Saturating directed spaces

TL;DR

This paper introduces a saturation condition for directed spaces in topology, refining the framework for analyzing concurrent processes by excluding exotic examples while preserving key properties and constructions.

Contribution

It proposes a local saturation condition for directed spaces, creating a well-behaved subcategory with desirable limits, colimits, and adjoint functors, enhancing the theoretical framework.

Findings

Saturation condition is satisfied by directed interval and circle.

The saturated subcategory is reflective, closed under limits, and has arbitrary colimits.

The forgetful functor has both left and right adjoints.

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.