The coarse shape

Abstract

Given a category C, a certain category pro*-C on inverse systems in C is constructed, such that the usual pro-category pro-C may be considered as a subcategory of pro*-C. By simulating the (abstract) shape category construction, Sh_(C, D), an (abstract) coarse shape category Sh*_(C, D) is obtained. An appropriate functor of the shape category to the coarse shape category exists. In the case of topological spaces, C=HTop and D=HPol or D=HANR, the corresponding realizing category for Sh* is pro*-HPol or pro*-HANR respectively. Concerning an operative characterization of a coarse shape isomorphism, a full analogue of the well known Morita lemma is proved, while in the case of inverse sequences, a useful sufficient condition is established. It is proved by examples that for C=Grp (groups) and C=HTop, the classification of inverse systems in pro*-C is strictly coarser than in pro-C. Therefore, the underlying coarse shape theory for topological spaces makes sense.