Abstract Homotopy and Simple Homotopy TheoryThe 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). |
Contents
Homotopical algebra པྲ | 76 |
Case studies | 145 |
Groupoid enrichment and track homotopy | 253 |
Homotopy coherence | 307 |
Abstract simple homotopy theories | 342 |
Injective simple homotopy theories | 402 |
Glossary of terms from category theory | 429 |
| 447 | |
| 455 | |
Other editions - View all
Common terms and phrases
abelian category abelian group abstract homotopy theory additive algebraic axioms bijection category of cofibrant Ch(A chain complexes cocylinder cofibrant objects colimits commutative diagram composition construction coproduct corresponding cotensored crossed complexes cubical set defined Definition denote Dold's theorem double mapping cylinder dual enriched categories eo(X equivalence relation exact sequence example Exercise factorisation fibration filler finite fundamental groupoid ƒ and g given gives Grpd hence Ho(C homotopy coherent homotopy commutative square homotopy equivalence homotopy inverse II(X induced isomorphism Lemma mapping cylinder module monomorphism morphism f natural transformation notation notion phism preserves pushouts Proof Proposition prove pullback pushout R-module 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 Top(X topological spaces trivial cofibration weak equivalence