Persistent homology of partially ordered spaces

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 of homological properties is 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.