Determine the homotopy type that captures intended concurrent-process behavior

Determine whether the homotopy type of directed path spaces or the homotopy type of trace spaces, in which changes of speed and waiting time are declared unobservable, better captures the intended behavior of concurrent processes.

Background

The paper constructs a saturated directed space whose directed path space and trace space have different weak homotopy types: the trace space is a square and therefore contractible, while the directed path space has a nontrivial fundamental group. This demonstrates that quotienting directed paths by reparametrizations can discard homotopical information.

The authors point out that this discrepancy creates an unresolved modeling question for applications in computer science. Directed paths retain distinctions involving changes of speed and waiting time, whereas traces identify such changes as unobservable. The paper does not determine which of these two homotopy types more accurately represents the intended behavior of concurrent processes.