Abstract Homotopy and Simple Homotopy Theory
The abstract homotopy theory is based on the observation that analogues of much of the topological homotopy theory and simple homotopy theory exist in many other categories (e.g. spaces over a fixed base, groupoids, chain complexes, 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. This demonstrates the logical interdependence of properties (in particular the existence of certain Kan fillers in associated cubical sets) and results (Puppe sequences, Vogt's Iemma, Dold's theorem on fibre homotopy equivalences, and homotopy coherence theory).
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abelian category abelian group abstract homotopy theory additive algebraic axioms bijection category of cofibrant Ch(A chain complexes Chapter cocylinder cofibrant objects colimits commutative diagram composition cone monad construction coproduct Corollary corresponding cotensored crossed complexes cubical set cylinder object defined Definition denote dimension Dold's theorem double mapping cylinder dual element enriched categories epimorphism equivalence relation exact sequence example Exercise exists factorisation fibration filler finite fundamental groupoid given gives hence Ho(C homotopy coherent homotopy commutative square homotopy equivalence homotopy inverse i?-module identity inclusion induced isomorphism left adjoint Lemma mapping cylinder module natural transformation notation obtain phism preserves pushouts Proof Proposition prove pullback pushout Qrpd relative injective Remark resp result S-category satisfies Simp Simp(A simple equivalence simple homotopy theory simplicial set simplicially enriched split monomorphism stably cofree structure Suppose tensored topological spaces track homotopy commutative trivial cofibration weak equivalence
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