In addition to inherent computational challenges, the absence of a canonical method for quantifying `persistence' in multi-parameter persistent homology remains a hurdle in its application. One of the best known quantifications of persistence for multi-parameter persistent homology is the rank invariant, which has recently evolved into the generalized rank invariant (GRI) by naturally extending its domain. This extension enables us to quantify persistence across a broader range of regions in the indexing poset compared to the rank invariant. However, the size of the domain of the GRI is generally formidable, making it desirable to restrict its domain to a more manageable subset for computational purposes. The foremost questions regarding such a restriction of the domain are: (1) How to restrict, if possible, the domain of the GRI without any loss of information? (2) When can we more compactly encode the GRI as a `persistence diagram'? (3) What is the trade-off between computational efficiency and the discriminating power of the GRI as the amount of the restriction on the domain varies? (4) What proxies exist for persistence diagrams in the multi-parameter setting that can be derived from the GRI? To address the first three questions, we generalize and axiomatize the classic fundamental lemma of persistent homology via the notion of M\"obius invertibility of the GRI which we propose. This extension also contextualizes known results regarding the (generalized) rank invariant within the classical theory of M\"obius inversion. We conduct a comprehensive comparison between M\"obius invertibility and other existing concepts related to the structural simplicity of persistence modules. We address the fourth question through the notion of motivic invariants. We demonstrate that many invariants from the literature can be both derived from the GRI and recast as motivic invariants.

翻译：除固有的计算挑战外，多参数持续同调中缺乏量化“持续性”的规范方法仍是其应用的主要障碍。秩不变量是多参数持续同调中最著名的持续性量化方法之一，近期通过自然扩展其定义域已演化为广义秩不变量（GRI）。相较于秩不变量，此扩展使我们能够在索引偏序集的更广泛区域中量化持续性。然而，GRI的定义域规模通常极为庞大，因此出于计算目的，将其定义域限制在更易处理的子集上是可取的。关于此类定义域限制的核心问题包括：（1）如何在可能的情况下限制GRI的定义域而不损失任何信息？（2）何时能将GRI更紧凑地编码为“持续图”？（3）随着定义域限制程度的变化，GRI的计算效率与判别力之间存在何种权衡？（4）在多参数设置中，存在哪些可从GRI导出的、可作为持续图替代方案的代理不变量？针对前三个问题，我们通过提出的GRI之Möbius可逆性概念，对经典持续同调基本引理进行推广与公理化。此扩展亦将关于（广义）秩不变量的已知结果置于经典的Möbius反演理论框架中。我们对Möbius可逆性与现有关于持续模结构简洁性的其他概念进行了全面比较。通过动机不变量的概念，我们探讨了第四个问题。我们证明文献中的许多不变量既可从GRI导出，亦可重构为动机不变量。