2-categorical aspects of strong shape (Q2502955)

For every g.e. pair \((C,K)\), that is a pair of 2-categories \(K\) in \(C\) enriched over the category \(GPD\) of groupoids, the author considers two related categories \(S(C,K)\) and \(SS(C,K)\). He shows that the category \([K,GPD]\) of 2-functors and lax natural transformations can be used to describe the strong shape category of compact metric spaces \(Ssh(CM)\) and the strong shape category of topological spaces \(Ssh(1)(Top)\) of height 1 as defined by Lisica and Mardesic. In the case \((C,K)=(Top,ANR)\) they are isomorphic, respectively, to the classical shape category \(Sh(Top)\) of Mardesic and Segal and to the strong shape category of topological spaces \(Ssh(1)(Top)\). For \((C,K)=(CM,ANR)\) there is an isomorphism between \(Ssh(CM)\) and \(SS(CM,ANR)\). A new characterization of topological strong shape equivalences is also given.