and there's the adjoint functors and connect these, which are already constructed, are going to give us the equivalence, complementarity principle in the quantum theory. And if I had more time, I'd show you what the models look like and show you why constructing a model, a quantum theoretic model on them, is very natural in both cases, although very different. But I was really strongly intimated that I should try to be early rather than late, and it's sort of a long story. But the thing is, nobody has ever looked at these

57:30 constructions and asked, how can we use them to do physics? It's brand new. It's as open as thinking about spin as a tensor category was when I first got into this subject, well, at least many years ago. But the thing I wrote on the board at the beginning, that when you a categorical description of the space-time, both Einstein's equation and quantum mechanics come naturally out of the algebraic. Here, the asymptotic formula for the 6J symbol gave you Einstein's equation, and the whole structure really made it into a quantum theory. That is going to recur. That is robust. The universe is made out of category theory. I don't know why, but it is. It's made out of pure mathematics. It's the only way you can understand it. And I think it's very hard for me to imagine that the categorification picture would ever have gotten off the ground because it's so technically difficult. Who would ever, ever believe that you could get through all those things and somehow come out the other side if you didn't have some sort of argument to make you think so, which didn't really come from the mathematics, but from physical intuition. so this is the other great principle that I'm always espousing that we really should think of mathematics and physics as very closely related we really ought to learn both subjects I know nobody ever does I think it's going to be important and so I believe that at the end of the day just as the discovery that Kepler's discovery that the planets move in olympic rather than circular orbits forced us to completely rewrite the foundations of mathematics in terms, even, you know, down to a different notion of what we meant by a real number, in order to make sense of calculus. And it took a very, very long time for mathematics to fully meet the challenge of classical mechanics. It was from, all the way from

1:00:00 Newton to Weierstrass before we knew what we were doing. We were in a similar picture. We have not not fully grasp the mathematical implications of quantum mechanics and relativity. And the Bekenstein bound is going to be a fundamental challenge, and we will change our ideas of what the foundations of geometry and topology are just as profoundly before we are through understanding this problem. I'm very convinced of that. Let me ask, how many people think I might be, I'm probably right? Okay. How many people think I'm completely out of my mind? Oh, you're so polite. Okay. Most people didn't want to vote, huh? That's okay. Okay, that's the end of my talk. You're probably right. Oh, thank you. Are there any questions? Any questions? So you said you like model categories for this Right. So some of the approaches to week-end categories actually have model structures. So if I work on those model categories, would I really get into it? I mean, that sounds very good. I have, the one thing I've discovered is that model categories interface very nicely with the other categorical, you know, there's actually a theorem that any construction of a model category is actually a functor that goes from topoid to model categories, which sounds very abstract, but when you think about it for a few seconds, it just says you can do any model construction inside any topos, it's not a big shot. So, there could be topos theoretic and model theoretic aspects, and there are clearly n-categorical aspects, which I think that's on. I didn't know that. I would like to know more about it. That sounds very interesting, not to tell me about it. Other questions? So what does the Beckman spelling bound have to do, you didn't actually say rational, the word is rational. I wrote the letters I was running out of time okay what it has okay the connection is if I'm going to actually instructing I need some hypothesis about

1:02:30 some description of the information that can come out of the region so I came up with a hypothesis called the lensing hypothesis which says you can saturate just by looking at all the multiple images of some light sorts in the past of the region. You know, this came out of reading books by astrophysicists about gravitational lenses, where they see multiple images of distant galaxies going to near galaxies, and learning there's a very rich mathematical structure there. Now, what I believe is that it's very implausible that a space-time would have a torsional, I suppose. constructions of space. It would be very hard to imagine. So I think the images are going to correspond to the generators and the rational cohomology of the path space of the region. And this fits very- so then I want to say I want to collapse the thing down. You know, Now, simplicial complexes, as you were so kind as to explain to everyone last hour, are a model category for spaces. So what I believe is that in order to pass the Beckenstein bound, we have to collapse the category. So these are simpler objects. They have very elegant descriptions that you're integrated algebraic structure. But what I believe is that if you can saturate the Beckenstein bound by describing all the multiple images that you would get going through the curve, the small curve region that we have to treat quantum mechanically, then it would be completely supported by its rational cohomology, and therefore it would be passed to its rational homotopy. Now, the reason I think this is so wonderful is because in this theory, we have something called minimal models, and the minimal models are unique. So where I might try to get away from this dichotomy a triangulation that's just big enough to contain all the information. If I were working on the level of triangulations, it would be a armature. So I can't really pin this down. But if I pass from triangulations to these wonderful things that I haven't explained to you, which are also kind of complex, there's a unique minimal. This is Sullivan's great.

1:05:00 One of his many, many great discoveries in the field. So there's a unique way of thinking or minimal category that would contain just the information that could pass out of the region which is described by an object in this model category. And therefore, since it contains just the right amount of information, if we can write a quantum area of gravity directly on it, then that would solve the problem. We'll find that you'll see the new barrier. Other questions? I have one thing? No? Okay. That was a good thanks. Do it again.