Metrization of Quasi-Uniformities, Powerset Monads, and Qualitative Robustness Analysis
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.
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BibTeX
@misc{dagnino2026metrization,
title = {{Metrization of Quasi-Uniformities, Powerset Monads, and Qualitative Robustness Analysis}},
author = {Francesco Dagnino and Amin Farjudian and Eugenio Moggi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02598},
eprint = {2609.02598},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02598},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Metrization of Quasi-Uniformities, Powerset Monads, and Qualitative Robustness Analysis
AU - Francesco Dagnino
AU - Amin Farjudian
AU - Eugenio Moggi
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02598
UR - https://arxiv.org/abs/2609.02598
AB - 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.
ER -