Enriched Pro-Categories and Shapes

Abstract

Given a category C and a directed partially ordered set J, a certain category pro^J-C on inverse systems in C is constructed such that the ordinary pro-category pro-C is the most special case of a singleton J ≡ {1}. Further, the known pro^*-category pro^*-C becomes pro^N-C. Moreover, given a pro-reflective category pair (C, D), the J-shape category Sh^J_{(C, D)} and the corresponding J-shape functor S^J are constructed which, in mentioned special cases, become the well known ones. Among several important properties, the continuity theorem for a J-shape category is established. It implies the “J-shape theory” is a genuine one such that the shape and the coarse shape theory are its very special instances.