  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 one-dimensional 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 one-parameter persistence is
carried out at the level of sections of correspondence modules that yield
sheaf-like 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. Applications include: (a) a
Mayer-Vietoris sequence that relates the persistent homology of sublevelset
filtrations and superlevelset filtrations to the levelset homology module of a
real-valued function and (b) the construction of slices of 2-parameter
persistence modules along negatively sloped lines.
