Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, others combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category so that functors to that category induce persistence functions. We port the interleaving and bottleneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we define colorable ranks, persistence diagrams and prove the equality between multicolored bottleneck distance and interleaving distance in semisimple Abelian categories. To illustrate our framework in practice, we give examples of multicolored persistent homology on filtered topological spaces with a group action and labeled point cloud data.
Keywords: rank, persistence, categorification, regular category, abelian category, semisimple category, classification, group action, point cloud, poset, bottleneck, interleaving
2010 MSC: 18E10, 18A35, 55N35, 68U05
Theory and Applications of Categories, Vol. 35, 2020, No. 9, pp 228-260.
Published 2020-02-24.
http://www.tac.mta.ca/tac/volumes/35/9/35-09.pdf
Note: when this article was originally published, the authors of the article and the handling editor were unaware of related work in this area. An addendum to this article has been published as Theory and Applications of Categories, Vol. 39, 2013, No. 14, pp 444-446, describing the similarities and differences between this article and these other works.
Revised 2023-03-31. Original version at
http://www.tac.mta.ca/tac/volumes/35/9/35-09a.pdf