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We build a functorial pipeline for persistent homology. The input to thispipeline is a filtered simplicial complex indexed by any finite metric latticeand the output is a persistence diagram defined as the M\"obius inversion ofits birth-death function. We adapt the Reeb graph edit distance to each of ourcategories and prove that both functors in our pipeline are $1$-Lipschitzmaking our pipeline stable. Our constructions generalize the classicalpersistence diagram and, in this setting, the bottleneck distance is stronglyequivalent to the edit distance.

Edit Distance and Persistence Diagrams Over Lattices