The String Coffee Table

As I have reviewed before ($\to$), we may think of the $n$-dimensional Kapranov-Voevodsky 2-vector spaces ${\mathrm{Vect}}^n$ as categories of (left, say) $A$-modules, where $A=K^{\oplus n}$ is the algebra of $K$-valued functions on the finite set $\overline{n}$ of cardinality $n$. So these are categories of vector bundles over finite sets.

Accordingly, the 1-morphisms (linear 2-maps) between these 2-vector spaces can be regarded as $K^n$-$K^m$ bimodules (aka vector bundles over $\overline{n}\times \overline{m}$, aka $n\times m$-matrices with entries being vector spaces).

Finally, the 2-morphisms between these are bimodule homomorphisms.

As I have said before ($\to$), it seems that several well-known constructions all somehow related to Fourier-Mukai transformations can be understood in terms of a continuous version of this setup, where finite sets are replaced by topological spaces, or by varieties, or schemes - you name it.

Given the above formulation in terms of bimodules, this should correspond to nothing but passing from modules of rings of functions over finite sets to (sheaves of) modules of (sheaves of) rings of continuous (or differentiable, or holomorphic, or …) functions on these more general spaces.

While I am not aware of any literature that would try to make this connection explicit, infinite dimensional versions of KV 2-vector spaces have been discussed in the categorification community:

In

L. Crane & D. Yetter
Measurable categories and 2-groups
math.QA/0305176

vector bundles over finite sets are preplaced by those over measurable spaces.

Clearly, while straightforward conceptually, this introduces one or two technicalities. But even without looking at these in detail I’d dare to guess that this corresponds, in the above bimodule-way of looking at things, to concentrating on modules for algebras which are commutative von Neumann algebras ($\to$).

Passing to such a setup corresponds, roughly, to looking at the categorification of $\mathrm{\infty }\times \mathrm{\infty }$ matrices with entries taking values in the integers.

Notice that the only invertible matrices in this setup are those which permute all entries of a vector, i.e. those which have precisely a single 1 in each line and all zeros everywhere.

So the only group that has interesting representations on these matrices is the symmetric group of permutations. This does not change in principle when we pass to infinite-dimensional matrices, but at least now the space of bijections becomes infinite dimensional and large enough to carry some interesting reps.

That’s essentially what Crane and Sheppeard make use of in the above paper. They are concentrating on a 2-group coming from the semidirect product of some group with an abelian group. The possibly nonabelian group $G$ is that of objects, while the abelian group $H$ labels morphisms.

2-representing such a 2-group on (measurably) infinite-dimensional KV-2-vector spaces hence amounts, by the above reasoning, to looking at (measure) spaces $X$ on which $G$ acts (measurably), and to assigning to each element $g\in G$ the corresponding action on $X$. A little reflection then shows that the 1-morphisms in the 2-group are represented by equivariant maps from $X$ to the space of characters of $H$.

The mechanism involved here is essentially not different from what you’d get for finite $G$ represented on finite-dimensional KV-2-vector spaces. The difference is that in the more general (infininite dimensional) case one finds a rich structure of irreducible 2-reps of this sort.

On the other hand, one is still restricted this way to the sub-bicategory of bimodules over commutative algebras. The 2-reps induced from ordinary reps of ordinary groups which I was talking about ($\to$, $\to$) can be thought of as replacing the vector bundles in the above setup by vector bundles over noncommutative spaces (just a fancy way to think about an elementary fact of algebra).

Passing to noncommutative algebra this way gives rise to the availability of many more invertible linear 2-maps, hence more interesting 2-representations.

In closing, I’ll mention that of course the main application that Crane and Sheppeard have in mind are certain state sum models, where you want to decorate simplicial complexes in 2-representations.

I don’t have the leisure to talk about that in detail here. But I’ll note that the formulas for these state sum models, for instance their equation (6.4), alway look suspiciously like the Gawedzki-Reis-like formulas for the surface holonomy of gerbes ($\to$). I think there is a systematic way to understand this: these “state sums” should arise from pulling back $n$-transport $n$-functors along certain injections. That’s at least true in $D=2$ ($\to$). Here we’d expect the relevant chain of injections to be something like