Time-reversal homotopical properties of concurrent systems

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.

Directed homology and persistence modules

“…Of course, this allows us also to define barcodes (a map from the set of intervals of Q to (relative) integers) by a Möbius inversion formula as in [27] for precubical sets. This should have relations also with the approach of [9]. This will be investigated and developed elsewhere, for practical applications.…”

“…Remark 17. Note that any interval I in F Q corresponds to the maximal chains of the trace poset of a partially-order space, as defined in [9]. These are the ones that define unidimensional persistence modules within natural homology (that we are going to recap in next Section).…”

“…First steps in that direction have been published in [9], where natural homology has been shown to be, for a certain class of directed spaces, a colimit of unidimensional persistence modules. Another step was made in [21] where the author constructed a directed homology theory based a the semi-abelian category of (unital) algebras.…”

Pair component categories for directed spaces

“…resp. only left extensions in order to distinguish clearly different d-spaces (for example the one arising by reversing all arrows from the original one); for a careful analysis, consult [4].…”

“…the fundamental category, with traces, resp. d-homotopy classes of such as objects; cf [21,3,4]. Although not essentially more difficult, we will not use factorization categories in the subsequent parts of this paper.…”