Generalized Persistence Diagrams for Persistence Modules over Posets
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When a category C satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors F:P→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:P→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 P of F: P→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 A persistence diagram to Lipschitz continuity theorem for the category of sets.
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