We give the reasons for embarking on the computation of crossed modules
as given in a recent GAP package XMOD.
We describe some of the facilities
in this package and discuss some of the computational issues involved in
computing crossed modules induced over subgroup inclusions.
Van Kampen theorems for categories of covering morphisms in
lextensive categories
Abstract:
We show that lextensive categories are a natural setting for statements and
proofs of the ``tautologous'' Van Kampen theorem, in terms of coverings of
a space.
Galois theory of second order covering maps of simplicial sets
Abstract:
We give a version for simplicial sets of a second order notion of covering
map, which bears the same relation to the usual coverings as do groupoids
to sets. The Generalised Galois theory of the second author yields a
classification of such coverings by the action of a certain kind of double
groupoid.
Groupoids and crossed objects in algebraic topology
Notes for lectures at the Summer School in Algebraic Topology,
Grenoble, June 15 - July 5, 1997. (66 pages).
Abstract:
The notes concentrate on the background, intuition,
proof and applications of the 2-dimensional
Van Kampen Theorem (for the fundamental crossed module of a pair),
with sketches of extensions to higher dimensions.
One of the points stressed is how the extension from groups to groupoids
leads to an extension from the abelian homotopy groups to
non abelian higher dimensional generalisations of the fundamental group,
as was sought by the topologists of the early part of this century.
This links with J.H.C. Whitehead's efforts to extend
combinatorial group theory to higher dimensions in terms of
combinatorial homotopy theory, and which analogously motivated his
simple homotopy theory.
Published in:
Homology, homotopy and applications 1 (1999) 1-78.
