Thickening of the diagonal and interleaving distance

Abstract

Given a topological space X, a thickening kernel is a monoidal presheaf on $({{\mathbb {R}}}_{\ge 0},+)$ 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 (Guillermou et al. in Duke Math J 161:201–245, 2012), 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.