Categorification & topology

0:00 Okay, ready for a second talk? The next figure is John Bias from Riverside, and he has a generic catamorification topology. Yes, so I thought instead of giving a very technical talk, I'd give a sort of historical overview of some ideas on categorification and topology, leading up to a very sketchy introduction to some new work of Jacob Lurie, where he's proving some old conjectures that have been around for a while, but as I say, it'll be quite sketchy. There are references on this web page here for further reading, so you can check that out if you want. Okay, so we begin at the beginning. Once upon a time, mathematics was all about sets, a remarkable The notable idea of late 1900s, early 20th century mathematics was to found all of mathematics upon structures such as that depicted above, where, of course, those little black dots could themselves be set, so they would be bags of dots themselves. In 1945, Eilenberg and McLean introduced categories, which really just amounts to introducing arrows between the dots and, of course, being able to compose the aerials in an associative way. It's rather remarkable, if you think about it, that a lot of mathematicians consider this sort of structure too abstract, whereas the one on top they consider acceptable, but the problem was that historically the first categories to be studied tended to be extremely large categories, like the category of all sets or the category of all groups, and it's only been more More recently, the categories have been treated as objects of mathematical study, rather than contexts within which to study mathematics. And quivers, for example, are categories that are small enough to draw on the blackboard. And so I think what's happening in the categorification industry is that people are losing their attitude

2:30 that categories are big, cloudy, misty entities and starting to take them as a very natural objects of mathematical study, in particular, from any category, you can get a set, namely the set of isomorphism classes. That's what I call decategorification, the process of watering down a category until it's just a set, and lots of mathematical structures that you think of as being sets with extra structure come from categories in this way. Of course, the most famous example is that the set of natural numbers comes from the category of finite sets by decategorification. But what we're now seeing is a wave of a study of categorifying other much more sophisticated mathematical structures than the natural numbers. A lot of these structures have linear algebra built into them so that the decategorification process isn't just taking the set of isomorphism classes, it's this growth in D group or K construction It's the process of going from the category to accept, which is a very similar construction. But that's really just the beginning of the story that I'm interested in telling here. So in 1967, when other people were doing other revolutionary things, Benavu introduced weak two categories, which amounts to throwing in the next level of morphism, two morphisms going between morphisms that have the same source and target. He called them bi-categories, which is a good kind of terminology as long as your n categories stay with n low enough that your knowledge of Greek is sufficient. But I don't know the Greek word for 37, so at some point I think we have to switch, so I'm going to do it right now, probably two categories. So many of you know this, but the idea is that you can compose two morphisms in two different ways, because they're two different, they're two-dimensional entities in a sense. You can compose them vertically, as drawn above, and also horizontally, as in the second line there. And I'm denoting that horizontal composition with a tensor product, just for fun, basically. And And various laws have to hold associativity of these forms of composition, but more importantly, or more excitingly, I should say, not more importantly, a kind of compatibility between

5:00 the vertical and horizontal composition, the interchange law that says when you have a diagram like this, you can either think of it as vertically composing both parts and then horizontally composing the result, which is this, I guess, or the other way around. And those are some, but not all, of the axioms. The reason for calling them weak, which is a crucial theme here, is that only the two-morphisms satisfy equations. The morphisms satisfy various laws, but only up to isomorphism, not only up to an invertible two-morphism, or two-isomorphism. So, for example, instead of the one-morphisms composing in an associative way, they compose associatively up to a specified isomorphism called the associative. Let's see if I can get this stick to operate properly. So that's the weakening of the associative law. But then whenever you have these specified isomorphisms, then they need to satisfy equations of their own, and the associator needs to satisfy this famous pentagon equation originally originally discovered by McLean and separately by Stasheff, which says that there are two different routes to re-parenthesizing this expression. Here all the parentheses are pushed over to the left as much as possible. Here they're pushed over to the right as much as possible. You can use the associator in two different ways to get from one to the other, and life would be too confusing if those ways were different, so we demand that they are equal. And then McLean's coherence theorem says that if you have any complicated parenthesized expression, maybe with like 50 morphisms all composed in some complicated way, and then any other way of parenthesizing it, then if you have two different ways to go from one to the other by repeatedly using the associator, those two ways will always agree. So it's not as if we're so fascinated by the Pentagon in itself, it's just that the Pentagon is way of stating the condition that when you use the associator to re-parenthesize, it doesn't matter how you go about it. You get the same result either way. That's called

7:30 a coherence law, and the idea in weak n-categories is that what once were equations now become isomorphisms, but the isomorphisms now satisfy equations, and those equations are called coherence laws. But when you get to n categories, that process gets iterated in a certain more sophisticated way. So Bertrand Russell got everybody scared about the set of all sets because of a certain paradox. There's actually a way to get around that paradox, which involves sets of various sizes, so the set of all small sets is a large set, but an even more beautiful idea is to combine that idea with the notion that the set of all sets isn't really what you're interested in. What you're really interested in is the category of all sets and functions. So there's a category of all sets and functions, and that process iterates. So there's not really a category of all categories unless you are a little bit careful, but what Well, what's much more interesting, anyway, is the two category of all categories. It has categories as the objects, functors as the morphisms, and natural transformations as the two morphisms. Here's a picture of a couple of categories, a couple of functors, and a natural transformation between them. And these squares have to commute, which is the naturality condition. Now, this is one of the most famous examples of a two-category, and this is probably the one that people thought of first. first, but it's misleading in a sense because it happens to be a strict two-category, meaning that all the laws, for example, associativity of composition of morphisms, i.e., of functors, all the laws hold exactly as equations, not just up to isomorphism. That led people down a certain, in some sense, a blind alley, although it's actually very productive in its own way, of thinking about strict n categories, but it turns out that the weak n categories become more important, especially as n goes up a little bit higher, and there are already, in fact, a lot of interesting examples of weak two categories or bi-categories. So, for example, here's an example of how this process goes of passing from categories to two categories. So every topological space has a group void associated to it called its fundamental

10:00 groupoid, which really just amounts to the idea of taking your topological space and thinking of it as a blackboard and drawing all possible diagrams on that blackboard. So, in other words, all the points will be the objects of this category, and roughly speaking, all paths will be the morphisms. Now, to make associativity of composition hold, you need to work with equivalence classes of paths, and we can use homotopy classes of paths if we like, and then you'll have a category. And then also all morphisms will have inverses, so your category will be a groupoid, and that's a fundamental invariant of your space. It contains the fundamental group as part of its structure, but it doesn't depend on choice of base point anymore, so in some sense it's more natural. And what's beautiful about it is that this invariant complete invariant of a certain class of spaces, the so-called homotopy one types. So you say a space is a homotopy one type if all its higher homotopy groups are trivial for all choices of base point. And it then turns out that if you have a homotopy one type, you can work out its fundamental groupoid, but then you can reconstruct that space from its up to homotopy equivalents. So, homotopy one-types up to homotopy equivalents are classified by group voids up to equivalents of group voids. So, in some sense, group voids know everything about topology up to dimension one. By dimension, you mean homotopy dimension. So, that pattern is a natural one to try to continue on. So, in fact, hemispace also has a fundamental two-group void, which is a weak two category, where now the points are the objects as before. The paths are the morphisms. Now we don't need to work with homotopy classes of paths. We can work with actual paths. They don't compose associatively, but they compose associatively up to reparametrization, and those re-parameterizations will be two-morphisms. What are our two-morphisms? Well, they'll basically roughly be paths of paths, that is one parameter families of paths as shown here. Again, to

12:30 make them satisfy laws on the nose, we'd better work with homotopy classes of them. So we only need to work with homotopy classes at the top level, that is in this case the two-morphism level to make our fundamental two-group void, or more generally, our fundamental n-group void satisfied equations at the top level. So if you work this out, you'll see that all the two-morphisms are invertible, and all the morphisms are weakly invertible, by which I mean they're invertible up to a two-morphism, up to an invertible two-morphism. Here are all two morphisms that are infertibles, so I don't need to say that. And so that's what we call a weak two-groupoid. And it turns out that that theorem I said about how homotopy one types are completely classified by grouploids as a two-dimensional analog that the fundamental two-groupoid of the space is a complete invariant of homotopy II types, meaning you can reconstruct a homotopy II type from its fundamental groupoid up to homotopy equivalents. A homotopy II type, by the way, just means that the homotopy groups above level II all vanish. So after you've seen these facts, you can't resist wanting to generalize them. And around 1975, I'm not sure really of the exact date, but he wrote a letter to Larry Green in 1975, in which he made the obvious generalization of what I've been discussing, which I'll call the homotopy hypothesis, which is that weak N-gruboids are equivalent to homotopy N-types. Now, I'm calling this a hypothesis instead of a conjecture because it's too vague to be a conjecture, namely, especially at the...well, because why? Because you have to make up a definition of weak N-gruboid. We know what the definition of homotopy N-type is, and we also really need You need to make precise this concept of equivalence. In what sense are they equivalent? Well, you should be able to go back and forth between the two items, and going back and then forth,

15:00 it won't get you back exactly where you started with. For example, if you start with a space, turn it into a n-groupoid, turn it back into space, you won't get a homeomorphic space, you should get a homotopy equivalent space. But you also need a similar kind of concept of equivalence for your n-group void. So if you start with an n-group void, turn it into a space and go back, you should get an equivalent weak n-group void, but you need to make that concept precise to make this hypothesis into a conjecture. Well, there are lots of ways to make this hypothesis into a conjecture, and in some of them it's now known to be true. In fact, in some sense, it was already known to be true when he brought the subject up in the simplicial approach. So I've been talking about the so-called globular approach so far, where your pictures look like this. So here's a picture of a 3-morphism. I haven't really gotten up to three categories yet, but here's a picture of a 3-morphism in the globular approach to weekend categories. So you have objects that are zero balls, morphisms that look like one ball, two morphisms that look like two balls, and three morphisms that look like three balls. And so that you're supposed to visualize this as a three-dimensional ball with these two morphisms being the front and the back. And if you take those pictures seriously, you can start imagining various ways of composing 3-morphisms by sticking these things together. But there's no reason why you have to use that kind of shape. You can use simplices, for example, and indeed most of the progress in homotopy theory has occurred from the simplicial point of view. and already at the time Grothinick was thinking about this, there were simplicial sets of a certain sort called con-complexes that we could think of as being a simplicial way of discussing weak infinity group voids. Infinity group voids because we are allowing simplices of arbitrary dimension. And so, in fact, it was shown and made quite precise by Quillen

17:30 in the formalism of model categories that you can do homotopy theory using spaces, or you can do homotopy theory using con-complexes, and the two approaches are equivalent, in a sense that Quillen made precise, and so we should now feel, I mean, after you've studied this for a while, you should feel comfortable with the idea of saying that the homotopy hypothesis is true if you work with the simplicial approach to weak end-group voids. Simplicial weak end-group voids would be a con-complex with some kind of cutoff on the homotopy groups it doesn't have homotopy groups above the level N. The homotopy hypothesis still hasn't been completely proved, at least to my satisfaction, in the globular approach, although there's been a lot of work on that recently. So right now the state of affairs is quite complicated, because there are a number of different approaches to N categories, in part of which I've just shown two of the various ones, and in part because there are other options besides just choice of shape involved in making up a definition of N category. And so what I'm going to do is not try to tell you about the definition of N category and all the theorems that are known for that. I'm just going to adopt a sort of vaguer approach where I'm trying to paint the big picture of what we would like to be true, because what we'd like to be true is actually much So, anyway, going back to the history, in 1995, Gordon Powered Street released a little booklet in which they gave some introduction to week three categories. The definition of week three categories was about a few pages long. So for example, instead of the Pentagon equation holding as an equation, now that holds only 3-morphism, and that has to satisfy an equation of its own, which you have to draw using a three-dimensional diagram. It's called the Stasheff polytope. It's a diagram that has some pentagonal faces, and it's also some square faces. So that approach of sort of

20:00 incrementally defining weekend categories, you know, first for two, then for three, and so on. That is fairly intimidating, and it makes it seem as if the subject may take quite a while to develop, especially if you think about how many decades per n we've been doing here. So there's almost 30 years to go from two to three. But surprisingly, people at this point suddenly realize that there are ways to not do it so incrementally, just to figure out a machine that gives you the definition for all n, or gives you the definition straight and then lets you chop down to finite N. And so in 1998, Batonin gave a globular definition of weak infinity categories. And there are lots of other approaches. I'm not going to try to review them all. And it's the simplicial approaches, as I mentioned, which have been the most successful in topology. So I want to tell you a little bit about what we think it may be true about weekend categories and categorification. So there are a lot of hypotheses, or a number of hypotheses, about how they work, which I find to be very helpful in understanding what should be going on. And many of these involve a concept which I call the periodic table. So let me introduce the idea a little bit at a time. Sorry, I don't want you to see that yet. So I wanted to tell you a word about what I mean by the periodic table. So N categories are complicated gadgets that have objects, morphisms, on up to N morphisms. Now it turns out to be very interesting to look at N categories that have just one object. And it's also very interesting to look at N categories that have just one object and one morphism. More generally, to look at N categories that are sort of trivial at the bottom few levels. If you look at an n category that's trivial at the bottom few levels, you can think of it as an n category for some lower value of n, n minus k, say. And it turns out that some very interesting patterns that emerge when you look at n categories of this special sort. So now let's see what happens. Let's just dive in and try some. So the simplest example of what I mean is, what's a category with just one object? Well, pleasantly, it's called a monoid. Monos means one. have had something in mind there. So, in other words, if you have a category that has just

22:30 one object, then what really matters are the morphisms. You can compose any one with any other. It's associative, and there's an identity morphism. So that's what we normally call a monoid. In other words, you forget the object and think about the set of morphisms, and that forms a monoid. So now let's try it again. Let's look at a two-category with only one Well, that's called a monoidal category because it has one object, a bunch of morphisms that you can compose with any other, and then also now a bunch of two morphisms going between those. So, if you play the same sort of trick that we did up here and forget the object and relabel everything by one, that is, now call these arrows objects and call these things here morphisms, you see that this is just a monoidal category, a category that has a tensor product. The tensor product will only be associated up to an associator. Similarly, the unit law will only hold up to some isomorphism. So it's a weakened version of a monoid. So the idea here is that when we play this game, we are always doing some relabeling, which can be incredibly confusing when you're getting going. So I'll just make it clear here that to think of a two-category with one object as a monoidal category, we ignore the object. We call what were the morphisms objects, and we call what were the morphisms, two-morphisms, we call them morphisms. And just to confuse ourselves even more, what had been vertical composition of morphisms, we call it composition, sorry, vertical composition of two morphisms, it's called composition of morphisms, and we may draw it in this string diagram way sometimes. And what had been horizontal composition, we now call tensoring of morphisms. So you can see that here things are being stuck together horizontally, and here they're being stuck together vertically. It's just a different style of drawing those globular diagrams. It's really a Poincaré sort of style of drawing the pictures. This way of drawing the pictures, however, is very suggestive because it suggests that something having to do with knot theory or tangles may be showing up pretty soon, and it will. So let's try the

25:00 next one. Let's look at a monoidal category with just one object. Well, for starters, it's a category with one object, so it has to be a monoid, but it's maybe your average monoid. So, let's see. So, it has one object, which is the unit for the tensor product. It'll have a bunch of morphisms from that object to itself, and we can compose them vertically, so to speak, and tensor them. That's the horizontal juxtaposition. And they're related by the interchange law. So, it sounds like a sophisticated structure, but it's actually not, because you can prove that, in fact, those two operations, the horizontal and vertical operations, have to be the same, and they both have to be commuted. And the proof is this famous Ekman-Hilton argument, which they invented, actually, when they were proving that the second homotopy group of a space was commuted. That's an example, in fact, of this. So the idea is that if you tensor horizontally compose these guys using the fact that the identity morphism of the unit object serves as the identity both for the composition but also the tensor product, you can insert these identities here like this, then use the interchange law to read this diagram a different way, then use the identity law again, now using the identity law vertically to say that it equals this, and continue until you get to beta times alpha. So we've proved that the two operations are the same and that they're both commutative. So the answer is that a monoidal category with one object is just a commutative object. But there's a moral to this, which is that adding the extra second dimension that is going to the level of two categories gives us enough room to move things around each other. And that's where commutativity comes from. Commutativity arises from having enough room to move things around each other. So we've seen, anyway, that a two category that has just one object, so it comes from an oil category, and just one morphism is a commutative

27:30 monoid. That's the beginning of this periodic table. So the pattern is that if you have an N plus K category that's boring at the bottom K levels, so it has only one J morphism for J less than K, you can reinterpret it by this re-indexing procedure as an N category, but it will be an N that has k different ways to multiply. We saw two ways to multiply in the previous, in this picture here, horizontal and vertical, but in general, there would be k ways to multiply, so I would call it a k-tupeli monoidal n-category. And when there are lots of ways to multiply, the Ekman-Hilton argument will kick in and give a kind of commutativity relation. So, here's what happens. This is the beginning of an infinite chart, which lists k-tickly monoidal end categories. So, this is just the beginning of it, just supposed to show you the pattern. So, I've shown you that categories with just one object are monoids. So, this two categories with just one object are monoidal categories, and the monoidal categories with just one object are commutative monoids. So we've basically just dealt with this little portion of it on the chart so far. But there's some patterns that show up as you continue playing this game further. So if I had started with a three-category and looked at it with just one object, it would be a two-category with some tensor product. We'll see in a minute that if you look at one of those with just one object, you get a braided monoidal category, and then you can check for yourself, if you know what a braided monoidal category is, that a braided monoidal category with one object is, again, just a commutative monoid. So we're beginning to see something interesting here, which is that things sort of acquire a multiplication on the first step down the periodic table, then it starts getting commutative, But in this first column, that's all. It just stays being commutative monoids from then on out. So it can't get any better than that. But at the next level here, the level of categories, it turns out that the concept of commutativity becomes more sophisticated. There are braided

30:00 monoidal categories, which are sort of commutative, but there are symmetric monoidal categories that are more commutative, in some sense, but then at that point, there's nothing better than that. So one of the hypotheses, which I'll be talking about, is the stabilization hypothesis, which says that there's a diagonal like this, and below that diagonal, things don't get any better. So this is quite interesting. I don't think I'll have time to talk about it too much, but Ross Street found out there's a nice word, selepsis, which is some kind of meta-braiding, and so he invented seleptic as the new concept before symmetric. An L comes right before M, but I don't advise you continuing that pattern. Syneptic and so on, it's not very, so we need, sorry to say, k2ple monoidal m-category, which doesn't sound so great either, but let me show you how the braided monoidal categories get into the act. So, if you take the argument that I gave you that got us to commutative monoids, and you say, now we are going to categorify that argument, in other words, now we're going to repeat that argument, but instead of working with sets, we're going to work with categories. Categorification, I should emphasize, is sort of the process of trying to move horizontally to the right in this chart. Why do we get braided monoidal categories? Well, roughly the idea is just that at each step, what had been an equation before is now a specified isomorphism. So we will prove that all these different things are not equal now, but isomorphic. The process of going from the top picture, or this picture, over to this picture, is a specific isomorphism from alpha tensor beta to beta tensor alpha. That we call the braiding because it looks just like an elementary crossing in a braid. So what has happened is that the process of proving an equation has become a specific isomorphism, and that's typical of categorification, of moving one step right in the periodic table. And indeed, there was another proof I could have

32:30 given you of the Ekman-Hilton theorem, where instead of moving the little boxes around clockwise, I moved them around counterclockwise. That different proof will now give us a different isomorphism, essentially the inverse braiding. But these are really different, and that's why there are knots in three dimensions. So in three dimensions, there's a big difference between this kind of crossing and this kind of crossing. You can't isotope from one to the other, and that's why I can never untangle a knot. And this is made more precise by a called Schum's theorem. So there's a category of one-dimensional tangles in a two plus one dimensional cube. Those are the tangles that people usually are talking about when they discuss tangles. And in fact, it's not just any old category, it's the free braided monoidal category with duals on one object. That object being a point, a positively oriented point. So the idea is that you could take the positively oriented point as an object, it will have a dual, which will be a negatively oriented point, duality will give you some of the features you expect in the theory of tangles, namely caps and cups, the braiding will also be there if you work with a braided monoidal category, but that this is, in fact, a category that's free. That is, if you have any other braided monoidal category with duels that contains some object, there'll be a functor, a braided monoidal functor, from this free one to that other one. And this is the origin of the quantum invariance of tangles. So, if you find me a braided monoidal category with duals, for example, a category of representations of a quantum group, you pick out an object in there, then I'll get an invariant of tangles. That's this, that was actually discussed, I guess, in the very first talk of the workshop part of this get-together, Cresci-Tegan-Toriath invariants of tangles. So what do I mean by duals a bit more precisely? has a dual, a alpha star, if there's a unit, which we can draw as a cap, and a co-unit,

35:00 which we draw as a cup, satisfying these two equations, the zigzag equations. So duality, duals in a sense, is a weakening of the concept of inverses. So I started out, you see, talking about n-group voids. Those are things where everything has an inverse. If alpha star was an inverse of alpha, what that would mean is that alpha tensor alpha star was isomorphic to I, and then you would have both this unit and co-unit. But here what we're doing is we're saying, no, they don't need to be isomorphisms, we just demand that there are morphisms, the unit and co-unit, but they shouldn't just be any morphisms, they should at least satisfy something nice, the zigzag equations. So knot theory, which we'll see is related to the is a relative of homotopy theory, which is the study of m-groupoid. It's a more sophisticated relative. Now, if we look at a triply monoidal one category that is on an extra dimension, extra codimension, that will be a symmetric monoidal category. So the idea is now we're looking at tangles where we have, say, one vertical dimension, which you can think of extra dimensions, horizontal dimensions, which you could call space, and in that type of situation, the braiding becomes equal to the inverse braiding. So if I was in four dimensions, in other words, I could untie knots. I could just take my arm here, push it slightly into the fourth dimension, and bring it back down, and it would be in front here. See, magic trick. And so you can untie all knots in the fourth dimension, and so in some sense, the commutative than in the braided case. In the braided case, you can switch two things, but there are two different ways to switch them that aren't equal. Now, when we go to the symmetric monoidal case, those become equal. And there's an analog of Schum's theorem here, which says that the category of one-dimensional tangles in a three plus one-dimensional cube is the free symmetric monoidal category with duals on one object. That's a less exciting theorem because you can't tie interesting knots in that dimension, but it suggests there's Well, as I suggested already, it seems that as you march down any column of the periodic

37:30 table, things become maximally commutative at a certain point. And if you stare at that chart, it's when k reaches n plus 2. And there's some good evidence for that. So in knot theory, it turns out that you can untie all n-dimensional knotted surfaces when you get enough dimensions, and you can do it when they're in a cube of dimension 2n plus 2. So having n plus 2 extra dimensions is enough to untie all nines. So that's an example of this, evidence of this. Having more dimensions doesn't change anything further. There's another much more classical result along this line called the Freudenthal suspension theorem that says that the homotopy n-type of a k-fold loop space, a space that's loops and loops and loops and loops in some space, it doesn't get any more sophisticated or more interesting when k surpasses m plus 2. So for these reasons, Larry Breen, James Dolan and I made a guess, which I'll call the stabilization hypothesis, that says that k tupole monoidal n-categories are equivalent to k plus 1 typically monoidal categories when k reaches n plus 2, greater than or equal to n plus 2. So let's temporarily assume that's true and let's call those stable n-categories. You could also maybe call them symmetric monoidal n-categories. So that's something we'll return to. Now I want to say a little bit more about the higher dimensional knot theory. So let's ponder the idea of n tang sub k, the n category of n dimensional surfaces tangled up in a k plus n dimensional cube. So I've been looking at cases where n was 1, but here's a case where n is 2, the simplest case where ends two, two tang sub one. So this is two-dimensional surfaces in a three-dimensional cube. Those are the top-level morphisms, the two morphisms. So the objects are collections of oriented points in the one cube, the one-morphisms are one-dimensional oriented tangles in the two cube, and then the two morphisms, and again we have to take equivalence classes at the top level,

40:00 of two-dimensional tangles in a three-cube. This will be a two-category if you get the details exactly right. So you can compose these morphisms by sticking squares on top of each other like this. You can compose these two morphisms by sticking cubes on top of each other. But it's a monoidal two-category because you can also stick these pictures side by side. But it has a lot of other interesting features. In particular, the objects have duals. In other words, you have this cap and cup, as I showed you before. But now, what had been the zigzag equation, which I discussed before, it gets weakened. It's now just a two-isomorphism. So in other words, this one-morphism here has a two-isomorphism, this surface, two-dimensional tangle going down to the identity. They're not equal, but they're isomorphic. This surface here is a famous surface. It's called the cusp catastrophe. It's part of Rene Tom's classification of catastrophes. And this two-isomorphism satisfies an equation of its own, as always happens when you weaken. And that equation is also a famous thing in catastrophe theory. I don't have a beautiful picture of it on me, but it's called the swallow tail catastrophe. So that suggests that as we go to this process of studying these higher dimensional tangles, the more sophisticated versions of duality that show up will have a whole lot to do with pretty interesting things, classification of singularities in differential topology. So we're trying to get an algebraic way of understanding that. There's more to it than that. It's not just the objects that have duals in n-tanks of K. It's everything. It's all the j-morphisms. So in this example that I'm talking about, two tangles in the 3Q, for example, there will be duals of morphisms. So, we have this co-unit of the point, otherwise known as a, sorry, I'm getting a little boggled here, the co-unit of the point, otherwise known as the cup. It will have a dual, which will be a cap-shaped thing, and duality just means reflection, so the dual of a cup will

42:30 look like a cap, and if you compose them, you get a circle, which is this morphism down here, but because they're duels, there will be a unit for that, so something going from the identity to that, and if you draw what that looks like, it looks like this. It looks like a higher dimensional version of a cap, the sort of cap that an actual three-dimensional person might actually wear. So this is sometimes called the birth of a circle, meaning if you read the story from top to bottom, at first there's nothing and then a little circle gets born. Similarly, there will be a co-unit of a unit, geomorphism, and that will be a kind of two-dimensional cup, and there will also be a unit of the unit and a co-unit of the co-unit, and if you work out what they look like and draw them, you see that those are two saddle-shaped things, and what we've gotten, you see, are all the different kind of critical points that you would expect in two-dimensional Morse theory. If you draw two-dimensional surface in generic position and slide a horizontal plane from top to bottom, occasionally something exciting happens, and that's one of these critical points of the sorts that I've listed here. And then it turns out that the zigzag equations for the duality of the one-morphisms gives you what you call cancellation of critical points. So last time I gave this talk, somebody said, that picture is terrible. All the rest are okay, but I can't understand that one. So if you can't understand that one, imagine you're in the southwest and you see one of these saguaro cacti, and then you come up to it, and it turns out it's hollow. It's just paper mache, and you cut it open, and you're left with something like this. So that's a certain kind of two-dimensional surface. It's a two-morphism in this two-category I'm talking about, and it has one of these caps in it and one of these saddles in it. And the zigzag equation that you can write down formally whenever you have duals will say that that two-morphisms should be equal to this one, and indeed, those are isotopic surfaces. So the point is, the differential topology seems to be following automatically from taking seriously this idea that we should describe these higher dimensional

45:00 with duals. It gets more exciting when you look at two-dimensional tangles in four dimensions. So in 1997, Laurel Langford and I proved the theorem, said that two tang sub two, which are, sorry, this is the two-category of two-dimensional tangles in a four-queue, is the free braided monoidal two-category with duals on one object, which is again the point. So this is categorifying theorem for describing the category of ordinary tangles, and this is where it connects to one of the main themes of this conference. So in the 1990s, the quantum group, quantum topology revolution involved ideas like this, that the category of representations of a quantum group is an example of a braided binoidal category with duals, the category of one-dimensional tangles in the 3-cube is the free such thing on one object. So as soon as you pick any object that is any representation of your quantum group, it will determine a braided monoidal functor which sends the point, the generating object in one tang two to the object you've chosen, the representation you've chosen. And that will be a tangle invariant. When your tangle is actually a knot, you get a number out of your knot that way. So this theorem here, similarly, provides a way of getting invariants of two tangles. So if somebody at this conference categorifies the braided monoidal category representations of some quantum group and can prove that it's a braided monoidal two category with duals, say c tilde, then this theorem will instantly kick into action and give you an invariant of two tangles in the exact same way. You just pick any object in your two category and you get a two tangle invariant. So that's why these types of algebraic descriptions of tangled categories are interesting. Let me say a little bit more about braided monoidal two categories. So on a braided monoidal category, there are various axioms. It's actually

47:30 not one of the axioms, but a very basic theorem is that the braiding satisfies a certain equation called the Yang-Baxter equation involving the braiding of three objects, also known as the third Breitemeister move. When you go up to braided monoidal two categories, that equation becomes a two-isomorphism, which I call by the barbaric name of the Yang-Baxter-Rader. So you should think of that thing as a two-dimensional surface in four-space that starts out at t equals zero looking like this, and at the end it looks like this. So it's the process of taking this top strand here and pushing it down, sorry, pushing it up, I guess, so it comes up above this crossing here. That is an example of weakening, replacing an equation by an isomorphism, so we expect that this Yang-Bachs creator should satisfy a coherence law, and from the definition of a brain admonoidal two-category, you can see that, yes, it does. It's slightly terrifying. It's also very beautiful. It's called the Zamelogikov tetrahedron equation. It seems that the higher these equations get and more complicated. It's also true that the name of invented them gets harder to pronounce. I don't know if that's a trend or not. So what's going on here, here what we have is up here we have a pattern that's just ripe for doing the Yang Daxterator, which in fact we do in this first step here. But you'll notice there's also a fourth strand which is going up in front, which is in front of everything else. So what we can do is we can do the Yang-Baxter-Aitor and then push this whole mess down below this top front strand, or we can take this mess here and push it, push the top strand in front of it first, and then do the Yang-Baxter-Aitor, and those agree. So, this is sort of the next thing after the Yang-Baxter equation, it involves four strands involved two strands, then the Yang-Baxter grader involved three, now this thing here involves four. This is just the beginning of an infinite sequence of more and more sophisticated laws, which are in fact all known, which govern braiding in braided monoidal N categories. So it shows that there's a lot packed into these concepts. In some sense, a lot of work

50:00 that needs to be done to make any of these hypotheses be theorems. So in 1995, James Dolan and I formulated something that I hope I've illustrated by examples, the tangle hypothesis, which says that n tang sub k is the free k-touple monoidal n category with duals on one object, a positively oriented point. So I've given you some examples of that for low values of n and k, where it's known. If you believe this, and if you believe the stabilization hypothesis, which says that the things stabilize when k gets large enough, you can combine the tangle hypothesis and the stabilization hypothesis to get this, which I call the cobordism hypothesis, which is just that k goes to infinity limit. When you take k goes to infinity, your tangles, which had been inside a cube of some dimension, are in a cube of a very high dimension, and when the dimension is high enough, it doesn't really matter what the dimension is anymore, it's just some high dimension, and then you might as well be working abstractly, in other words, not thinking about tangles embedded in a cube, but just thinking about cobordisms. So let n-cob be the n-category of cobordisms, which is supposed to be defined as a kind of limit of these. The hypothesis would be that that would be the free stable n-category with duals on one object, where stable just means k-tipoly monoidal with k big enough. Now to make... Are you assuming somehow we're on line here to get all possible manifolds this way? Yeah, that's right. Well, we have to be a little careful here, but roughly right. So now if you looked at cobordisms that start and end at the empty set, that would just be all manifolds. Now, we have to be careful because there are lots of different flavors of manifolds, piecewise, linear, smooth, and so on. And there's a little piece of fine print that's very crucial for making these hypotheses have a chance of being true as stated, which is that we need a framing on the normal bundle of our tangles. So we should be working with framed tangles here, a detail that I suppressed up till now, and then here we'd be working with framed cobordisms. There are other versions of these hypotheses for

52:30 different kinds of cobordisms, so this is actually not the most general version of these hypotheses, but it's gentle enough to get the point of time. So if the cohort of some hypothesis is true, it gives us a way to construct what you could call n-dimensional extended TQFTs. What do I mean by that? Well, this is a stable n-category. If you give me any other stable n-category, a n-dimensional extended TQFT would be a stable n-functor, stable n categories. So in lower dimensions, you might take C to be vect, the category of vector spaces, that's for ordinary TQFDs, or you might take it to be two vect, there's something called two vector spaces, which form a two category, and that's a stable two category, and when we get up to higher dimensions, it gets to be more work to find good examples of these Cs, but there probably is an example called n vector spaces. But regardless of that, if you have one, because n-prob is the free stable n category with duals on one object, you just need to pick any object in C, and then you should get an extended TQFT determined by its value on the point. So the amazing thing is that as soon as you know where the point goes, it's supposed to determine everything by means of the machinery of stable n categories. So, apparently Jacob Lurie explains this by saying, extended TQFTs are easy to get. So, yeah, they're easy to get if you know what all this stuff means and can actually prove this theorem, which guess is easy for him. So here's a little picture of the periodic table drawn by Aaron Lauda, illustrating the examples of tangles in various dimensions. So here's a zero-dimensional tangle in zero-dimensional space, a rather pathetic entity. Here's a one-dimensional tangle in one-dimensional space, a one-dimensional tangle in two, one-dimensional tangle in three. Well, now you have to believe me. This is a four-dimensional cube. Your eyes just aren't You see that factor in five dimensions, as soon as we get up to four dimensions, you can untangle all crossings until it stabilizes.

55:00 Here are two tangles, two-dimensional tangles in various dimensions. In four dimensions, you have the interesting case of braided monoidal two categories. In five dimensions, there's actually enough room to pass crossings through each other, but there are, in fact, two different ways to do it, up or down. in six dimensions, those become equal, and it's stabilized. So recently, Mike Hopkins and Jacob Lurie have been working on making these ideas precise and proving them. So Jacob Lurie says that he can prove the cabortism hypothesis. He has a manuscript on his website, which is a partially completed proof of this, which he keeps updating as time goes on. He's trying to, so far he's aiming for proving the cavortism hypothesis, although he also says he's going to go for the tangle, the more general tangle hypothesis. He reformulated the hypothesis a little bit, which is perfectly fair. He reformulated it using the concept of infinity and category. Let me roughly say what that is. So it's an infinity category, but where all the j-morphisms above dimension n are invertible, or weakly invertible, invertible up to a morphism one level higher up. So in fact, n-cob, you should really be able to think of it that way if you like. So the idea is you start out doing the usual thing. You take collections of oriented points as objects, framed one-dimensional cobordisms between those as the morphisms, framed two-dimensional cobordisms between those as the two morphisms, and so on, up to level N. But then you add a bunch of higher levels using the fact that you can talk about bithyomorphisms of cobordisms. So you let those be the N plus 1 morphisms. Notice that those are invertible, so that so that we're getting invertibility of these higher levels. Then you can look at smooth paths of dipheomorphisms as the N plus 2 morphisms and so on, paths of paths and so on. So up above here in the top dimensions, we're working with a kind of N-groupoid type situation or infinity-groupoid type situation, and that means that you can use homotopy theory up top there,

57:30 and that's crucial to his arguments. And Ulrika Tillman will be talking about similar ideas later this week. So a little bit more formally, an infinity zero category would be just an infinity groupoid. Everything would be invertible at all levels. You can formulate those simplifiably as con-complexes, and the homotopy hypothesis can be proved in that framework. This sentence I should say, the model category of simplicial sets, where the vibrant objects are the con-complexes, is equivalent to the model category of topological spaces. That's a way of trying to make precise the homotopy hypothesis, and it's been a theorem for a long time. You can try to do something similar and take a simplicial approach to infinity one categories. There's a number of different ways to do that. The one that Jacob Lurie is now using in his manuscript are called Complete Siegel Spaces. And there's a version of the homotopy hypothesis known for those, too. Namely, the model category of those things is equivalent to a model category of what you could call topological categories. So those are categories where the Homsets are actually topological spaces and composition is continuous. So, if you have a topological category, you've got objects, you've got morphisms, which don't need to be invertible, but then you have a space of morphisms from X to Y, so you have a concept of a path of morphisms from one morphism to another, and you can think of that as a two-morphism, but it will be invertible, because paths have inverses. So the idea is that the fact that your Homsets are topological spaces is giving you the concept of homotobies homotopies between homotopies between paths of objects, and giving you this infinity here where all its higher levels are invertible. So in 2005, Clark Barwick, in a, as far as I can tell, unreleased thesis, generalized complete Siegel spaces to define infinity n categories, and apparently that's the kind of infinity n category that Jacob Lurie wants to use, the kind of formalization he wants he used to prove his result. And his result... Do you believe Laurie's result? Well, I believe this result for ages.

1:00:00 Well, I believe his proof. There's not enough of his proof on the written down yet to really say about that. I believe that he will prove it, but that's sort of more like trust in him and his intelligence rather than there being enough written down for me to verify a lot of things. It's not really written down enough. So he's going around saying he knows how to prove it or he's going to prove it. If he doesn't, well, he'll be embarrassed. Then someone else says he better prove it, right? He has to prove it now. this is the statement that he gives So he says, give me any stable infinity n category, then there will be a bijection between, well, equivalence classes of, on the one hand, these stable infinity n functors, think of those as your extended TQFTs, and on the other hand, well, equivalence classes of objects that are fully dualizable. So you'll notice that he's not requiring that C be a stable infinity-end category with duals. Not everything needs to have duals, he's just saying, if this particular object needs to have a dual, and all the morphisms associated with that, like its unit and co-unit, they need to have duals, and so on and so on. So everything having to do with A has to have duals, so to speak. So it's a reformulation of the idea that makes it a little bit more convenient. And so the way the bijection works is that if you have such a fully dualizable object, there will be your extended TQFD that sends the point to that object. Pure equivalence may make you nervous, but the point is that when you have objects in an n category, there's a concept of equivalence between them, meaning that there's a morphism from one to the other that's invertible up to a 2-morphism, that's invertible up to a 3-morphism, and so on. And similarly, there's a concept of equivalence for these functors. And so you shouldn't expect to get a bijection except to work at the level of equivalence. So that's a rough statement of what he's trying to prove.

1:02:30 And he's also working on a version of the tangle hypothesis. So it seems like there's a preponderance of algebraists in this crowd, so I'll issue a challenge to algebraists. challenges that this network of ideas poses is, first of all, start to construct examples of K2-plemonoidal n-categories with duals, or similar structures, and use them to get invariants of higher dimensional knots, or, in a stable case, manifolds. There are a lot of other questions. One that puzzles me immensely is, why are we able to categorify quantum groups and their representations. Well, we're in the midst, we, meaning you, are in the midst of trying to do that. So assuming you succeed and get a braided monoidal two-category duals, there's still the question of why was that possible? What about the concept of, you know, a simple Lie algebra lets you not only define the quantum groups, but these categorified versions, and we can make that question a little bit more pressing by saying, can you go on and to categorify again. Can you get braided monoidal three categories in some interesting way? You can always do it in some trivial way. I'm just throwing an identity three morphisms or something, but that's not what I mean. So in other words, is this thing that you guys are all busy discovering, is it just the tip of an even bigger iceberg, or does it just stop there? There should be some way to get a handle on that question. I don't know what it is, though. And more generally, what kinds of algebraic gadgets are going to have representations that form K2, Glee, Monoidal, and categories with duels. What's the pattern? What's going on here? I think I'll quit there. That's a tough question. Thank you. Any questions? Yes? That's a good question, it has an answer. So the Yang-Baxter equation really has to do

1:05:00 I'm backing up to the Yang-Baxter equation. I'll show you what that has to do with a triangle. So the Yang-Baxter equation really comes from this fact that if you generically draw three lines in a plane, it looks like that. Generically, they don't all hit at one point. But then as you start moving them around, you will generically hit these catastrophes like that. And then if you go past that catastrophe, well, you get something like this, which is sort of a triangle, it's sort of inside-out, a backward version of the triangle. So the Yang-Baxter equation really is about the process of going from this kind of picture over to that kind of picture. Here I'm drawing them in the plane, and I guess I actually did draw the Yang-Baxter equation in a plane here. So this is the same thing, but drawn in a more curvy, artistic fashion. But the Angbacher's equation is really not about lines in a plane, it's about lines in three-dimensional space, projected onto a plane and the catastrophes that occur as you move them around. So, the next level up, the tetrahedron equation, is about what happens when you have planes in four-space. So you just add one to the dimension of everything. So when you have planes in four-space, as opposed to... Sorry, I'm getting maybe a little bit confusing here. These lines are really in three-space, but you project them down to the plane. You can look at planes in four-space, but then you can project them down to three-space. And the fun happens when you have four such planes. So a generic configuration of four planes in three-space, which I'm too lazy to draw, will intersect in certain ways, and you'll get a little tetrahedron. So think of this tetrahedron as being a small portion of the picture where each of these triangles you can extend out to a plane in three space. You'll generically get a tetrahedron like this when you have four planes in three space. Then as you start moving them around, at certain moments they will all hit. And then when you pass that moment, you'll get an inside-out version of that tetrahedron. And if you stare really hard, and, you know, it's basically impossible to do it without drawing lots of pictures,

1:07:30 but if you stare really hard at this stuff and think about it, you'll see that this movie here, this sort of time evolution here, is really, if you could visualize it in four dimensions, it would be four planes in four dimensions. What? We can't see it. Well, even if you saw it, it wouldn't help. But okay, you want to see it, but it's not going to help. So here you have a bunch of lines moving around as time passes, so what you really have are a bunch of planes in four space, and you have four of them. And if you projected it down to 3-space, you'd see a tetrahedron in there. And over here, you'd see another such thing, but it would be sort of the inside-out version of that tetrahedron. So there's a subject called the theory of hyperplane arrangements, where they study this kind of stuff. And that's why they know the generalization of this to all the higher dimensions. That's a rough answer. Any other questions? Yeah? I have one. So, you mentioned Hilton and I find all these higher and higher commutativity versions, so to say, which you get from that picture. Right. And yesterday evening we talked about a few theories where algebra can vary from all categories. And then one version of commutativity would be very commutative. And that played a role, for example, in your brain . Right. Yeah. And then the next step up would be ribbon algebra in modular tensor pedigrees, where you would sort of say, oh, I can braid twice and do these tangled bits and then multiply and this should be the same and just multiply. Does the structure also come up with such a version of cognitive ? Sorry, the ribbon structure you mean? The ribbon algebra. I think the ribbon structure is actually a different origin. And that really comes from the framing issue, which I was suppressing at the beginning of my talk. So I've been drawing these tangles here as little line segments, but you should, to get these hypotheses to all work out nicely in an algebraical way, their normal bundle should have a trivialization. That's what homotopy theorists mean by framing. And so you could, in this particular dimension, you can just draw that by drawing a little arrow coming out and then think of that as a ribbon.

1:10:00 So to get the algebra to work out right, you should really work with ribbons rather than just strands. It's the beginning of a long description, right? Anyone else? Let's thank John Hanks. Thank you. Thank you.