From Zero-Dimensional to Continuous Dualities: A Double-Categorical Account
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.
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BibTeX
@misc{kurz2026from,
title = {{From Zero-Dimensional to Continuous Dualities: A Double-Categorical Account}},
author = {Alexander Kurz and M. Andrew Moshier and Achim Jung},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07695},
eprint = {2610.07695},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07695},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Zero-Dimensional to Continuous Dualities: A Double-Categorical Account
AU - Alexander Kurz
AU - M. Andrew Moshier
AU - Achim Jung
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07695
UR - https://arxiv.org/abs/2610.07695
AB - 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.
ER -