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When a category $\mathcal{C}$ satisfies certain conditions, we define thenotion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P}\rightarrow \mathcal{C}$ from a category theory perspective. This generalizesthe 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 canbe 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'sgeneralized persistence diagram of $F$. Of particular importance is the factthat the generalized persistence diagram of $F$ is defined regardless ofwhether $F$ is interval decomposable or not. By specializing our idea to zigzagpersistence modules, we also show that the barcode of a Reeb graph can beobtained in a purely set-theoretic setting without passing to the category ofvector spaces. This leads to a promotion of Patel's semicontinuity theoremabout type $\mathcal{A}$ persistence diagram to Lipschitz continuity theoremfor the category of sets.

Generalized Persistence Diagrams for Persistence Modules over Posets