Temporalize structured decompositions for cumulative narratives

Determine whether dualizing the assumptions used to temporalize spined structured-decomposition categories yields a spined sd-category of cumulative narratives of the form \((\mathsf{Cu},G,\check{\Omega})\), and derive the corresponding proof or identify the necessary modifications.

Background

The paper constructs temporalized spined sd-categories for persistent narratives under hypotheses involving pullbacks, cocompleteness, sheafification, and pushouts along monomorphisms. It then defines an analogous filtration $\check{\Omega}$ for cumulative narratives using a pullback construction over a sub-time-category.

Whether the dual hypotheses suffice to reproduce the spined sd-category structure on cumulative narratives is left unresolved, as is the manner in which the proof must be adapted from the persistent setting.