  A persistence module is a functor $f: \mathbf{I} \to \mathsf{E}$, where
$\mathbf{I}$ is the poset category of a totally ordered set. This work
introduces saecular decomposition: a categorically natural method to decompose
$f$ into simple parts, called interval modules. Saecular decomposition exists
under generic conditions, e.g., when $\mathbf{I}$ is well ordered and
$\mathsf{E}$ is a category of modules or groups. This represents a substantial
generalization of existing factorizations of 1-parameter persistence modules,
leading to, among other things, persistence diagrams not only in homology, but
in homotopy.
  Applications of saecular decomposition include inverse and extension problems
involving filtered topological spaces, the 1-parameter generalized persistence
diagram, and the Leray-Serre spectral sequence. Several examples -- including
cycle representatives for generalized barcodes -- hold special significance for
scientific applications.
  The key tools in this approach are modular and distributive order lattices,
combined with Puppe exact categories.
