Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Institutions: University of Genoa, University of Birmingham
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207829970