Approximation in quantale-enriched categories

TL;DR

This paper develops a unified theoretical framework for approximation and continuity in quantale-enriched categories, encompassing domain theory, metric, and topological spaces, by introducing new concepts like Scott-continuous distributors.

Contribution

It introduces auxiliary and Scott-continuous distributors, characterizes continuous V- and (U,V)-categories, and unifies various approaches to quantitative domain theory.

Findings

Characterizes continuous V-categories among cocomplete V-categories.

Defines Scott-continuous distributors and way-below distributors.

Provides a flexible theory of continuity applicable to orders, metrics, and topologies.

Abstract

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.