AI 中文总结
本文开发了用赋值构造阿贝尔范畴对象上伪度量的通用框架，针对多参数持续模的自然设置，证明这类距离由轮廓支配，还给出伪度量成为度量的条件并识别有限表示表示的丰富紧子集族。
AI 中文摘要
我们开发了一个通用框架，用于使用赋值（为态射分配非负代价）在阿贝尔范畴的对象上构造伪度量。针对偏序集[0,∞)^ℓ的有限表示向量空间表示（这是多参数持续模的自然设置），我们表明这类距离中的一大类由轮廓（即[0,∞)在参数偏序集上的右连续 lax 作用）支配。轮廓的性质可转化为诱导距离的几何性质，包括所得伪度量成为度量的条件，还能让我们识别有限表示表示的丰富紧子集族。
英文摘要
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.
Comments28 pages
