Homotopy, homology, and persistent homology using closure spaces (Q6645902)

This paper develops persistent homology in the setting of filtered closure spaces. A closure space is a set \(X\) together with a set-endomorphism of its power set, called a \emph{closure operator}, that satisfies three very simple axioms. Closure spaces, defined originally by Cech, form a category that includes in particular topological spaces, graphs and directed graphs as subcategopries. They have been used in various applications since the 1980s. The contribution of this paper is in importing homotopy theoretic ideas into the context of closure spaces, and then introducing filtrations of closure spaces so that the resulting theory can be applied in topological data analysis.\N\NMore specifically, interval objects in closure spaces are used to develop homotopy theories and homology theories for closure spaces. In particular the paper develops homotopy theories and homology theories for graphs and directed graphs. It introduces a category of filtrations of closure spaces and natural transformations, which contains the categories of metric spaces, filtered topological spaces, weighted graphs, and weighted directed graphs and their respective morphisms as subcategories.\N\NNaturally each homology theory for closure spaces gives rise to a persistent homology of filtrations of closure spaces. Functorial definitions of Vietoris-Rips and Cech complexes for closure spaces are given, which together with simplicial homology also produce persistent homologies of filtrations of closure spaces. Importantly, stability theorem for all these new persistent homologies are proven, which include and extend several of the existing stability theorems as special cases, including stability of many variants of persistent homology for metric spaces and for weighted directed graphs.