Exact sequences of fibrations of crossed complexes,
homotopy classification of maps, and nonabelian extensions of groups
Summary:
Higher Homotopy van Kampen Theorems allow the computation as colimits
of certain homotopical invariants of glued spaces.
One corollary is to describe homotopical excision in critical dimensions
in terms of induced modules and crossed modules over groupoids.
This paper shows how fibred and cofibred categories give an overall context
for discussing and computing such constructions,
allowing one result to cover many cases.
A useful general result is that the inclusion of a fibre of a fibred category
preserves connected colimits.
The main homotopical application are to pairs of spaces
with several base points,
but we also describe briefly the situation for triads.
