Abstract: Classical persistent homology takes as input a one-parameter filtration of a space and produces a so called persistence diagram. The persistence diagram is a combinatorial summary of homological features that appear and disappear along the filtration. An important property of the persistence diagram is that it is stable to perturbations to the filtration. This means persistent homology can handle uncertainty in data. However, there are two major drawbacks to one-parameter persistent homology: (1) it requires field coefficients so it misses torsion in data, and (2) it is highly sensitive to outliers in data. It is widely accepted that a solution to the second problem requires a theory of persistent homology for two-parameter filtrations.
The conventional interpretation of a persistence diagram as a list of indecomposable is limited to 1D filtrations and homology with field coefficients. However, if one interprets a persistence diagram as the Möbius inversion of a rank or a birth-death function, then all sorts of new possibilities emerge. First, the generalized persistence diagram is well defined for filtrations indexed over any finite metric lattice and it is stable. Second, the generalized persistence diagram is well defined for coefficients in any abelian category and it is stable. Thirdly, the construction of the generalized persistence diagram is functorial, which is new even to classical persistent homology. I will give an introduction to generalized persistence diagrams with an emphasis on functorilaity and open problems.