Exact weights and path metrics for triangulated categories and the derived category of persistence modules
Peter Bubenik, Jose A. Velez-Marulanda
TL;DR
The paper develops exact weights on triangulated categories, defines corresponding path metrics via cones in a fixed subcategory, and shows how additive weights or cohomological functors induce exact weights whose path metrics are preserved by triangle-equivalences. It connects these constructions to Brown representability to express metrics in large, generated categories, and introduces Wasserstein distances for triangulated settings. The framework is then applied to persistence modules, deriving quantitative bounds for homotopy and derived categories, and to continuous quivers of type A, where derived-equivalence classifications and Wasserstein distances between simple objects are analyzed. Together, these results provide a robust, computable metric theory for triangulated categories with concrete stability guarantees and broad applicability to persistence theory and representation theory.
Abstract
We define exact weights on a triangulated category to be nonnegative functions on objects satisfying a subadditivity condition with respect to exact triangles. Such weights induce a metric on objects in the triangulated category, which we call a path metric. Our exact weights generalize the rank functions of J.\ Chuang and A.\ Lazarev and are analogous to the exact weights for an exact category given by the first author and J.\ Scott and D.\ Stanley. We show that cohomological functors from a triangulated category to an abelian category with an additive weight induce an exact weight on the triangulated category. We prove that triangle equivalences induce an isometry for the path metrics induced by cohomological functors. In the perfectly generated or compactly generated case, we use Brown representability to express the exact weight on the triangulated category. We give three characterizations of exactness for a weight on a triangulated category and show that they are equivalent. We also define Wasserstein distances for triangulated categories. Finally, we apply our work to derived categories of persistence modules and to representations of continuous quivers of type $\mathbb{A}$.