Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem

Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of intervals on the real line. Recent work of Edelsbrunner, Jablonski, and Mrozek suggests an equivalent description of barcodes as functors R →Mch, where R is the poset category of real numbers and Mch is the category whose objects are sets and whose morphisms are matchings (i.e., partial injective functions). Such functors form a category MchR whose morphisms are the natural transformations. Thus, this interpretation of barcodes gives us a hitherto unstudied categorical structure on barcodes. We show that this categorical structure leads to surprisingly simple reformulations of both the well-known stability theorem for persistent homology and a recent generalization called the induced matching theorem. These reformulations make clear for the first time that both of these results can be understood as the preservation of certain categorical structure. We also show that this perspective leads to a more systematic variant of the proof of the induced matching theorem.