
First, the astronomy picture of the week:
1) NASA Photojournal, 'Victoria Crater' at Meridiani Planum, http://photojournal.jpl.nasa.gov/catalog/PIA08813
This is the crater that NASA's rover called Opportunity has been exploring. It's 800 meters across. I like this picture just because it's beautiful. It was taken by the High Resolution Imaging Science Experiment on NASA's Mars Reconnaissance Orbiter.
Now, on to business! I want to talk about this paper, which took over 2 years to write:
2) John Baez, Aristide Baratin, Laurent Freidel and Derek Wise, Representations of 2groups on infinitedimensional 2vector spaces, available as arXiv:0812.4969.
We can dream up the notion of "2vector space" by pondering this analogy chart:
numbers vector spaces addition direct sum multiplication tensor product 0 the 0dimensional vector space 1 the 1dimensional vector spaceJust as you can add and multiply numbers, you can add and multiply vector spaces  but people call these operations "direct sum" and "tensor product", to make them sound more intimidating. These new operations satisfy axioms similar to the old ones. However, what used to be equations like this:
x + y = y + x
now become isomorphisms like this:
X + Y ≅ Y + X.
This means we're "categorifying" the concepts of plus and times.
The unit for addition of vector spaces is the 0dimensional vector space, and the unit for multiplication of vector spaces is the 1dimensional vector space.
But here's the coolest part. Our chart is like a snake eating its own tail. The first entry of the first column matches the last entry of the second column! The set of all "numbers" is the same as "the 1dimensional vector space". If by "numbers" we mean complex numbers, these are both just C.
This suggests continuing the chart with a third column, like this:
numbers (C) vector spaces (Vect) 2vector spaces (2Vect) addition direct sum direct sum multiplication tensor product tensor product 0 C^{0} Vect^{0} 1 C^{1} Vect^{1}Here C^{0} is short for the 0dimensional vector space, while C^{1} is short for the 1dimensional vector space  in other words the complex numbers, C. Vect is the category of all vector spaces. So, whatever a "2vector space" is, to make the chart nice we'd better have Vect be the 1dimensional 2vector space. We can emphasize this by calling it Vect^{1}.
In fact, about 15 years ago Kapranov and Voevodsky invented a theory of 2vector spaces that makes all this stuff work:
3) Mikhail Kapranov and Vladimir Voevodsky, 2categories and Zamolodchikov tetrahedra equations, in Algebraic Groups and Their Generalizations: Quantum and InfiniteDimensional Methods, Proc. Sympos. Pure Math. 56, Part 2, AMS, Providence, RI, 1994, pp. 177259.
They mainly considered finitedimensional 2vector spaces. Every finitedimensional vector space is secretly just C^{n}, or at least something isomorphic to that. Similarly, every finitedimensional 2vector space is secretly just Vect^{n}, or at least something equivalent to that.
(You see, when we categorify once, equality becomes isomorphism. When we do it again, isomorphism becomes "equivalence".)
What's Vect^{n}, you ask? Well, what's C^{n}? It's the set where an element is an ntuple of numbers:
(x_{1}, ..., x_{n})
So, Vect^{n} is the category where an object is an ntuple of vector spaces:
(X_{1}, ..., X_{n})
It's all pathetically straightforward. Of course we also need to know what's a morphism in Vect^{n}. What's a morphism from
(X_{1}, ..., X_{n})
to
(Y_{1}, ..., Y_{n})?
It's just the obvious thing: an ntuple of linear operators
(f_{1}: X_{1} → Y_{1}, ..., f_{n}: X_{n} → Y_{n})
And we compose these in the obvious way, namely "componentwise".
This may seem like an exercise in abstract nonsense, extending formal patterns just for the fun of it. But in fact, 2vector spaces are all over the place once you start looking. For example, take the category of representations of a finite group, or the category of vector bundles over a finite set. These are finitedimensional 2vector spaces!
Here I can't resist a more sophisticated digression, just to impress you. The whole theory of Fourier transforms for finite abelian groups categorifies nicely, using these examples. Any finite abelian group G has "Pontryagin dual" G* which is again a finite abelian group. I explained how this works back in "week273". The Fourier transform is a map from functions on G to functions on G*. So, it's a map between vector spaces. But, lurking behind this is a map between 2vector spaces! It's a map from representations of G to vector bundles over G*.
You can safely ignore that last paragraph if you like. But if you want more details, try section 6.1 of this old paper:
4) John Baez, Higherdimensional algebra II: 2Hilbert spaces, Adv. Math. 127 (1997), 125189. Also available as arXiv:qalg/9609018.
As you can see from the title, I was trying to go beyond 2vector spaces and think about "2Hilbert spaces". That's because in quantum physics, we use Hilbert spaces to describe physical systems. Recent work on physics suggests that we categorify this idea and study 2Hilbert spaces, 3Hilbert spaces and so on  see "week58" for details. In the above paper I defined and studied finitedimensional 2Hilbert spaces. But a lot of the gnarly fun details of Hilbert space theory show up only for infinitedimensional Hilbert spaces  and we should expect the same for 2Hilbert spaces.
After these old papers on 2vector spaces and 2Hilbert spaces, various people came along and improved the whole story. For example:
5) Martin Neuchl, Representation Theory of Hopf Categories, Ph.D. dissertation, University of Munich, 1997. Chapter 2: 2dimensional linear algebra. Available at http://math.ucr.edu/home/baez/neuchl.ps
6) Josep Elgueta, A strict totally coordinatized version of Kapranov and Voevodsky 2vector spaces, to appear in Math. Proc. Cambridge Phil. Soc. Also available as arXiv:math/0406475.
7) Bruce Bartlett, The geometry of unitary 2representations of finite groups and their 2characters, available as arXiv/0807.1329.
In the last of these, Bruce worked out how finitedimensional 2Hilbert spaces arise naturally in certain topological quantum field theories!
Just as we can study representations of groups on vector spaces, we can study representations of "2groups" on 2vector spaces:
8) Magnus ForresterBarker, Representations of crossed modules and cat^{1}groups, Ph.D. thesis, Department of Mathematics, University of Wales, Bangor, 2004. Available at http://www.maths.bangor.ac.uk/research/ftp/theses/forresterbarker.pdf
9) John W. Barrett and Marco Mackaay, Categorical representations of categorical groups, Th. Appl. Cat. 16 (2006), 529557. Also available as arXiv:math/0407463.
10) Josep Elgueta, Representation theory of 2groups on finite dimensional 2vector spaces, available as arXiv:math.CT/0408120.
A group is a category with one object, all of whose morphisms are invertible. Similarly, a 2group is a 2category with one object, all of whose morphisms and 2morphisms are invertible. Just as we can define "Lie groups" to be groups where the group operations are smooth, we can define "Lie 2groups" to be 2groups where all the 2group operations are smooth. Lie groups are wonderful things, so we can hope Lie 2groups will be interesting too. There are already lots of examples known. You can see a bunch here:
11) John Baez and Aaron Lauda, Higherdimensional algebra V: 2groups, Theory and Applications of Categories 12 (2004), 423491. Available at http://www.tac.mta.ca/tac/volumes/12/14/1214abs.html and as arXiv:math/0307200.
However, Barrett and Mackaay discovered something rather upsetting. While Lie groups have lots of interesting representations on finitedimensional vector spaces, Lie 2groups don't have many representations on finitedimensional 2vector spaces!
In fact, the problem already shows up for representations of plain old Lie groups on 2vector spaces. A Lie group can be seen as a special sort of Lie 2group, where the only 2morphisms are identity morphisms.
The problem is that unlike a vector space, a 2vector space has a unique basis  at least up to isomorphism. In C^{n} there's an obvious basis consisting of vectors like
(1,0,0,...)
(0,1,0,...)
(0,0,1,...)
and so on, but there are lots of other bases too. But in Vect^{n} the only basis goes like this:
(C^{1},C^{0},C^{0},...)
(C^{0},C^{1},C^{0},...)
(C^{0},C^{0},C^{1},...)
Well, I'm exaggerating slightly: we could replace C^{1} here by any other 1dimensional vector space, and C^{0} by any other 0dimensional vector space. That would give other bases  but they'd still be isomorphic to the basis shown above.
So, if we have a group acting on a finitedimensional 2vector space, it can't do much more than permute the basis elements. So, any representation of a group on a finitedimensional 2vector space gives an action of this group as permutations of a finite set!
That's okay for finite groups, since these can act in interesting ways as permutations of finite sets. But it's no good for Lie groups. Lie groups are usually infinite: they're manifolds. So, they have lots of actions on manifolds, but not many actions on finite sets.
This suggests that to study representations of Lie groups (or more general Lie 2groups) on 2vector spaces, we should invent some notion of "infinitedimensional 2vector space", where the basis can be not a finite set but an infinite set  indeed, something more like a manifold.
Luckily, such a concept was already lurking in the mathematical literature!
In the categorification game, it's always good when the concepts you invent shed light on existing issues in mathematics. And it's especially fun when you categorify a concept and get a concept that turns out to have been known  or at least partially known  under some other name. Then you're not just making up new stuff: you're seeing that existing math already had categorification built into it! This happens surprisingly often. That's why I take categorification so seriously.
The concept I'm talking about here is called a "field of Hilbert spaces". Roughly speaking, the idea is that you pick a set X, possibly infinite. X could be the real line, for example. Then a "field of Hilbert spaces" assigns to each point x in X a Hilbert space H_{x}.
As I've just described it, a measurable field of Hilbert spaces is an object in what we might call Hilb^{X}  a hairier, scarier relative of the tame little Vect^{n} that I've been talking about.
Let's think about how Hilb^{X} differs from Vect^{n}. First, the the finite number n has been replaced by an infinite set X. That's why Hilb^{X} deserves to be thought of as an infinitedimensional 2vector space.
Second, Vect has been replaced by Hilb  the category of Hilbert spaces. This suggests that Hilb^{X} is something more than a mere infinitedimensional 2vector space. It's closer to an infinitedimensional 2Hilbert space! So, we've departed somewhat from our original goal of inventing a notion of infinitedimensional vector space. But that's okay, especially if we're interested in applications to quantum physics that involve analysis.
And here I must admit that I've left out some important details. When studying fields of Hilbert spaces, people usually bring in some analysis to keep the Hilbert space H_{x} from jumping around too wildly as x varies. They restrict attention to "measurable" fields of Hilbert spaces. To do this, they assume X is a "measurable space": a space with a sigmaalgebra of subsets, like the Borel sets of the real line. Then they assume H_{x} depends in a measurable way on x.
The last assumption must be made precise. I won't do that here  you can see the details in our paper. But, here's an example of what I mean. Take X and partition it into countably many disjoint measurable subsets. For each one of these subsets, pick some Hilbert space H and let H_{x} = H for points x in that subset. So, the dimension of the Hilbert space H_{x} can change as x moves around, but only in a "measurable way". In fact, every measurable field of Hilbert spaces is isomorphic to one of this form.
So, a measurable field of Hilbert spaces on X is like a vector bundle over X, except the fibers are Hilbert spaces and there's no smoothness or continuity  the dimension of the fiber can "jump" in a measurable way.
If you've studied algebraic geometry, this should remind you of a "coherent sheaf". That's another generalization of a vector bundle that allows the dimension of the fiber to jump  but in an algebraic way, rather than a measurable way. One reason algebraic geometers like categories of coherent sheaves is because they need a notion of infinitedimensional 2vector space. Similarly, one reason analysts like measurable fields of Hilbert spaces is because they want their own notion of infinitedimensional 2vector space. Of course, they don't know this  if you ask, they'll strenuously deny it.
We learned most of what we know about measurable fields of Hilbert spaces from this classic book:
12) Jacques Dixmier, Von Neumann Algebras, NorthHolland, Amsterdam, 1981.
This book was also helpful:
13) William Arveson, An Invitation to C*Algebra, Chapter 2.2, Springer, Berlin, 1976.
As you might guess from the titles of these books, measurable fields of Hilbert spaces show up when we study representations of operator algebras that arise in quantum theory. For example, any commutative von Neumann algebra A is isomorphic to the algebra L^{∞}(X) for some measure space X, and every representation of A comes from a measurable field of Hilbert spaces on X.
The following treatment is less detailed, but it explains how measurable fields of Hilbert spaces show up in group representation theory:
14) George W. Mackey, Unitary Group Representations in Physics, Probability and Number Theory, BenjaminCummings, New York, 1978.
I'll say a lot more about this at the very end of this post, but here's a quick, rough summary. Any sufficiently nice topological group G has a "dual": a measure space G* whose points are irreducible representations of G. You can build any representation of G from a measurable field of Hilbert spaces on G* together with a measure on G*. You build the representation by taking a "direct integral" of Hilbert spaces over G*. This is a generalization of writing a representation as a direct sum of irreducible representation. Direct integrals generalize direct sums  just as integrals generalize sums!
By the way, Mackey calls measurable fields of Hilbert spaces "measurable Hilbert space bundles". Those who like vector bundles will enjoy his outlook.
But let's get back to our main theme: representations of 2groups on infinitedimensional 2vector spaces.
We don't know the general definition of an infinitedimensional 2vector space. However, for any measurable space X, we can define measurable fields of Hilbert spaces on X. We can also define maps between them, so we get a category, called Meas(X). Crane and Yetter call these "measurable categories".
I believe someday we'll see that measurable categories are a halfway house between infinitedimensional 2vector spaces and infinitedimensional 2Hilbert spaces. In fact, when we move up to nvector spaces, it seems there could be n+1 different levels of "Hilbertness".
The conclusions of our paper include a proposed definition of 2Hilbert space that can handle the infinitedimensional case. So, why work with measurable categories? One reason is that they're they're well understood, thanks in part to the work of Dixmier  but also thanks to Crane and Yetter:
15) David Yetter, Measurable categories, Appl. Cat. Str. 13 (2005), 469500. Also available as arXiv:math/0309185.
16) Louis Crane and David N. Yetter, Measurable categories and 2groups, Appl. Cat. Str. 13 (2005), 501516. Also available as arXiv:math/0305176.
The paper by Crane and Yetter studies representations of discrete 2groups on measurable categories. Our paper pushes forward by studying representations of topological 2groups, including Lie 2groups. Topology really matters for infinitedimensional representations. For example, it's a hopeless task to classify the infinitedimensional unitary representations of even a little group like the circle, U(1). But it's easy to classify the continuous unitary representations.
A group has a category of representations, but a 2group has a 2category of representations! So, as usual, we have representations and maps between these , which physicists call "intertwining operators" or "intertwiners" for short. But we also have maps between intertwining operators, called "2intertwiners".
This is what's really exciting about 2group representation theory. Indeed, intertwiners between 2group representations resemble group representations in many ways  a fact noticed by Elgueta. It turns out one can define direct sums and tensor products not only for 2group representations, but also for intertwiners! One can also define "irreducibility" and "indecomposability", not just for representations, but also for intertwiners.
Our paper gives nice geometrical descriptions of these notions. Some of these can be seen as generalizing the following paper of Crane and Sheppeard:
17) Louis Crane and Marnie D. Sheppeard, 2categorical Poincare representations and state sum applications, available as arXiv:math/0306440.
Crane and Sheppeard studied the 2category of representations of the "Poincare 2group". It turns out that we can get representations of the Poincare 2group from discrete subgroups of the Lorentz group. Since the Lorentz group acts as symmetries of the hyperbolic plane, these subgroups come from symmetrical patterns like these:
18) Don Hatch, Hyperbolic planar tesselations, http://www.plunk.org/~hatch/HyperbolicTesselations/
But Crane and Sheppeard weren't just interested in beautiful geometry. They developed their example as part of an attempt to build new "spin foam models" in 4 dimensions. I've talked about such models on and off for many years here. The models I've discussed were usually based on representations of groups or quantum groups. Now we can build models using 2groups, taking advantage of the fact that we have not just representations and intertwiners, but also 2intertwiners. You can think of these models as discretized path integrals for gauge theories with a "gauge 2group". To compute the path integral you take a 4manifold, triangulate it, and label the edges by representations, the triangles by intertwiners, and the tetrahedra by 2intertwiners. Then you compute a number for each 4simplex, multiply all these numbers together, and sum the result over labellings.
Baratin and Freidel have done a lot of interesting computations in the CraneSheppeard model. I hope they publish their results sometime soon.
To wrap up, I'd like to make a few technical remarks about group representation theory and measurable fields of Hilbert spaces. In "week272" I talked about a class of measurable spaces called standard Borel spaces. Their definition was frighteningly general: any measurable space X whose measurable subsets are the Borel sets for some complete separable metric on X is called a "standard Borel space". But then I described a theorem saying these are all either countable or isomorphic to the real line! They are, in short, the "nice" measurable spaces  the ones we should content ourselves with studying.
In our work on 2group representations, we always assume our measurable spaces are standard Borel spaces. We need this to get things done. But standard Borel spaces also show up ordinary group representation theory. Let me explain how!
To keep your eyes from glazing over, I'll write "rep" to mean a strongly continuous unitary representation of a topological group on a separable Hilbert space. And, I'll call an irreducible one of these guys an "irrep".
Mackey wanted to build all the reps of a topological group G starting from irreps. This will only work if G is nice. Since Haar measure is a crucial tool, he assumed G was locally compact and Hausdorff. Since he wanted L^{2}(G) to be separable, he also assumed G was second countable.
For a group with all these properties  called an "lcsc group" by specialists wearing white lab coats and big hornrimmed glasses  Mackey was able to construct a measure space G* called the "unitary dual" of G.
The idea is simple: the points of G* are isomorphism classes of irreps of G. But let's think about some special cases....
When G is a finite group, G* is a finite set.
When G is abelian group, not necessarily finite, G* is again an abelian group, called the "Pontryagin dual" of G. I talked about this a lot in "week273".
When G is both finite and abelian, so of course is G*.
But the tricky case is the general case, where G can be infinite and nonabelian! Here Mackey described a procedure which is a grand generalization of writing a rep as a direct sum of irreps.
If we choose a sigmafinite measure μ on G* and a measurable field H_{x} of Hilbert spaces on G*, we can build a rep of G. Here's how. Each point x of G* gives an irrep of G, say R_{x}. These form another measurable field of Hilbert spaces on G*. So, we can tensor H_{x} and R_{x}, and then form the "direct integral"
∫_{x} (H_{x} ⊗ R_{x}) dμ(x)
As I already mentioned, a direct integral is a generalization of a direct sum. The result of doing this direct integral is a Hilbert space, and in this case it's a rep of G. The Hilbert spaces H_{x} specify the "multiplicity" of each irrep R_{x} in the representation we are building.
The big question is whether we get all the reps of G this way.
And the amazing answer, due to James Glimm, is: yes, if G* is a standard Borel space!
In this case we say G is "type I". People know lots of examples. For example, an lcsc group will be type I if it's compact, or abelian, or a connected real algebraic group, or a connected nilpotent Lie group. That covers a lot of ground. However, there are plenty of groups, even Lie groups, that aren't type I. The representation theory of these is more tricky!
If you want to know more, either read Mackey's book listed above, or this summary:
19) George W. Mackey, Infinitedimensional group representations, Bull. Amer. Math. Soc. 69 (1963), 628686. Available from Project Euclid at http://projecteuclid.org/euclid.bams/1183525453
The most fascinating thing about algebra and geometry is the way they struggle to help each other to emerge from the chaos of nonbeing, from those dark depths of subconscious where all roots of intellectual creativity reside.  Yuri Manin
© 2008 John Baez
baez@math.removethis.ucr.andthis.edu
