John Baez
Categorical Groups,
Institut de Matemàtica, Universitat de Barcelona
June 16, 2008
Classifying Spaces for Topological 2Groups
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 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, BaasBökstedtKro, Stevenson and
others generalizing this result to topological 2groups. We explain
various viewpoints on topological 2groups and the Cech cohomology
H^{1}(M,G) with coefficients in a topological 2group
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,BG],
where BG is the classifying space of the geometric realization
of the nerve of G.
Click on this to see the transparencies of the talk:

Classifying Spaces for Topological 2Groups  in
PDF
and
Postscript
For a less technical version with more applications, try this:

Classifying Spaces for Topological 2Groups  in
PDF
and
Postscript
These talks summarize the following paper:
which in turn is based on the following work:

Nils Baas, Marcel Bökstedt and Tore Kro,
2Categorical Ktheories

John Baez, Alissa Crans, Urs Schreiber and Danny Stevenson,
From Loop
Groups to 2Groups

John Baez and Urs Schreiber,
Higher Gauge Theory

Toby Bartels,
Higher Gauge
Theory I: 2Bundles

Lawrence Breen,
Notes on 1 and 2Gerbes

John Duskin,
Simplicial Matrices and the Nerves of Weak nCategories I:
Nerves of Bicategories

Manuel Bullejos and Antonio Cegarra,
On the
Geometry of 2Categories and their Classifying Spaces

Manuel Bullejos, Emilio Faro and Victor Blanco,
A Full and
Faithful Nerve for 2Categories

Branislaw Jurco, Crossed
Module Bundle Gerbes; Classification, String Group and Differential
Geometry

André Henriques,
Integrating
L_{∞}Algebras
See also these related talks, which cover other aspects of the
big picture:
© 2008 John Baez
baez@math.removethis.ucr.andthis.edu