We introduce the concept of a persistence diagram (PD) bundle, which is the space of PDs for a fibered filtration function (a set \(\{f_p: {\mathcal {X}}^p\rightarrow \mathbb {R}\}_{p\in B}\) of filtrations that is parameterized by a topological space \(B\) ). Special cases include vineyards, the persistent homology transform, and fibered barcodes for multiparameter persistence modules. We prove that if \(B\) is a smooth compact manifold and \(\mathcal {X}^p \equiv \mathcal {X}\) is a simplicial complex, then for a generic fibered filtration function, \(B\) is stratified such that within each stratum \(Y \subseteq B\) , there is a single PD “template” (a list of “birth” and “death” simplices) that can be used to obtain the PD for the filtration \(f_p\) for any \(p\in Y\) . If \(B\) is compact, then there are finitely many strata, so the PD bundle for a generic fibered filtration on \(B\) is determined by the persistent homology at finitely many points in \(B\) . We also show that not every local section can be extended to a global section (a continuous map s from \(B\) to the total space E of PDs such that \(s(p) \in \text {PD}(f_p)\) for all \(p\in B\) ). Consequently, a PD bundle is not necessarily the union of “vines” \(\gamma : B\rightarrow E\) ; this is unlike a vineyard. When there is a stratification as described above, we construct a cellular sheaf that stores sufficient data to obtain the PDs for each \(p \in B\) , to connect \(\text {PD}(f_p)\) to \(\text {PD}(f_{p'})\) for nearby \(p, p'\) , and to calculate sections when they exist.
