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Ephemeral persistence modules and distance comparison

Nicolas Berkouk and François Petit

Algebraic & Geometric Topology 21 (2021) 247–277

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 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.

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persistent homology, interleaving distance, sheaf theory
Mathematical Subject Classification 2010
Primary: 35A27, 55N99
Secondary: 18A99
Received: 26 June 2019
Revised: 26 April 2020
Accepted: 11 May 2020
Published: 25 February 2021
Nicolas Berkouk
Inria Saclay-Ile-de-France
François Petit
Université de Paris