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.