The Classifying Space of a Topological 2-Group

By John C. Baez, Danny Stevenson. 2008

52 citations recorded by the source corpus.

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 Ĥ1(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–Bokstedt–Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and the Cech cohomology Ĥ1(M, G) with coefficients in a topological 2-group G, also known as “nonabelian cohomology”. Then we give an elementary proof that under mild conditions on M and G there is a bijection Ĥ1(M, G) ≡ [M,B|G|]G] where B|G| is the classifying space of the geometric realization of the nerve of G. Applying this result to the “string 2-group” String(G) of a simply-connected compact simple Lie groupG, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of H*(BG,Q)/‹c›, where c is any generator of H 4(BG,Q).

Subfields: Algebra and Number Theory, Mathematical Physics, Pathology and Forensic Medicine

View Paper: https://www.semanticscholar.org/paper/31448cabcf8b201a22b3bdf074363894e6589cf3

Citations in the corpus

• Principal 2-bundles and their gauge 2-groups (2008, unknown): “homotopy classes [M,B|NG|] and consequently with Hˇ1 top(M,|NG|), where |NG| is a topological group, associated to G (the geometric realisation of the nerve of the category G). This has been shown in [BS08], cf. also [Jur05] (note that G is always well-pointed in our case, for we are only dealing wi”