Filtered cospans and interlevel persistence with boundary conditions

Abstract

We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is expressed via a functor from a category of filtered cospans to a category of persistence modules that arise in Bauer-Botnan-Fluhr's study of relative interlevel set homology. We associate a filtered cospan to a Morse function $f:X\to [-\Lambda,\Lambda]$ such that $\partial X$ is the union of the regular level sets $f^{-1}(\{\pm\Lambda\})$; this allows us to capture the interlevel persistence of such a function in terms of data associated to Morse chain complexes. Similar filtered cospans are associated to simplicial and singular chain complexes, and isomorphism theorems are proven relating these to each other and to relative interlevel set homology. Filtered cospans can be decomposed, under modest hypotheses, into certain standard elementary summands, giving rise to a notion of persistence diagram for filtered cospans that is amenable to computation. An isometry theorem connects interleavings of filtered cospans to matchings between these persistence diagrams.