Relative Interlevel Set Cohomology Categorifies Extended Persistence Diagrams

TL;DR

This paper introduces a categorical framework that categorifies extended persistence diagrams using relative interlevel set cohomology, linking topological invariants with algebraic structures in sheaf theory.

Contribution

It defines an abelian Frobenius category of presheaves that categorifies extended persistence diagrams, connecting them with sheaf-theoretic and derived category approaches.

Findings

RISC categorifies extended persistence diagrams.

The Grothendieck group captures the persistence diagram.

Links derived level set persistence with sheaf theory.

Abstract

The extended persistence diagram introduced by Cohen-Steiner, Edelsbrunner, and Harer is an invariant of real-valued continuous functions, which are $\mathbb{F}$-tame in the sense that all open interlevel sets have degree-wise finite-dimensional cohomology with coefficients in a fixed field $\mathbb{F}$. We show that relative interlevel set cohomology (RISC), which is based on the Mayer--Vietoris pyramid by Carlsson, de Silva, and Morozov, categorifies this invariant. More specifically, we define an abelian Frobenius category $\mathrm{pres}(\mathcal{J})$ of presheaves, which are presentable in a certain sense, such that the RISC $h(f)$ of an $\mathbb{F}$-tame function $f \colon X \rightarrow \mathbb{R}$ is an object of $\mathrm{pres}(\mathcal{J})$, and moreover the extended persistence diagram of $f$ uniquely determines - and is determined by - the corresponding element $[h(f)] \in K_0 (\mathrm{pres}(\mathcal{J}))$ in the Grothendieck group $K_0 (\mathrm{pres}(\mathcal{J}))$ of the abelian category $\mathrm{pres}(\mathcal{J})$. As an intermediate step we show that $\mathrm{pres}(\mathcal{J})$ is the abelianization of the (localized) category of complexes of $\mathbb{F}$-linear sheaves on $\mathbb{R}$, which are tame in the sense that sheaf cohomology of any open interval is finite-dimensional in each degree. This yields a close link between derived level set persistence by Curry, Kashiwara, and Schapira and the categorification of extended persistence diagrams.