Categorification of persistent homology
Abstract
We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving distance, which we show generalizes the previouslystudied bottleneck distance. To illustrate the utility of this approach, we greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.
 Publication:

arXiv eprints
 Pub Date:
 May 2012
 arXiv:
 arXiv:1205.3669
 Bibcode:
 2012arXiv1205.3669B
 Keywords:

 Mathematics  Algebraic Topology;
 Computer Science  Computational Geometry;
 Mathematics  Category Theory;
 55N99;
 68W30;
 18A25;
 18E10;
 54E35
 EPrint:
 27 pages, v3: minor changes, to appear in Discrete &