从零维到连续对偶：一个双范畴解释
From Zero-Dimensional to Continuous Dualities: A Double-Categorical Account
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中文总结 AI 辅助
本文提出一种基于关系 Stone 对偶和双范畴的三步构造方法，系统地从零维对偶（如 Stone 空间）构造连续对偶（如紧 Hausdorff 空间），并统一处理函数与关系态射。
中文摘要 AI 辅助
我们研究如何利用代数、拓扑、范畴论和域论中的众所周知的方法，系统地由零维对偶构造连续对偶。尽管我们的方法是通用的，但本文聚焦于从 Stone 空间到紧 Hausdorff 空间的过渡，以及从 Priestley 空间到紧有序 Hausdorff 空间的过渡。我们方法的引擎是关系的 Stone 对偶：在空间一侧，通过预序取商将零维空间变为连续空间，而带邻近关系的分配格则是它们的代数对偶。我们的关系对偶本质上是序 enriched 的。双范畴将函数态射和关系态射组织在同一个结构中。从零维到连续对偶的过渡因此是一个三步构造：将对偶从函数态射扩展到关系态射，分裂幂等元，并限制到映射。
英文摘要
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.
