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.