Enriched pro-categories and shapes

Abstract: Given a category $\mathcal C$ and a directed partially ordered set $J$, a certain category $pro^J -\mathcal C$ on inverse systems in $\mathcal C$ is constructed such that the ordinary pro-category $pro-\mathcal C$ is the most special case of a singleton $J \equiv {1}$. Further, the known $pro^*$-category $pro ^*-\mathcal C$ becomes $pro ^{\mathbb N }-\mathcal C$. Moreover, given a pro-reflective category pair $(\mathcal C, \mathcal D)$, the $J$-shape category $Sh^J_{(C,\mathcal D)}$ 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 examples.