Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

Time-reversal homotopical properties of concurrent systems

Pages: 31 – 57

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a2


Cameron Calk (LIX, École Polytechnique, CNRS, Institut Polytechnique de Paris, Palaiseau, France)

Eric Goubault (LIX, École Polytechnique, CNRS, Institut Polytechnique de Paris, Palaiseau, France)

Philippe Malbos (Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France)


Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which reflect this directedness have been introduced to classify directed spaces. In this work we study the properties of such invariants with respect to the reversal of the flow of time in directed spaces. Known invariants, natural homotopy and homology, have been shown to be unchanged under this time-reversal.We show that these can be equipped with additional algebraic structure witnessing this reversal. Specifically, when applied to a directed space and to its reversal, we show that these enhanced invariants yield dual objects. We further refine natural homotopy by introducing a notion of relative directed homotopy and showing the existence of a long exact sequence of natural homotopy systems.


directed spaces, concurrent systems, time-reversibility, natural homology and natural homotopy

2010 Mathematics Subject Classification

18D35, 55U99, 68Q85

Received 27 January 2019

Received revised 3 July 2019

Accepted 6 August 2019

Published 26 February 2020