Correspondence modules and persistence sheaves: a unifying perspective on one-parameter persistent homology

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