2-Categorical Aspect of Strong Shape

For every pair (C,K) of groupoid enriched categories, we define two related categories S(C,K) and SS(C,K). In the case (C,K)=(TOP,ANR) they are isomorphic, respectively, to the classical shape category Sh(TOP) of Mardešić and Segal and to the strongshape category of topological spaces SSh(1)(TOP) of height 1, defined by Lisicá and Mardešić. Moreover, for (C,K)=(CM,ANR), where CM is the category of compact metric spaces, there is an isomorphism SSh(CM)≅SS(CM,ANR). A new characterization of topological strongshape equivalences is also given.