Approximation in quantale-enriched categories

Our work is a foundational study of the notion of approximation in Q-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Q- and (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (resp. (U,Q)-categories) in the same ways as continuous domains are characterized among all dcpos. By varying the choice of the quantale Q and the notion of ideals, and by further allowing the ultrafilter monad to act on the quantale, we obtain a flexible theory of continuity that applies to partial orders and to metric and topological spaces. We demonstrate on examples that our theory unifies some major approaches to quantitative domain theory.