Enriched pro-categories and shapes

Given a category $\mathcal{C}$ and a directed partially ordered set $J$, a certain category $pr{o}^J-\mathcal{C}$ 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 $pr{o}^{\ast }$-category $pr{o}^{\ast }-\mathcal{C}$ becomes $pr{o}^{\mathbb{N}}-\mathcal{C}$. Moreover, given a pro-reflective category pair $(\mathcal{C},\mathcal{D})$, the $J$-shape category $S{h}_{(C,\mathcal{D})}^J$ 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.