Seminário de Teoria Quântica do Campo Topológica  

14/07/2008, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
John Baez, Department of Mathematics, University of California at Riverside

Topological 2-Groups and Their Classifying Spaces

Categorifying the concept of topological group, one obtains the notion of a topological 2-group. This in turn allows a theory of "principal 2-bundles" generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group $G$ and a space $M$, principal $G$-bundles over $M$ are classified by either the Cech cohomology $H^1(M,G)$ or the set of homotopy classes $[M,BG]$, where $BG$ is the classifying space of $G$. Here we review work by Bartels, Jurco, Baas-Bökstedt-Kro, and others generalizing this result to topological 2-groups. We explain various viewpoints on topological 2-groups and the Cech cohomology $H^1(M,G)$ with coefficients in a topological 2-group $G$, also known as "nonabelian cohomology". Then we sketch a proof that under mild conditions on $M$ and $G$ there is a bijection between $H^1(M,G)$ and $[M,B\mid G\mid ]$, where $B\mid G\mid$ is the classifying space of the geometric realization of the nerve of $G$. Applying this result to the ''string 2-group" $\mathrm{String}(G)$ of a simply-connected compact simple Lie group $G$, we obtain a theory of characteristic classes for principal $\mathrm{String}(G)$-2-bundles.