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24 January 2021 

   

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The Classifying Space of a Topological 2Group  John C. Baez
; Danny Stevenson
;  Date: 
24 Jan 2008  Abstract:  Categorifying the concept of topological group, one obtains the notion of a
’topological 2group’. This in turn allows a theory of ’principal 2bundles’
generalizing the usual theory of principal bundles. It is wellknown that under
mild conditions on a topological group G and a space M, principal Gbundles
over M are classified by either the first Cech cohomology of M with
coefficients in G, or the set of homotopy classes [M,BG], where BG is the
classifying space of G. Here we review work by Bartels, Jurco,
BaasBokstedtKro, and others generalizing this result to topological 2groups
and even topological 2categories. We explain various viewpoints on topological
2groups and Cech cohomology with coefficients in a topological 2group C, also
known as ’nonabelian cohomology’. Then we give an elementary proof that under
mild conditions on M and C there is a bijection between the first Cech
cohomology of M with coefficients in C and [M,BC] where BC is the
classifying space of the geometric realization of the nerve of C. Applying this
result to the ’string 2group’ String(G) of a simplyconnected compact simple
Lie group G, it follows that principal String(G)2bundles have rational
characteristic classes coming from elements of the rational cohomology of BG
modulo the ideal generated by c, where c is any nonzero element in the 4th
cohomology of BG.  Source:  arXiv, 0801.3843  Services:  Forum  Review  PDF  Favorites 


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