Persistence Diagrams over Lattices

In this paper by Alexander McCleary and Amit Patel, they 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 Mobius 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.

Persistence diagrams are related to the complexes I construct in my paper “Thin Posets, CW Posets, and Categorification”. One of my goals for the next year is to better understand this connection. Furthermore, Turner and Everitt generalize my construction to one which works for any poset. In the case of a geometric lattice, one gets a particularly interesting result which could be compared to the lattice construction coming from the paper above.