John Baez
Categorical Groups,
Institut de Matemàtica, Universitat de Barcelona
June 16, 2008
Classifying Spaces for Topological 2-Groups
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
H1(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, Stevenson and
others generalizing this result to topological 2-groups. We explain
various viewpoints on topological 2-groups and the Cech cohomology
H1(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 H1(M,G) and [M,B|G|],
where B|G| is the classifying space of the geometric realization
of the nerve of G.
Click on this to see the transparencies of the talk:
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Classifying Spaces for Topological 2-Groups - in
PDF
and
Postscript
For a less technical version with more applications, try this:
-
Classifying Spaces for Topological 2-Groups - in
PDF
and
Postscript
These talks summarize the following paper:
which in turn is based on the following work:
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Nils Baas, Marcel Bökstedt and Tore Kro,
2-Categorical K-theories
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John Baez, Alissa Crans, Urs Schreiber and Danny Stevenson,
From Loop
Groups to 2-Groups
-
John Baez and Urs Schreiber,
Higher Gauge Theory
-
Toby Bartels,
Higher Gauge
Theory I: 2-Bundles
-
Lawrence Breen,
Notes on 1- and 2-Gerbes
-
John Duskin,
Simplicial Matrices and the Nerves of Weak n-Categories I:
Nerves of Bicategories
-
Manuel Bullejos and Antonio Cegarra,
On the
Geometry of 2-Categories and their Classifying Spaces
-
Manuel Bullejos, Emilio Faro and Victor Blanco,
A Full and
Faithful Nerve for 2-Categories
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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
