Strong shape in categories enriched over groupoids

Abstract

For any pair of categories \((\mathsf {C,K})\) enriched over the category \(\mathsf {Gpd}\) of groupoids, it is possible to define a strong shape category \(SSh(\mathsf {C,K})\) in such a way that, for \(\mathsf {C}\) the category of topological spaces and \(\mathsf {K}\) its full subcategory of spaces having the homotopy type of absolute neighborhoods retracts for metric spaces, one obtains the strong shape category \(SSh(\mathsf {Top})\), as defined by Marde?i?. We also introduce a new category \(SS_{\tiny \mathsf K}\) with the same objects as \(\mathsf {C}\) and morphisms given by suitable pseudo-natural transformations into the category of groupoids. The main result is then that such a category \(SS_{\tiny \tiny \mathsf K}\) is isomorphic to the strong shape category \(SSh(\mathsf {C,K})\), when \(\mathsf {C}\) is also a proper model category.