A property of the interleaving distance for sheaves

Abstract

Let $X$ be a real analytic manifold endowed with a distance satisfying suitable properties and let $k$ be a field. In [Petit and Schapira, Selecta Math. 29 (2023), no. 70, DOI 10.1007/s00029-023-00875-6], the authors construct a pseudo-distance on the derived category of sheaves of $k$-modules on $X$, generalizing a previous construction of [Kashiwara and Schapira, J. Appl. Comput. Math. Topol. 2 (2018), 83–113]. We prove here that if the distance between two constructible sheaves with compact support (or more generally, constructible sheaves up to infinity) on $X$ is zero, then these two sheaves are isomorphic, answering a question of [Kashiwara and Schapira, J. Appl. Comput. Math. Topol. 2 (2018), 83–113]. We also prove that our results imply a similar statement for finitely presentable persistence modules due to Lesnick.