Interleaving distances from height-difference functions on posets

Interleaving distances provide a fundamental tool for comparing persistence modules and have been widely used in topological data analysis. Their definitions are typically based on translation structures (shift operations) on the indexing poset, but on general posets such structures can be scarce, making this framework restrictive. In this paper, we introduce a new interleaving-type distance for functor categories over arbitrary posets, induced by a height-difference function $\rho$. The key idea is to associate to $\rho$ an $\mathbb{R}_{\geq 0}$-indexed family of adjoint endofunctors on $\mathrm{Fun}(P,\mathcal{C})$, which play the role of generalized translations and allow us to formulate interleavings in a purely categorical manner and define the distance $d_{\rho}$, called the height-interleaving distance. In particular, any height function (i.e., a real-valued order-preserving map) canonically induces such a height-difference function, so our framework remains useful on finite posets. Moreover, when $P=\mathbb{R}^d$ and $\rho=\rho_{\mathrm{diag}}$, the resulting distance coincides with the classical interleaving distance for multiparameter persistence modules. However, in general, $d_{\rho}$ need not satisfy the triangle inequality. Under suitable hypotheses (e.g. CIP for $(P,\rho)$; this condition is automatic when $P$ is a tree poset), we prove a triangle inequality up to an additive defect bounded by a constant $c(\rho)$; in particular, when $c(\rho)=0$ this yields an extended pseudo-distance. We also establish a stability property with respect to perturbations of height-difference functions: small changes in $\rho$ induce small changes in the associated height-interleaving distance. Finally, we study an analogue of the erosion-type constructions from classical interleavings within our framework.