Persistence of Morse Decompositions over Grid Resolution for Maps and Time Series

The paper defines grid map morphisms (Definition 4.1) and proves that refinement yields a canonical morphism between grids whose composition is functorial (Proposition 4.2), then lifts this to augmented Morse decompositions via a standard SCC argument (Lemma 4.5), culminating in the functorial persistence result (Theorem 4.6). The candidate constructs the same containment map ι_{k2,k1} (fine cell ↦ unique coarser cell) and shows it is a grid map morphism, induces order-preserving maps on SCCs with componentwise restrictions, and composes as expected. The candidate’s Lemma 1 (uniqueness of the containing coarser cell) follows directly from the grid axioms (nonempty interiors and disjoint interiors) in Definition 2.1, which the paper assumes implicitly. Hence the model’s proof is essentially the same as the paper’s, with a few details made explicit. See the paper’s statements and proofs for Definition 4.1 and Proposition 4.2, including the composition argument, and Lemma 4.5 and Theorem 4.6 for the induced augmented Morse decomposition morphisms and functoriality; the grid axioms used in the candidate’s uniqueness lemma are given in Definition 2.1 (fine cells have nonempty interior and distinct cells have disjoint interiors) .