When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P} \rightarrow \mathcal{C}$ from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules $F:\mathbf{P} \rightarrow \mathbf{vec}$ of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset $\mathbf{P}$ of $F: \mathbf{P} \rightarrow \mathcal{C}$ in defining Patel's generalized persistence diagram of $F$. Of particular importance is the fact that the generalized persistence diagram of $F$ is defined regardless of whether $F$ is interval decomposable or not. By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type $\mathcal{A}$ persistence diagram to Lipschitz continuity theorem for the category of sets.

翻译：当某类 $\ mathcal{C} 符合某些条件时, 我们定义了任意制表成指数式运算器的等级变异值概念 $F:\ mathbf{P}\rightrow\mathcal{C} 美元 从分类理论角度对任意制成指数式运算器的等级变异值 $F:\ mathbf{C} 从分类理论角度对任意制成指数制成指数式的等级变异值定义 $F:\ mathbf{C} 我们定义 Patel 通用的坚持性图表 $Frightror literbbbf{vec} 。 普通化这个概念允许自由选择制成指数式 $F:\ mathbf{Ptel 以及 Patel 最近的扩展。 具体而言, 在定义 Pateltel 通用的 定义 $F$F$ 的坚持性图表时, 尤其重要的是, $F$ 的通用持续性图表是定义的, 不论关于 $F$ $ rof$ liental developality tyality tyality tylate tyleglemental ty rolate rolate rodudeme roductions roductions mations rolate maus the liver maclate macol max max mausm max or decol maus or decilental decild max max or decol max max max max max max max max max max max max max max max max max max ro ro or ro ro ro routal or ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ma ro ro routal de ro ro ro ro ro ro ro ro ro