  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 real intervals. 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 Mch^R whose
morphisms are the natural transformations. Thus, this interpretation of
barcodes gives us a hitherto unstudied categorical structure on barcodes. The
aim of this note is to 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.
