Re: Schreier theory

[Kirill Mackenzie <K.Mackenzie@sheffield.ac.uk>]

On Thu, 17 Nov 2005, Ronald Brown wrote:

>
> An interesting problem is to classify extensions of crossed complexes!
>
> There is an interesting account  of extensions of principal bundles and
> transitive Lie groupoids by Androulidakis, developing work of Mackenzie,
> at math.DG/0402007 (not using crossed complexes).
>

Apropos the classification of extensions of principal bundles
(equivalently, of locally trivial Lie groupoids) :

The 1989 paper on extensions of principal bundles which Iakovos
Androulidakis developed,

@article { ,
     AUTHOR = {Mackenzie, Kirill},
      TITLE = {Classification of principal bundles and {L}ie groupoids with
               prescribed gauge group bundle},
    JOURNAL = {J. Pure Appl. Algebra},
   FJOURNAL = {Journal of Pure and Applied Algebra},
     VOLUME = {58},
       YEAR = {1989},
     NUMBER = {2},
      PAGES = {181--208},
       ISSN = {0022-4049},
      CODEN = {JPAAA2},
    MRCLASS = {58H05 (20L05 22E99 55R15 58A99)},
}

actually set out to demonstrate that non-abelian cohomology
is _not_ needed for the classification.

The standard classification of principal bundles P(M,G) with
M, G given is by nonabelian H^1. My point was that it is at
least equally interesting to classify possible P(M,G) with
not just G prescribed, but the bundle of Lie groups associated
to P through the inner automorphism action. This is variously
called the gauge group bundle or (I think misleadingly) the
adjoint group bundle.

Given M and a bundle of Lie groups B on M, there is an
obstruction class to the existence of a principal bundle
P(M,G) with gauge group bundle B. If such a P exists then
all possible such P are classified in the usual way by abelian
cohomology.

This approach extends in principle to general extensions of
principal bundles.

This approach arose because in the corresponding problem on
the infinitesimal level, it is certainly more natural to
classify transitive Lie algebroids with prescribed adjoint
bundle. (The adjoint bundle of a transitive Lie algebroid is
the kernel of the anchor map. For the Atiyah sequence of a
principal bundle it is the bundle associated to P through the
adjoint representation.)

Kirill

=============================================
Kirill C H Mackenzie
Department of Pure Mathematics
University of Sheffield
Sheffield S3 7RH
United Kingdom
http://www.shef.ac.uk/~pm1kchm/
=============================================