Poset functor cocalculus and applications to topological data analysis

TL;DR

This paper introduces poset cocalculus, a new functor calculus tool for topological data analysis, providing stable approximations of persistence modules and linking filtrations to functor approximations.

Contribution

It develops poset cocalculus for functors from lattices to model categories, with applications to stability of multipersistence modules and relations between filtrations and functor approximations.

Findings

Codegree n approximation of multipersistence modules is stable.

Vietoris-Rips filtration is the codegree 2 approximation of Čech filtration.

Codegree 1 approximation relates to the space of continuous maps.

Abstract

We introduce a new flavor of functor cocalculus, called \emph{poset cocalculus}, as a tool for studying approximations in topological data analysis. Given a functor from a distributive lattice to a model category, poset cocalculus produces a Taylor telescope of codegree $n$ approximations of the functor, where a codegree $n$ functor takes strongly bicartesian $(n+1)$--cubes to homotopy cocartesian $(n+1)$--cubes. We give several applications of this new functor cocalculus. We prove that the codegree $n$ approximation of a multipersistence module is stable under an appropriate notion of interleaving distance. We draw connections to filtrations of simplicial complexes, and show that the Vietoris-Rips filtration is precisely the codegree 2 approximation of the \v{C}ech filtration. We demonstrate that the codegree 1 approximation of the space of simplicial maps between two simplicial complexes is in some sense the space of continuous maps between their realizations, and that this statement can be made precise.