The n-Category Café

Abstract: For 2-categories $2C$ we define a notion of associated charted $2C$-bundles and $2K$-theories. For the 2-category of 2-vector spaces in the sense of M. Kapranov and V. Voevodsky this gives the $2K$-theory of Baas, Dundas and Rognes. When the 2-category is a nice 2-groupoid we prove that its 2-nerve is a classifying space for the associated charted $2C$-bundles. In the process of proving this result we develop a lot of powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. As another corollary it follows that if we take the 2-category to be the 2-vector spaces introduced by J. Baez, then the associated $2K$-theory is just two copies of ordinary $K$-theory.

The last result, that <Baez-Crans 2-vector>-bundles give rise to a classification by two copies of ordinary $K$-theory (instead of one copy of “2-$K$-theory”, as one might expect) has surfaced before, but, as far as I am aware, only now appears in (pre-)print.

So, Baez-Crans 2-vector spaces, while useful for many purposes, are not what one wants to use as the typical fiber in an interesting 2-<vector bundle>.

Hence, instead, Baas, Dundas, Rognes & Kro use Kapranov-Voevodsky 2-vector spaces.

But - evidently - these are still not quite what we want as fibers of really interesting 2-vector bundles. Because ultimately one wants a geometric realization (in terms of classes of 2-bundles) not of $K$-theory of $K$-theory, but of elliptic cohomology proper.

In case anyone cares, as I have remarked before (also here), I think there are several indications that we want to pass to a 2-category of 2-vector spaces which is somewhat larger than that of KV 2-vector spaces.

Let $K$ be some field. I believe Baez-Crans 2-vector spaces are module categories for the monoidal category $\mathrm{Disc}(K)$, the discrete category of the set $K$.

On the other hand, KV 2-vector spaces are a very special case of categories that are $\mathrm{Vect}_K$-modules

A “$n$-dimensional” KV 2-vector space is, in fact, a category of modules for the algebra $K^{\oplus n}$.

Being careful about distinguishing the two different levels of “modules” that play a role here, it is noteworthy that every category of right modules for an ordinary $K$-algebra is itself a module for a left action by $\mathrm{Vect}_K$.

In fact, there is an embedding

which sends $K$-algebras $A$ to their categories of $A$ modules

which sends $A$-$B$ bimodules to functors

and so on.

The way $KV2\mathrm{Vect}$ sits inside ${}_{\mathrm{Vect}_K}\mathrm{Mod}$ is just a special case of that

But $KV2\mathrm{Vect}$ is too small. It doesn’t have sufficiently many multiplicative inverses. That’s what makes constructing $KV2\mathrm{Vect}$-bundles a little unwieldy: the transitions on double intersections are not in general (not even weakly) invertible.

$\mathrm{Bim}(\mathrm{Vect}_K)$ is larger. It has plenty of inverses (all weakly invertible bimodules). Is it large enough?

At least, there are several indications that $\mathrm{Bim}(\mathrm{Vect}_K)$ 2-vector bundles are the 2-vector bundles that appear in physics.

A $U(1)$-gerbe with curving is a 2-functor to bimodules, taking values in bimodules of algebras of compact operators. Since these are Morita equivalent to the ground field, that’s a “1-dimensional” 2-vector bundle. A line 2-bundle.

Quite similarly, a $\mathrm{String}(n)$-bundle with connection is a 2-functor with values in bimodules for the representation algebra of a loop group.

It is quite straightforward to write down local descriptions in terms of transitions for these bimodule 2-vector bundles. This I can do.

What I cannot do right now without investing a forbidding amount of time it to work out the classification of the bimodule 2-vector bundles.

But I am hoping that somebody will.