Saturating directed spaces

Abstract

Directed topology is a refinement of standard topology, where spaces may have non-reversible paths. It has been put forward as a candidate approach to the analysis of concurrent processes. Recently, a wealth of different frameworks for, i.e., categories of, directed spaces have been proposed. In the present work, starting from Grandis's notion of directed space, we propose an additional condition of saturation for distinguished sets of paths and show how it allows to rule out exotic examples without any serious collateral damage. Our saturation condition is local in a natural sense, and is satisfied by the directed interval (and the directed circle). Furthermore we show in which sense it is the strongest condition fulfilling these two basic requirements. Our saturation condition selects a full, reflective subcategory of Grandis's category of d-spaces, which is closed under arbitrary limits of d-spaces, has arbitrary colimits (obtained by saturating the corresponding colimits of d-spaces), and has nice cylinder and cocylinder constructions. Finally, the forgetful functor to plain topological spaces has both a right and a left adjoint.