(untitled) (Contemporary Mathematics, 2022)
Given a category C and a directed partially ordered set J, a certain category proJ-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 proN-C. Moreover, given a pro-reflective category pair (C, D), the J-shape category ShJ(C, D) and the corresponding J-shape functor SJ 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.