Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph properties. Precisely, given a feature of interest on graphs, it is possible to build two types of persistence (steady and ranging persistence) that follow the evolution of the feature along graph filtrations. This study extends steady and ranging persistence to other objects using category theory and investigates the stability of such persistence. In particular, a characterization of the features that induce balanced steady and ranging persistence is provided. The main results of this study are illustrated using a practical implementation for hypergraphs.

翻译：持续同调是一种拓扑数据分析工具，已被广泛推广，其应用范围超越了拓扑学领域。在其扩展中，稳态持续性和范围持续性被开发用于研究各种图属性。具体而言，给定图上感兴趣的某种特征，可以构建两种类型的持续性（稳态持续性与范围持续性），以追踪该特征沿图过滤结构的演变过程。本研究利用范畴论将稳态持续性和范围持续性推广至其他对象，并探究此类持续性的稳定性。特别地，我们给出了诱导平衡稳态持续性与范围持续性的特征刻画。本研究的主要结果通过超图上的实际实现加以展示。