Thickening of the diagonal and interleaving distance

TL;DR

This paper develops a theoretical framework for constructing and analyzing interleaving distances on derived categories of sheaves using thickening kernels, with applications to metric spaces and manifolds.

Contribution

It introduces the existence and uniqueness of thickening kernels on intervals containing zero, enabling new constructions of interleaving distances in geometric and topological contexts.

Findings

Existence and uniqueness of thickening kernels on suitable intervals.

Interleaving distances satisfy stability and Lipschitz properties.

Bi-thickening kernels coincide on certain Riemannian manifolds.

Abstract

Given a topological space $X$, a thickening kernel is a monoidal presheaf on $(\mathbb{R}_{\geq0},+)$ with values in the monoidal category of derived kernels on $X$. A bi-thickening kernel is defined on $(\mathbb{R},+)$. To such a thickening kernel, one naturally associates an interleaving distance on the derived category of sheaves on $X$. We prove that a thickening kernel exists and is unique as soon as it is defined on an interval containing $0$, allowing us to construct (bi-)thickenings in two different situations. First, when $X$ is a ``good'' metric space, starting with small usual thickenings of the diagonal. The associated interleaving distance satisfies the stability property and Lipschitz kernels give rise to Lipschitz maps. Second, by using [GKS12], when $X$ is a manifold and one is given a non-positive Hamiltonian isotopy on the cotangent bundle. In case $X$ is a complete Riemannian manifold having a strictly positive convexity radius, we prove that it is a good metric space and that the two bi-thickening kernels of the diagonal, one associated with the distance, the other with the geodesic flow, coincide.