October 19, 2008

Morton on 2-Vector Spaces and Groupoids

Posted by John Baez

My student Jeffrey Morton has come out with a paper based on his thesis:

• Jeffrey Morton, 2-vector spaces and groupoids.

Abstract: This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation — a weak functor — from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail. It has applications in constructing quantum field theories, among others.

Jeffrey Morton’s paper builds a 2-functor

$\Lambda : 2Span(Gpd) \to 2Vect$

Here 2Span(Gpd) is the 2-category of:

• finite groupoids,
• spans of finite groupoids, and
• (equivalence classes of) spans of spans of finite groupoids

while 2Vect is the 2-category of:

• Kapranov–Voevodsky 2-vector spaces,
• 2-linear maps between such 2-vector spaces, and
• natural transformations between linear maps between such 2-vector spaces.

Let me explain some of this stuff.

A (Kapranov–Voevodsky) 2-vector space is a category equivalent to $Vect^n$ for some finite $n$, where $Vect$ is the category of finite-dimensional vector spaces. A 2-linear map between 2-vector spaces is a functor that’s linear on homsets and preserves direct sums. More concretely, we can think of any linear map like this:

$f : Vect^n \to Vect^m$

as an $m \times n$ matrix of finite-dimensional vector spaces. So, we’re doing categorified quantum mechanics: matrix mechanics with vector spaces replacing complex numbers! This is an old idea, promoted by Louis Crane.

From a finite groupoid $X$, Morton constructs the 2-vector space

$\Lambda(X) = Vect^X$

whose objects are functors $\psi : X \to Vect$, and whose morphisms are natural transformations between these. Mathematicians should think of $\psi: X \to Vect$ as a representation of $X$, since that’s all it is when $X$ is a group. Physicists should think of $\psi : X \to Vect$ as a categorified ‘wavefunction’, since in quantum mechanics a wavefunction is a function $\psi : X \to \mathbb{C}$ where $X$ is a mere set.

From a span of finite groupoids:

$X \leftarrow S \rightarrow Y$

Morton constructs a 2-linear map:

$\Lambda(S) : Vect^X \to Vect^Y$

This map takes any functor $\psi: X \to Vect$, pushes it forward from $X$ to $S$, and then pulls it back from $S$ to $Y$. This is one of the ‘push-pull’ constructions we see so often on this blog.

Finally, from a span of spans of groupoids — it’s tough for me to draw such a thing beautifully here, so look at the picture at the beginning of Section 5 of the paper — Morton constructs a natural transformation between linear maps between 2-vector spaces. The interesting thing about this step of the construction is that it makes essential use of groupoid cardinality!

What’s the point of all this business? One point is that it lets Morton construct the Dijkgraaf–Witten model as an extended topological quantum field theory. For that, see his thesis and also the paper where he constructs a weak 2-category nCob2 consisting of:

• compact $(n-2)$-manifolds,
• cobordisms between such manifolds,
• (equivalence classes) of cobordisms between cobordisms between such manifolds.

Given any finite group $G$, Morton gets a weak 2-functor

$Z : n Cob_2 \to 2 Vect$

This is the untwisted Dijkgraaf–Witten model, viewed as an extended TQFT. He builds $Z$ in two stages. First he constructs a weak 2-functor

$Z_0 : n Cob_2 \to 2Span(Gpd)$

Then he composes this with the weak 2-functor I just described:

$\Lambda : 2 Span(Gpd) \to 2 Vect$

So, $\Lambda$ — short for linearization — is the algebraic essence of the Dijkgraaf–Witten model!

Posted at October 19, 2008 12:00 AM UTC

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Cobordisms

Thanks for commenting on that. I should point out that the previous paper you linked to has a new version, since I decided to clarify part of it and it grew into a two-part story. The old version contains the cobordism stuff.

I should also point out the thing which this paper corrects that was flawed in my thesis, in that the construction of the natural transformation for a span of spans wasn’t properly defined before. There is some essential use of the groupoid cardinality, which was there before, but also some representation theory which I hadn’t dealt with properly. Specifically, it involves the representations of the various automorphism groups of the objects - it’s what degroupoidification into a vector space discards.

Posted by: Jeffrey Morton on October 19, 2008 1:54 AM | Permalink | Reply to this

Re: Cobordisms

Cool! I’m glad you’re getting this rather large set of ideas nicely organized and polished.

It must because my dad was an editor, but now I can’t help noticing that in the abstract of version 2 of Double bicategories and double cospans, the very first character is a typo. Also, you misspelled Dominic Verity’s first name. Ain’t it annoying?

Posted by: John Baez on October 19, 2008 2:14 AM | Permalink | Reply to this

Re: Cobordisms

John wrote:

It must because my dad was an editor, but now I can’t help noticing

to which I respond ;-)

It must BE because my dad was in effect an editor as an English teacher including script writing, but now I can’t help noticing

I must have inherited my infamous red pen from him.

Posted by: jim stasheff on October 19, 2008 2:06 PM | Permalink | Reply to this

Re: Cobordisms

The First Law of Pedantry says that any internet post correcting a spelling or grammar mistake has to contain such a mistake itself. I don’t see one in yours, Jim — what’s wrong?

Posted by: John Baez on October 20, 2008 6:01 PM | Permalink | Reply to this

Re: Cobordisms

Jeffrey wrote:

I should also point out the thing which this paper corrects that was flawed in my thesis, in that the construction of the natural transformation for a span of spans wasn’t properly defined before.

You mentioned a problem along these lines before. I thought we just hadn’t carefully worked out the correct normalizations required when turning a span of groupoids into a linear operator. There are various normalizations that ‘work’: that map composite spans to composite operators. But, there’s one very obvious-looking normalization that doesn’t work.

But now I fear something even subtler is going on:

Specifically, it involves the representations of the various automorphism groups of the objects - it’s what degroupoidification into a vector space discards.

Did you need this adjustment to make $\Lambda$ preserve vertical composites of spans-of-spans? Or to make $\Lambda$ preserve horizontal composites?

This would be a very helpful clue. I’m dying to understand exactly what issue is at stake here.

Posted by: John Baez on October 19, 2008 5:10 PM | Permalink | Reply to this

Re: Cobordisms

The issue about normalizations is okay - that would be a minor change. The change is needed to make the natural-transformations well-defined. It comes from the way the 2-linearization of spans of groupoids differs from the 1-linearization.

So remember that we can write the 2-linear map (up to natural isomorphism) as a span of groupoids $A \leftarrow X \rightarrow B$ as a matrix of vector spaces, which act on 2-vectors in the expected way. The bases in the 2-vector spaces $\Lambda(A)$ and $\Lambda(B)$ consist of pairs $([a],V)$ and $([b],W)$, where $[a]$ is an isomorphism class in the groupoid $A$, and $V$ is an irreducible representation of $Aut(a)$, and likewise for $B$.

Then the coefficient in the $(([a],V),([b],W))$ position of the matrix for $\Lambda(X)$ is a direct sum (over objects $x$ mapping to both $a$ and $b$) of a bunch of spaces like $hom(s^{\star}V,t^{\star}W)$ (by Schur’s lemma and Frobenius reciprocity, if you want to put things in representation-theory terms, or just by Schur’s lemma and the definition of the pushforward if you don’t).

Then given a span of spans, say $X_1 \leftarrow Y \rightarrow X_2$ (all over $A \times B$), you have to pull-and-push an intertwiner in that space (looking at just one of those hom-spaces for the moment, over one particular $x_1 \in X_1$ - the whole space is a direct sum of them), through the corresponding hom-space from $Y$ and into the one for $X_2$.

Now the thing is, the corresponding hom-space over $Y$ (actually, there’s one for each object $y \in Y$ restricting to $x_1$) may be bigger: since the pullback (“restricted”) representations $s^{\star}V$ and $t^{\star}W$ at $y$ are also pullbacks from the $s_1^{\star}V$ and $t_1^{\star}W$ over $x_1$, they may break down irreducible reps of $Aut(x_1)$ into a sum over irreps of $Aut(y)$, but not the reverse. So $hom(s_1^{\star}V,t_1^{\star}W) \subset hom(s^{\star}V,t^{\star}W)$, and similarly on the $X_2$ side. So pulling back intertwiners is easy: just keep the same linear operator, and it’s still an intertwiner between the restricted reps. But when you push forward, this may not be true: so you have to project onto the space of intertwiners over $Aut(x_2)$.

Now, this issue doesn’t come up at all if you only consider trivial representations: if all the $V$ and $W$ were one-dimensional (so all intertwiners are just scalar multiples), all you’d really need would be those normalization factors. When you take basis 2-vectors corresponding to non-trivial representations, you need this projection to make the pull-push of an intertwiner well-defined.

Okay, I hope that clears up the issue.

Posted by: Jeffrey Morton on October 20, 2008 2:04 AM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

Hi Jeffrey,

Yes congrats on writing up this new stuff and I look forward to reading it. I noticed this paper when it came out a few days ago; there have been so many of interest for the n-category cafe recently but another one that comes to mind right now is Michael Shulman’s Set theory for category theory, of which I have only had time to read the introduction so far but it looks like just the kind of thing that will really help me.

John wrote:

This is the untwisted Dijkgraaf–Witten model, viewed as an extended TQFT. He builds Z in two stages. First he constructs a weak 2-functor

(1)$Z_0:nCob 2 \rightarrow 2Span(Gpd)$

Then he composes this with the weak 2-functor I just described:

(2)$\Lambda: 2Span(Gpd) \rightarrow 2Vect$

So, $\Lambda$ — short for linearization — is the algebraic essence of the Dijkgraaf-Witten model!

Do you think so? I see it a bit differently. I’m not sure it is helpful to separate the two processes in that way (Jeffrey didn’t separate them explicitly that way in his thesis, for example).

I will probably embarrass myself here since I haven’t thought about this precise point for a while, but my thoughts are as follows: the essence of the Dijkgraaf-Witten model is the same as that of Chern-Simons theory, namely it’s all about higher line bundles and their spaces of sections.

I’m hoping Urs will back me up on this point :-)

In particular, the geometric input one works with is not quite the groupoid $P_M$ of connections on a manifold $M$ — it’s some or other higher line bundle over $P_M$. So, it seems to me that right from the very beginning, the “classical theory” as Freed calls it already has a linear aspect to it. You can’t separate it into a ‘purely groupoid’ part followed by a ‘linearization’ part.

Of course, it does depend on how one wants to approach the notion of a ‘higher line bundle’. You may want to think of a 2-line bundle as an $S^1$-gerbe, for instance. But, even in this picture we are no longer purely in the groupoid world; there’s an $S^1$-action to take care of and it can be twisted, etc. In other words, I’m trying to stress that the complex numbers enter from the very beginning, even in the classical theory… they are the action for the theory! I may well be confused about what you meant though, and as a result I’m taking issue with a straw man.

Posted by: Bruce Bartlett on October 20, 2008 12:33 AM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

As to whether the processes should be separated: I think it’s not necessary if your main purpose is to think about the Dijkgraaf-Witten model. I didn’t separate the two pieces in my thesis partly because it wasn’t necessary to articulate the difference between them until I went to break the thesis up into smaller portions to publish as papers. But the process I described in the thesis does factor into these two parts. I think it’s a useful breakdown because the 2-functor $\Lambda$ a model for a categorified “quantization” operation, and it’s useful to think about what that is generally.

Now, all that is based on my way of looking at the DW model as based on the groupoid of flat connections, which from what you’re saying is not quite right. So the gap there may be real, and the fact that what’s missing is connected to $S^1$, the circle (or, to put it another way, $U(1)$) reminds me of the gap that has to be papered over between the categorically nice spans-of-groupoids picture for the harmonic oscillator and the real quantum mechanical guy that uses complex numbers. There I had to go over to $U(1)$-groupoids, where there is an action of the circle group…

This $\Lambda : Span(Gpd) \rightarrow 2Vect$ is supposed to stand on its own two feet (hence the picture? no, it’s named that because $\Lambda$ looks like a span). It’s a simple kind of categorified “quantization” of a system if you know its pure states and their symmetries. (In this case, connections and gauge symmetries; for the harmonic oscillator, numbers of quanta, and the boson symmetries, namely the symmetric group).

But $\Lambda$ has a couple of shortcomings: one is that it doesn’t really take proper account of an action principle (in the case of connections, in taking the groupoid of flat connections, we’re a-priori saying the exponential of the action is zero). So maybe it needs to deal with those $U(1)$-groupoids again? Is this the same as the issue you’re raising? If so, I’ve thought about it some under the guise discussed above, but in the present paper, it’s not there.

The other shortcoming is that I’ve only described it for finite-dimensional 2-vector spaces, so that $\Lambda$’s infinite-dimensional-2-Hilbert-space analog can act on the smooth groupoids that come up as moduli stacks for flat connections in a gague group like $SU(2)$.

Though, again, it seems from what you’re saying that this is not quite the right input, so I’m hoping the two pictures fit together properly.

Posted by: Jeffrey Morton on October 20, 2008 2:31 AM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

Let me just add that I might take that phrase “the algebraic essence of the Dijkgraaf-Witten model” with a grain of salt, in that the geometric essence is almost all in the other half: the groupoids of connections encode all the topological data the theory uses, as I’m talking about it. By developing the algebraic part separately, I can focus on the geometric part when I write up the paper which explains the ETQFT. So I think I agree with both John and Bruce about where the interesting geometric and algebraic aspects live (modulo what I said above).

Posted by: Jeffrey Morton on October 20, 2008 7:50 AM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

John wrote:

So, $\Lambda$ — short for linearization — is the algebraic essence of the Dijkgraaf-Witten model!

Bruce wrote:

Do you think so?

I was just trying to explain why Jeffrey wrote a paper on this $\Lambda$ business, which at first glance must seem a bit esoteric… not to you, but to ordinary mortals.

I see it a bit differently. I’m not sure it is helpful to separate the two processes in that way (Jeffrey didn’t separate them explicitly that way in his thesis, for example).

Regardless of whether he made a big deal of it, in Jeff’s thesis the construction of the untwisted Dijkgraaf–Witten model factors neatly as a process that turns manifolds into groupoids:

$Z_0 : nCob_2 \to 2Span(Gpd)$

followed by this ‘purely algebraic’ construction:

$\Lambda : 2Span(Gpd) \to 2Vect$

The first step turns any manifold into the groupoid of principal $G$-bundles over that manifold, where $G$ is some finite group. The second step is worthy of attention in itself, and that’s what Jeff is talking about here.

That’s all I was trying to say.

But I agree with what you’re hinting, namely that a full-fledged approach to this subject should handle the twisted Dijkgraaf–Witten model with equal ease. Here complex phases must be built into the game somehow. There must be lots of ways to go about it. You have yours, using gerbes. Can we also do it using $\mathrm{U}(1)$-groupoids, as Jeff suggests? I don’t know — but that would be really cool, since these are what Jeff used to categorify the perturbed harmonic oscillator. To have a chance at categorifying all of quantum physics, we probably need a formalism that neatly handles all these toy models.

I am very happy to see that both you and Jeffrey have good ideas on this fascinating question: how should we generalize groupoidification to build in the complex numbers?

All the time while you were writing your thesis on the Dijkgraaf–Witten model and Jeff was writing his thesis on the same thing, I was looking forward to the day where you’d combine your ideas and figure out the incredibly cool big picture which must be lurking here. You’re using 2-Hilbert spaces; he’s just using 2-vector spaces. You’re focused on the 3d case; he’s doing the $n$d case with a real definition of $n Cob_2$. You’re paying great attention to gerbes, while his thesis does not… and so on. Have at it! The implications will be awesome.

Posted by: John Baez on October 20, 2008 6:41 PM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

Jeffrey writes:

But Λ has a couple of shortcomings: one is that it doesn’t really take proper account of an action principle (in the case of connections, in taking the groupoid of flat connections, we’re a-priori saying the exponential of the action is zero). So maybe it needs to deal with those U(1)-groupoids again? Is this the same as the issue you’re raising?

Now I think it is. The twisted Dijkgraaf–Witten model assigns to each flat $G$-bundle over spacetime a phase, which we think of as the exponential of the action. Isomorphic flat $G$-bundles get the same phase. So, instead of merely forming a groupoid, the flat $G$-bundles form what we call a $\mathrm{U}(1)$-groupoid!

At least this is true for closed spacetimes, i.e. without boundary or corners. The fun will be generalizing this to spacetimes with boundary or corners.

If it works (and I now feel it must), we’ll get a nice generalization of the story in your thesis. With luck, the twisted Dijkgraaf–Witten model will give a weak 2-functor

$Z_0 : nCob 2 \to 2Span(U(1)Gpd)$

and then a generalization of your construction will give a weak 2-functor

$\Lambda : 2Span(U(1)Gpd) \to 2Vect$

Composing these, we get our extended TQFT

$Z : nCob_2 \to 2Vect$

This would be great!

(Technical note: since $G$ is a finite group, every principal $G$-bundle has a unique flat connection. So, the word ‘flat’ is redundant when I speak about ‘flat $G$-bundles’ above. I’m just trying to act like this is a warmup for Chern–Simons theory, where $G$ isn’t finite.)

Posted by: John Baez on October 21, 2008 7:11 AM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

The twisted Dijkgraaf–Witten model assigns to each flat G-bundle over spacetime a phase, which we think of as the exponential of the action. Isomorphic flat G-bundles get the same phase. So, instead of merely forming a groupoid, the flat G-bundles form what we call a $U(1)$-groupoid!

The twisted Dijkgraaf-Witten model is a $\sigma$-model whose target space is the groupoid $\mathbf{B}G$ (one object, $G$ worth of morphsims) on which the background field is a $U(1)$-3-bundle (the “Chern-Simons 2-gerbe”) represented by a cocycle aka $\omega$-anafunctor $\alpha : \mathbf{B}G \to \mathbf{B}^3 U(1)$ (= strict 3-functor out of cofibrant replacement = free 3-categoriecal resolution of $\mathbf{B}G$).

This $\alpha$ is nothing but the group 3-cocycle that governs the model. For Chern-Simons it’s the same, only with the finite $\mathbf{B}G$ replaced by something bigger and smooth.

For any piece of worldvolume $\Sigma$ of the membrane (a boundary surface of a 3-dimensional trajectory) the parameter “space” of the theory is the fundamental $\infty$-groupoid of $\Sigma$, $\Pi_\omega(\Sigma)$. If we want to stay in the finite context we can replace this with the finite groupoid with one object $x$ per connected component of $\Sigma$ and $\pi_1(\Sigma,x)$ as automorphisms of that object. Call that finite approximation $P_1(\Sigma)$.

Then the configuration space, the space of fields, is $hom(parameter space, target space) = hom(P_1(\Sigma), \mathbf{B}G)$ which is the groupoid of flat $G$-bundles on $\Sigma$ (up to equivalence). This should be the assignment of groupoids to cobordisms which Jeffrey mentions: it is the assignment to each piece of worldvolume of the space of fields living there.

Now there are some choices involved in obtaining the quantum theory from here. I’ll concentrate for simplicity now on the 1-functorial QFT version.

To get that, fix a surface $\Sigma$ and transgress the background field to the configuration space $hom(P_1(\Sigma),\mathbf{B}G)$ by

a) homming $P_1(\Sigma)$ into everything

b) taking equivalence classes of 1-morphisms of the resulting 3-groupoidal structure.

I have described the result of this (simple) computation in a bit of detail in this entry. The resulting transgression of the original background field is indeed the action functional (as a 1-functor with values in $\mathbf{B}U(1)$) of the Dijkgraaf-Witten theory.

What remains is hence to do the path integral over this action 1-functor from this point of view. For that first choose your favorite rep $\rho$ of $U(1)$ (take the standard rep for the standard result – in an extended QFT version we would have chosen a 3-rep of $\mathbf{B}^2 U(1)$ before transgression) and then plug everything into the discussion at An exercise in groupoidification: the path integral #.

I am pretty sure that the result is indeed the $\alpha$-twisted Dijkgraaf-Witten model (albeit described above just for the simple case where all cobordisms are of the form $\Sigma \times I$, which is of course a bit boring, but can be generalized).

As I mentioned # on Jeffrey’s blog, one can pass from the description used at “exercise in groupoidification” to something with closer appearance to Jeffrey’s picture by reversing perspective between fibered groupoids and fiber-assigning functors.

I am metnioning this again because there is such a great relation between “higher gauge theory” aka “parallel transport theory” aka “nonabelian differential cohomology” which I am probably mostly thinking about, and the quantization of the “transport functors” aka “differential cocycles” appearing there to (extended) QFT that it is a pity that we don’t have more interaction on this.

As Bruce emphasizes: the “twist” in Dijkgraaf-Witten theory is indeed a “2-gerbe”: better a $U(1)$-3-bundle with connection on $\mathbf{B}G$, hence something living in “nonabelian differential cohomology”. This is slightly invisible for DW theory since everything is so degenerate, but becomes manifest if you take the formula for the background field 3-anafunctor $\mathbf{B}G \to \mathbf{B}^3 U(1)$ and replace the left by the path 3-groupoid of a smooth target space $P_3(X) \to \mathbf{B}^3 U(1)$ whence it describes a “2-gerbe with connection” there. $X$ may be a smooth model of $\mathbf{B}G$ for $G$ Lie, of course.

Posted by: Urs Schreiber on October 21, 2008 10:21 AM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

Well, I think we are all basically on the same page here… I don’t really know what to say! I think we should all drink a toast to Dan Freed though, especially with regard to his recent Remarks on Chern-Simons Theory, a nice summary of the State of the Nation.

John wrote:

…was looking forward to the day where you’d combine your ideas and figure out the incredibly cool big picture which must be lurking here.

Sure I’m definitely up for that, but I think we need to wait for the higher category gurus to do their jobs and explain to us exactly what the n-category of cobordisms really is, and what the best language for dealing with it is!

Posted by: Bruce Bartlett on October 22, 2008 1:29 AM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

Bruce wrote:
what the n-category of cobordisms really is

in what sense? in other words, what is the meaning of `is’ :-)

Posted by: jim stasheff on October 22, 2008 2:18 PM | Permalink | Reply to this

Re: Morton on 2-Vector Spaces and Groupoids

Dear Jeffrey,
just a small comment. I think a 2-linear between 2-vector spaces can be just defined as a k-linear functor between them. Such a functor automatically preserves biproducts (each equation describing a biproduct is preserved). Actually for functors between Ab-categories being additive and preserving biproducts are equivalent.

Posted by: Josep Elgueta on October 20, 2008 10:48 AM | Permalink | Reply to this