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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.11623 |
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| _version_ | 1866915448647319552 |
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| author | Dagnino, Francesco Farjudian, Amin Moggi, Eugenio |
| author_facet | Dagnino, Francesco Farjudian, Amin Moggi, Eugenio |
| contents | We define a (preorder-enriched) category $\mathsf{Met}$ of quantale-valued metric spaces and uniformly continuous maps, with the essential requirement that the quantales are continuous. For each object $(X,d,Q)$ in this category, where $X$ is the carrier set, $Q$ is a continuous quantale, and $d: X \times X \to Q$ is the metric, we consider a topology $τ_d$ on $X$, which generalizes the open ball topology, and a topology $τ_{d,R}$ on the powerset $\mathsf{P}(X)$, called the robust topology, which captures robustness with respect to small perturbations of parameters. We define a (preorder-enriched) monad $\mathsf{P}_S$ on $\mathsf{Met}$, called the Hausdorff-Smyth monad, which captures the robust topology, in the sense that the open ball topology of the object $\mathsf{P}_S(X,d,Q)$ coincides with the robust topology $τ_{d,R}$ for the object $(X,d,Q)$. We prove that every topology arises from a quantale-valued metric. As such, our framework provides a foundation for quantitative reasoning about imprecision and robustness in a wide range of computational and physical systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_11623 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Robust Topology and the Hausdorff-Smyth Monad on Metric Spaces over Continuous Quantales Dagnino, Francesco Farjudian, Amin Moggi, Eugenio Logic in Computer Science 06B35, 06F07 F.3.2 We define a (preorder-enriched) category $\mathsf{Met}$ of quantale-valued metric spaces and uniformly continuous maps, with the essential requirement that the quantales are continuous. For each object $(X,d,Q)$ in this category, where $X$ is the carrier set, $Q$ is a continuous quantale, and $d: X \times X \to Q$ is the metric, we consider a topology $τ_d$ on $X$, which generalizes the open ball topology, and a topology $τ_{d,R}$ on the powerset $\mathsf{P}(X)$, called the robust topology, which captures robustness with respect to small perturbations of parameters. We define a (preorder-enriched) monad $\mathsf{P}_S$ on $\mathsf{Met}$, called the Hausdorff-Smyth monad, which captures the robust topology, in the sense that the open ball topology of the object $\mathsf{P}_S(X,d,Q)$ coincides with the robust topology $τ_{d,R}$ for the object $(X,d,Q)$. We prove that every topology arises from a quantale-valued metric. As such, our framework provides a foundation for quantitative reasoning about imprecision and robustness in a wide range of computational and physical systems. |
| title | Robust Topology and the Hausdorff-Smyth Monad on Metric Spaces over Continuous Quantales |
| topic | Logic in Computer Science 06B35, 06F07 F.3.2 |
| url | https://arxiv.org/abs/2508.11623 |