  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.
