# Ephemeral persistence modules and distance comparison

## TL;DR

This paper introduces ephemeral multi-persistent modules, establishes their relation to $mma$-sheaves, and compares different distances used in persistent homology, providing a unified categorical framework.

## Contribution

It defines ephemeral modules, proves their quotient category is equivalent to $mma$-sheaves, and establishes isometry theorems for interleaving distances.

## Key findings

- Quotient of persistent modules by ephemeral modules is equivalent to $mma$-sheaves.
- In 1D, the definition aligns with classical persistent homology.
- Isometry theorems relate interleaving distances between categories.

## Abstract

We provide a definition of ephemeral multi-persistent modules and prove that the quotient of persistent modules by the ephemeral ones is equivalent to the category of $\gamma$-sheaves. In the case of one-dimensional persistence, our definition agrees with the usual one showing that the observable category and the category of $\gamma$-sheaves are equivalent. We also establish isometry theorems between the category of persistent modules and $\gamma$-sheaves both endowed with their interleaving distance. Finally, we compare the interleaving and convolution distances.