Algebraic colimit calculations in homotopy theory using fibred and cofibred categories

Details Algebraic colimit calculations in homotopy theory using fibred and cofibred categories Algebraic colimit calculations in homotopy theory using fibred and cofibred categories

2008-09-24

arxiv.org/abs/0809.4192v2

Mathematics

Algebraic Topology

31 pages, xy, tac.cls v2. 29/09/08. 32 pages. This now contains a detailed proof of Theorem 4.4, on the computation of colimit

Scientific paper

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.

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