# Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem

arXiv: https://arxiv.org/abs/1610.10085 · HTML: https://mlsanity.com/paper/1610.10085/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2016.

## Built on by (most cited first, 3 of 3)

- [Topologically Faithful Multi-class Segmentation in Medical Images](https://mlsanity.com/paper/2403.11001/) (2024)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [The Universal ℓp-Metric on Merge Trees](https://mlsanity.com/paper/2112.12165/) (2021)

## Abstract (arXiv)

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.