Edit Distance and Persistence Diagrams over Lattices

We build a functorial pipeline for persistent homology. The input to this is filtered simplicial complex indexed by any finite metric lattice, and the output persistence diagram defined as Möbius inversion of its birth-death function. adapt Reeb graph edit distance each our categories prove that both functors in are 1-Lipschitz, making stable. Our constructions generalize classical diagram, setting, bottleneck strongly equivalent distance.