Generalized persistence diagrams for persistence modules over posets

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