Thickening of the diagonal and interleaving distance

Given a topological space X, thickening kernel is monoidal presheaf on $$({{\mathbb {R}}}_{\ge 0},+)$$ with values in the category of derived kernels X. A bi-thickening defined {R}}},+)$$ . To such kernel, one naturally associates an interleaving distance sheaves We prove that exists and unique as soon it interval containing 0, allowing us to construct (bi-)thickenings two different situations. First, when X “good” metric space, starting small usual thickenings diagonal. The associated satisfies stability property Lipschitz give rise maps. Second, by using (Guillermou et al. Duke Math J 161:201–245, 2012), manifold given non-positive Hamiltonian isotopy cotangent bundle. In case complete Riemannian having strictly positive convexity radius, we good diagonal, distance, other geodesic flow, coincide.