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We develop a unifying framework for the treatment of various persistenthomology architectures using the notion of correspondence modules. In thisformulation, morphisms between vector spaces are given by partial linearrelations, as opposed to linear mappings. In the one-dimensional case, amongother things, this allows us to: (i) treat persistence modules and zigzagmodules as algebraic objects of the same type; (ii) give a categoricalformulation of zigzag structures over a continuous parameter; and (iii)construct barcodes associated with spaces and mappings that are richer ingeometric information. A structural analysis of one-parameter persistence iscarried out at the level of sections of correspondence modules that yieldsheaf-like structures, termed persistence sheaves. Under some tamenesshypotheses, we prove interval decomposition theorems for persistence sheavesand correspondence modules, as well as an isometry theorem for persistencediagrams obtained from interval decompositions. Applications include: (a) aMayer-Vietoris sequence that relates the persistent homology of sublevelsetfiltrations and superlevelset filtrations to the levelset homology module of areal-valued function and (b) the construction of slices of 2-parameterpersistence modules along negatively sloped lines.

Correspondence Modules and Persistence Sheaves: A Unifying Perspective on One-Parameter Persistent Homology