The observable structure of persistence modules

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.

[1]  Carl Faith,et al.  Rings, modules, and categories , 1973 .

[2]  Michael Lesnick,et al.  The Optimality of the Interleaving Distance on Multidimensional Persistence Modules , 2011, ArXiv.

[3]  W. Crawley-Boevey.  Decomposition of pointwise finite-dimensional persistence modules , 2012, 1210.0819.

[4]  Michael Lesnick,et al.  The Theory of the Interleaving Distance on Multidimensional Persistence Modules , 2011, Found. Comput. Math..

[5]  Jean-Marie Droz.  A subset of Euclidean space with large Vietoris-Rips homology , 2012, 1210.4097.

[6]  Afra Zomorodian,et al.  Computing Persistent Homology , 2004, SCG '04.

[7]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[8]  Peter Bubenik,et al.  Categorification of Persistent Homology , 2012, Discret. Comput. Geom..

[9]  Peter Gabriel,et al.  Calculus of Fractions and Homotopy Theory , 1967 .

[10]  P. Gabriel,et al.  Des catégories abéliennes , 1962 .

[11]  Herbert Edelsbrunner,et al.  Topological persistence and simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[12]  Peter Hilton,et al.  Introduction to the Theory of Categories and Functors , 1968 .

[13]  Bodo Pareigis,et al.  Categories and Functors , 1970 .

[14]  Steve Oudot,et al.  The Structure and Stability of Persistence Modules , 2012, Springer Briefs in Mathematics.

[15]  Steve Oudot,et al.  Persistence stability for geometric complexes , 2012, ArXiv.

[16]  Cary Webb.  Decomposition of graded modules , 1985 .

[17]  Jean-Pierre Serre,et al.  Groupes d'homotopie et classes des groupes abeliens , 1953 .

[18]  A. Holme.  Categories and Functors , 2012 .

[19]  Afra Zomorodian,et al.  The Theory of Multidimensional Persistence , 2007, SCG '07.