Homotopical approach to strong shape or completion theory
Strong shape theory has been introduced in the seventies by different authors and in different settings e.g. by Edwards and Hastings and by the reviewer. The main idea is embodied in the use of specific homotopies between mappings into ''good'' spaces satisfying certain coherence conditions (rather than merely employing homotopy classes which leads to ordinary shape theory). So, in both cases a strong shape category is determined by a pair of categories \(M\subset C\) (M standing for ''good'' objects). Based on this scheme there have appeared different models for strong shape categories for topological spaces [cf. for example the reviewer, Lect. Notes Math. 1060, 119-128 (1984; Zbl 0553.55007); the authors, Topology Appl. 15, 119-130 (1983; Zbl 0505.55012)]. In this paper the authors propose an entirely categorical approach to strong shape theory: Given a pair of categories \(M\subset C\), being equipped with a ''simplicial enrichment of C'' (generalizing the structure entering into Top or pro-Top by the existence of n-homotopies) the authors offer various systems of conditions which lead to a strong shape structure on a purely categorical basis. Most examples deal with pro- categories. One of the central aims is to prove the properties of closed model categories. There are, apart from (topological) strong shape theory, other interesting examples like R-completion theories (Bousfield- Kan) and the case where M consists of Eilenberg-MacLane spaces. Other objectives of the paper are: (1) Representation of the final category as a quotient category, (2) enlarging a given category M (of ''good'' objects) in such a way that the required conditions for the validity of the main theorems of the paper are fulfilled; (3) investigation of the relationship between strong and ordinary shape (defined in the categorical terms of the authors). Furthermore the paper contains some interesting results about topological strong shape theory for compacta.
topological strong shape theory for compacta
scientific article; zbMATH DE number 3944836
scientific article