Mathematics Preprints 2008
Category Theory & Homotopy Theory
08.06 : BROWN, R. & SIVERA, R.
Algebraic colimit calculations in homotopy theory
using fibred and cofibred categories
Higher Homotopy van Kampen Theorems allow some colimit calculations 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 applications are to pairs of spaces
with several base points;
we also describe briefly applications to triads.
Theory & Applications of Categories
22 (2009) 222-251.