### What Does the Classifying Space of a 2-Category Classify?

#### Posted by Urs Schreiber

My personal spy has just returned from the Nordic Conference in Topology that took place last week.

I hear that Tore A. Kro has new notes on his work with N. Baas and M. Bökstedt available online

N. Baas, M. Bökstedt, T. A. Kro

2-categorical K-theories.

They try to answer the question: *What does the classifying space of a 2-category classify?*
Their answer is: for sufficiently well behaved topological 2-categories $C$, the nerve of $C$ is the classifying space for *charted $C$-bundles*.

Here a charted $C$-bundle is essentially like what one would call the transition data for a 2-groupoid bundle #. The only difference is that no invertibility in $C$ is assumed. As a consequence, transition functions may go from patch $i$ to patch $j$, but not the other way around.

The main application of this theory in these notes is a proof of the previously announced claim, that for $C$ the 2-category of Kapranov-Voevodsky 2-vector spaces the classifying space is the 2K-theory introduced by Baas, Dundas and Rognes. For $C$ the 2-category of Baez-Crans 2-vector spaces the classifying space is two copies of ordinary K-theory.

Posted at December 4, 2006 3:28 PM UTC
## Re: What does the Classifying Space of a 2-Category classify?

The Baas, Dundas, and Rognes paper is here.