Characterize rigidity classes of narratives

Determine which left-rigid, right-rigid, rigid, and loose narrative classes arise in a given ambient category, and characterize these classes intrinsically in categorical terms, including their behavior under products, limits, colimits, and functorial changes of the ambient category.

Background

The persistence–accumulation adjunction generally loses information when a persistent narrative is converted to a cumulative narrative and then reconstructed, or vice versa. The paper introduces left rigidity, right rigidity, rigidity, and looseness according to whether the unit and counit of the adjunction are isomorphisms. Beyond examples in adhesive categories and simple posets, the structural conditions governing these classes are not established.

The proposed problem seeks both a classification of which rigidity behaviors can occur in a target category and intrinsic categorical criteria for recognizing them. It also asks whether rigidity is preserved or reflected by standard categorical constructions and by changes of the ambient data category.

References

A first natural question is to determine which rigidity classes arise in a given ambient category and how they reflect its structural properties. More generally, one may seek intrinsic categorical characterizations of left, right, and rigid narratives, as well as investigate how rigidity behaves under products, limits, colimits, functorial changes of the ambient category, and other categorical constructions.

— Time-Varying Data as Sheaves: an Invitation to Narratives  (2609.09056 - Leal et al., 8 Sep 2026) in Conclusion and future directions, paragraph “Towards a characterization of the fixed points of the adjunction”