extlat2014

THE ORDER STRUCTURE OF EXTENDED PERSISTENCE

Joao Pita Costa, Maria Joao Gouveia and Primož Škraba

Abstract. Given a real-valued function f ∶ X → R, extended persistence is the collection of pairs arising from a sequence of absolute and relative homology groups. The corespondent pairs in the extended persistence diagram keep track on the changes in the homology of the input function. Every class which is born at some point of the two-stage process will eventually die, being associated with a pair of critical values. The ordered pairs now fall into three classes: ordinary, relative and extended. Due to the need of their representation in the same persistence diagram, we assume the extended persistence diagram, from Edelsbrunner et al., and describe an extended underlying lattice structure that is compatible with the lattice presented as the internal logic for the topos of persistence in earlier works. We also describe the Alexandrov space of irreducibles, and the technique to recover the algebraic structure from it.