# Level-sets persistence and sheaf theory

## TL;DR

This paper establishes a deep connection between level-sets persistence in topological data analysis and derived sheaf theory, providing categorical equivalences and stability results.

## Contribution

It constructs functors linking 2-parameter persistence modules and sheaves, and proves classification, barcode decomposition, and stability theorems for Mayer-Vietoris systems.

## Key findings

- Establishes a pseudo-isometric equivalence of categories between sheaves and Mayer-Vietoris systems.
- Provides classification and stability theorems for Mayer-Vietoris systems.
- Shows a functorial equivalence between level-sets persistence and derived pushforward.

## Abstract

In this paper we provide an explicit connection between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over $\mathbb{R}$, as well as a functor in the other direction. We also observe that the 2-parameter persistence modules arising from the level sets of Morse functions carry extra structure that we call a Mayer-Vietoris system. We prove classification, barcode decomposition, and stability theorems for these Mayer-Vietoris systems, and we show that the aforementioned functors establish a pseudo-isometric equivalence of categories between derived constructible sheaves with the convolution or (derived) bottleneck distance and the interleaving distance of strictly pointwise finite-dimensional Mayer-Vietoris systems. Ultimately, our results provide a functorial equivalence between level-sets persistence and derived pushforward for continuous real-valued functions.