Homology, Homotopy and Applications

Volume 18 (2016)

Number 2

The observable structure of persistence modules

Pages: 247 – 265

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n2.a14


Frédéric Chazal (DATASHAPE Research Group, INRIA Saclay, Île-de-France, Palaiseau, France)

William Crawley-Boevey (Department of Pure Mathematics, University of Leeds, United Kingdom; and Fakultät für Mathematik, Universität Bielefeld, Germany)

Vin de Silva (Department of Mathematics, Pomona College, Claremont, California, U.S.A.)


In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence modules indexed over a totally ordered finite set or the natural numbers, such diagrams do not provide a complete invariant of q-tame modules. The purpose of this paper is to show that the category of persistence modules can be adjusted to overcome this issue. We introduce the observable category of persistence modules: a localization of the usual category, in which the classical properties of q-tame modules still hold but where the persistence diagram is a complete isomorphism invariant and all q-tame modules admit an interval decomposition.


persistence module, persistent homology

2010 Mathematics Subject Classification


Published 29 November 2016