In week241 of This Week’s Finds, you can follow me on my tour of the Laser Interferometry Gravitational-Wave Observatory in Louisiana:
Also hear some tales of the dodecahedron… from the pyritohedron and Neolithic carved stone spheres, through the Pariocoto virus and dodecahedrane, all the way to its relation with the exceptional Lie group E8!
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1042
David wrote:
On a tangential note, reading what you say about singularities, in what ways might we expect theories concerning them - resolution, quivers, MacKay correspondence, etc. - to have something to say about the singularities which appear in monoidal -categories with duals?
Great question! I’ve thought about it, of course, but I don’t have much to say yet, since I don’t actually understand most of these theories concerning singularities. But, I’m trying to.
James Dolan and Todd Trimble have been working hard on Hecke algebras and flag geometries, Hall algebras and quiver representations, and the relation between the two. They’ve made a lot of progress understanding this stuff by categorifying it. I want to explain that sometime, though I’m hoping Todd will write up some of what they’ve done.
In parallel, Jim and I (and maybe Todd, I don’t know) have been thinking about Kleinian singularities, their resolutions, and their relation to quivers via the geometric McKay correspondence.
All this stuff is related. To organize my thoughts, I listed a bunch of facts about quivers in week230. Now I’m trying harder to understand why these facts are true. Until I do, it’s hard to guess where these facts will take us.
On the other hand, the basic relation between n-categories with duals and singularity theory is perfectly sensible and intuitive! The basic idea is that -categories with duals describe the geometry of how things happen. When something drastic happens, we call it a catastrophe. So, catastrophe theory is related to -categories with duals.
The key idea - I guess I’ll give it away, in case I die before getting time to write it up properly - is that a stratified space has a “fundamental -category with duals”, which generalizes the fundamental -groupoid of a plain old space. When a path crosses a codimension 1 stratum, “something interesting happens” - i.e., a catastrophe. So, we say such a path gives a noninvertible morphism. The idea is that going along such a path and then going back is not “the same” as having stayed put. So, going back along such a path is not its inverse, just its dual!
It’s like the difference between milling around in your room all day, and a day where you walk out the door, pick up the mail, and walk back in. Here the boundary of your house is the codimension 1 stratum; when you cross it, we say something interesting happened. Otherwise, it’s just a dull day: “nothing much happened”.
More generally, when a -dimensional path-of-path-of-path-of-…-paths crosses a stratum of codimension , we say something interesting has happened, so we say this kind of thing gives a noninvertible -morphism. Otherwise, we say it’s invertible.
So, when your stratified space has just one stratum - the top-dimensional stratum, of codimension 0 - all paths, paths-of-paths, and so on have inverses. So in this case, your -category with duals reduces to an -groupoid: the fundamental -groupoid of an ordinary space!
I hope this is sort of cryptic because I’ll be mildly miffed if someone formalize it all before I do. But, I hope at least David understands it.
(People are already catching up, in work on directed homotopy theory - but there they get a fundamental category, not necessarily with duals. In my house analogy, you’d get that kind of thing from a room with a one-way door. Get it?)
That’s really interesting.
Is there a good notion of a representation of the fundamental n-category of a stratified space? And if so, is there a relation between such representations and perverse sheaves with respect to the stratification? One would hope so. By now it’s pretty well accepted, I think, that perverse sheaves give the right notion of locally constant family of vector spaces on a stratified space. Although, I think it’s also pretty well accepted that the concept of perverse sheaf itself is still pretty mysterious. Could your stuff resolve the mystery?
Is the fundamental n-category of, say, a CW-complex stratified by sub-CW-complexes always finitely presented, in some suitable sense? Can you give us an example? You probably won’t want to say, alas.
For me, this has the potential to be the most interesting thing I’ve heard about n-categories in a long time.
The CW-example would not generally be
finitely presented as it may have infinitely many cells in infinitely many dimensions.
Right. I should have said finite CW complex.
James wrote:
For me, this has the potential to be the most interesting thing I’ve heard about n-categories in a long time.
Great! You’ll have to work out the answers to most of those interesting questions you asked, though. So far the definition of -categories with duals is only understood up to braided monoidal 2-categories with duals. There’s a lot of stuff to be done even down in these low dimensions!
Paul Gunnels has a nice introduction to stratified spaces.
So you might have a fundamental category with duals of say the disk with boundary and interior as strata. Then classes of paths from the centre to itself are in correspondence with the number of times the boundary is hit.
I can see quite a bit of scope there. Never one to shy away from a grand vision, are you.
Doesn’t that Schubert stratification stuff on p. 4 of Gunnells’ talk have some bearing on the incidence relations we want to categorify? You could kind of imagine that stratified space being acted upon to form a groupoid with arrows between points in the same stratum.
David writes:
So you might have a fundamental category with duals of say the disk with boundary and interior as strata. Then classes of paths from the centre to itself are in correspondence with the number of times the boundary is hit.
So, you’re getting .
I see what you’re saying, and it sounds right, but this example is trickier than my favorite examples. I’d prefer to think about cases like this. Take our space to the plane. Let the plane minus the unit circle be the top stratum, — the codimension-0 stratum. Let the unit circle be the next stratum, — the codimension-1 stratum.
What’s the big difference?
Well, you’re taking a manifold with boundary and making the boundary into the codimension-1 stratum. In the cases I’m imagining, a generic path will cross the codimension-1 stratum transversely - not tangentially, and not at zero speed. But you can’t do this when the codimension-1 stratum is on the boundary: the path has to be tangent when it hits, or else slow down to zero speed!
One can probably straighten this out, but I prefer to start with the easiest examples.
The general idea is something vaguely like this. We form an -category out of a stratified space by taking
and so on.
By generic points, I mean points that live in the top-dimensional stratum — the codimension 0 stratum, . Such points are “generic” because they form an open dense set in . So, any point can in can be slightly perturbed to be of this form, but a point of this form will stay of this form whenever you perturb it sufficiently little!
In the cases I’m imagining, generic paths are paths that start and end in the codimension-0 stratum , and cross the codimension-1 stratum transversely (nontangentially, at nonzero velocity), and never touch the strata , , etc. Such paths are “generic” because they form an open dense set in the space of paths in . So, any path can in can be slightly perturbed to be of this form, but a path of this form will stay of this form whenever you perturb it sufficiently little!
(In your example, generic paths won’t cross transversely - they’ll do something else instead.)
And so on… It gets more complicated as we climb up, and I don’t claim to have figured it out, but maybe someone has. The path space of a stratified space should again be a stratified space, but of some infinite-dimensional sort. If someone has made this precise, that will let us define generic paths-of-paths-of-paths… in a stratified space , and perhaps define the -category we’re after: the ‘fundamental -category’ of a stratified space.
But, for now, it’s much safer to truncate the construction, and work with the fundamental category. For this we need the morphisms to be equivalence classes of generic paths in .
I’m actually a bit confused about the equivalence relation here, though I know in my gut what it’s supposed to look like. It’s certainly not “homotopy between generic paths”. I don’t think it’s even the obvious alternative: “generic homotopy between generic paths”. For some reason I think it’s “generic homotopy between paths, where at each stage of the homotopy the path remains generic”.
This example is helpful: is the plane, is the interior and exterior of the unit circle, is the unit circle, and no lower-dimensional strata. Then a path which starts inside the unit circle, goes outside by crossing the circle transversely, and then goes back inside by crossing the circle transversely is not equivalent to a path that stays inside!
More or less by definition, there’s a generic homotopy from this path to a path that stays inside the circle. But, at some stage of this generic homotopy, the path must be tangent to the circle - so it’s not generic!
(The point is, any sort of process will generically go through a few moments when the situation is not generic: the catastrophes.)
In short, we want “walking outside and then walking back inside” to count as different from “staying indoors all day”.
I can see quite a bit of scope there. Never one to shy away from a grand vision, are you.
Thanks! As so often the case, this grandiose vision was developed jointly by James Dolan and me, but he’s not to blame for my description of it.
You’re right, there’s a lot more to say about this. I’ve been waiting for years to write a paper about it, but right now I’ve decided it’s better to spill the beans and just get the ideas out into the world. Life is too short.
Talking to Chris Lee on the beach yesterday, I realized I want to switch over to even more of a gift economy approach to research - just giving away lots of ideas, instead of officially publishing them. I already give away This Week’s Finds for free, and that’s paid back immensely. So, giving away more will probably make me even better off.
Hmm. Now you might think there’s a good story here which goes: homotopy theorists have taught n-category theorists a lot because they were already working on n-groupoids, a special case of n-category, without realising it. So now we find that those working on stratified spaces can teach us a lot because they are already on a special case of n-category (one with duals) without realising it.
OK, the definition of an ordinary stratified space doesn’t mention smoothness considerations, so all seems quite similar to the other two classes: homotopy n-types; stratified homotopy n-types; directed homotopy n-types. But now you’re telling us our paths between paths between … have to have some generic features, expressible in terms of differentials. And maybe this taps into lots of differential topology, so we should be learning all about Morse theory, stratified Morse theory, equivariant Morse theory (and probably equivariant stratified Morse theory).
But I’m left wondering why this special kind of n-category, i.e., ones with duals, relate to rather special kinds of spaces, i.e., smooth real manifolds. Why smooth? And why real? Is there a special kind of n-category modelled on complex manifolds? On almost complex manifolds? Kahler manifolds? Symplectic? etc.
Something relevant, I think, is described on p. 159 of my book where I explain how a lemma in Poincaré’s first proof of his duality theorem was found not to work, by himself after a counterexample by Heegard, because he hadn’t controlled the way his subvarieties intersected.
By the way, here’s a fun puzzle: which stratified space has as its “fundamental category with duals” the category of tangles?
There are slightly different answers for framed/unframed oriented/unoriented tangles, so for full credit, please say which kind you’re getting.
The configuration space of particles and antiparticles on the plane as oriented unframed tangles?
David wrote:
The configuration space of particles and antiparticles on the plane as oriented unframed tangles?
That’s the right space. 
But, how are you going to stratify it?
We mainly need the codimension-0 and codimension-1 strata to get the fundamental category with duals. I guess we also need the codimension-2 stratum, just in order to make our paths avoid those situations.
But, how are you going to stratify it?
I’d try to stratify it by collisions. Codimension-k is where k particle-antiparticle pairs collide. If we imagine the path in this configuration space as providing a movie as we go up the tangle, such collisions are cups or caps in the tangle. We want to make sure that (a) only one cup or cap exists on a given slice, and (b) we don’t have a particle and antiparticle just passing through each other, which would be a cup and cap at the same point (an even higher order stratum).
John Armstrong wrote:
I’d try to stratify it by collisions. Codimension- is where particle-antiparticle pairs collide.
Right! Or at least almost right: your reply actually revealed a confusion in my own thinking. In the configuration space of particles and antiparticles in the plane, the subspace where particle-antiparticle pairs collide has codimension .
So, the top stratum has codimension 0, and the next stratum, with a single collision, has codimension 2. As you note, a path that hits this codimension-2 stratum once is a tangle with a single cap or cup.
Digressing slightly: note also that we must topologize our configuration space of particles and antiparticles in such a way that a continuous path can go from a configuration with particles and particles, to one with particles and particles. This requires a bit of care, but it’s familiar from the description of as a configuration space of particle-antiparticle pairs on the Riemann sphere.
Right now I’m a bit flummoxed by the fact that we don’t have a codimension-1 stratum, and what this means for the big picture I was trying to paint. But, I’m sure it all works out somehow - I’ll just need to sleep on it. 
Ah, I suppose I was just counting “what can go wrong” rather than a proper codimension.
So why do we need a codimension-1 stratum? The “2-cone” in 3-dimensional space is a good example of a stratified space with only codimension-0 and codimension-2 strata, isn’t it?
Was there an earlier discussion of
configurations of particles without anti-particles?
Then there is a stratum of each codim and some
nice pictures or rather of the corresponding
compactified moduli space.
But how can a path with only one end
not in the top stratum give a tangle?
Both ends - fine. Or is this some arcane
use of the standard language?
cf. braids as paths in config space in
R2
With self-dual particles we get unoriented tangles. Imagine a tangle sitting in and consider plane slices parametrized by the interval.
Generically (in the top stratum) we’ll have a bunch of points where the tangle hits the plane. If the tangle is oriented we consider an upwards-oriented point as a particle and a downwards-oriented point as an antiparticle. Without orientation there’s just one kind of point.
The tangle describes these points moving around as we walk up the interval. Sometimes we hit a cup where the movie sees a particle-antiparticle pair being created, or a cap where we see them annihilating. In the unoriented case there’s just pairs of identical particles being created or annihilating.
Actually, we specifically don’t want a cup or a cap at the top or bottom of the slab. A tangle has just a collection of (oriented) points at the top and bottom of the slab, so both endpoints of the path should lie in the top stratum.
As for braids, that’s exactly the right viewpoint, but note that there are never any collisions for braids, and orientation doesn’t ever matter since different strands don’t interact.
Ah, I suppose I was just counting “what can go wrong” rather than a proper codimension.
Exactly - and there seems to be something right about that here, but for other reasons it makes me unhappy.
So why do we need a codimension-1 stratum?
Well, a stratified space may certainly have an empty codimension-1 stratum. But, I was trying to define the fundamental -category of a stratified space in a systematic way, something like this:
etcetera, with a subtle sort of cutoff at level , as sketched above.
Then, we can see what this definition gives! Generically, a point will lie in the codimension-0 stratum. Generically, a path will start and end in the codimension-0 stratum, but cross through the codimension-1 stratum at some finite set of points. Generically, a path-of-paths will start and end with a generic path, but do some trickier things in the middle, including hitting the codimension-2 stratum at some finite set of points. Etcetera.
(I hope somebody has worked this all out for all , or at least up to or so. Does anyone know? It’s not too hard for low .)
This philosophy works very nicely in some examples, but apparently not so nicely in the first example I decided to inflict on David! Generically, a bunch of particles and antiparticles tracing out paths in the plane will not collide! It’s only when we get to paths-of-paths that collisions are generic.
I’m still hoping that by some shift of viewpoint I get this example to fit in my grand scheme, e.g. by thinking of the braided monoidal category of tangles as a 3-category with one object and one morphisms. I feel sure it used to work back when I used to think about this stuff, years ago.
In fact, now I think I remember how it works! It requires the notion of ‘Thom space’. It’s all a continuation of the stuff about tangles and the Thom-Pontryagin theorem at the end of section 7 of HDA0.
I should explain this more sometime, and make sure I know what I’m talking about.
But, it’s interesting that we went about it ‘wrong’ this time, and got something funny: we seem to want to build a category from a configuration space of ‘particles in the plane’ where
and the equivalence relation presumably involves strata of even higher codimension.
I don’t know what this means in the grand scheme of things, but it’s interesting.
But how can a path with only one end not in the top stratum give a tangle? Both ends - fine
I’m only talking about paths with both ends in the top stratum. Perhaps I confused you by telling the story of a “cup”, or collision:
\ / \__/
and quitting the story right when the particles collided - the catastrophe! At this instant the configuration is not in the top stratum. But, life goes on, rather boringly, without any particles - and then we’re in the top stratum again.
It sort of like those silly action movies that build to a tremendous climax: you feel like leaving the theater just then, but you have to stick around through some denoument where nothing much happens, just to make sure the movie is officially over.
David Corfield wrote:
But I’m left wondering why this special kind of -category, i.e., ones with duals, relate to rather special kinds of spaces, i.e., smooth real manifolds. Why smooth? And why real?
Well, this is the mystery that makes the Tangle Hypothesis so interesting! You start with an algebraically natural sort of -category, namely -tuply monoidal -categories with duals, and you discover that the free such gadget on one object describes -dimensional surfaces in !
(People unfamiliar with this might take a peek at section 7 of HDA0. When I say “discover”, I’m referring to the discovery of the Tangle Hypothesis. It hasn’t actually been proved yet, except in low-dimensional cases.)
More precisely, -morphisms in this -category are framed -dimensional tangles in codimension .
The framing here shows we’re looking at framed cobordism theory. There are other versions of the tangle hypothesis for other kinds of cobordism theory. And, ultimately, there’s one kind of cobordism theory for each generalized cohomology theory, or spectrum! Framed cobordism theory corresponds to the sphere spectrum.
Jargon aside, what we’re really doing in the tangle hypothesis — it turns out — is examining the fundamental -categories of the spheres , where we stratify these spheres with just two strata: a codimension-0 stratum which include everything but the basepoint, and a codimension- stratum which is just the basepoint.
What I’m saying should be fairly cryptic, even to David. But, we can unravel it by doing this puzzle, which should be closely related to the first puzzle I inflicted on David:
If that seems a bit intimidating, we can warm up with this one:
Or, if even that’s too scary:
Don’t be scared by all the stuff about generalized cohomology theories, spectra, and so on - these are fun problems.
There are other versions of the tangle hypothesis for other kinds of cobordism theory. And, ultimately, there’s one kind of cobordism theory for each generalized cohomology theory, or spectrum!
I can’t let this just slip by. Are you saying I can take any generalized cohomology theory, let us say the theory of topological modular forms. Then, ‘ultimately’, this corresponds to a cobordism theory. Now that first sentence is a little ambiguous, but it might mean that there is a tangle hypothesis for each cobordism theorem. If so, then there’s a tangle hypothesis associated to topological modular forms.
What does ‘a tangle hypothesis’ mean here? That there’s a special kind of n-category associated to topological modular forms?
I’d also love to find out more about spectra, n-categories with duals, and generalized cohomology theories. For one thing, it will help me convince my local algebraic topologists that ‘n-categories with duals’ are a great construct. Relating it to topological modular forms? Just the stuff I need!
To learn about generalized cohomology theories and spectra, warm up with week149 and week150, and then read Frank Adams’ book Infinite Loop Spaces - it’s old but it’s packed with insights and jokes, and if you read it you’ll see -categories everywhere between the lines.
If you fail to see the -categories between the lines, read my paper with Jim on categorification, paying special attention to the stuff on -fold loop spaces, the little -cubes operad, tangle -categories, and cobordism -categories.
If you want a local expert on spectra and generalized cohomology theories, you’re in luck at Sheffield - there’s Neil Strickland, and also others who have not shown up on this blog.
I’ll warn you now, though: generalized cohomology and spectra are all about “stable” phenomena. To the -category theorist, this means stuff like -tuply monoidal -categories where . In fact, it means -tuply monoidal -categories, where the first should be thought of as at least 2 more than the second ! This is exactly the sort of stuff -category theorists are not yet ready to handle. Where we have a chance to shine is in the “unstable range”, where , so far for low values of and . For example, quantum group knot invariants are about , . Khovanov homology seems to be about , .
Most homotopy theorists do stable homotopy theory these days. But, see if you can find an unstable homotopy theorist in your vicinity. Or, find a stable one and pester him with so many questions that he becomes unstable. 
I suppose you might have meant every generalized cohomology theory is ‘contained’ in a cobordism theory, e.g., complex cobordism theory sits above topological modular forms in the family of all complex oriented generalized cohomology theories. But even so, there are many cobordism theories:
piecewise-linear cobordism theory, smooth cobordism theory, oriented cobordism theory, spin cobordism theory, complex cobordism theory, symplectic cobordism theory, stable homotopy theory, and so on. TWF 150
So that’s still a heap of tangle hypotheses.
David wrote:
I suppose you might have meant every generalized cohomology theory is ‘contained’ in a cobordism theory, e.g., complex cobordism theory sits above topological modular forms in the family of all complex oriented generalized cohomology theories.
No, that’s not what I meant. It’s true that complex cobordism theory is the universal complex oriented generalized cohomology theory. But I was talking about all generalized cohomology theories. I believe the Baas-Sullivan construction lets us see any generalized cohomology theory as a cobordism theory where we allow our cobordisms to have singularities (!) of a specified sort. Even ordinary cohomology can be seen as a cobordism theory this way.
To learn about generalized cohomology theories, and what it means for one of these guys to be complex oriented, folks should read week149, week150, and the references therein. I’m afraid I don’t know a good online reference for Baas–Sullivan theory. Someday I should read these:
Here’s a review by Stong:
This series of papers is a discussion of the notion of bordism theory using manifolds with singularity. This concept was first introduced by Sullivan, who generalized the notion of oriented bordism with coefficients.
The difficulties that arise are the necessity of keeping track of the nature of the singularities allowed and the necessity of being sufficiently rigorous in laying out the foundations of the theory. In addition, one would like to know the multiplicative behavior of the resulting theories.
The first and second papers are essentially the same. The first is an Aarhus preprint, which has been readily available, while the second is the published version, which is largely unrevised. Primarily this provides the formal and rigorous treatment needed. The author considers manifolds that are decomposed in certain ways as a presentation of a manifold with singularity. The author’s basic results are that he obtains a homology theory, and that there are exact sequences relating the theories for different classes of singularities.
The third paper is concerned primarily with singularities in complex bordism, to obtain a tower of multiplicative theoreis between complex cobordism and cohomology. Such towers are constructed using the Quillen-Adams formal group techniques. The crucial point is that one must localize at the different primes if one is to obtain an adequate theory, as was pointed out in the work of Johnson and Wilson.
I should also read Dennis Sullivan’s work on this stuff, but I don’t know which papers of his are relevant.
My limited understanding of this theory is mainly derived from a few conversations with Nils Baas. There could be lots of fine print I don’t know about. But he’ll be at the n-category workshop in January, so maybe I’ll ask him about this.
Various comments:
1) Baas-Sullivan theory allows you to start with a cobordism spectrum R and introduce singularities to construct R-module spectra that can be thought of as R/(x1,…,xn), where xi ∈ π*R.
2) This is computationally tractable when the elements xi form a regular sequence. You can construct connective Morava K-theories from complex cobordism this way, for example. You can also get ordinary homology, as the cobordism theory of complexes that are allowed arbitrary singularities of codimension at least two.
3) The original Baas-Sullivan framework is quite technical, and combinatorially complex. It is now easier to use the framework developed in the book by Elmendorf, Kriz, Mandell and May.
4) This procedure always gives R-modules, so if you start with MU (= complex cobordism) or MSO or MO, you will always end up with something complex orientable. In particular, you cannot get tmf or KO or the sphere spectrum from MU.
5) You can get more things if you do cobordism of manifolds with extra structure, such as a spin bundle or framing, for example. It is probably possible to get kO from MSpin. It might even be possible to get tmf from MString.
6) There is a theorem that I think appears in an old book by Buoncristiano, Rourke and Sanderson, showing that any generalised homology theory is a cobordism theory of manifolds with some kind of extra structure and singularities. I don’t think that they were able to given any nonobvious concrete examples other than ordinary homology, and I don’t think that anyone else has managed to go anywhere with this theory. But perhaps it would be worth taking another look.
Neil
Thanks for all the info, Neil… this is great stuff! Homotopy theory seems like an endless mountain to climb, for a dilettante like me. But, it’s beautiful.
(By the way, if you choose the text filter itex to MathML with parbreaks before posting your comments, your TeX will be rendered as math symbols. Unfortunately there’s no way for me to change your choice of text filter after the fact, except by copying your post and deleting the original. So, I’ve used some other trick to prettify the formula in your post.)
Homotopy theory seems like an endless mountain to climb
Not endless. You just have to get to the first limit ordinal :D
David wrote:
I can’t let this just slip by. Are you saying I can take any generalized cohomology theory, let us say the theory of topological modular forms… then, ‘ultimately’, this corresponds to a cobordism theory?
I think this is true, but I’m not sure. If I really knew Baas–Sullivan theory, I’d know for sure.
I believe we should be able to get the generalized cohomology theory ‘tmf’ (‘topological modular forms’, or roughly speaking, ‘elliptic cohomology’) by taking complex cobordism theory and tweaking it a bit by letting our complex cobordisms have singularities of a certain sort.
Unfortunately, Baas–Sullivan theory is ‘out of fashion’: the experts believe that geometrical approaches to cobordism theory are less powerful than algebraic approaches. So, it’s hard to obtain information about this subject. But, Baas is working on elliptic cohomology, and he likes -categories, so he’d surely know something about this.
Of course, if the relation between -categories with duals and stratified spaces is clarified, the difference between ‘geometric approaches’ and ‘algebraic approaches’ to cobordism theory will diminish.
So what kind of thing will one of these generalized Tangle Hypotheses say? Are you expecting an algebraic characterisation of what extra the k-monoidal n-category relevant to each type of codimension-k n-space must have?
David wrote:
So what kind of thing will one of these generalized Tangle Hypotheses say? Are you expecting an algebraic characterisation of what extra the -monoidal -category relevant to each type of codimension- -space must have?
Yeah. You can guess some of these yourself. You presumably know the vanilla Tangle Hypothesis, which deals with framed -dimensional tangles in codimension . Let’s chop it down to the classic case , :
The category of framed oriented tangles in 3 dimensions is the free braided monoidal category with duals on one object.
Note the shift in terminology here. Homotopy theorists use “framing” to mean a trivialization of the normal bundle of our tangle, but now I’ve switched to knot theorist’s terminology, where such a thing is called a “framing and orientation”.
A knot theorist’s “orientation” is a field of little arrows pointing tangent to our tangle, i.e. a trivialization of its tangent bundle. A knot theorist’s “framing” is a field of little arrows pointing normal to our tangle. Taking these together, and using the cross product, we get another field of little arrows pointing normal to our tangle - and thus a trivialization of its normal bundle. This is a homotopy theorist’s framing. Conversely, a homotopy theorist’s framing gives a knot theorist’s framing and orientation.
It took me about a year to understand that.
Anyway, maybe you can guess some other versions of the tangle hypothesis in this , case. Now let’s use knot theorist’s terminology:
The category of framed, unoriented tangles in 3 dimensions is the ???? braided monoidal category with duals on ????.
The category of unframed, oriented tangles in 3 dimensions is the ???? braided monoidal category with duals on ????.
The category of unframed, unoriented tangles in 3 dimensions is the ???? braided monoidal category with duals on ????.
If you don’t do these in a couple of days, I bet John Armstrong will jump in and give them a try.
The ‘unframed, oriented’ case is a baby version of what homotopy theorists call ‘oriented cobordism theory’, corresponding to the spectrum . The ‘unframed, unoriented’ case is a baby version of what homotopy theorists call ‘unoriented cobordism theory’, corresponding to the spectrum . This mysterious ‘’ construction is due to Thom; it’s called the ‘Thom space’ or ‘Thom spectrum’ construction. But don’t let that worry you - it’s not necessary to understand any of that to tackle these puzzles.
If you don’t do these in a couple of days, I bet John Armstrong will jump in and give them a try.
Well, I’ll try, but I’m more used to thinking of these things combinatorially. If I can fit it in between preparing talks for two of our seminars I’ll see what I can come up with.
My basic intuition, though, is to use the (almost) universality of framed, oriented tangles (there I go, tipping my hand as a knot theorist) and the fact that these three are (equivalent to) the first three “quotients” one considers.
Incidentally, I went back to add that “almost” comment because a lot of people (myself included) forget that you have to explicitly add the cancellation of opposite kinks as a relation – it doesn’t follow from the braiding relations. As such, tangles as knot theorists consider them aren’t quite as universal as we’d like.
John Armstrong wrote:
I’ll try, but I’m more used to thinking of these things combinatorially.
You mean, as opposed to categorically? The category theory is just an efficient way of discussing the combinatorics; when I said:
The category of framed, oriented tangles in 3 dimensions is the free braided monoidal category with duals on one object.
I was really just saying that such tangles can be drawn as pictures using caps, cups and crossings, satisfying relations like the Reidemeister moves (but using the framed version of the first Reidemeister move).
So, when I posed this puzzle:
The category of unframed, oriented tangles in 3 dimensions is the ???? braided monoidal category with duals on ????.
I was really just asking what extra relations our category of tangles satisfies when we ignore the framing. And similarly with my other two puzzles:
The category of unframed, oriented tangles in 3 dimensions is the ???? braided monoidal category with duals on ????.
The category of unframed, unoriented tangles in 3 dimensions is the ???? braided monoidal category with duals on ????.
You clearly know this game:
My basic intuition, though, is to use the (almost) universality of framed, oriented tangles (there I go, tipping my hand as a knot theorist) and the fact that these three are (equivalent to) the first three “quotients” one considers.
So, I’m not trying to torture you (or anyone) — I’m just trying to get someone to say what relations actually hold in these “quotients” of the category of framed oriented tangles.
Incidentally, I went back to add that “almost” comment because a lot of people (myself included) forget that you have to explicitly add the cancellation of opposite kinks as a relation – it doesn’t follow from the braiding relations. As such, tangles as knot theorists consider them aren’t quite as universal as we’d like.
I get your point, but the zig-zag identity and the framed first Reidemeister move do follow from the definition of “braided monoidal category with duals”, so the category of frame tangles is universal enough to be the free such gadget on one object. It’s the duals that get the job done.
By the way, both the zig-zag identity
| /\ /\ | | | / \ / \ | | | / \ / \ | | \ / | = | \ / = | \ / | | \ / | \/ | | \/ |
and the framed first Reidemeister move look a bit like a “cancellation of kinks”, so I’m not quite sure which you were referring to, but anyway, they both hold when you’ve got duals.
Here are some definitions that may help.
Suppose is a braided monoidal category. Then we say:
is closed if the functor of tensoring with any object : has a right adjoint, the internal hom In other words, we have a natural isomorphism Here HOM denotes the usual set of morphisms from one object to another, while hom is the “internal hom”, which is an object in our category.
has duals for objects if it is closed and for any object there is an object , the dual of , such that for all . If has duals for objects, it’s common for category theorists to say is compact or compact closed; algebraic geometers say it’s rigid.
has duals for morphisms if for any morphism there is a morphism such that and all the structural isomorphisms (the associator, the left and right unit laws, and the braiding and balancing) are unitary, where a morphism is unitary when is the inverse of .
has duals if it has duals for objects and morphisms.
You mean, as opposed to categorically?
What I mean is that I default to thinking of the equivalent category of tangle diagrams presented by generators and relations. You’ve got the cup, cap, and both crossings as generators, and the relations are the five “Reidemeister moves”, three of which are the regular ones (use framed R1) and two of which handle new quirks of the diagrams introduced by the height function. In fact, one of the nicest things I’ve seen in the tangle approach is the way the Reidemeister theorem turns into an equivalence of categories.
If this approach is what you meant, then I can easily say what the new relations are. I wrote them out explicitly in my dissertation, in fact. The thing is, I’m not sure that “free braided monoidal category with duals” by itself includes framed R1. You need to add it by hand, otherwise you can just cancel opposite curls on opposite sides of the strand with the Whitney trick.
Okay, that said. Starting with FrOTang add a “reverser” arrow from the generating object to its dual and insist that it be its own inverse. Then you can pass to the equivalent category FrTang. Or you could add R1, which implies framed R1. That gets you to OTang. Once you’ve passed to either of those quotients you can do the other and get Tang.
What I don’t know is how to fit these desciptions into your given sentence structures. I don’t know what replaces the “????”. That’s ultimately the language mismatch: I think directly in terms of the relations, while you’re thinking in terms of universal properties which imply those relations. I’m treating tangles “like an algebra, but more so”, while you’re treating them in a much more categorical style.
John Armstrong wrote:
The thing is, I’m not sure that “free braided monoidal category with duals” by itself includes framed R1.
It does if you use the definition in, say HDA0 (see the remarks on the bottom of page 25 and the top of page 26).
Namely: using duals for objects and duals for morphisms, we can define for any object a morphism which some folks call the balancing. Diagrammatically the balancing looks like this:
| /\
| / \
\ / \
\ / |
/ |
/ \ |
/ \ /
| \ /
| \/
|
As you note, the Whitney trick shows this is invertible. However, in the definition of ‘braided monoidal category with duals’ I also include an extra axiom saying this morphism is unitary: its inverse equals its dual, , which is drawn like this:
| /\
| / \
\ / \
\ / |
\ |
/ \ |
/ \ /
| \ /
| \/
|
This is just a way of asserting that this:
| /\
| / \
\ / \
\ / |
/ |
/ \ |
/ \ /
| \ /
| \/
|
| /\
| / \
\ / \
\ / |
\ |
/ \ |
/ \ /
| \ /
| \/
|
can be pulled tight to give a straight vertical string. And that’s what knot theorists call the framed Reidemeister 1 move.
Demanding that the balancing be unitary is a bit ad hoc, but it somehow goes along with the fact that in a braided monoidal category with duals we must also demand that all the other structural isomorphisms - the left/right unitors, the associator and the braiding - be unitary.
Okay, that said. Starting with FrOTang add a ‘reverser’ arrow from the generating object to its dual and insist that it be its own inverse. Then you can pass to the equivalent category FrTang.
Exactly.
What I don’t know is how to fit these desciptions into your given sentence structures […] That’s ultimately the language mismatch.
Right. It’s no fair demanding you express something you already know in my own favored jargon — but if that’s what it takes to get people to read my papers, that’s what I’ll do! 
So, let me just translate what you said into category theory. We start with this theorem:
If we then throw in an isomorphism , we get the category of framed unoriented tangles. An object equipped with an isomorphism is said to be self-dual. So, we say:
If alternatively we impose a relation saying the balancing is the identity, we get unframed tangles. For this reason, an object with is called unframed. So:
We can also do both things, and get:
The next fun thing would be to cook up stratified spaces that have these various braided monoidal categories as their ‘fundamental 3-categories’. So far we just did the basic case of framed oriented tangles, getting . A sphere! By no coincidence, the sphere spectrum is the spectrum for framed cobordism theory — now using ‘framed’ in the homotopy theorist’s sense, which means ‘framed oriented’ to you.
The other cases will be related to various other spectra, which are a wee bit more complicated. But, it’s a good way to see how these three are related:
What completes the correspondence?
John said:
The next fun thing would be to cook up stratified spaces that have these various braided monoidal categories as their ‘fundamental 3-categories’.
Wildly guessing, what happens if we look at the fundamental 3-category of the stratified spaces and ?
Neil Strickland has a very good introductory piece about MO and other things, and here for general research interests.
Someone’s been thinking about framed unoriented tangles for a long time - at least 14 years:
quantum gravity in the loop representation has a lot to do with framed unoriented tangles, and the group of framed braids on acts as symmetries.
See ‘Tangled up in Blue’ here.
This isn’t quite right. A framing automatically gives you an orientation. Thus, under one reasonable interpretation, (framed oriented) = (framed unoriented) and in both cases the relevant cobordism spectrum is S. Alternatively, you could say that a “framed oriented” manifold is a manifold with a specified framing and a specified orientation, which might or might not be the one arising from the framing. In that case the cobordism spectrum would be .
Is John’s disambiguation relevant here?
Homotopy theorists use “framing” to mean a trivialization of the normal bundle of our tangle, but now I’ve switched to knot theorist’s terminology, where such a thing is called a “framing and orientation”.
A knot theorist’s “orientation” is a field of little arrows pointing tangent to our tangle, i.e. a trivialization of its tangent bundle. A knot theorist’s “framing” is a field of little arrows pointing normal to our tangle. Taking these together, and using the cross product, we get another field of little arrows pointing normal to our tangle - and thus a trivialization of its normal bundle. This is a homotopy theorist’s framing. Conversely, a homotopy theorist’s framing gives a knot theorist’s framing and orientation.
It’s not only relevant, it’s exactly what the previous commenter seems to have ignored.
If we then throw in an isomorphism , we get the category of framed unoriented tangles. An object equipped with an isomorphism is said to be self-dual.
Of course, you could also call such an object unoriented, but it’s nice to have a directly categorial name. (Then you can later tell people the big secret, that “self-dual” in category theorists’ language equal “unoriented” in knot theorists’ language.)
If alternatively we impose a relation saying the balancing is the identity, we get unframed tangles. For this reason, an object with is called unframed.
Now we don’t have a directly categorial name; have you tried calling such an object balanced? It’s a fine word that seems quite appropriate here. (Then you get another big secret later on.)
For what it’s worth, there’s another usage of ‘balanced’ in category theory. A category is traditionally called ‘balanced’ if every arrow in it that’s both monic and epic is an isomorphism.
This doesn’t seem a great name to me, but it does seem to have been used for quite a long time.
I’m a tad busy at the moment, so John Armstrong is very welcome to the questions. But here’s a couple of questions of my own:
Will we start to see more mixed flavours of category, like the quasi--categories of Joyal, which are like ordinary -categories but with invertible -morphisms after a certain point? Why not have ones which have duals after a certain point? Or dual for a range, then invertible thereafter?
One of the first stratified spaces I heard about was the space of immersions of the circle in . Connected components of generic points correspond to knots, and the walls of the ‘discriminant’ - the complement of the top stratum - correspond to passages of one piece of a knot through another. Is there something special about this stratified space from the -category point of view?
I guess its zeroth homotopy is isomorphic to the 3-morphisms which run between trivial 2-morphisms in the fundamental 3-category of .
Hmm. Maybe my 2 questions aren’t unrelated. We could have Hom sets enriched over stratified spaces.
Can there be Tangle hypotheses for equivariant cobordism theories?
Sure - everything that anyone can do, someone else can do equivariantly. This is one of those things that keeps topologists busy (because it’s often far from straightforward).
But, instead of generalizing hypotheses that are already unproved and vaguely stated, I’d like to keep on going with a Socratic dialog about the “the fundamental -category of a stratified space” and the Tangle Hypothesis for arbitrary cobordism theories. The two topics are closely related, but I’m not sure I’ve completely managed to show you how. I think if we tackle a few puzzles, we can make real progress.
(Here “we” means anybody reading this!)
David wrote:
One of the first stratified spaces I heard about was the space of embeddings of the circle in R3. [Later David corrected “embeddings” to “immersions”.]
Actually this sits inside the space of tangles. These are the paths in the configuration space we were just discussing from the generic no particles configuration to itself…
Hang on a minute… I just noticed something odd. There’s only one point in the space corresponding to “no particles”, but it should be generic. How are we supposed to topologize things so that a single point is its own connected component of the codimension-0 stratum?
Better get our homework done to keep the teacher happy. So setting out from:
The category of framed oriented tangles in 3 dimensions is the free braided monoidal category with duals on one object.
We have:
The category of framed unoriented tangles in 3 dimensions is the free braided monoidal category with duals on one self-dual object.
The category of unframed oriented tangles in 3 dimensions is the free braided monoidal category with duals on one unframed object.
The category of unframed unoriented tangles in 3 dimensions is the free braided monoidal category with duals on one unframed self-dual object.
Now rather than give a definition of what an unframed object is, I’ll refer teacher to his own paper with Laurel Langford Higher-Dimensional Algebra IV: 2-Tangles, where we find out the conditions for unframedness, at least in the case of self-dual objects, in Definition 14, p. 41.
Sadly, doing homework this way is rather too much like the way my kids do their school assignments. Need to ‘Research the life and work of Picasso’, head straight for Wikipedia.
Let me then at least try and ask a sensible question. Does this framedness/unframedness distinction tally with a feature of the algebraic objects used to represent the category? Perhaps characterising representations of quantum groups?
Re: This Week’s Finds in Mathematical Physics (Week 241)
On a tangential note, reading what you say about singularities, in what ways might we expect theories concerning them - resolution, quivers, MacKay correspondence, etc. - to have something to say about the singularities which appear in monoidal n-categories with duals? You showed us the swallowtail catastrophe here, and I seem to recall in Carter and Saito’s work, which you used in your paper on 2-tangles, seeing other ones. Oh yes, there’s the swallowtail on p. 67 of their paper with Kauffman, Diagrammatics, Singularities, and Their Algebraic Interpretations, followed by many others.