  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.
