Theory of interleavings on categories with a flow

Abstract

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, ∞). In this way, the class of objects in such a category acquires the structure of a Lawvere metric space. Functors that are colax [0, ∞)-equivariant yield maps that are 1-Lipschitz. This leads to concise proofs of various known stability results from TDA, by considering appropriate colax [0, ∞)-equivariant functors. Along the way, we show that several common metrics, including the Hausdorff distance and the L ∞ -norm, can be realized as interleaving distances in this general perspective.

Beyond topological persistence: Starting from networks

“…Persistence diagrams are but a shadow of much more general and powerful tools: persistence modules and further [3,24,11], on which the interleaving distance plays a central role. It is necessary to connect the ideas of the present paper to that research domain.…”

“…We are aware of the recent categorical generalizations of persistence [3,24,26,11]. Nonetheless, they seem to be rather far from the agile tool for applications we want to make available to the scientific community.…”

Steady and ranging sets in graph persistence

“…St. Louis (17) St. Paul-Minneapolis (38) Tampa ( 19) Toronto (26) Tucson (10) Vancouver (18) Washington (32)…”

“…We are therefore rather far from the categorifications of [5,17,19,10], in that we aim to provide a simpler and more agile tool for a direct use on graphswithout a passage through simplicial complexes-and possibly on other structures naturally arising from applications.…”

Erosion distance for generalized persistence modules

“…For more on the natural pseudo-distance, see e.g. [DF04], [DF07], [DF09], and to see how the natural pseudo-distance can be interpreted as an interleaving distance, see [dSMS17,Section 3.3].…”