发表机构
University of Genoa; University of Birmingham（热那亚大学; 伯明翰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究建立了拟一致空间范畴与 quantale 值度量空间范畴的度量化等价定理，细化了拓扑空间的 quantale 度量化构造，将定量鲁棒性分析所用的 Hausdorff-Smyth 单子提升至拟一致空间，构建了三者的统一范畴框架。
AI 中文摘要
我们研究拟一致空间、拓扑空间与 quantale 值度量空间之间的关系。主要结果是一个度量化定理，确立了拟一致空间范畴与 quantale 值度量空间范畴之间的等价性。我们还得到了适用于任意拓扑空间的基于 quantale 的度量化定理，该定理对现有构造进行了细化。这些结果表明拟一致结构是 quantale 值度量的恰当定性对应物。基于这种对应关系，我们证明了用于定量鲁棒性分析的 quantale 值度量空间上的 Hausdorff-Smyth 单子，可通过等价关系提升为拟一致空间上的对应单子。这提供了一个连接拓扑学、拟一致结构与定量鲁棒性分析的统一范畴框架。
英文摘要
We study the relationship between quasi-uniform spaces, topological spaces, and quantale-valued metric spaces. Our main result is a metrization theorem establishing an equivalence between the category of quasi-uniform spaces and a category of quantale-valued metric spaces. We also obtain a quantale-based metrization theorem for arbitrary topological spaces that refines existing constructions. These results identify quasi-uniformities as the appropriate qualitative counterpart of quantale-valued metrics. Building on this correspondence, we show that the Hausdorff-Smyth monad on quantale-valued metric spaces, which is used in quantitative robustness analysis, arises as a lifting of a corresponding monad on quasi-uniform spaces along the equivalence. This provides a unified categorical framework connecting topology, quasi-uniformity, and quantitative robustness analysis.
Comments24 pages
