Algebraic colimit calculations in homotopy theory using fibred and cofibred categories

TL;DR

This paper uses fibred and cofibred categories to unify and extend algebraic colimit calculations in homotopy theory, enabling broader computations of homotopical invariants for complex spaces.

Contribution

It introduces a categorical framework that generalizes homotopical colimit calculations, covering multiple cases with a single result, and applies to spaces with multiple base points.

Findings

Fibred category fibers preserve connected colimits.

Homotopical excision described via modules over groupoids.

Framework applies to pairs of spaces and triads.

Abstract

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.