2-Categorical aspects 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 strong shape 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 strong shape equivalences is also given.