Stable components of directed spaces

TL;DR

This paper introduces stable component systems for directed spaces without loops, creating invariants that capture the structure of directed paths and associating enriched categories to these systems.

Contribution

It defines stable component systems and categories for directed spaces, providing a new framework for analyzing their path structures and invariants.

Findings

Finite pre-cubical sets with no loops have unique minimal stable component systems.

Stable component categories are enriched in monoidal categories like the homotopy category.

These constructions yield new invariants for directed spaces.

Abstract

In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry information about the spaces of directed paths between particular points. It is shown that the geometric realizations of finite pre-cubical sets with no loops admit the unique minimal stable (future/past/total) component systems. These constructions provide a new family of invariants for directed spaces.