# Time-reversal homotopical properties of concurrent systems

## TL;DR

This paper investigates how algebraic invariants in directed topology, used to model concurrent systems, behave under time-reversal, revealing dualities and refining the understanding of directed homotopy and homology.

## Contribution

It demonstrates that natural homotopy and homology invariants can be equipped with structures reflecting time-reversal, and introduces a relative directed homotopy with a long exact sequence.

## Key findings

- Invariants are dual under time-reversal.
- Refined invariants reveal additional algebraic structure.
- A new relative directed homotopy concept with a long exact sequence.

## Abstract

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 respect 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 refined 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.