Subject: categories: Schreier theory

From: Ronald Brown <[email-masked]>
Date: Thu, 17 Nov 2005 14:32:24 +0000

John Baez gave an interesting account of nonabelian cohomology and
extension theory. Here are a few more references with which I have been
involved,  all using crossed complexes and free crossed resolutions:

1) (with P.J. HIGGINS), ``Crossed complexes and non-abelian
extensions'', {\em Category theory proceedings, Gummersbach},
1981, Lecture  Notes in Math. 962 (ed. K.H. Kamps et al, Springer,
Berlin, 1982), pp.  39-50.

This generalises classical Schreier theory to extensions of groupoids.

2)   (with O. MUCUK), ``Covering groups of non-connected
topological  groups revisited'',  {\em Math. Proc. Camb. Phil.
Soc},  115 (1994) 97-110.

This relates the theory of covering topological groups of non connected
topological groups to the classical theory of extensions and
obstructions to a Q-kernel with an invariant in H^3. It uses the
properties of the internal hom in crossed complexes CRS(F,C) , and exact
sequences derived from a fibration C \to D and the induced fibration on
CRS(F, -).

3) (with T. PORTER), ``On the Schreier theory of non-abelian
extensions: generalisations and computations''. {\em Proceedings
Royal Irish Academy} 96A (1996) 213-227.

This establises a generalisation of the Schreier theory in two ways (but
only for groups). One is using coefficients in a crossed module,
following Dedecker's key ideas, as in the references in John's account.
Second it shows how to compute with such extensions

A \to E \to G

in terms of presentations of the group G.  This involves identities
among relations for the presentation, as shown originally by Turing in

Turing, A. 1938. The extensions of a group. {\em Compositio
Mathematica.} {\bf 5 }  357-367

The advantage of this method is that one can actually do sums, even when
the group G may be infinite. The example given by us is G= trefoil group
with two generators x,y and relation x^3=y^2. This presentation has no
identities among relations, and so the calculation is especially simple.
Equivalence of extensions is described in terms of homotopies of
morphisms of crossed complexes, and this relates the ideas to other
forms of homological or homotopical algebra.

An advantage of this approach is the ability to calculate some small
free crossed resolutions of some groups: this is one reason for using
crossed complexes. Note that a convenient monoidal closed structure on
the category of crossed complexes has been explicitly written down, and
this allows convenient calculation and representations of homotopies,
using the `unit interval' groupid, as a crossed complex.

One of the problems I have with the globular approach is the difficulty
of writing down homotopies, and higher homotopies. For example, Ilhan
Icen and I found it difficult to rewrite in terms of group-groupoids the
well known Whitehead theory of automorphisms of crossed modules,
explained for the crossed modules of groupoids case in

(with \.I. \.I\c cen ), `Homotopies and automorphisms of crossed modules
of groupoids', Applied Categorical Structures,  11 (2003) 185-206.

It looks as if it would be better expressed in terms of automorphisms of
2-groupoids: good marks to anyone who writes it down in that way!

One knows such homotopies  of globular infinity groupoids exist because
globular  infinity-groupoids are equivalent to crossed complexes

 (with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids and
crossed  complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981) 371-386.

(This paper contains an early definition of a (strict, globular)
infinity category.)

This raises a question: what is the crossed complex associated to the
free globular groupoid on one generator of dimension n? I have a
round-about sketch proof, using also cubical theory, and a van Kampen
theorem,  that it *is* the fundamental crossed complex of the n-globe.
Does anyone have a purely  algebraic proof?

The idea of singular nonabelian cohomology of a space X with
coefficients in a crossed complex C is given in

(with P.J.HIGGINS), ``The classifying space of a crossed complex'',
{\em Math. Proc. Camb. Phil. Soc.} 110 (1991) 95-120.

This cohomology is give by [\Pi SX, C], homotopy classes of maps from
the fundamental crossed complex of the singular complex of X, to C.
There is also a Cech version (current work with Jim Glazebrook and Tim
Porter).

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).

Ronnie Brown
www.bangor.ac.uk/r.brown

Ronnie Brown

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From: jim stasheff <[email-masked]>
Date: Fri, 18 Nov 2005 08:21:39 -0500

> at math.DG/0402007 (not using crossed complexes).
This brings to mind an interesting (but insufficiently well known)
treatment by I htink) Dennis Johnson of  extensions
A \to E \to G
of topological groups which are also principal bundles

the classification of such extensions fits into an exact sequence

X --> Y  --> Z
where Z classifies the bundle - forgetting the group structure
and  X is the appropriate H^2 classifying split topolgocical
group extensions i.e. A x G as bundles

jim

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From: Kirill Mackenzie <[email-masked]>
Date: Fri, 25 Nov 2005 16:36:59 +0000 (GMT)

> 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/
=============================================