2 research outputs found
Approximation in quantale-enriched categories
Our work is a fundamental study of the notion of approximation in
V-categories and in (U,V)-categories, for a quantale V and the ultrafilter
monad U. We introduce auxiliary, approximating and Scott-continuous
distributors, the way-below distributor, and continuity of V- and
(U,V)-categories. We fully characterize continuous V-categories (resp.
(U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in
the same ways as continuous domains are characterized among all dcpos. By
varying the choice of the quantale V 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.Comment: 17 page