Choosing the Geometry: valuations, norms, and contours

Jens Agerberg; Wojciech Chachólski; Christina Kapatsori

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arXiv · 2608.14915

Choosing the Geometry: valuations, norms, and contours

Jens Agerberg·Wojciech Chachólski·Christina Kapatsori

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Abstract

We develop a general framework for constructing pseudometrics on objects of Abelian categories using valuations, which assign non-negative costs to morphisms. Specializing to finitely presented vector space representations of the poset $[0,\infty)^\ell$, a natural setting for multiparameter persistence modules, we show that a large class of these distances is governed by contours, right-continuous lax actions of $[0,\infty)$ on the parameter poset. Properties of contours translate into geometric properties of the induced distances, including conditions under which the resulting pseudometrics are metrics and allowing us to identify a rich family of compact subsets of finitely presented representations.

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Jens Agerberg, Wojciech Chachólski, Christina Kapatsori. 2026-08-14. Choosing the Geometry: valuations, norms, and contours. https://arxiv.org/abs/2608.14915

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