The n-Category Café

Now, a paper out today, Two-vector bundles define a form of elliptic cohomology, pushes further the BDR-2K project.

There are two (old) claims: that the “classifying space” (the (group completion, really, of the) realization of the nerve of the topological 2-category

a) of Baez-Crans 2-vector spaces

b) of Kapranov-Voevodsky 2-vector spaces

“is” (homotopy theorist’s “is”)

a) two copies of the ordinary K-theory spectrum

b) the algebraic K-theory spectrum (that of a ring) of the ordinary K-theory spectrum, with the latter regarded as a ring (since it is a ring spectrum) (in short: “K-theory of K-theory”).

For quite a while, it had not been fully clear to me to what degree these claims have been supplied with complete proofs.

It was only recently that the proofs had been completed. I mentioned this in What Does the Classifying Space of a 2-Category Classify?

(Birgit Richter played a crucial role in that. Her office happens to be right next to mine.)

What do these nerves of categories have to do with 2-vector bundles?

The idea is this: as we have discussed at length here at the blog (and are still discussing), ordinary bundles are given by isomorphism classes of functors (the “transition functions”) $$g : Y^{[2]} \to \Sigma G \,,$$ where $Y^{[2]} \stackrel{\to}{\to} Y \to X$ is the groupoid coming from a good cover $Y$ of $X$ by open sets, and where $\sigma G$ is the structure group, regarded as a category.

By applying the nerve realization functor $$|\cdot| : \mathrm{Cat}_{\mathrm{Top}} \to \mathrm{Top}$$ from topological categories to topological spaces, we find that such functors are the same as homotopy classes of maps $$|g| : |Y^{[2]}| \to |\Sigma G| \,.$$ But $|Y^{[2]}| \simeq X$. Moreover, $|\Sigma G| := B G$, the classifying space of $G$-bundles (or, equivalently, “of the category $\Sigma G$”).

So $$|g| : X \to B G \,.$$

This is the familiar classification of principal 1-bundles.

Now, these authors take the point of view that suich transition data is all there is to $n$-bundles. Hence what they do is computing the nerves of higher categories which are to be thought of as categorifications of $\Sigma G$.

The original hope was that 2-vector bundles would have a “categorified K-theory” which is directly related to elliptic cohomology. This idea dates back to

Baas, Dundas & Rognes, 2-Vector bundles and forms of elliptic cohomology.

I have tried to give a rather detailed discussion of this literature and lots of trelated things in Seminar on 2-Vector bundles and Elliptic Cohomology, I, II, III.

Early on it was rather clear that by using Baez-Crans 2-vector spaces and KV 2-vector spaces does not really yield the classifying spaces one was hoping to see.

As you mention, I believe there is a pretty plausible reason for that: both these notions of 2-vector spaces exhaust only a tiny fraction of the 2-category of all 2-vector spaces.

Probably one doesn’t need the full thing, but I am pretty sure that one does need at least all 2-vector spaces which admit a 2-basis: this are those equivalent to module categories of algebras.

One reason for this is the existence of the “canonical 2-rep” of any strict 2-category, and hence in particular the existence of String-2-bundles (which are to be thought of a 2-Spin-bundles). This I describe in Connections on String-2-Bundles.

Now who were those experts Urs persuaded to work on the first n-category Café millennium prize?

I don’t want to talk about names here - yet. Let’s see how things proceed!