Ephemeral persistence modules and distance comparison

Abstract

We provide a definition of ephemeral multipersistent modules and prove that the quotient category 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.