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