Correspondence Modules and Persistence Sheaves: A Unifying Framework for OneParameter Persistent Homology
Abstract
We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the onedimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of oneparameter persistence is carried out at the level of sections of correspondence modules that yield sheaflike structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions of persistence sheaves. Applications include: (a) a MayerVietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a realvalued function and (b) the construction of slices of 2parameter persistence modules along negatively sloped lines.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.08557
 Bibcode:
 2020arXiv200608557H
 Keywords:

 Mathematics  Algebraic Topology;
 55N31 (Primary);
 18F20 (Secondary)