The interleaving distance was originally defined in the field of Topological Data Analysis (TDA) by Chazal et al. as a metric on the class of persistence modules parametrized over the real line. Bubenik et al. subsequently extended the definition to categories of functors on a poset, the objects in these categories being regarded as `generalized persistence modules'. These metrics typically depend on the choice of a lax semigroup of endomorphisms of the poset. The purpose of the present paper is to develop a more general framework for the notion of interleaving distance using the theory of `actegories'. Specifically, we extend the notion of interleaving distance to arbitrary categories equipped with a flow, i.e. a lax monoidal action by the monoid $[0,\infty)$. In this way, the class of objects in such a category acquires the structure of a Lawvere metric space. Functors that are colax $[0,\infty)$-equivariant yield maps that are 1-Lipschitz. This leads to concise proofs of various known stability results from TDA, by considering appropriate colax $[0,\infty)$-equivariant functors. Along the way, we show that several common metrics, including the Hausdorff distance and the $L^\infty$-norm, can be realized as interleaving distances in this general perspective.
Keywords: Topological Data Analysis, Persistent Homology, Category Theory, Lawvere Metric Spaces
2010 MSC: 18C10, 18D05, 18D10, 18D20 and 55N10
Theory and Applications of Categories, Vol. 33, 2018, No. 21, pp 583-607.