Persistence Diagram Bundles: A Multidimensional Generalization of Vineyards

I introduce the concept of a persistence diagram (PD) bundle, which is the space of PDs for a fibered filtration function (a set $\{f_p: \mathcal{K}^p \to \mathbb{R}\}_{p \in B}$ 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. I prove that if $B$ is a smooth compact manifold, 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 $B$. I 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 $s(p) \in \textrm{PD}(f_p)$ for all $p \in B$). Consequently, a PD bundle is not necessarily the union of "vines" $\gamma: B \to E$; this is unlike a vineyard. When there is a stratification as described above, I construct a cellular sheaf that stores sufficient data to construct sections and determine whether a given local section can be extended to a global section.