Generalized Persistence Diagrams for Persistence Modules over Posets

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