Ephemeral persistence modules and distance comparison

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 γ-sheaves. In the case of one-dimensional persistence, our definition agrees with the usual one showing that the observable category and the category of γ-sheaves are equivalent. We also establish isometry theorems between the category of persistent modules and γ-sheaves both endowed with their interleaving distance. Finally, we compare the interleaving and convolution distances.