Persistent homology of partially ordered spaces

TL;DR

This paper connects natural homology and persistent homology in directed spaces, showing how natural homology can be computed as a colimit of one-dimensional persistent homologies, with implications for analyzing concurrent programs.

Contribution

It establishes a novel link between natural homology and persistent homology, enabling the computation of natural homology as a colimit of persistent homologies along traces.

Findings

Natural homology can be viewed as a persistence object.

Natural homology can be computed as a colimit of persistent homologies.

The approach offers new avenues for analyzing directed spaces and concurrent programs.

Abstract

In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was developed in the context of topological data analysis, in which topological properties of point-cloud data sets are extracted while eliminating noise. In both approaches, the evolution homological properties are tracked through a sequence of inclusions of usual topological spaces. Exploiting this similarity, we show that natural homology may be considered a persistence object, and may be calculated as a colimit of uni-dimensional persistent homologies along traces. Finally, we suggest further links and avenues of future work in this direction.