In week275 of This Week’s Finds, read about progress towards proving the Cobordism Hypothesis.
Here’s a nice ‘generators and relations’ description of the symmetric monoidal bicategory of 2d oriented cobordisms, from Chris Schommer-Pries’ thesis:
This can be translated into a purely algebraic description of this bicategory… and the Cobordism Hypothesis generalizes this idea to arbitrary dimensions!
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Yikes! I don’t know why I made that mistake. I’ve fixed everywhere I can; unfortunately I read your comment after posting This Week’s Finds to a couple of newsgroups. I’ll do my best to correct it there too, but if anyone ever calls you “Christian”, you’ll know who to blame.
Great thesis, by the way!
Hi, this is a very nice generators and relations description of 2d oriented Cobordisms.
Is there a similar description of 3d oriented Cobordisms? In principel this should be implied by the cobordism hypotheses but I don’t knwo how it looks like exactly.
Is there additionally a description of the category with
objects: 1 dimensional manifolds
morphisms: 2 dimensional cobordisms
2-morphisms: 3 dimsensional cobordisms between cobordisms.
There should be something known, because there is this theorem due to Turaev, that a functor from this category to 2Vect (modelled as complex-enriched additive categories) is the same as a modular tensor category.
Dear Thomas,
Chris Schommer-Pries can say things about the 3d case too. It looks important. Hold on to your hats! I for one never thought that something akin to a generators and relations description of 3 manifolds was possible, but apparantly it becomes more and more tractable when one extends down low enough.
Thomas wrote approximately:
Is there additionally a description of the bicategory with
- objects: 1 dimensional manifolds
- morphisms: 2 dimensional cobordisms
- 2-morphisms: 3 dimensional cobordisms between cobordisms.
We’ve discussed this question here and here.
Briefly, there should be a very nice algebraic answer to this question. As you note, the theory of modular tensor categories provides a huge clue. Also, the answer should be a categorification of the category with:
which is well-known to be the free symmetric monoidal category on a Frobenius monoid.
So: as soon as Chris or someone does the hard topological work of giving a generators-and-relations description of the bicategory you mention, the -category theorists should be able to polish it up into something very slick.
But we should start with a similar polishing treatment of Chris’ thesis. He almost showed that the bicategory with
is the ‘free symmetric bimonoidal category on an unframed fully dualizable object’. The main thing he didn’t do is define all the words in the quoted phrase.
Hi John, thanks a lot for the links to the other discussions.
It seems to me, that there are much conjectures around in this topics which are semi-formal or hard to state in formal terms. But as you said, it would be extremely nice to condense the topological statement to a clear categorial statement. This would help people like me, which are ignorant about the topological subtleties, to use and manipulate TQFTs.
So, let me maybe restate my questions:
1) Does anyone exactly know the categorification of the “well-known […] the free symmetric monoidal category on a Frobenius monoid.”. I recently got through the long book of turaev and tried to find out what he exactly classified but didn’t suceed in relation this to higher categories. Dan Freed says for example in “http://www.ma.utexas.edu/users/dafr/MSRI_25.pdf” on page 18 and in some papers,
that a functor from this 1-2-3-Cobordism category is exactly a modular tensor category.
2) Is it possible to say modular tensor category in a diagram language?
3) Is it possible to say fully dualizable object in diagrams?
4) If one of the last two answers is affirmative, doesn’t this imply a generators and relations description of the corresponding categories?
Sorry for such a lot of questions, but maybe someone knows the answers.
Hi Thomas!
I am trying to see what exactly you are asking here.
Is it possible to say modular tensor category in a diagram language?
At least all the structure of a rigid ribbon category has an obvious diagrammatic and even geometrical interpretation in terms of tangles.
The crucial modularity condition says that certain such diagrams read as endomorphisms of the tensor unit define the components of an invertible linear map.
I gather you are looking for a diagrammatic description such that categorifying the notion of MTC would amount to internalizing these diagrams in some higher category?
doesn’t this imply a generators and relations description of the corresponding categories?
What if we looked at the simplest nontrivial example of an MTC, the category of finite dimensional vector spaces. Would you be looking for a generators and relations description of that?
I thought Thomas was talking about a generators and relations description of the concept of a modular tensor category, not of a specific example of one.
Very roughly, the idea is supposed to be that is “the free symmetric monoidal bicategory on blah-di-blah”, while a modular tensor category is “a blah-di-blah in ”, and a “blah-di-blah” is something you can define purely diagrammatically in any symmetric monoidal bicategory.
Then it’ll instantly follow that any modular tensor category gives a symmetric monoidal functor
which is an once extended TQFT. We already know something like this is true; we want to make it pretty.
For more details on the “blah-di-blah” part, see this.
Thomas Nikolaus wrote:
Does anyone exactly know the categorification of the “well-known […] the free symmetric monoidal category on a Frobenius monoid.”
No, alas!
We have a hypothesis, and I’ve been wanting someone to make it precise and prove it for years. But nobody has proved a theorem giving a purely algebraic description of the bicategory I call , for which (ignoring some important subtleties):
That’s why I gave you references to blog entries on the subject: that’s the best publicly available information I know! Jamie Vicary explained the hypothesis:
should be the free symmetric monoidal 2-category on a braided Frobenius pseudomonoid, which has multiplication left- and right-adjoint to its comultiplication, and unit left- and right-adjoint to its counit.
It’ll take a bit of work to make this precise. But the really hard part is proving it. One way is to use Cerf theory, but people say this is tough to learn.
I suppose I can now reveal that for several years I was hoping Aaron Lauda would write a paper making this hypothesis precise and proving it. We worked on it quite a bit, and he wrote two preliminary papers that tackle simpler related problems:
But then he got distracted by his work with Khovanov, which has been extremely successful — and uses some of the same ideas.
I’m now hoping that Chris Schommer–Pries can prove the hypothesized description of — or at least do the hard part, which is using Cerf theory to get a complete set of ‘generators and relations’ for . Chris knows Cerf theory! He’s trying to understand ! And after the hard part is done, it’s a fun purely algebraic problem to make the hypothesis precise and prove it.
Is it possible to say modular tensor category in a diagram language?
Yes, it’s possible, but it hasn’t been done. At least, not in the way I want.
In particular, I believe that a modular tensor category should be a braided Frobenius pseudomonoid in , with multiplication left- and right-adjoint to its comultiplication, and unit left- and right-adjoint to its counit.
Combining this with the hypothesis about , it should then be instantly obvious that given a modular tensor category, we get a symmetric monoidal 2-functor
which is our 3d once-extended TQFT.
By the way, a lot of this stuff should eventually work in arbitrary dimensions, not just 3 dimensions — and it’ll be a consequence of Lurie’s work. But it might be possible, and even useful, to do the 3d case before Lurie gets around to it.
I don’t think it’s modular tensor categories that this should classify! It should only be the braided ones (i.e., the braided fusion *-categories over the complex numbers.)
This is easy to see: just look at the category of representations of , seen as a symmetric pseudomonoid in 2Hilb. This can’t be modular, since it’s symmetric and nontrivial, and it’s certainly pseudo-Frobenius with respect to the adjoint of its multiplication and the adjoint of its unit.
I can understand why you would expect modularity to be built-in, because the modularity condition is important for constructing certain invariants in the ‘standard’ (i.e. Turaev) approach to TQFTs. I don’t know why there’s this mismatch; this would be a fun thing to discuss if anybody has any ideas.
The modularity condition — that the centre of the monoidal category consists of direct sums of the identity object — would be best expressed by some sort of limit notion, probably. But I wouldn’t expect it to be definable just by some equations.
Jamie wrote:
I don’t think it’s modular tensor categories that this should classify!
Okay, good point. But once we can say “ is the free symmetric monoidal bicategory on a blah-di-blah”, we should see that “blah-di-blahs in ” give 3d once-extended TQFTs. So then I’ll be more interested in these “blah-di-blahs in 2Vect” than in modular tensor categories, which will be just a special case.
John said:
we should see that “blah-di-blahs in 2Vect” give 3d once-extended TQFTs … I’ll be more interested in these “blah-di-blahs in 2Vect” than in modular tensor categories
Me too! But they’ll just be braided fusion -categories over the complex numbers, and nobody’s described a way to get 3-manifold invariants out of such a thing. Of course, you can use the conjecture we’re talking about to do it ‘manually’ — and I bet if you do this for a braided fusion -category over that happens to be modular, you won’t get the usual invariants that arise from modular tensor categories!
Jamie wrote:
But they’ll just be braided fusion *-categories over the complex numbers…
I should warn you that I have my doubts about this. What I mean is: if that’s all we get from one of your “blah-di-blahs in ”, it’s possible we’ve left out some important clauses in the definition of “blah-di-blah”.
(Here “blah-di-blah” is supposed to be whatever makes this sentence true: is the free symmetric monoidal bicategory on a blah-di-blah. We have a proposed definition of “blah-di-blah” that’s supposed to make this sentence true, but we might have left out some clauses.)
Of course, you can use the conjecture we’re talking about to do it ‘manually’ — and I bet if you do this for a braided fusion *-category over that happens to be modular, you won’t get the usual invariants that arise from modular tensor categories!
That makes me unhappy. It’s not supposed to work like that.
Do you want to think about this now? I haven’t thought about this stuff for a long time, so it’ll take me a while to get back up to speed.
Let’s see. Say we’ve got a once-extended 3d TQFT
and let
is our “blah-di-blah in .” It’s definitely a braided pseudomonad in — is that all you mean by a ‘braided fusion category’? Anyway, it has a tensor product
and unit
Next, what about the comultiplication and the counit? How do you get ahold of those? Suppose for example that is just a ‘braided fusion *-category’ — how do you use that to get ahold of the comultiplication
and counit
Sorry to go slow here — it’ll take a while to get to the real point at this rate, and you may not have the patience, but I think we can get there. We have to reach the point where the modularity condition becomes important in the usual Reshetikhin–Turaev construction of TQFTs.
Of course, you can use the conjecture we’re talking about … [but] you won’t get the usual invariants that arise from modular tensor categories!
That makes me unhappy. It’s not supposed to work like that.
Well, maybe you’re right. All I know is that the process we’re trying to describe couldn’t care less if our category isn’t modular, but if you apply the Turaev construction, you end up trying to divide by zero. So it’s possible that the invariants are the same, but I don’t know why one would assume that they are. Plus — and I’m sure this is just my ignorance talking — I always thought the definition of the Turaev invariants was kind of ugly! This more purely categorical approach should be much more elegant.
(Disclaimer: my knowledge of Turaev’s method of calculating invariants from modular tensor categories has been gathered from a few afternoons reading his book, so I really know very little about it.)
So: by “braided fusion -category over ” I mean a braided monoidal 2–Hilbert space with duals. (Let’s stick with Hilbert spaces rather than vector spaces, and require our TQFTs to be unitary, otherwise it doesn’t work as nicely.) I reckon you’re right, you do need to specify more than this to describe a functor , but not much more! In fact, I think the extra data you need to specify is just a finite list of nonzero real numbers. This should then define your TQFT uniquely — or at least up to a unitary monoidal equivalence which is unique up to unique monoidal unitary isomorphism! :)
We’ve got the multiplication and the unit . The comultiplication and counit are the adjoints to these, and to witness the fact that these are indeed adjoints, we need s and s for the adjunctions (I’m desperately trying not to overuse the word “unit” here.) It’s in defining these that you use these real numbers.
You then need to check that the pseudo-Frobenius conditions hold: the one that says , and its mirror-image. It’s actually pretty easy to show that if is the multiplication vertex of a monoidal 2-Hilbert space with duals, then it’ll satisfy this, for any choice of adjoint .
Conjecturally, we can build any closed 3-manifold out of the s and s defining our adjunctions, and the Frobenius isomorphisms. For example, take the unit natural transformation . Then . What does Turaev get for ?
“You then need to check that the pseudo-Frobenius conditions hold”
But isn’t this to less just checking this condition. It semms to me, that this just gives a functor from the category:
1-manifolds
2-cobordsims
diffeomorphisms between cobordisms
because there are just the relation we had in 2Cob(2) but now not in Diffeoclasses of cobordisms but as the categorified way. I think this is the way like for example Ulrike Tillmann doest it. She comes to the conclusion that a functor from this categorie to k-linear categories is a semisimple tensorcategory with duals (what she calls semisimple artinian frobenius category).
But take in contrast the category 3Cob(2) with
Object: 1 manifolds
Morphisms: 2 cobordism
2-Morphisms: 3 cobordisms
There should be more 2-morphisms giving rise to a richer strucutre on this category?
Is the difficulty here that you are dealing with a continuum? If you were dealing with finite cell complexes, it seems like things should be pretty trivial, e.g.
Then, if you used the “right” definition of a cell complex (maybe a diamond?), you could take the “continuum limit” (if you wanted to).
I’m sure what I just wrote down was explained 100 years ago. Any references you can point me to?
What is the difficulty with starting finite and constructing a continuum limit?
By the way, even a keyword to send me off to google would be appreciated, e.g. what is that finite thing I described usually called?
I don’t believe anyone has worked out a bicategory of the sort you’re alluding to — even though they should — so I can’t point you anywhere. If you’ll settle for a mere category where
then you might be interested in this:
where however cell complexes are replaced by something even simpler: chain complexes.
Also try the the more expository
But you’ve probably seen this stuff before.
Thanks!
Yes, I have seen and enjoyed Derek’s stuff. It is always good to be reminded though (especially as I age and my already feeble mind gets feebler).
If I ever get some strength, maybe I’ll try to work out that bicategory on the nLab. It’ll do me good.
I’m working on this here:
If it turns into anything worthy enough, I’ll transfer it over to the “main grid”.
Comments and contributions welcome.
I’m surprised people haven’t tried this before! The bicategory you get should certainly be equivalent to the one built from smooth manifolds, and so just as good for TQFT purposes. Maybe it doesn’t make anything fundamentally any easier doing it this way.
My view is that the Witten-Reshethikin invariant is secretly an invariant of 4-manifolds. From the point of view of skein theory you have an algebra associated to a surface, and a vector space associated to a 3-manifold. This is a cobordism invariant and so only depends on the signature. Therefore modular tensor categories should be two levels up from a story about surfaces. The 3-manifold story is Turaev-Viro invariants.
He says he’s using a definition of -categories due to Clark Barwick. Since I haven’t seen anything in writing about this definition,
Hm, but it seems Jacob Lurie gives a pretty detailed description on pages 31-34 of On the classification of topological field theories:
He says that an -category is an -fold complete Segal space (def 2.1.38). This in turn is a rather straightforward generalization of a simplicial space that is a complete Segal space to an -fold simplicial space.
The crucial definitions are on page 33.
Maybe what Jacob Lurie does not spell out in detail is what a symmetric monoidal such gadget is precisely. But given his definition of symmetric monoidal -category it seems there is a more or less obvious definition.
I don’t really know Lurie’s precise formulation of the cobordism hypothesis, much less his proof.
On the other hand one should maybe say that up to the symmetric monoidalness Jacob Lurie’s statement of theorem 2.4.6, page 43 is pretty explicit.
My impression is that the details not mentioned here are of the non-mysterious sort that are straightforwardly filled in (maybe I am wrong, let me know).
In the same vein, even though it is not the full proof, there is a tremendous amount of detail in the 30 pages(! in small print, with small margin) (pages 51 to 85) that Jacob Lurie spends on spelling out the proof.
I am just saying this since your comment looked like it would convey to people who haven’t read the article the idea that Jacob Lurie just presents claims without any substantial attempt to back them up.
It seems that Lurie’s proof uses at least the following two non-trivial ingredients:
a) the Galatius-Madsen-Tillmann-Weiss theorem (thm 2.7.4, page 50) which characterizes the geometric realization in terms of the suspension of the Thom spectrum;
b) Igusas connectivity result which he uses to show that putting “framed Morse functions” on cobrdisms doesn’t change their homotopy type (theorem 3.4.7, page 73)
(It seems that these page and theorem numbers haven’t changed since the Jan 2009 version.)
All in all this is clearly not the full proof, but quite a bit of detail already, certainly enough to see what kind of proof he is up to.
I agree with Urs’ comment, but would add that Jacob is no longer using GMTW but rather it follows as a special case from his proof.
Thanks for the clarifications; I haven’t been following this subject in detail like you two have. I’ll fix up “week275” so it gives a more accurate picture.
… add that Jacob is no longer using GMTW but rather it follows as a special case from his proof.
Well, that’s remarkable. I feel like I’m in a repeat of the old story about Eddington and Ludwik Silberstein and the ‘three people in the world who understood relativity’. No doubt there are more than three people in the world who understand Lurie’s proof, but I think it’s clear it’s no mean feat to do so!
The classification of polytopes becomes boring beginning at dimension 6.
Does it mean that the construction of any kind of polytope is linked to 3-categories, according to the periodic table?
The classification of regular polytopes stabilizes at dimension 5, as you can see from the Wikipedia page you cite or my own webpage on this subject.
I hadn’t known that convex tesselations stabilize at dimension 6.
I see no relation between this stuff and -categories, but who knows? I wouldn’t have guessed a relation between the series of Lie algebras and del Pezzo surfaces, either.
my own webpage on this subject
Are there pictures missing from that webpage? It reads a little weird now.
Actually, I figured it out: You are a victim of the English Wikipedia's ongoing systematic mismanagement of its images. You should be able to fix this by changing the broken image URIs from http://upload.wikimedia.org/wikipedia/en/… to http://upload.wikimedia.org/wikipedia/commons/…. (Or, once you find the images, use your own copies to be safe.)
Thanks, Toby. I’ve fixed that page on regular polytopes, at least for now.
(I should keep my own copies of the pictures, just to be safe, but I may be too tired. It’s 11:15 pm, and I still haven’t had dinner. Hmm, but Lisa says she wants to work 15 more minutes. Okay, so maybe I’ll do it.)
The link is just for regular polytopes.
Yes, sure, I know, I’m sorry. I forgot to type that… And I wish I could fix that.
I’ve noticed more and more references to articles on the arXiv that even years later have not made it into print! Since they are already on the arXiv, it would be trivial to submit them for publication. JHRS even asks for them to be on the arXiv.
Morse Generators!
I suspect that they may be related to Morse Theory cited as one of the reasons for awarding a Fields Medal to Witten.
Could the M in M-theory stand for Morse?
Morse-Smale complexes are also interesting.
I was trying to figure out the categorical periodic table, and an idea came up to me: Although it is table with a pattern, there is no cyclical period , as in the original.
So, the “periodic” in the “periodic table” expression is just there because it sounds cool? Or am I missing something?
We called it the ‘periodic table’ not because it’s periodic (though Bott periodicity should show up somewhere), but because it looks a bit like the periodic table — and more importantly, because like the original periodic table, it lets you guess stuff based on patterns in the entries.
Mendeleev guessed the existence and properties of an element he called eka-silicon by extrapolating from the properties of carbon and silicon. Similarly, we can guess some properties of 3-dimensional surfaces in (3+2)-dimensional space (for example) by looking at the , entry in the periodic table — braided monoidal 3-categories — and extrapolating their properties from what we know of commutative monoids, braided monoidal categories and braided monoidal 2-categories.
This is just one of millions of examples of how we can use the periodic table to guess things about mathematics. And many of these guessed patterns are starting to become theorems.
Yes, I know the story. But given that this is mathematics, I’d like to know what is a “guess”. In the case of the Mendeleev’s table, guessing about the electronic structure, cristal structure, is ok, but for 19th century standards. Every chemical element has lots self interaction, mainly due extremely complicated corrections of the valence shell with the inner ones. But I think mathematical structures shouldn’t have corrective effects like in physics, so I don’t understand the meaning of “guess”… What is “guess” here?
You mean you never guessed the answer to a math problem before? Okay, try one now. Suppose I have two 30-dimensional spheres embedded in 80-dimensional space. Can I always unlink them? The answer is pretty easy to guess using the periodic table.
(In this particular case, the answer was known before the periodic table. But it illustrates the point I’m trying to make.)
Well, you said above:
“we can guess some properties of 3-dimensional surfaces in (3+2)-dimensional space (for example) by looking at the n=3, k=2 entry in the periodic table — braided monoidal 3-categories — and extrapolating their properties from what we know of commutative monoids, braided monoidal categories and braided monoidal 2-categories.”
So, we have two 30-spheres in an (30+50)-dimensional space, so n=30, k=50. Since k is bigger than n, we have a symmetric monoidal 30-category. It says here that “Intuitively this says that switching things twice has no effect”, so I guess it doesn’t matter what we do, we cannot tie them because, otherwise there would be some kind of anti symmetry.
So, yes, we can always unlink them.
Right! Great!
Mind you, historically people didn’t need the periodic table to guess — and prove — that two -spheres are always unlinked in -dimensional space when . In fact, Jim and I used this known fact to guess the existence of the periodic table!
But, it’s the easiest example of how to use the periodic table to guess stuff.
But are there non abelian guesses for the periodic table? Because in that case it seems an existencial proof for something that depended in an abelian relation, but if it were a 30 sphere in less than 60 dimensions, I would be lost…
Another thing, this periodic table also seems to be also aestheticaly related to a Pascal Triangle:
-Each level seems to depend on 2 parameters,
-Each level new level inherits a inherits all properties from the former.
-Each level adds a new property contained in a new term which is equivalento to both binomial adition property and the sum of a new term in the Pascal case.
-Since the pacal behavior is linked to the power of a binomial, the dull behavior could be equivalent to put a 0 after the expanded binomial. Eg.:
(x+1)^2=x^2 + 2*x +1 + 0 + 0 + 0 + 0 +… infinity. Where these 0, would be related to the dull symmetrical monoidal 2 category down to infinity. The same would be valid for relating symmetrical monoidal n-category to an n expanded polinomial with its infinite zeroes.
Maybe there is a way to automaticaly and recursively find the properties of (r,s)-categories, just like the Pascal binomials.
One more thing, the triangle pasca also has geometrical properties.
Quoting from wikipedia:
“Pascal’s triangle can be used as a lookup table for the number of arbitrarily dimensioned elements within a single arbitrarily dimensioned version of a triangle (known as a simplex).”
So, remember when talked about finding the number of regular polytopes? Maybe it has has some relation to that.
Note that these comments are mere aesthetical… I cannot see beyond this.
I do not know about a relation of periodic table to Pascal triangle but there is a clear connection to another wonderful object from combinatorics: Fibonacci numbers.
If you place at the place of (k, n) the number of n-dimensional coherence laws the k-monoidal category should satisfy then this number is F_{k,n} — the generalised Fibonacci number (usual Fibonacci numbers are F_{2,-}) . This can be used to prove stabilisation hypothesis in appropriate setting. John and Jim know it very well, I think.
People can see Jim’s work on Fibonacci numbers and the cohomology of Eilenberg–Mac Lane spaces here and here. Lurking behind here is a lot of information about -tuply monoidal -categories and the periodic table. This set of ideas deserves to be popularized! Clemens Berger has been giving some talks about it recently.
is there a distinction between 2 - stage trees and 2-level trees?
See also
Configuration spaces from combinatorial, topological and categorial perspectives
for my lecture about coherence laws. The periodic table of trees is at the end.
It seems the tinyurl link is not working. Let me see if the link to wikipedia works now
I’ve written about this before - with some egg on my face, I might add, since I hadn’t read properly what Lurie had written, which is no doubt the case again - but darn it, I still feel that my only beef with Lurie’s brilliant paper is that he does not pay enough homage to the concept of “duals”.
Lurie’s paper argues that an extended TQFT is determined by a fully dualizable object in the target higher category — the gismo the TQFT assigns to the point.
My beef is with the phrase “fully dualizable object”. Essentially (see Definition 2.3.21) he defines a fully dualizable object to be one which can be equipped with a duality structure at all levels.
This doesn’t sound right to me… the duality should be an integral part of the structure of the object, and not a property.
Let’s look at extended 2d TQFT’s. The Baez-Dolan hypothesis says that a unitary 2d extended TQFT
should be determined by the 2-Hilbert space it assigns to the point. This means that there is supposed to be information in a 2-Hilbert space — enough information to have a 2d TQFT! And indeed, this was one of the things I tried to point out in my thesis: the information that a 2-Hilbert space carries are(is? help!) precisely the real scalar factors
the inner products of the identity morphisms on the simple objects with themselves. Different numbers? Then you’ve got a different 2-Hilbert space.
Why is this important? Because those numbers are precisely the numbers we need to define an ordinary (i.e. not extended) unitary 2d TQFT
This goes back to the early TQFT days of 1993 from a paper by Durhuus and Jonsson: a unitary 2d TQFT corresponds to a “unitary Frobenius algebra”. Since it’s unitary, it must be semisimple, and so the only information is the “eigenvalues of the handle operator”, which must be real numbers since the theory is unitary.
So it really works! A 2-Hilbert space (the gismo the extended TQFT assigns to a point) gives us precisely the right information to construct the rest of the TQFT, such as the numbers it will assign to closed 2-manifolds.
So the point I’m trying to make is: the information we need to determine a unitary extended 2d TQFT is not a “dualizable object” — which would be just a “2-vector space”, i.e. something of the form , but rather “an object equipped with a duality structure” — a 2-Hilbert space.
In the derived setting, the replacement for “2-Hilbert space” in the sense of the linear category assigned to the point in an extended 2d TQFT should probably be a “Calabi-Yau category” as in the work of Costello. And a Calabi-Yau category isn’t just a category which “is dualizable” — rather, the data of the Serre traces is an integral part of the structure of the category. A 2-Hilbert space is roughly the same thing as a linear category equipped with a trivial Serre functor. A Calabi-Yau category is basically a -category equipped with a trivial Serre functor. (Unfortunately this terminology is bad: a trivial Serre functor is nontrivial in general :-) The real information lies not in the Serre functor itself but in the Serre trace! Heh, lol. In a 2-Hilbert space, the Serre trace is given by taking the inner product with the identity morphism).
I have other issues along the same lines, for instance Definition 2.3.11 about a 2-category having “adjoints for 1-morphisms”. I agree that this definition is along the right lines, but it doesn’t go far enough… there is structure in the duality for 1-morphisms: the right adjoints need to be related to the left adjoints. For instance, in the 2-category Var, whose objects are derived categories of coherent sheaves, whose morphisms are integral kernels, and whose 2-morphisms are natural transformations, this relationship between right adjoints and left adjoints is given by the Serre functor. The same thing holds for duality for 1-morphisms in the 2-category of 2-Hilbert spaces.
Lurie does give a very interesting explanation of Serre functors in Remark 4.2.4. Maybe I am wrong on this… but his treatment makes it seem as if the Serre functor is canonical, whereas I am trying to say it is additional structure. Could someone kindly help me out on this? I might be misguided. It is certainly additional structure in the context of a semisimple linear category. Maybe not in the context of derived categories of coherent sheaves?
The same thing can be seen when you think of a moniodal category as a 2-category with one object. Then having duals for 1-morphisms in a 2-category corresponds to having duals for objects in the monoidal category. And the structure of relating “right duals” to “left duals” is nontrivial and interesting: it’s called a “pivotal structure” and there is still an outstanding conjecture by Etingof, Nikshych and Ostrik that such a structure exists! I’ve written about this before (I, II), so I apologize to regular denizens of the n-category cafe who are surely rolling their eyes at this stage if they bothered to read this far.
Of course Lurie’s paper is amazing. I’m just registering a little stumbling block I have about this issue of duals. I believe his paper already has the appropriate technology to talk about “object with fully dualizable structure” instead of “fully dualizable object”, and perhaps that is morally the way he treats it anyway! So maybe I’m just having a pedantic argument about terminology.
I had intended to reply to previous comments as follows.
Thomas wrote:
Is it possible to say fully dualizable object in diagrams?
As my post indicated above, I belive this is indeed an important task… at least, if you allow me to make the terminology change “fully dualizable object” -> “object equipped with a fully dualizable structure”. Nontrivial too, I might add. All I know is that duals for 1-morphisms in a 2-category seems to involve “Serre functors”. Who knows what data will be involved when we go further up?
Bruce Westbury wrote:
My view is that the Witten-Reshethikin invariant is secretly an invariant of 4-manifolds.
Hmm, interesting. I guess that sort of relates somehow with the Freed-Teleman-Hopkins-Lurie paper where they argue that the Witten-Reshethikin theory (or some kind of -version of it) is really a trivialization of the Crane-Yetter 4d theory, in the sense of a transformation of 4-functors (gulp!). This somehow accounts for the anomaly.
Hi Bruce,
It is indeed true that being fully dualizable is a property and not a structure, but it’s a weaker property than what you have in mind. Fully dualizable objects only define a TFT on framed manifolds; to get rid of the dependence on framings you need to supply homotopy fixed point data, and that’s the extra structure you’re thinking of. For example, to get a TFT on oriented manifolds you need to supply homotopy fixed point data for the natural action of SO(n) on n-dualizable objects. When n = 2 the natural action of SO(2) on 2-dualizable objects is essentially the Serre functor (whose existence is therefore a property, not a structure), so the data needed in this case is precisely a trivialization of the Serre functor (which is a structure).
Yes indeed, if one only wants a framed tqft, then a fully dualizable object is sufficient. But if one wants an oriented tqft, as I was implicitly wanting above, then one needs to work out those homotopy fixed points. Which I agree seems to boil down to the data of a fully dualizable object, plus all the coherence data I was alluding to. So those coherence issues ultimately need to be tackled.
I am hoping that we will eventually jointly move this stuff to the Lab and create a web of useful entries there.
Even though I ought to be doing something else, I gave it a start today:
added to [[cobordism hypothesis]] a section Lurie’s formulation and proof of the cobordism hypothesis.
(this entry and [[generalized tangle hypothesis]] is in need of somebody finding the time to spell out some of the basic ideas more coherently, like the difference between cobordism and tangle hypothesis, etc.)
would it be okay with you, John, if we copy-and-pasted material from TWFs into Lab entries, for a start? Which ones would be the relevant ones here?
created [[2-category of 2-dimensional cobordisms]] for discussion of Chris Schommer-Pries’s work on this for
created [[QFT with defects]] to host, among other things, Chris’s ground-breaking description of defect TQFts in terms of transformations between QFT -functors.
Nothing of this is even close to being satisfactory. But I already bought more Futures than I can afford to write even that little bit. If you (anyone of you) have five minutes, add or polish a little bit and eventually it’ll converge to something more satisfactory.
It’s fine with me if people take stuff from This Week’s Finds and stick it in the Lab. I don’t have the energy right now myself.
I imagine my first big Lab projects will be 1) to write about the algebraic geometry project Jim Dolan and I are doing (we’ve barely begun to explain it) and 2) to take some papers like the ‘prehistory of -categorical physics’ and maybe the ‘Rosetta stone’ and put material from them into the Lab.
I hope project 2) will be part of a bigger project, to write some books about categories and -categories.
But there’s also a smaller, easier, project: to take the preface and bibliography and table of contents of my book with Peter May and put it on the Lab. I’ll do this as soon as I know the title of the book!
It’s fine with me if people take stuff from This Week’s Finds and stick it in the nLab. I don’t have the energy right now myself.
Okay, thanks for letting me know.
I imagine my first big nLab projects will be
I am hoping that apart from plans on big projects there, you’ll find time, frome time to time, to add little snippets of insights that you post here anyway.
For: the Lab is the project. :-)
But there’s also a smaller, easier, project: to take the preface and bibliography and table of contents of my book with Peter May and put it on the nLab. I’ll do this as soon as I know the title of the book!
Since recently, thanks to Jacques Distler, we have the functionality that allows us to change Lab page titles.
So I went ahead and created [[Approaching Higher Category Theory]] and poured your blog entry material into that.
We can still change the entry title should you settle on a different book title.
Re: This Week’s Finds in Mathematical Physics (Week 275)
Hi John,
Just a small typo. My name is “Chris” or “Christopher”. Not “Christian”.
I just wanted to set the record straight.