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!
Posted at November 20, 2006 9:53 AM UTCTrackBack URL for this Entry: https://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?
I’ll refer teacher to his own paper
Grr.. I knew that I should have forced myself to work through the rest of that series. Maybe on the flight to and from Nawlins.
I should have known this too. I was very impressed by the paper on 2-tangles.
Better get our homework done to keep the teacher happy.
Yes, sorry, I’m becoming a bit of the schoolmaster here…
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.
Right! But…
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.
Yes, with the attendant perils:
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.
If you’re going to copy your homework from the web, don’t do it from the teacher’s own paper!
The definition you cite is for an unframed object in a braided monoidal 2-category, since we were studying 2-tangles: 2d surfaces in 4d space. This is vastly more complicated than the question I actually asked, which was about an unframed object in a braided monoidal category.
In a braided monoidal category, an object is unframed if its balancing:
| /\
| / \
\ / \
\ / |
/ |
/ \ |
/ \ /
| \ /
| \/
|
equals the identity. For example, in physics a boson is unframed, while a fermion or more general anyon is not: turning it around 360 degrees does something nontrivial.
In a braided monoidal 2-category, an object is unframed if its balancing is isomorphic to the identity… via some 2-morphism which satisfies a complicated law of its own!
Of course in a braided monoidal 3-category this law would get promoted to a 3-morphism, etc. But, as far as I can tell, nobody gone this far. Maybe topologists have thought about it: we’re talking about the proces of untwisting the process of untwisting the process of untwisting… a twist in a string.
Baas–Sullivan theory is ‘out of fashion’
This paper speaks on page 2 of a ‘modernised’ version of Baas-Sullivan theory.
I don’t think we can construct a Baas-Sullivan presentation of TMF.
Just like line bundles over are isomorphic to maps from , elliptic curves over the scheme are isomorphic to maps , and elliptic curves over the derived scheme are isomorphic to maps from .
Naively, we might expect , such that
How do we define the global sections of ? We take the homolopy limit of , where is an affine cover of . We see immediately that our naive desire doesn’t work – global section of this sheaf depends on taking a homotopy limit. Do do this, we need the target category to be enriched in spaces, so we redefine our functor :
, s.t. .
Now we can take the homotopy limit, and makes sense.
Really locally (over a point), looks like either height 1 or height 2 Morava E-theory, both of which can certainly be presented via the Baas-Sullivan construction (Morava E-theories are constructed by starting with BP and killing off generators).
We don’t have a method to check that these presentations of Morava E-theories (as bordism theories with singularities) are -ring spectra, so I don’t think we can take a homotopy limit of these Baas-Sullivan presentations and get TMF.
Thanks for the update!
The fundamental 1-category of should be sequences of +s and -s, marking how a path passes through *.
The fundamental 2-category of should have generic paths as 1-morphisms, and 2-morphisms describing the creation and annihilation of adjacent ‘+ -’ or ‘- +’ pairs.
The fundamental 3-category of should have generic paths (so don’t pass through *) as 1-morphisms, 2-morphisms sweeping between them crossing * in a sequence of +s and -s. Then 3-morphisms capturing the creation and describing the creation and annihilation of adjacent ‘+ -’ or ‘- +’ pairs.
Maybe I should have been more precise on this last case. I suppose we ought to care how far into the sweeping and at what point of the path being swept, * gets hit. Although there would be invertible 3-morphisms between two similar configurations. Oh, so it’s a bit like particles and antiparticles in a square.
And ditto for the second case except on the line.
This PhD project, announced by Jon Woolf, looks relevant:
A new project involves developing a refinement of homotopy theory which is adapted to stratified spaces (or more generally to any space with a partial order). The idea is to only allow homotopies which respect the partial order, and in this way to obtain a finer invariant.
Perhaps I should try to explain what I envisage this PhD project being about, and how it relates, or not, to this discussion!
The idea, like a lot of things, goes back to MacPherson. A local system on a manifold is a representation of (or perhaps on this post that should be the fundamental 1-category). On a stratified space a natural generalisation is a constructible sheaf i.e. a sheaf which is locally constant on each stratum or, more simply, a collection of local systems, one for each stratum, which fit together nicely. Constructible sheaves are representations of the ‘partially-ordered fundamental 1-category’ whose objects are points and whose morphisms are homotopy classes of paths which ‘wind out’ from the singular (higher codimension) strata. A nice way to describe these paths is to put a partial-order on the stratified space in which if where is in the closure of . A path which ‘winds out’ is just an order-preserving path from . The notion of homotopy here is (unordered) homotopy through ordered paths.
This construction can be applied to any space with a partial order but the result is a bit hard to understand (for me at least) in general. For a stratified space the partially-ordered fundamental 1-category is fairly tractable; one can give a pretty explicit description of a skeleton for instance. It’s a refinement of in the sense that inverting non-isomorphisms allows us to recover the fundamental group.
Everything is nicely geometric: another way of describing a constructible sheaf is as a stratified etale space over (etale in the sense of topology not algebraic geometry; by stratified I mean the projection to is a stratified map). Contravariant representations of the po-fundamental 1-category can be interpreted as stratified branched covers over . For example if is the real line with the origin as a codimension stratum then the non-Hausdorff line with two origins is a stratified etale space over and two lines intersecting at the origin is a stratified branched cover over .
The idea for the PhD project is to try to develop these definitions and compute some nice examples (arising in singularity theory). Everything is functorial under stratified maps so one goal would be to develop an analogue of the LES of homotopy groups. The first part is easy enough to build but I don’t yet understand how the higher po-fundamental categories should be defined (there are lots of choices for what should be ordered and what not) to obtain this.
So in brief, it’s a more geometric take on some of the work in directed homotopy that’s been going on. It differs from John’s ideas in that there’s no notion of dual and no need to worry about genericity (because paths never cross strata, only leave them).
Having said that, I’ve been thinking about trying to introduce duality into the picture too as it would be much more interesting to have perverse sheaves as representations, and duality should somehow be built in for this. The natural setting for this would be spaces with only even codimension strata where the theory of perverse sheaves works nicely. A good example is . Constructible sheaves don’t capture all the topology in this example, in particular they don’t see the fundamental class in , but perverse sheaves do (which is why they are ‘better’). In this case there are explicit descriptions of the constructible sheaves as representations of the quiver
and of perverse sheaves as representations of a quiver with two objects and two arrows with . (Here the objects correspond to the nearby and vanishing cycles, to can and to if that helps.) It would be really nice if the latter quiver naturally arose out of the fundamental n-category with duals of ! A (possibly relevant) observation against this is that the duality for perverse sheaves is Poincaré-Verdier duality in the constructible derived category. This is not a strongly compact closed category but rather a *-autonomous category (at least this my understanding) so maybe this notion of duality is a little different from what you have in mind?
One more comment, and then I’ll stop as this post is overlong! There are lots of notions of stratified space but most of them are not that well adapted for homotopy theory. In particular they often lack coning constructions (e.g. the mapping cone on a nicely stratified map is not necessarily nicely stratified - yuk). Frank Quinn has a notion of homotopically stratified space which is a bit subtle but seems to be better adapted to homotopy theory. On the other hand, working out what generic means for these might be tricky so maybe it’s not such a good route!
Jon
PS Thanks for the stimulating discussion.
Thanks very much for this, Jon. It’s great to have different angles on this material.
Does anyone know of a very gentle introduction to perverse sheaves, or could anyone jot down the ‘simplest nontrivial example’, to use a phrase of Gelfand?
Unfortunately there’s a bit of a gap in the market for gentle introductions to perverse sheaves (at least as far as I know). The standard approach involves a fairly hefty diet of homological algebra and sheaf theory. Reasonably gentle, but not painless, introductions in this line can be found for example in Reitsch’s notes or in Massey’s. There are numerous books too which take a similar approach. MacPherson has some great unpublished notes which take a completely different approach motivated by stratified Morse theory. Well worth reading if you can find a copy but I don’t know where you can get hold of one.
The simplest examples are when the stratified space is a manifold with the trivial stratification i.e. everything in the codimension 0 stratum. Then a perverse sheaf is a local system a.k.a. representation of the fundamental groupoid (there’s a choice of what to represent into, the most common place is vector spaces over Q or C).
If there are higher codimension strata a perverse sheaf will be an extension of a local system on one of the strata to the closure of that stratum. `Extension’ here means `extension as an object in the constructible derived category’, this is where the homological algebra comes in. But very loosely you should think of it as an extension of a local system which records useful data about the nature of the singularities along that stratum, in particular information about what has collapsed to form the singularity (i.e. information about the vanishing cycles).
Perverse sheaves on a stratified space form an abelian category. If all the strata have even codimension then there is a duality which extends the obvious duality on local systems given by dualising the stalks. In some simple cases it’s possible to describe the abelian category of perverse sheaves as representations of a quiver with relations. This is nice because it’s fairly concrete and avoids the homological algebra and sheaf theory.
The simplest nontrivial case was in my previous post where perverse sheaves on are representations of the quiver with two objects and two composable arrows c and v with cv=0.
Here’s a second example. If we add another point stratum P’ to the sphere then perverse sheaves will be representations of a quiver with three objects, one for each stratum. Let’s label them P, P’ and U corresponding to the open stratum which is a twice punctured sphere. There are arrows and from U to P and P’ respectively and arrows and from P and P’ to U (which should be thought of as dual to and ). In addition each object has a nontrivial isomorphism (arising from a monodromy action of ) and these commute with the arrows defined above. Finally we have relations and where is the nontrivial isomorphism at U and its inverse.
Hope this is some help!
Jon Woolf has a paper out today The fundamental category of a stratified space.
The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified Treumann, based on unpublished ideas of MacPherson, constructed an ‘exit category’ (in the terminology of this paper, the ‘fundamental category’) which classifies constructible sheaves, equivalently stratified etale covers. This paper generalises this construction to homotopically stratified sets, in addition showing that the fundamental category dually classifies constructible cosheaves, equivalently stratified branched covers.
The more general setting has several advantages. It allows us to remove a technical ‘tameness’ condition which appears in Treumann’s work; to show that the fundamental groupoid can be recovered by inverting all morphisms and, perhaps most importantly, to reduce computations to the two stratum case. This provides an approach to computing the fundamental category in terms of homotopy groups of strata and homotopy links. We apply these techniques to compute the fundamental category of symmetric products of , stratified by collisions.
Remember that this is a different approach from John’s fundamental category with duals, since here “paths never cross strata, only leave them”.
Cool! This may help me finally understand what ‘constructible sheaves’ are. Or ‘constructible cosheaves’, at least.
David almost wrote:
Morphisms in the fundamental 1-category of should be sequences of +s and -s, marking how a path passes through *.
Right! So, a morphism in this category is a 0-dimensional framed tangle in 1 dimension.
(Here I’m using “framing” in the theorist’s sense, to mean a trivialization of the normal bundle. So, a framed 0-dimensional manifold in is a bunch of points in the interior of the interval, with signs attached. A “0-dimensional framed tangle in 1 dimension” is an equivalence class of these: just a string like +-++-.)
The fundamental 2-category of should have generic paths as 1-morphisms, and 2-morphisms describing the creation and annihilation of adjacent ‘+ -’ or ‘- +’ pairs.
Right! More precisely, a 2-morphism in this 2-category is a 1-dimensional framed tangle in 2 dimensions.
The fundamental 3-category of should have generic paths (so don’t pass through *) as 1-morphisms, 2-morphisms sweeping between them crossing * in a sequence of +s and -s. Then 3-morphisms capturing the creation and describing the creation and annihilation of adjacent ‘+ -’ or ‘- +’ pairs.
Right! More precisely, a 3-morphism in this 3-category is a 1-dimensional framed tangle in 3 dimensions.
One needs to think carefully to understand the framing information in this case. For the 2-morphisms this information is in the sign + or - labelling each point. But in the 3-morphisms, there’s extra information saying how each of these points rotates! I.e., ‘framing twist’ information. It all falls out from the definitions if you think about it hard.
Maybe I should have been more precise on this last case. I suppose we ought to care how far into the sweeping and at what point of the path being swept, * gets hit. Oh, so it’s a bit like particles and antiparticles in a square.
Yes, in this last case a 2-morphism is a framed 0-dimensional manifold in . So, it’s a bunch of points in the interior of the square, with signs attached.
In short, your first answer to my original question was wrong! (The italics here are my tribute to Toby’s bluntness.)
But, my question was also wrong. The correct question is:
Which stratified space has the braided monoidal category of framed oriented tangles as its fundamental 3-category?
and the correct answer is: .
But, the first time around we were close enough to being right to enjoy some productive confusion.
Now I hope we see the relation between the wrong approach and the right approach.
Is it possible to pick out the n-groupoidal part of the fundamental n-category with duals in a natural way?
For example, the fundamental 3-category with duals of (S^2,*) is supposed to be the category of tangles, viewed as a braided monoidal category with duals. It seems reasonable to me that the 3-groupoidal part of that would be the category of braids, viewed as a braided groupoidal category with duals. And this is the fundamental 3-category of some simply connected space G(S^2,*). However, this space is not well-defined, only its homotopy 3-type is.
My question is Is is possible to make it well-defined in a natural way? In other words, is there a functor G from stratified spaces to spaces such that the fundamental 3-groupoid of G(X) is naturally the 3-groupoidal part of the fundamental 3-category with duals of the stratified space X?
If so, you’d hope you could say quite explicitly what G(X) is. Maybe people who are good at realizing braid groups as fundamental groups of configuration spaces can guess the answer.
There are also the same questions for general n, instead of n=3, and you might even hope the family of functors G_n fit together in a nice way. Actually, at the limit n=infinity, there might be nothing to do: just set G(X) to be “the” homotopy type associated to the infinity-groupoidal part of the fundamental infinity-category with duals of X.
Actually, at the limit n=infinity, there might be nothing to do: just set to be “the” homotopy type associated to the infinity-groupoidal part of the fundamental infinity-category with duals of X.
Do you need to mention the “infinity-groupoidal part” when an -category with all duals is an -groupoid?
David said:
‘Do you need to mention the “infinity-groupoidal part” when an ω-category with all duals is an ω-groupoid?’
Is that true? Is it easy to say why? Assuming it is, then I guess you wouldn’t.
Actually, even in the infinite case, you might want to understand G(X) in a way that has nothing to do with fundamental anythings (ie groups, categories, etc). So there might even be something to do in that case.
The reason I am wondering about all this is because it would be nice to understand what the fundamental n-category with duals *does*, rather than what it *is*. For example, what usual fundamental groups do is classify covering spaces, whereas what they are is a bunch of (classes of) paths. I think it’s important to understand both points of view. And it might be that understanding the groupoidal part will give us a hint.
Eugenia Cheng has a paper An -category with all duals is an -groupoid. Don’t I recall this has something to do with the infinite-dimensional sphere being contactible?
I thought that if we consider the triple (braided monoidal category, braided monoidal category with duals, braided groupal category) then it’s the first which corresponds to braids rather than the third.The passage from second to third is a form of group completion, in this case to .
I don’t know how this fits in with the stratified space picture.
John wrote:
a day where you walk out the door, pick up the mail, and walk back in
You have to walk out the door to pick up your mail? How uncivilized!
Next thing you’ll be telling us that you only have an intermittent power supply.
:-)
Tom wrote:
You have to walk out the door to pick up your mail? How uncivilized!
Well, if you’ve ever seen Steve Martin’s movie “L.A. Story”, you’ll know that most of us around here hop in the car, press the magic button the opens the garage door, back out to the street, press the button that rolls down the car window, take the mail out of the mailbox, and then drive back inside. But I’m sort of low-tech.
I’m going to pop a few parentheses and continue our highly nested conversation down here. This is probably a good excuse to summarize what we’ve seen about the relation between:
We looked at an example and saw these three are the same:
We can see the pattern better if recall that a braided monoidal category is secretly a special sort of 3-category: one that’s boring on the bottom 2 levels. The objects of our braided monoidal category are secretly 2-morphisms in this 3-category. The morphisms in our braided monoidal category are secretly 3-morphisms in this 3-category.
This is why we call a braided monoidal category a ‘doubly monoidal category’. In general, we define -tuply monoidal -categories to be -categories with only one -morphism for . These are extensively discussed here and here.
Phrased this way, we’ve seen these three are the same:
This suggests a generalization to higher dimensions and codimensions - a version of the Tangle Hypothesis.
But first, we must address an annoying little issue of terminology. As already mentioned, a knot theorist says a tangle is ‘framed and oriented’ if it is equipped with a trivialization of its normal bundle, while a homotopy theorist would call such a tangle merely ‘framed’. As we ascend to higher dimensions and codimensions, the homotopy theorist’s terminology becomes more convenient. So, let’s switch now!
Then the Tangle Hypothesis says these three are the same:
There’s a lot of evidence for this, but proving it would first require a precise definition of ‘-tuply monoidal -categories with duals’. So far this has been achieved only for low .
A strong version of the Stabilization Hypothesis says that for any , the fundamental -category of approaches a limit as — in fact, it stabilizes as soon as . We can then take a limit as . Without going into any details yet, it seems that the result is none other than the ‘sphere spectrum’ S, much loved by homotopy theorists. This is the spectrum for ‘framed cobordism theory’ — where a framed cobordism is just a framed -dimensional tangle whose codimension is so big that making it bigger doesn’t change anything.
Such thoughts were surely running through David’s mind when he guessed a pattern relating other kinds of tangle to other stratified spaces and other cobordism theories.
Knowing that the Thom spaces and give spectra and just as the spheres give the sphere spectrum S, and knowing that is the spectrum for oriented cobordism theory and is the spectrum for unoriented cobordism theory, he presumably guessed that these three are the same:
He also guessed that these three are the same:
And, I agree that these are plausible conjectures! They may need some fine-tuing, but they’re on the right track. So, let’s take a look at one of them, for very low .
David also raised another puzzle, but I’m too tired to address that right this instant. It’s about a certain gap in the terminology between knot theorists and homotopy theorists.
Okay, I’ll unmask as a tangle-theorist rather than as a knot-theorist a bit here. n.b.: I think I’ve got the numbering convention down, but I might be wrong. n is the dimension of the tangle itself and n+k is the dimension of the embedding cube, right?
The picture for n=1 comes to mind easily. If k>2, everything’s pretty boring. All crossings are the same since we’ve got “enough room” to move any strands past each other. The same is true for all (n,k) with k>2: you’ve got a bunch of n-manifolds floating around with their boundaries at the top and bottom of an (n+k)-cube.
For (n,k) = (1,1) the picture is almost as boring, but not completely so. There’s no space in the square to cross strands, so we’ve just got a bunch of arcs crossing the square and a bunch of free-floating loops. If we linearize this category and set a loop equal to some value we get the Temperley-Lieb category with that loop value. I’ve been spending a bunch of time on this one lately, so I’ll mention in passing that its representation theory turns out to be deeply connected with the theory of bilinear forms. Basically, this part of the structure of the (1,2) case is what the bracket polynomial teases out for us, though even I don’t really know what the bracket “means” in a topological sense.
Anyhow, there’s some beautiful combinatorial descriptions of the (1,1) case. With no loops we have Catalan((a+b)/2) tangles from a points to b points. Each carves the square into (a+b+2)/2 regions, into which we can put configurations of loops. The difference between here and the case (1,k) for k>2 is that we can nest loops inside each other.
For (n,1) the picture should be similar, but now there are more manifolds than just arcs and loops. We’ve got a bunch of n-manifolds floating around in the (n+1)-cube, but now they’re unknotted not because there’s too much space, but because there’s not enough space for them to pass around each other. There are a bunch of manifolds with boundaries on the top and bottom, and the top and bottom sheets are just codimension-1 tangles one dimension down. Between these are a bunch of (possibly nested) closed n-manifolds.
This picture is more complicated than for n=1, but only because n-manifolds are more complicated. It’s still pretty much just a list, though now there’s a bit of ordering information to keep track of.
The really interesting things all seem to happen at the boundary case, where k=2, just before the stabilization happens. Codimension 1 is too tight, codimension 3 and up is too loose, but codimension 2 is just right.
So back to the tangle hypothesis: what’s so special about 2-monoidal n-categories? To play the game back, 1-monoidal n-categories are ?????, 3-monoidal n-categories are ?????, but 2-monoidal n-categories are ????? (= “just right”).
what’s so special about 2-monoidal n-categories?
Is this right? Stabilization occurs for fixed when , i.e., -monoidal -categories are the ones just before stabilization.
Can we say which out of, say,
2-categories; monoidal 2-categories; braided monoidal 2-categories; weakly involutory monoidal 2-categories; strongly involutory monoidal 2-categories
is the most interesting?
John Armstrong wrote:
So back to the tangle hypothesis: what’s so special about 2-monoidal -categories?
Nothing. I think you’re biased by the case .
Knot theory is boring when there aren’t enough extra dimensions for the knots to get tangled in complicated ways. But, it’s also boring when there are too many extra dimensions — since then we can untie them too easily.
In general, the Tangle Hypothesis says that -tuply monoidal -categories are related to -dimensional knots in a space with extra dimensions. In other words: -dimensional submanifolds of .
For any , as we increase starting from , knot theory will gradually get more and more interesting, and then gradually get more and more boring. By the time , we can untie all -dimensional knots in , so the fun is over.
The concept of “linking number” makes sense only when the game is almost over: when .
Of course when , the game is practically over as soon as it gets started!
But, there’s usually a big range of interesting dimensions. High-dimensional knot theorists know a lot about this. Here are some examples of how complicated things get:
Very impressive! I don’t understand this stuff very well. But consider the case .
Surfaces can be knotted in 3-space: if you hand me a knotted rope, its surface is a knotted torus in 3-space! So, the case is already very interesting — just as interesting as ordinary knot theory.
Surfaces can also be knotted in 4-space; in fact there are lots of fun ways to tie a 2-sphere in knots in 4-space. So, is also interesting, and this is the case I’ve examined in detail.
Spheres can be linked in 5-space; since 2+2+1 = 5, this is the dimension where surfaces have a linking number.
When we get to 6-space, the fun is over: all surfaces can be unknotted.
As David points out, this is all about the Periodic Table of -categories, specialized to -categories with duals.
Knot theory is boring when there aren’t enough extra dimensions for the knots to get tangled in complicated ways. But, it’s also boring when there are too many extra dimensions – since then we can untie them too easily.
Yeah, I’m not sure why I was thinking codimension 3 is enough to untie things in general…
Is the given link for “1048319 non-trivial ways to link two 102-dimensional spheres in 182-space” and the like, broken?
http://www.pims.math.ca/knotplot/links/sphere.html
Fixed.
Wow this is amazing; I must have missed this before. I didn’t know higher dimensional knots were so intricate. Maths will truly never end.
As we ascend to higher dimensions and codimensions, the homotopy theorist’s terminology becomes more convenient.
But if we’re finding it can’t be used to make the 4 distinctions (framed/unframed oriented/unoriented) necessary to treat tangles in 3 dimensions, won’t it prove to be even less convenient in higher dimensions?
Won’t we need at least 4 terms for -manifolds in -space: orientation or not on the manifold; trivialization or not of the normal bundle? And we might want more, such as a vector field, or independent vector fields, on the manifold.
To test out my conjectures about (MO(), *) for our tangle example, we need MO(2), which from this seems to be to do with the space of (, ) a point on plane through the origin in a high-dimensional space, completed by a point at infinity. That point looks like a good choice for the 0-dimensional stratum.
David wrote:
That point looks like a good choice for the 0-dimensional stratum.
I thought so too at first, but now I realize that was wrong. We don’t want a dimension-0 stratum; we want a codimension- stratum, because our tangles have codimension . And, this stratum is not the ‘point at infinity’ in — it’s the ‘points at zero’!
I’ve made a start here. I didn’t get around to actually explaining how the Thom spaces are related to tangles. They look scary at first, but they’re actually very simple and visually intuitive. When we get to the point of seeing this, the correct choice of codimension- stratum should be obvious.
You can already see from what I wrote that we want a codimension- stratum, not a dimension-0 stratum. The special case of -framed tangles, where , is misleading. In this case codimension- and dimension-0 just happen to be the same thing!
I wrote:
As we ascend to higher dimensions and codimensions, the homotopy theorist’s terminology becomes more convenient.
David wrote:
But if we’re finding it can’t be used to make the 4 distinctions (framed / unframed, oriented / unoriented) necessary to treat tangles in 3 dimensions, won’t it prove to be even less convenient in higher dimensions?
Okay, now I’ve had some sleep and I can get back to another hard day’s work of blogging.
Here’s the wise way to do things. Consider a smooth -dimensional tangle in the -cube. Say that it’s -framed if it’s equipped with linearly independent normal vector fields. Say that it’s oriented if it’s oriented as a manifold in its own right — or equivalently, if its normal bundle is oriented.
Knot theorists focus on the case , . Then they say their tangle is
Stable homotopy theorists focus on the stable case, where is arbitrary and . This is the case where making bigger ceases to change anything. In this case they call a tangle a cobordism. They say their cobordism is
Get it? I hope you agree that my new terminology is nicer if we’re trying to understand the full range of cases.
In all these cases, what we’re doing is equipping the normal bundle of our -tangle with some extra structure. So, we’re reducing its structure group from to some smaller group G.
We can painlessly reduce the structure group to the orthogonal group using the god-given inner product on tangent vectors in the cube, so it’s harmless to assume — except in funky cases where our tangle is not even smooth, or we’re equipping its normal bundle with extra stuff. But let’s not talk about those funky cases now.
Given , let’s say an -dimensional tangle in the -cube is G-structured if the structure group of its normal bundle has been reduced to G.
So:
In all these cases — and many others — it’s crucial to introduce the Thom space of our subgroup . We build this by forming the universal -bundle then forming the associated vector bundle then taking the one-point compactification of each fiber to get a sphere bundle and finally collapsing all the points at infinity to a single point , getting with its basepoint .
Clearly this needs some explanation. But for now let me just say what it does for us! That will motivate the explanation.
In nice situations — I may be forgetting some hypotheses — we can get any -structured -tangle in the -cube from a generic map from the -cube to : Our -tangle is just the inverse image where is the codimension- subspace of coming from the zero section of our vector bundle .
And, in really nice situations, we get this:
Generalized Tangle Hypothesis. The -tuply monoidal -category of G-structured -tangles in the -cube is the fundamental -category of .
This is implicit in the work of Thom and Pontryagin — but of course, they didn’t formulate their work using -categories. Also, they focused on the stable case. Even now, nobody knows how to make this beautiful hypothesis precise and proving it. I hope some kid reading this succeeds.
I agree with you, David, that everyone here should try to work out some spaces like and .
But first, maybe people should work out , where is the trivial group! It’s a useful warmup for people who have never seen Thom spaces.
Here I am wide awake in the early hours of a Vancouver Sunday morning. What better than some Café life?
So an initial thought: why limit extra structure to the normal bundle? Why not besides your -framing also specify some independent vector fields on the tangent bundle (with hopes of connecting to that result of Adams about vector spaces on spheres?
What extra insight does this formulation of the Generalized Tangle Hypothesis provide? Does its potential lie mostly in being a launch pad for categorification concerning various embeddings of smooth 2-spaces having structure on their normal 2-bundles. Or can we already see benefits at the 1-level, perhaps prompting us to look for algebraic objects which form categories with the right structure to yield -structured -tangle invariants?
So in the case of -framed tangles with , i.e, trivial structure group, and are contractible. is the pointed -sphere, and the Generalized Tangle Hypothesis works fine.
It would be good to see where the non-orientedness of the tangles of the fundamental -category of comes from. sits inside , the compactified -dimensional bundle over -planes in a high dimensional space (a Grassmanian), as the zero section. In the case , why does a line sweeping through a point on the zero section count the same as one sweeping the other way? Must be linked to the way you can swing a 2-plane about. Yes, I can convince myself of that.
David wrote:
So in the case , i.e, trivial structure group, and are contractible. is the pointed -sphere, and the Generalized Tangle Hypothesis works fine.
Right!
Or, at least it reduces to something very plausible - maybe something like this:
This is consistent with the ordinary Tangle Hypothesis:
and this:
We’ve got quite a stack of ‘hypotheses’ here, but they make intuitive sense once you understand them, they’re consistent with each other, and so far they all hold in cases where we can test them — that is, low enough and . So, it seems worthwhile to go further out on this limb and consider tangles other than framed tangles. So, next let’s take a look at the Generalized Tangle Hypothesis:
in the special case where is itself, rather than the trivial group. Now we’re looking at smooth tangles with the minimum amount of structure on their normal bundle, instead of the maximum.
I’ll do this somewhere else…
We can now answer my question about what completes the correspondence?
Let be the subgroup of with elements and . Then we’re after . Hmm, what’s the universal -bundle for ? It needs to know that there are two bundles over (the boundary of the Moebius strip, and two copies of the circle). So the first homotopy group of had better be . And its connected.
The universal -bundle for what I called above is the map from the infinite-dimensional sphere to , isn’t it? Then one needs to multiply by , and compactify.
What is known about maps between and if there are maps between and ? The sphere spectrum is initial. Is this because is terminal? Is a contravariant functor?
I’m about to catch a plane so I only have time to answer a few of David’s questions.
David wrote:
What is known about maps between MG and MH if there are maps between G and H? The sphere spectrum is initial. Is this because 1 is terminal? Is M a contravariant functor?
The group 1 is not only terminal, it’s also initial!
Like the classifying space functor B, the Thom space functor M is covariant.
Note also that we’ve been working unstably, using a Thom space construction that gives a covariant functor from [subgroups of ] to [topological spaces]. It’s only after you stabilize that Thom spectra get into the game.
So for us, M1 is , not the sphere spectrum. Only after taking a clever limit would we get the sphere spectrum.
Finally, I doubt the sphere spectrum is initial. Spectra are like infinitely categorified, infinitely stabilized versions of abelian groups. Via this analogy the sphere spectrum is analogous to . This is why Joyal called the sphere spectrum “the true integers”. In the category of abelian groups, is neither initial nor terminal - it’s the trivial abelian group that’s both initial and terminal. The significance of lies elsewhere: it’s the unit for the tensor product of abelian groups. Similarly, the sphere spectrum is the unit for the tensor product of spectra.
So I should have said the sphere spectrum is the initial -ring.
I think I understood why this business is about structure on the normal bundle. So might we want to put complex or spin structures on it? It would be good to get at structure on the tangent space too.
David Corfield wrote:
So I should have said the sphere spectrum is the initial -ring.
You’re a quick study. I think the usual term is ‘ ring spectrum’. Just as is the initial ring, we could hope the sphere spectrum is the initial ring spectrum. We can hope some expert like Neil Strickland takes pity on us and tells us if this is actually true.
(Since stuff in homotopy theory tends to hold just ‘up to homotopy’, it took a lot of work for people to find a symmetric monoidal category of spectra in which ring spectra are simply commutative monoid objects - commutative on the nose. This is what symmetric spectra and various other equivalent constructions accomplish, and this led to the birth of Brave New Algebra.)
I think I understood why this MG business is about structure on the normal bundle. So might we want to put complex or spin structures on it?
A complex structure on the normal bundle is a very nice thing; here we are reducing the structure group from to . In the stable limit this gives complex cobordism theory, corresponding to the Thom spectrum .
We can even do quaternionic cobordism theory, corresponding to the Thom spectrum ! People usually call this symplectic cobordism theory, since the quaternionic unitary group is just another real form of the symplectic group - part of some extensive magic relating ‘symplectic’ and ‘quaternionic’.
But, a so-called ‘spin structure’ is not actually structure: it’s stuff! That’s because the spin group is not a subgroup of : there’s a homomorphism
but it’s not 1-1.
This doesn’t stop us; we can still form the Thom spaces and the Thom spectrum , which is the spectrum for spin cobordism theory.
So, when I said the Thom space construction starts with a subgroup of , I was lying: any group equipped with a homomorphism to will do!
It’s just good to keep in mind the distinction between structure and stuff. Whether a map preserves structure is a yes-or-no question; not so for stuff.
It would be good to get at structure on the tangent space too.
That’s a wild and crazy new idea - it could be very interesting, but I have only one thing to say about it right now. John ‘beautiful mind’ Nash proved that given a manifold , we can get any Riemannian metric on from some embedding of in Euclidean , for sufficiently large.
We can hope some expert like Neil Strickland takes pity on us and tells us if this is actually true.
We could also read Jacob Lurie’s A Survey of Elliptic Cohomology (4 lines from bottom of page 19).
There’s also something on -manifolds on p. 37 of Lurie’s survey. But why does he speak of a -structure as a reduction of the structure group of the (stabilized) tangent bundle of the manifold, when we’ve been talking about the normal bundle?
-structure as a reduction of the structure group of the (stabilized) tangent bundle
That’s at least a common use of the term “-structure on a manifold”. It says that the group of holonomies along loops in the manifold is the group .
Much as I regret it, I did not really follow the discussion you are having here with John. Part of the reason is that much of what I saw you saying I only understood half-way.
David wrote:
But why does he speak of a G-structure as a reduction of the structure group of the (stabilized) tangent bundle of the manifold, when we’ve been talking about the normal bundle?
I don’t know, but I would not be shocked if any structure on the tangent bundle can be thought of as some sort of structure on the stable normal bundle.
(Note: the tangent bundle of a submanifold of a hypercube doesn’t depend on the dimension of the hypercube, but the normal bundle does. So, taking the limit as this dimension goes to infinity — so-called stabilization— is something we do to the normal bundle, not the tangent bundle.)
The simplest case is that an orientation on the tangent bundle gives an orientation on the stable normal bundle, and vice versa, using the standard orientation on the tangent space of the hypercube.
But, it seems the ‘vice versa’ doesn’t usually work: I don’t see how to interpret a trivialization of the stable normal bundle (a framing in the homotopy theorist’s sense) as some structure on the tangent bundle.
I’ve always been a bit confused about this stuff, and I guess I still am.
Picking this conversation up after nearly two years, we’d got to the stage of the Generalized Tangle Hypothesis, which was all about -tangles with structure on their normal bundles. In the comment above, you’re convincing me that this covers cases where there’s structure on a stabilised tangent bundle. (The conversation was also carried over to another thread.)
Now, what’s going on in Ayala’s Geometric Cobordism Categories? He claims to be generalising from results of Galatius et al. in The homotopy type of the cobordism category, mentioned in TWF255. Apparently,
[They] studied cobordism categories of manifolds with fiber-wise structures on their tangent bundles. The results of this paper expand upon this in that the structures at hand are geometric and not just tangential.
Does this ‘geometric’ take us beyond structure and stuff on the normal bundle? Is there then a Yet-More-Generalized Tangle Hypothesis?
I would love to understand how even ordinary cohomology fits in with Generalized Tangle Hypothesis thinking. Remember your claim is:
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!
For ordinary cohomology theory with coefficients in , we should be thinking of the spectrum of Eilenberg-MacLane spaces, , see TWF 149. So, is there some kind of class of tangles linked to the fundamental -category of these spaces, presumably one for which cobordisms are restricted?
is the configuration space of integer-valued points on the sphere, or . This is not too far from other configurations spaces we’ve mentioned, such as the fundamental 2-category of .
Hi guys,
I’ve got a question about duality for 2-categories. John Baez and Laurel Langford defined what a “monoidal 2-category with duals” was in HDA IV. Well, they only needed a reasonably strict notion there, but the concept is easy to understand. A monoidal 2-category with duals is a 2-category with duals on all levels : duals for objects, morphisms and 2-morphisms.
Thus every 2-morphism has a dual , every morphism has a dual and every object has a dual . That’s the basic picture.
For our purposes here, we can ignore the tensor product and the duals for objects side of things, so don’t worry about that.
Since this is a long post, for the experts I’ll state my question right up. Can anyone help me understand the equation
Ok, lets explain this. Imagine we have a 2-category where every 2-morphism has a dual , satisfying the obvious axioms like
and compatibility with vertical and horizontal composition:
where and denote vertical and horizontal composition respectively.
You want to be thinking of examples like “2-tangles” at this point, or “2-Hilbert spaces”. Or even “hermitian coherent sheaves” - if there are such a thing.
Ok, lets draw all of this in string diagrams [don’t try upload this post at home!!]. So for every 2-morphism ,
,
we have a dual 2-morphism ,
.
The arrows are there for our next step. Suppose that every morphism in our category has a left adjoint . As usual, we draw these in string diagrams by giving orientations to the edges:
The unit and counit maps are drawn as
and they satisfy the snake diagrams
.
If is a 2-morphism, we can use the left adjoints for and to make a 2-morphism [Ed : some people call this the “star”]:
.
Ok. Standard stuff so far. Now, the -structure on the 2-morphisms allows us to make also into a right adjoint of , with unit and counit .
This means we could also have used the right adjoints to define the “daggers”. Do we get the same answer?
If you draw these out, you’ll see that’s the same question as asking whether
In string diagrams,
.
In the case of 2-tangles, John and Laurel proved that these define the same 2-morphism. In the case of 2Hilb, I am unable to prove it elegantly except by brute force calculation! One has to choose a basis (2-Hilbert spaces are semisimple categories), etc.
Can anyone (John?) explain the significance of this equation, ? In the context of coherent sheaves, it seems that this equation has to do with Serre duality (though I don’t understand this stuff). More precisely, “Serre duality is the failure of this equation to hold” - one has to twist the left or right hand sides first.
Anyhow, in the stuff I do with 2-characters of 2-representations, I need this equation to hold, eg. in 2Hilb. I’d like to understand it more deeply!
It might make sense to post questions like that as a new entry, for two reasons:
1) if they don’t trigger replies, it might be because people who would reply don’t see it here, buried in this long thread.
2) if they do trigger replies, and maybe even many replies, we produce quite an intimidating comment thread here.
Anyone agrees? Bruce doesn’t mind? Then I (or David or John) could turn this into a separate entry.
I stumbled upon another use for dodecahedrons in:
Encoding, decoding, examples, generalizations, conclusions and references are provided.
Octads, Steiner system, Mathieu groups, Leech lattice, hexacode and parity conditions are included in the discussion.
Some addenda to week241:
Someone with the handle “Dileffante” has found another nice example of the dodecahedron in nature - and even in Nature:
While perusing a Nature issue I found this short notice on a paper, and I remembered that in your talk (which I saw online) you mentioned that the dodecahedron was not found in nature. Now I see in “week241” that there are some things dodecahedral after all, but nevertheless, I send this further dodecahedron which was missing there.
Nature commented in issue 7075:
15) The complete Plato, Nature 439 (26 January 2006), 372-373.
According to Plato, the heavenly ether and the classical elements - earth, air, fire and water - were composed of atoms shaped like polyhedra whose faces are identical, regular polygons. Such shapes are now known as the Platonic solids, of which there are five: the tetrahedron, cube, octahedron, icosahedron and dodecahedron. Microscopic clusters of atoms have already been identified with all of these shapes except the last.
Now, researchers led by José Luis Rodríguez-López of the Institute for Scientific and Technological Research of San Luis Potosé in Mexico and Miguel José-Yacamén of the University of Texas, Austin, complete the set. They find that clusters of a gold-palladium alloy about two nanometres across can adopt a dodecahedral shape.
The article is in:
16) Juan Martín Montejano-Carrizales, José Luis Rodríguez-López, Umapada Pal, Mario Miki-Yoshida and Miguel José-Yacamán, The completion of the Platonic atomic polyhedra: the dodecahedron, Small, 2 (2006), 351-355.
Here’s the abstract:
Binary AuPd nanoparticles in the 1-2 nm size range are synthesized. Through HREM imaging, a dodecahedral atomic growth pattern of five fold axis is identified in the round shaped (85%) particles. Our results demonstrate the first experimental evidence of this Platonic atomic solid at this size range of metallic nanoparticles. Stability of such Platonic structures are validated through theoretical calculations.
Either there is some additional value in the construction, or the authors (and Nature editors) were unaware of dodecahedrane.
Dodecahedrane is a molecule built from carbon and hydrogen - a bit different from an “atomic cluster” of the sort discussed here. It’s a matter of taste whether that’s important, but I bet these gold-palladium nanoparticles occur in nature, while dodecahedrane seems to be unstable.
My friend Geoffrey Dixon contributed these pictures of Platonic life forms:
They look a bit like Ernst Haeckel’s pictures from his book Kunstformen der Natur (artforms of nature), like these:
Here’s an interesting error in the original version of “week241”:
I wrote:
Since pyrite is fundamentally a cubic crystal, the pyritohedron is basically made out of little cubic cells […]
It has 12 pentagonal faces, orthogonal to these vectors:
(0,1,2) (0,-1,-2) (0,2,1) (0,-2,-1) (1,0,2) (-1,0,-2) (1,2,0) (-1,-2,0) (2,0,1) (-2,0,-1) (2,1,0) (-2,-1,0)
In fact the 12 vectors are these:
(2,1,0) (2,-1,0) (-2,1,0) (-2,-1,0) (1,0,2) (-1,0,2) (1,0,-2) (-1,0,-2) (0,2,1) (0,2,-1) (0,-2,1) (0,-2,-1)
I gradually realized my original guess made no sense, and then I confirmed this new guess by looking at this webpage recommended by CarlB:
If your webbrowswer can handle Java, go to this webpage and click on “Pyrite no. 7” to see a rotating pyritohedron. Then, while holding your left mouse button down when the cursor is over the picture of the pyritohedron, type “m” to see the vectors listed above. They’re called “Miller indices”.
If you think of these 12 vectors as points in space, they’re the corners of three 2×1 rectangles: a rectangle in the xy plane, a rectangle in the xz plane, and a vector in the yz plane.
These points are also corners of an icosahedron! It’s not a regular icosahedron, though. It’s probably the “pseudoicosahedron” shown here:
Building isometric crystals with unit cells
To get the corners of a regular icosahedron, we need to replace the number 2 by the golden ratio Φ:
(Φ,1,0) (Φ,-1,0) (-Φ,1,0) (-Φ,-1,0) (1,0,Φ) (-1,0,Φ) (1,0,-Φ) (-1,0,-Φ) (0,Φ,1) (0,Φ,-1) (0,-Φ,1) (0,-Φ,-1)
The number 2 thus deserves to be called the “fool’s golden ratio”.
Just as the regular docahedron is dual to the regular icosahedron - the vertices of the regular icosahedron give the normal vectors to the faces of the regular dodecahedron - I bet the pyritohedron is dual to the pseudoicosahedron.
So, we could call the pyritohedron the “fool’s dodecahedron”, and the pseudoicosahedron the “fool’s icosahedron”. Fool’s gold may have fooled the Greeks into inventing the regular dodecahedron, by giving them an example of a fool’s dodecahedron!
As pointed out by Noam Elkies and James Dolan, there is a sequence of wiser and wiser dodecahedra whose faces have normal vectors
(B,A,0) (B,-A,0) (-B,A,0) (-B,-A,0) (A,0,B) (-A,0,B) (A,0,-B) (-A,0,-B) (0,B,A) (0,B,-A) (0,-B,A) (0,-B,-A)
where A and B are the nth and (n+1)st Fibonacci numbers, respectively. As n → ∞, these dodecahedra approach the regular dodecahedron in shape, because the ratio of successive Fibonacci numbers approaches the golden ratio.
When A = 1 and B = 2, we get the fool’s dodecahedron, since only a fool would think 2/1 is the golden ratio.
However, this is not the most foolish of all dodecahedra! The case A = 1 and B = 1 gives the rhombic dodecahedron, which doesn’t even have pentagonal faces:
So, the rhombic dodecahedron deserves to be called the “moron’s dodecahedron” - at least for people who think it’s actually a regular dodecahedron.
But actually, even this dodecahedron isn’t the dumbest. The Fibonacci numbers start with 0:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
So, even more foolish is the case A = 0 and B = 1. Here our 12 vectors reduce to just 6 different ones:
(1,0,0) (1,0,0) (-1,0,0) (-1,-0,0) (0,0,1) (-0,0,1) (0,0,-1) (-0,0,-1) (0,1,0) (0,1,-0) (0,-1,0) (0,-1,-0)
These are normal to the faces of a cube. So, the cube deserves to be called the “half-wit’s dodecahedron”: it doesn’t even have 12 faces, just 6.
Some pyrite crystals are cubes. Half-wit Greeks mistook these for regular dodecahedra.
Moving in the direction of increasing wisdom, we can consider the case A = 2, B = 3. This gives a dodecahedron which is closer to regular than the pyritohedron. And, apparently it exists in nature! It shows up in this list of crystals:
They also call this one a pyritohedron, so presumably some pyrite forms these less foolish crystals! You can compare it with the A = 1, B = 2 case here:
It’s noticeably better!
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.