Generalized persistence diagrams for persistence modules over posets

Abstract

When a category $\\mathcal{C}$ satisfies certain conditions, we define the notion of \\emph{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 $F$. 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.