From Zero-Dimensional to Continuous Dualities: A Double-Categorical Account

2026-10-05

AI Summary

This study addresses the challenge of systematically extending zero-dimensional dualities, such as Stone spaces, to continuous dualities, including compact Hausdorff spaces. To this end, it proposes a relation-based bicategorical framework that integrates distributive lattices, domain theory, and topological algebra, unifying the treatment of functional and relational morphisms by combining algebraic proximity relations with topological preorders. The transformation is achieved through a three-step procedure comprising the extension of relational morphisms, the splitting of idempotents, and the restriction of mappings. This work successfully establishes a systematic duality translation mechanism from Priestley and Stone spaces to their corresponding continuous counterparts, thereby providing a unified paradigm for duality theory bridging discrete and continuous structures.

Abstract

We investigate how to systematically construct continuous dualities from zero-dimensional dualities, employing well-known methods from algebra, topology, category theory, and domain theory. While our method is general, this paper focusses on the move from Stone spaces to compact Hausdorff spaces and the move from Priestley spaces to compact ordered Hausdorff spaces. The engine of our approach is Stone duality for relations: on the space side quotienting by a preorder turns zero-dimensional spaces into continuous ones, while distributive lattices with a proximity relation are their algebraic duals. Our duality for relations is inherently order-enriched. Double categories organise both functional and relational morphism in the same structure. The move from zero-dimensional to continuous dualities is then a three-step construction: extend a duality from functional to relational morphism, split idempotents, restrict to maps.

Problem

• continuous dualities
• zero-dimensional dualities
• Stone spaces
• compact Hausdorff spaces
• double categories

Innovation

• Stone duality
• Double categories
• Continuous dualities
• Idempotent splitting
• Proximity relation