toposys2014

Given a real-valued function f ∶ X → R, extended persistence looks at the collection of pairs arising from a sequence of absolute and relative homology groups. 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 corespondent pairs in the extended persistence diagram keep track on the changes in the homology of the input function. Due to the need of their representation in the same persistence diagram, we describe an extended underlying lattice structure that is compatible with the lattice correspondent to the internal logic for the topos of persistence. We also describe the Alexandrov space of irreducibles, and the technique to recover the algebraic structure from it.

Key Words: lattices, ideals and filters, extended persistence, barcodes.