Categorifying the Dijkgraaf-Witten model

August 24, 2006

Categorifying the Dijkgraaf-Witten model

Posted by John Baez

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The Dijkgraaf-Witten model is a simple sort of topological quantum field theory where the only field is a gauge field, and the gauge group G is finite:

Robbert Dijkgraaf and Edward Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990), 393.

Martins and Porter have a new paper on how to categorify this model, replacing the group G by a categorical group, or “2-group”.

I wrote about this stuff eleven years ago in week54 of This Week’s Finds - so if you want an elementary intro to these ideas, start there.

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In the simplest version of the Dijkgraaf-Witten model, the path integral is just an integral over the moduli space of principal G-bundles, using the simplest possible measure on that space. It’s nice to formulate this theory on a triangulated manifold, where we assign a group element to each edge, and require that these group elements multiply to 1 around each triangle. This formulation makes it clear that we can also “twist” the Dijkgraaf-Witten model, which in n dimensions amounts to changing the action by any element of the nth cohomology of the gauge group.

I explained this stuff in the Winter 2005 Quantum Gravity Seminar. I also discussed the generalization where G is a Lie group - this gives 3d quantum gravity. And, I explained how the path integral in such theories can be rewritten as a sum over spin foams.

For people who like higher gauge theory, it’s tempting to “categorify” the Dijkgraaf-Witten model by replacing the gauge group by a 2-group. There’s already been some work on this, going back to a paper by Yetter:

David Yetter, TQFT’s from homotopy 2-types, Journal of Knot Theory and its Ramifications 2 (1993), 113-123.

and you can see more recent papers online:

Hendryk Pfeiffer, Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics.
Timothy Porter and Vladimir Turaev, Formal homotopy quantum field theories, I.
Timothy Porter and Vladimir Turaev, Formal homotopy quantum field theories, II.

Anyway, here’s a new one!

João Faria Martins and Timothy Porter, On Yetter’s invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups.

A categorical group or “crossed module” is the same as a strict 2-group. Note that this paper “twists” the categorified Dijkgraaf-Witten model using an element of the nth cohomology of the classifying space of our 2-group. So, it’s using the obvious generalization of group cohomology to 2-groups.

It would be nice to generalize this work from finite (or discrete) 2-groups to Lie 2-groups, and that’s sort of what I’m doing with Freidel and Baratin - we’re focusing on the case of the Poincaré 2-group.

Note that in all these theories, the connection or 2-connection is flat - it has to be when G is discrete, but it still is in these theories when we generalize G to a Lie group or Lie 2-group. Flat connections are a wee bit boring in physics, but good for getting TQFTs. Urs is busy working on more exciting theories that involve 2-connections or 3-connections with nontrivial curvature.

One can try to go further, replacing the group in the Dijkgraaf-Witten model by an $n$ -group, but the Homotopy Hypothesis conjectures that such things are the same as pointed connected homotopy $n$ -types, so at this point it’s more efficient to use simplicial techniques rather than $n$ -categories to make the ideas precise. For work along these lines, try:

Timothy Porter, Interpretations of Yetter’s notion of G-coloring: simplicial fibre bundles and non-abelian cohomology.
Timothy Porter, TQFTs from homotopy $n$ -types.

Posted at August 24, 2006 7:16 AM UTC

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Re: Categorifying the Dijkgraaf-Witten model

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Thanks, John, for inserting into my entry on Martins & Porter a link back to this entry here, which I forgot to point to.

At one point I want to get back and explain and emphasize again how really cool things happen when we replace here the finite group $G$ by our strict Fréchet Lie 2-group $\mathrm{tar} = \mathbf{B}(\Omega G \to P G)$ without the central extension (so this is still equivalent to $\mathbf{B}G$ ) and then realize that putting in the central extension to the strict string 2-group $(\hat \Omega G \to P G)$ amounts to introducing the “twist” here in the guise of a weak 2-functor $\mathrm{tra} : \mathbf{B}(\Omega G \to P G) \to \mathbf{B}^2 U(1) \,.$

Transgressing this setup to loop space produces the Lie-analog of Simon Willerton’s baby FHT theorem as I once described in the entry with the funny title 2-Monoid of Observables on String-G.

In particular, the representations of the loop groupoid that we transgress to are twisted equivariant vector bundles on $G$ and we make contact with non-baby FHT.

Back then I was stopped by the fact that I wasn’t sure if I could handle the smooth structure on the quotient $\Lambda(\Omega G \to P G) := \mathrm{Hom}(\mathbf{B}\mathbb{Z}, \mathbf{B}(\Omega G \to P G))/\sim \,,$ where “ $/\sim$ ” is supposed to denote the operation of identifying isomorphic 1-morphisms.

But now I think I actually can handle this, essentially by the standard smooth space Yoga.

I should try to find the time to do that. There is something interesting lurking here, which will connect all this combinatorial and homotopy theoretic reasoning to the real thing.

Posted by: Urs Schreiber on January 7, 2008 7:46 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

A twist is a twist is a…
unfortunately not
only slightly better than weak

or to paraphrase A:
Give me a twist to lean on, and I will move the world

Posted by: jim stasheff on January 7, 2008 11:26 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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A twist is a twist is a… unfortunately not only slightly better than weak

I know exactly what you mean and I am all in favor of not using the term “twist”.

In fact, if you read what I wrote above and at the link given, you’ll see that I say:

hey, what these guys call the “twist” in Dijkgraaf-Witten theory is really to be thought of as a Line $n$ -bundle over the target space $\mathbf{B} G$ of the theory.

This point of view is rather common in the Lie-analog of Dijkgraaf-Witten, namely Chern-Simons theory, where everybody is familiar with the idea that the “twist” there is really the canonical 2-gerbe over $\mathbf{B} G$ . But the point is: if we regard higher $n$ -bundles in terms of their transport $n$ -functors, then the same kind of statement goes through also for the discrete Dijkgraaf-Witten theory.

And for its categorifications.

Posted by: Urs Schreiber on January 10, 2008 10:56 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Perhaps I can ask a question about this, which I haven’t been able to answer by browsing through the references provided. I thought that conventional finite-group Dijkgraaf-Witten only gave you 2+1 dimensional TQFTs. What about this categorified case? Does that give you 3+1 dimensional TQFTs, or doesn’t it work like that? I suspect it doesn’t, as John didn’t mention any dimensionality in his original post.

Posted by: Jamie Vicary on May 21, 2008 8:22 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Jamie,

When Joao and I were working on that paper we did think about this. It did not seem clear. In my earlier stuff generalising Yetter’s construction there was no restriction on the dimensionality in principle. I tried last year to see what the connection between the two approaches was but got snarled up in some related things (basically looking at the Puppe exact sequences of sheaves of simplicial groups as in Breen’s bitorsors paper as it seemed that understanding that might give the key to twisting in more generality.)

I then spent time translating something relevant from Flemish (very slow going) and your query suggests that I look at back at our ideas again!!! Any helpful ideas would be much appreciated. (If Joao sees this blog, he might make some comment here.)

In other threads I have hinted at the need to look at several of these HQFT/ETQFT notions as they change along `change of groups’ i.e. if we have $G$ -based HQFT as in Turaev and then say an epimorphism from $G$ to $H$ there will be morphisms both ways. (This is consistent with thinking of HQFTs etc as representations.) The interesting case where $G$ is a central extension of $H$ is relevant to what Urs was trying to do I think.

Posted by: Tim Porter on May 22, 2008 9:23 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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DW theory, while you could consider it in any dimension, is 3-dimensional because the “twist” is a 3-cocycle: there is a 3-bundle over “target space” (where “target space” here is just a point with $G$ -worth of automorphisms).

The Yetter model allows twists of one dimension higher. Just think of the special case there the 2-group involved is just the shifted version of an abelian group. So it will admit 4-bundles over its “target space” and hence be inherently 4-dimensional.

But now I have a question: isn’t it precisely this 4-dimensionality which made people conjecture that this is related to 4d quantum gravity?

Posted by: Urs Schreiber on May 22, 2008 9:58 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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I’m not sure what you mean by the Yetter model — his version of the Dijkgraaf–Witten model with a finite 2-group replacing the finite group $G$ ?

That’s an interesting theory, but it’s the Crane–Yetter model that has tantalizing relations to 4d quantum gravity. You can loosely think of this model as a 4d version of the Dijkgraaf–Witten model with a quantum group replacing the finite group $G$ . Like the Dijkgraaf–Witten model, it’s a topological quantum field theory.

A while back I conjectured that the Crane–Yetter model is the quantum version of 4d $BF$ theory with cosmological term, with action:

$S = \int tr(B \wedge F + \lambda B \wedge B)$

Here $\lambda$ is related to $q$ . As $\lambda \to 0$ we have $q \to 1$ , and the Crane–Yetter model reduces to the Ooguri model (the 4d version of the Dijkgraaf–Witten model with a Lie group replacing the finite group $G$ ). This conjecture seems to be widely accepted, but I haven’t seen a completely satisfactory argument for it.

Posted by: John Baez on May 24, 2008 1:37 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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I’m not sure what you mean by the Yetter model — his version of the Dijkgraaf–Witten model with a finite 2-group replacing the finite group $G$ ?

Yes, that’s what I mean.

It seems I had thought that the Crane-Yetter model is essentially the same. Sorry for the confusion.

Posted by: Urs Schreiber on May 25, 2008 3:52 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Jamie wrote:

I thought that conventional finite-group Dijkgraaf-Witten only gave you 2+1 dimensional TQFTs.

No, the idea behind the Dijkgraaf-Witten model works in any dimension. The idea is to build a field theory from a finite group $G$ where a ‘field’ on a closed manifold $M$ is a $G$ -bundle over $M$ , and the action is $0$ . So, the partition function of a closed manifold $M$ just counts isomorphism classes of $G$ -bundles over $M$ , where the counting is done in a suitable ‘groupoid cardinality’ sense: a $G$ -bundle with lots of automorphisms counts less. The details are a bit subtler for manifolds with boundary, but not in a bad way.

There are lots of ways to get your hands on the Dijkgraaf–Witten model, but many of these seem to have only been worked out in low dimensions.

For example, in weeks 5–10 of the Fall 2004 quantum gravity seminar I showed how Fukuma, Hosono and Kawai built a 2d TQFT from any finite-dimensional semisimple algebra. It’s a state model idea where you triangulate your manifold and label the edges by basis vectors in your algebra. If you apply this idea to the group algebra $\mathbb{C}[G]$ you get the 2d Dijkgraaf–Witten model, as I explained in week 10 of that course.

Then in the Winter of 2005 I sketched how to get 3d TQFTs from ‘semisimple 2-algebras’. In particular, this construction gives the 3d Dijkgraaf–Witten model when we use $Vect[G]$ as our semisimple 2-algebra.

I see no reason in principle why this idea can’t work in higher dimensions too, though it’s not the most efficient way to construct the Dijkgraaf–Witten model in arbitrary dimensions. My student Jeffrey Morton constructed the Dijkgraaf–Witten model as a ‘once extended TQFT’ in arbitrary dimensions in his thesis. By a ‘once extended TQFT’, I mean that he constructed a bicategory $nCob_2$ of:

closed $(n-2)$ -manifolds,
$(n-1)$ -dimensional cobordisms between these, and
$n$ -dimensional cobordisms between those,

and showed that the Dijkgraaf–Witten model gives a functor

$nCob_2 \to 2Vect$

All this so far is for the ‘untwisted’ Dijkgraaf–Witten model. We can also choose a slightly more interesting action for the $n$ -dimensional model, which is called ‘twisting’ it by an $n$ -cocycle. There are lots of ways of thinking about this; I presented a fun $n$ -categorical way in weeks 8–10 of the Winter 2005 seminar.

Briefly: a 2-cocycle modifies the multiplication in $\mathbb{C}[G]$ , a 3-cocycles modifies the associator in $Vect[G]$ , etc… You get some very pretty pictures involving Pachner moves this way. But again, this is not the most efficient approach.

Posted by: John Baez on May 23, 2008 11:16 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Thanks, everybody, for your useful replies. It’s a bit frustrating that there isn’t a better universal description of TQFTs. We know that in 1+1 dimensions they’re commutative $\dagger$ -Frobenius algebras in Hilb, and I know what all of these are. But I get the impression from the literature that even for 2+1 dimensions, there’s no equally useful description of what a TQFT is. So we can come up with clever ways to generate them — like Dijkgraaf-Witten models — but have no reason to believe we’re obtaining all the 2+1 TQFTs in this way.

It seems obvious to wonder whether 3Cob ${}_2$ is the free strongly symmetric monoidal 2-category on a symmetric $\dagger$ -Frobenius pseudomonoid… but I’m sure this isn’t true, or somebody would have proved it and I would have read it somewhere!

Posted by: Jamie Vicary on May 24, 2008 9:21 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Jamie Vicary wrote:

It seems obvious to wonder whether $3Cob_2$ is the free strongly symmetric monoidal 2-category on a symmetric $\dagger$ -Frobenius pseudomonoid… but I’m sure this isn’t true, or somebody would have proved it and I would have read it somewhere!

Huh? Are you serious? There aren’t many people who could prove a theorem like this, much less understand all the words you just wrote!

In fact, for a long time I’ve believed a result like this was true. For a while I was trying to get Aaron Lauda to prove it. In August 2005 he got as far as giving a similar description of a simpler 2-category called $2Thick$ , where the 2-morphisms are things like this:

A bit more precisely, $2Thick$ has

collections of open strings embedded in the line as objects,
open string worldsheets embedded in the plane as morphisms,
isotopy classes of 3-dimensional manifolds with corners defined by diffeomorphisms of these open string worldsheets as 2-morphisms.

He showed that $2Thick$ is the free monoidal 2-category on a Frobenius pseudomonoid. But then he got involved in Khovanov homology and stopped working on this project.

So, I’m afraid it’s up to you to give a nice category-theoretic description of $3Cob_2$ .

Posted by: John Baez on May 25, 2008 2:36 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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This is wonderful! I love the pictures, especially the one of the Frobeniusator. Not sure how I’ve managed to miss this paper for so long… this is really fantastic stuff.

I’m a bit surprised about something, though. If we’re looking for an equivalence with 3Cob ${}_2$ , wouldn’t we expect the objects to be 2-dimensional topological spaces? He has them as open strings, which are 1-dimensional topological spaces.

Posted by: Jamie Vicary on May 25, 2008 10:20 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Hang on, that’s not right. The 2-category 3Cob ${}_2$ presumably has objects which are 2-dimensional topological spaces which are compact without boundary. For the purposes of Aaron’s paper we would probably be more interested in an ‘open’ version of this, that has objects as 2-dimensional topological topological spaces that are compact with boundary.

Posted by: Jamie Vicary on May 25, 2008 10:30 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Jamie wrote:

I’m a bit surprised about something, though. If we’re looking for an equivalence with $3Cob_2$ , wouldn’t we expect the objects to be 2-dimensional topological spaces?

Yikes! We’d better agree on how to count before we keep talking.

For me, $3Cob_2$ is the 2-category where:

objects are compact oriented 1-manifolds,
morphisms are cobordisms between these — thus 2-manifolds with boundary,
2-morphisms are cobordisms between those — thus 3-manifolds with corners.

This 2-category has been intensively studied in work on extended TQFTs, since it’s implicit in how people get their hands on certain 3d TQFTs, especially the Witten–Reshetikhin–Turaev theory. So, it deserves a purely algebraic description!

More generally, I define $n Cob_k$ to the $k$ -category where the top-dimensional morphisms are $n$ -dimensional cobordisms between cobordisms between… compact $(n-k)$ -manifolds.

So, the category $n Cob_1$ is just what folks usually call $n Cob$ .

Now, a famous fact is that $2 Cob_1$ is the free symmetric monoidal category on a commutative Frobenius monoid. This Frobenius monoid is just the circle.

And, it’s easy to see that when we decategorify $3 Cob_2$ we get $2 Cob_1$ .

So, we naively expect $3 Cob_2$ to be something like the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid — namely, the circle.

This is close to true, but not quite: we need some extra bells and whistles on our Frobenius pseudomonoid to deal with the extra layer of duality in our 2-category! Drawing a bunch of pictures would make this clear…

Anyway: before tackling this problem — which he never got around to — Aaron Lauda tried something easier.

There’s a category $1 Thick$ with:

collections of open strings embedded in the line as objects,
isotopy classes of open string worldsheets embedded in the plane as morphisms.

And, it’s easy to see that this is the free monoidal category on a Frobenius monoid. This Frobenius monoid is just the closed interval.

So, Aaron categorified this result. He considered the 2-category $2 Thick$ with

collections of open strings embedded in the line as objects,
open string worldsheets embedded in the plane as morphisms,
isotopy classes of 3-dimensional manifolds with corners defined by diffeomorphisms of these open string worldsheets as 2-morphisms.

And, Aaron showed this is the free monoidal 2-category on a Frobenius pseudomonoid. This Frobenius pseudomonoid is just the closed interval.

But, I wish some young and energetic person would give a purely algebraic characterization of $3 Cob_2$ in terms of Frobenius pseudomonoids!

By the way, the following paper is extremely helpful for anyone interested $3 Cob_2$ :

Ulrike Tillman, Discrete models for the category of Riemann surfaces, Math. Proc. Cambridge Philos. Soc. 121 (1997), 39–49.

Here’s part of the description in Math Reviews, written by my old schoolmate Kathryn Hess:

Segal’s category of Riemann surfaces, $M$ , in which the objects are oriented 1-manifolds up to homotopy and the morphisms are Riemann surfaces, is the basis of any conformal field theory [G. B. Segal, in Differential Geometrical Methods in Theoretical Physics (Como, 1987), 165–171, Kluwer, Dordrecht, 1988]. It can be difficult to work in $M$ , however, due to the analytic description of its morphisms. It is therefore of interest to find a discrete, algebraic category that models $M$ in some appropriate sense.

The goal of the research presented in this article is to define and study a particular discrete, algebraic model of $M$ , thus establishing a theoretical framework for results the author has since proved about 3-manifold invariants, as well as for the proof that the stable mapping class group is an infinite loop space after group completion.

The author begins with a concise and clearly presented overview of the theory of 2-categories. She succeeds in defining concepts as complex as that of a symmetric, strict monoidal 2-category without overwhelming the reader with notation and axioms. She also reviews the construction and properties of the classifying space of a category and its specialization to the classifying category of a 2-category.

In the second part of the article, the author defines a strict 2-category, $S_D$ , from which she derives her model of $M$ . In $S_D$ , (0) the 0-morphisms are in one-to-one correspondence with the natural numbers; (1) a 1-morphism from $n$ to $m$ consists essentially of a smooth, oriented cobordism whose boundary is composed of $n$ circles on the “bottom” and $m$ circles on the “top”; (2) the set of 2-morphisms between two cobordisms is the group of all orientation-preserving diffeomorphisms between them (which is empty if they are not diffeomorphic). The discrete category $S_\Gamma$ that models $M$ is then the quotient of $S_D$ that identifies two diffeomorphisms if they belong to the same path component in the space of all diffeomorphisms.

So, you see $S_D$ doesn’t have all the 2-morphisms in $3 Cob_2$ : only the cobordisms that come from diffeomorphisms! In this respect it resembles $2Thick$ , where the 2-morphisms come from diffeomorphisms of the square:

But, it’s a good step towards $3Cob_2$ .

By the way, the higher category theory in this paper is a bit sketchy: personally I’d feel more confident in the results after being ‘overwhelmed with notation and axioms’.

Posted by: John Baez on May 26, 2008 1:04 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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John said:

But, I wish some young and energetic person would give a purely algebraic characterization of $3\mathrm{Cob}_2$ in terms of Frobenius pseudomonoids!

OK, so first you’d get an algebraic characterisation of the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid. But to then prove an equivalence to $3\mathrm{Cob}_2$ … wouldn’t that be rather a stupendous result, given all the problems there are with classifying 3D manifolds?

Posted by: Jamie Vicary on May 28, 2008 11:16 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Jamie wrote:

OK, so first you’d get an algebraic characterisation of the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid.

That is an algebraic characterization, and that’s really all we need on the algebraic end of things. The work is showing that $3Cob_2$ admits an algebraic characterization sort of like this.

But to then prove an equivalence to $3Cob_2$ …

By the way, I never claimined that $3Cob_2$ is the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid! We need a symmetric Frobenius pseudomonoid with a few extra bells and whistles, which I could describe if you bought me enough coffee. I don’t want people trying to prove the wrong conjecture.

…wouldn’t that be rather a stupendous result, given all the problems there are with classifying 3D manifolds?

No, you wouldn’t need to classify 3-manifolds to prove this sort of result. An algebraic description of $3Cob_2$ of the sort we’re contemplating amounts to a ‘generators and relations’ description of 3d manifolds with corners in terms of basic building blocks. There are already plenty of results like this in the literature, obtained using Morse theory and Cerf theory; you can see some nice ones in the book by Kerler and Lyubashenko. They just haven’t been massaged into the right form. Doing so won’t be trivial, but I think it’s within reach.

The point is this: knowing a bunch of generators and relations for an algebraic structure doesn’t mean you can tell when two elements of this algebraic structure are equal. Indeed, there’s a finitely presented group where it’s algorithmically undecidable when two elements of this group are equal!

In fact, the Poincaré conjecture was long known to be equivalent to various purely algebraic questions about group presentations… but these turned out not to be helpful for people trying to prove that conjecture.

Here’s an algebraic statement equivalent to the Poincaré conjecture. It’s not very relevant to what we’re talking about, but it’s awfully cute. I just found it in a talk by Dale Rolfsen.

Say two group homomorphisms $h_1, h_2 : G \to H$ equivalent if there is an automorphism $\alpha : G \to G$ such that $h_1 \alpha = h_2$ .

Let $G$ be the group with generators $x_1,y_1, \dots, x_g, y_g$ and one relation:

$[x_1, y_1] \cdots [x_g, y_g] = 1$

where $[x,y] = x y x^{-1} y^{-1}$ . Topologists will recognize this group as the fundamental group of the $g$ -holed torus.

Let $F_1$ be the free group generated by $x_1, \dots, x_g$ and let $F_2$ be the free group generated by $y_1, \dots, y_g$ .

There’s an obvious surjective homomorphism

$\phi : G \to F_1 \times F_2$

since in $F_1 \times F_2$ all the $x_i$ ’s commute with all the $y_i$ ’s, which implies the relation shown above.

Theorem (Hempel, Perelman): Up to equivalence, $\phi$ is the only surjective homomorphism from $G$ to $F_1 \times F_2$ .

Hempel showed this fact was equivalent to the Poincaré conjecture. Later, Perelman proved the Poincaré conjecture! So, it’s an example of an elementary-sounding result about group theory whose proof currently involves lots of hard analysis and Riemannian geometry.

Posted by: John Baez on May 29, 2008 6:24 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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John said:

That is an algebraic characterization

I was implying was that it’s not immediately obvious to me what a symmetric Frobenius pseudomonoid should actually be. For example, I think there are some choices to make about how coherent we want things to be. But I know that an answer to this is given in that impressive-looking book of Kerler and Lyubashenko that you mention.

By the way, I never claimed that ${}3\mathrm{Cob}_2$ is the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid! We need a symmetric Frobenius pseudomonoid with a few extra bells and whistles, which I could describe if you bought me enough coffee.

Coffee is fast becoming the currency of choice for trading mathematical favours. I suppose we’re really after the free monoidal stable weak 2-category with duals on one object, right? I know what all those words mean, but I’m sure it would take me a good while to turn them into an algebraic presentation. Of course, it would be good for me!

An algebraic description of ${}3\mathrm{Cob}_2$ of the sort we’re contemplating amounts to a ‘generators and relations’ description of 3d manifolds with corners in terms of basic building blocks. There are already plenty of results like this in the literature

Oh, right — that’s better than I thought, then. Thanks for making this clear. The example you give using the Poincaré theorem is great!

Posted by: Jamie Vicary on May 30, 2008 11:02 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

The example you give using the Poincaré theorem is great!

I agree. It rattled my bones when I read it!

Posted by: Bruce Bartlett on May 30, 2008 11:05 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Hempel showed this fact was equivalent to the Poincaré conjecture.

Do you have a way of seeing heuristically why this might be true?

I suppose $G$ here I should think of as a model for the fundamental groupoid $\Pi_1(S_g)$ , of the genus $g$ surface $S_g$ . So the statement is somehow about maps out of genus $g$ surfaces.

Posted by: Urs Schreiber on May 30, 2008 1:20 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Urs wrote:

Do you have a way of seeing heuristically why this might be true?

Unfortunately I don’t think Dale Rolfsen’s talk says how Hempel proved the equivalence of this algebraic result and the Poincaré conjecture. It might be discussed in Hempel’s book on 3-manifolds, which has a chapter on the Poincaré conjecture. I’m too lazy to check.

Naively I’d guess it has something to do with Heegaard splittings. You can always split a compact oriented 3-manifold into two solid handlebodies glued together along a $g$ -holed torus. You can do it lots of ways, and there are ‘Heegard moves’ that take us between any two such splittings.

So, here’s my wild guess. Maybe any Heegard splitting of a homotopy 3-sphere can be coaxed to give a surjection $\phi: G \to F \times F$ . I don’t see how. But maybe when such a surjection is equivalent to the obvious one, our homotopy 3-sphere is homeomorphic to the usual 3-sphere. That would give Hempel’s result.

Posted by: John Baez on May 30, 2008 8:32 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Just 30 minutes ago I talked about this with Soren Galatius over coffee. He also pointed me to Heegard splittings.

One thought I had:

the homeomorphism with which we glue the two handlebodies is relevant only up to isotopy. This probably corresponds to the notion, that you mentioned, of equivalence of two functors out of $\Pi_1(S)$ , for $S$ the $g$ -holed torus if they differ by precomposition with an automorphism of $\Pi_1(S)$ .

Posted by: Urs Schreiber on May 30, 2008 8:51 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Jamie wrote:

I was implying was that it’s not immediately obvious to me what a symmetric Frobenius pseudomonoid should actually be.

Oh, okay. Right, this is part of the problem: figuring out the correct definitions.

For example, I think there are some choices to make about how coherent we want things to be.

Right. In particular, while we more or less know what a ‘symmetric pseudomonoid’ in a 2-category is (just copy the definition of symmetric monoidal category, and quickly make up your mind whether you want associators and unitors or not), and we more or less know what a ‘Frobenius pseudomonoid’ is, we need scan for coherence laws involving both the symmetry and the Frobeniator.

But I know that an answer to this is given in that impressive-looking book of Kerler and Lyubashenko that you mention.

The warning light is blinking on my feeble American-made irony detector. I’m quite sure the answer is not given in this book. However, it’s full of useful stuff.

I suppose we’re really after the free monoidal stable weak 2-category with duals on one object, right?

No, $3Cob_3$ should be the free stable monoidal weak 3-category with duals on one object. The unit object $1 \in 3Cob_3$ should be the empty 0-manifold. And we’re looking to understand

$3Cob_2 = hom(1,1)$

which should be roughly the 2-category of:

compact oriented 1-manifolds,
cobordisms between these (which are 2-manifolds with boundary), and
cobordisms between those (which are 3-manifolds with corners).

Every object in this 2-category should be a tensor product of copies of the circle. So, we focus on the circle, which should be a categorified version of a commutative Frobenius object — with extra bells and whistles. And, we hope that $3Cob_2$ is the free symmetric monoidal 2-category on this gadget.

All this is motivated by the story one dimension down, so let me review that:

$2Cob_2$ should be the free stable monoidal weak 2-category with duals on one object. The unit object $1 \in 2Cob_2$ should be the empty 0-manifold. And now we’re looking to understand

$2Cob_1 = hom(1,1)$

which is the 1-category of:

compact oriented 1-manifolds,
cobordisms between these (which are 2-manifolds with boundary).

Every object in this category is a tensor product of copies of the circle. So, we focus on the circle, which is a commutative Frobenius object. And, we discover to our delight that $2Cob_1$ is the free symmetric monoidal category on a commutative Frobenius object.

Even in this lower-dimensional case, nobody has gotten around to characterizing $2Cob_2$ as a 2-category! That’s why I propose shirking the job of characterizing $3Cob_3$ , and going for the easier $3Cob_2$ .

I know what all those words mean, but I’m sure it would take me a good while to turn them into an algebraic presentation. Of course, it would be good for me!

It would be good for the world.

By the way, to be perfectly clear: I write ‘should be’ above for things I think are true, and ‘is’ for things that have been proved.

Posted by: John Baez on May 30, 2008 8:54 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

It’s a bit late for me… but I can assure you that your

feeble American-made irony detector

was misfiring! I don’t know why I trigger it so often… :) I was referring to Aaron Lauda’s paper, lemma 32, which directly cites Kerler and Lyubashenko. I’m a bit too tired to establish whether it’s concerned with exactly the same 2-category that I was talking about, but frankly, 20 coherences are enough for anyone. I’ll say more in the morning.

Posted by: Jamie Vicary on May 31, 2008 12:26 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Jamie wrote:

I can assure you that your

feeble American-made irony detector

was misfiring!

Whoops! I need to turn down the sensitivity level. I thought if you said you “knew” that an “impressive-looking book” you hadn’t actually read contained the answer to some interesting question, that was a British way of saying the book was useless pompous piffle.

Anyway, I don’t believe that book mentions categorified Frobenius algebras. I think Aaron Lauda took a long list of diagrammatic equations from that book and realized they were the definition of a Frobenius pseudomonoid, as part of his proof that 2Thick is the free (semistrict) monoidal 2-category on a Frobenius pseudomonoid.

I don’t know if anyone has defined symmetric Frobenius pseudomonoids yet, which is part of what we’d need to describe $3Cob_2$ .

Posted by: John Baez on May 31, 2008 8:14 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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John wrote:

So, we naively expect 3Cob 2 to be something like the free symmetric monoidal 2-category on a symmetric Frobenius pseudomonoid — namely, the circle.

This is close to true, but not quite: we need some extra bells and whistles on our Frobenius pseudomonoid to deal with the extra layer of duality in our 2-category!

Bruce Bartlett pointed out a mistake here: I should have said braided Frobenius pseudomonoid!

Thanks to Aaron Lauda and Jamie Vicary for helping think about this.

Posted by: John Baez on July 30, 2008 1:54 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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It’s a bit frustrating that there isn’t a better universal description of TQFTs. We know that in 1+1 dimensions they’re commutative $\dagger$ -Frobenius algebras in Hilb,

The thing which stokes my interest about TQFTs is not a universal “top-down” description of this form, but rather a geometric “bottom-up” description. In other words, it’s true that a (not-necessarily unitary) 2d TQFT gives rise to a commutative Frobenius algebra in Vect - but for me the key words there are gives rise to which I would be very reluctant to change to is.

I sort of prefer to take the idea of a TQFT at face value… it is a gadget which takes geometric data and transforms it into higher algebraic structures. How it does this is a geometric question - it could be via principal bundles, or gerbes, or stacks, and so on. Those are the things I find cool. I think this is also the “ $\Sigma$ -model” philosophy of Urs, which I am an adherent of.

Having said that, I must say that the top-down approach is also very important and will be extremely useful when it is achieved. I spend my lunch breaks there, but the rest of the time I spend down in the engine-room in the basement, trying to learn about the geometric gismos which make this thing tick.

Posted by: Bruce Bartlett on May 25, 2008 12:18 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

Bruce wrote:
I sort of prefer to take the idea of a TQFT at face value; it is a gadget which takes geometric data and transforms it into higher algebraic structures.

Is it clear what sort of higher algebraic structures should be called a TQFT? Namely?

Posted by: jim stasheff on May 25, 2008 2:34 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

Surely the ultimate goal is to understand these TQFTs. So if there was a straightforward characterisation of all Frobenius algebras, would you be happy with that? Or is there another reason why we are interested in the more exciting bundly-gerby-stacky power-tools that you mention?

Posted by: Jamie Vicary on May 28, 2008 12:16 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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Is it clear what sort of higher algebraic structures should be called a TQFT? Namely?

Dear Jim: I guess the answer is no; for instance I don’t think we even yet have a good grip on the kind of ‘2-vector spaces’ we will need… and that’s only at codimension 2. 2-Hilbert spaces don’t seem to fit the examples coming from the ‘derived world’, where a ‘2-vector space’ is more akin to a category of the from $D(X)$ for some smooth space $X$ . But can one characterize these categories intrinsically in a simple manner?

So if there was a straightforward characterisation of all Frobenius algebras, would you be happy with that?

Dear Jamie: Yes indeed… with the disclaimer that for example semisimple Lie algebras are classified in terms of roots and weights - but somehow that’s not always the most alluring viewpoint (see this for example).

Posted by: Bruce Bartlett on May 28, 2008 1:46 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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The reason to be interested in the

bundly-gerby-stacky power-tools that you mention

is that it is what connects the abstract Frobenius algebra which you may find your TQFT to be encoded by to geometric data which the TQFT can then be thought of as coming from by “quantization”.

For instance, just from abstract TQFT reasoning you find that Chern-Simons theory assigns vector spaces with certain properties to surfaces. But the particularly interesting statements only arise after you can also identify these abstract vector spaces with spaces of holomorphic sections of line bundles that are obtained from transgressing a certain 2-gerbe to some mapping space.

Posted by: Urs Schreiber on May 29, 2008 4:42 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

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I’m having quite a lot of luck banishing some of my higher-categorical gremlins here, so I’ll try again with one more.

I’ve lurked around enough by now to discover that a fundamental motivation for higher category theory is the relationship between $n$ -groupoids and the homotopies of compact topological spaces. For example, we can think of a 2-group $G$ as a connected compact topological space with $\pi_3$ , $\pi_4$ and so on all trivial.

But surely, the same topological space would be just as well-represented categorically by a 3-group created from $G$ by adding trivial 3-cells. Let’s call this $T(G)$ , where the process $T$ adds trivial $n+1$ cells to an $n$ -category. It would also be just as well represented by $T^m(G)$ , or even by $T ^{\infty}(G)$ .

Something else that’s of fundamental importance for $n$ -groups is their representation theory, given by unitary functors into $n\!\mathrm{Hilb}$ . If $G$ and $T(G)$ essentially represent the same topological space, wouldn’t we expect there to be a correspondence between the representations of $G$ and of $T(G)$ , and in fact of $T^m(G)$ and $T ^{\infty} (G)$ , for any $n$ -groupoid $G$ ? However, it seems to me that this isn’t the case, which I’ve tested at all the levels for which I know what ‘functor’ means.

Does this bother anybody else, or is there some deeper philosophy of $n$ -groupoids by which it all makes sense? The obvious conclusion is that a representation of an $n$ -groupoid doesn’t tell you something about the underlying topological space, but is sensitive to the way you’ve chosen to describe it as an $n$ -category… this seems a shame to me!

Posted by: Jamie Vicary on May 25, 2008 10:59 AM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

A related point:

The n-cat language seems to index according to the first n homotopy groups NOT by
the first non-trivial homotopy groups,
what in homotopy theory might be called n-stage.

Posted by: jim stasheff on May 25, 2008 2:38 PM | Permalink | Reply to this

Re: Categorifying the Dijkgraaf-Witten model

Yes, I suppose that’s right. But I don’t think it would help if we changed to indexing by the first non-trivial homotopy groups, unfortunately.

Posted by: Jamie Vicary on May 28, 2008 11:09 AM | Permalink | Reply to this

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