Whole-space analog of Gubler–Jell–Rabinoff coefficients for tropical intersection homology

Ascertain whether the coefficient system F_{p,w} used to define tropical intersection homology on the subset X_sm (the union of polyhedra of dimension at least d−1 with dense intersection in N_R) can be extended to the entire tropical variety X so that it constitutes a genuine analog of the coefficients introduced by Gubler–Jell–Rabinoff for Berkovich analytic spaces and weighted metric graphs, preserving the intended duality and functorial properties.

Background

In constructing tropical intersection homology, the paper defines a coefficient system F_{p,w} tailored to the subset X_sm of the tropical variety X (regular at infinity), motivated by achieving Poincaré–Verdier duality near strata that play the role of codimension-one singularities. This choice is inspired by coefficients developed by Gubler–Jell–Rabinoff for Berkovich analytic spaces and weighted metric graphs, which are known to yield robust duality properties in one-dimensional tropical settings.

The author notes uncertainty about whether these coefficients are true analogs of Gubler–Jell–Rabinoff’s system when considered beyond X_sm, i.e., on the whole tropical variety X. Clarifying this would strengthen the conceptual foundations of the theory and potentially broaden the scope of the duality results established in the paper.