Duality theory for enriched Priestley spaces

TL;DR

This paper extends Stone-type dualities to enriched Priestley spaces and quantale-enriched categories, focusing on duality restrictions to functions and exploring algebraic properties of these enriched structures.

Contribution

It advances duality theory by restricting dualities to functions and analyzing algebraic features of quantale-enriched Priestley spaces.

Findings

Duality results for [0,1]-enriched Priestley spaces are restricted to functions.

The category of quantale-enriched Priestley spaces is characterized.

Properties that identify algebraic duals are investigated.

Abstract

The term Stone-type duality often refers to a dual equivalence between a category of lattices or other partially ordered structures on one side and a category of topological structures on the other. This paper is part of a larger endeavour that aims to extend a web of Stone-type dualities from ordered to metric structures and, more generally, to quantale-enriched categories. In particular, we improve our previous work and show how certain duality results for categories of [0,1]-enriched Priestley spaces and [0,1]-enriched relations can be restricted to functions. In a broader context, we investigate the category of quantale-enriched Priestley spaces and continuous functors, with emphasis on those properties which identify the algebraic nature of the dual of this category.