Abstract Homotopy And Simple Homotopy Theory

Abstract homotopy theory is based on the observation that analogues of much of topological homotopy theory and simple homotopy theory exist in many other categories, such as spaces over a fixed base, groupoids, chain complexes and module categories. Studying categorical versions of homotopy structure, such as cylinders and path space constructions enables not only a unified development of many examples of known homotopy theories, but also reveals the inner working of the classical spatial theory, clearly indicating the logical interdependence of properties (in particular the existence of certain Kan fillers in associated cubical sets) and results (Puppe sequences, Vogt's lemma, Dold's Theorem on fibre homotopy equivalences, and homotopy coherence theory).