A convenient category for directed homotopy

TL;DR

This paper introduces a new category of preordered topological spaces generated by cubes, which is locally presentable and facilitates the study of directed homotopy in a more manageable framework.

Contribution

It defines a convenient, locally presentable category for directed homotopy based on cube-generated preordered topological spaces, extending existing simplicial approaches.

Findings

The category is shown to be locally presentable.

It provides a structured framework for directed homotopy theory.

The approach parallels the simplicial methods used in classical topology.

Abstract

We propose a convenient category for directed homotopy consisting of preordered topological spaces generated by cubes. Its main advantage is that, like the category of topological spaces generated by simplices suggested by J. H. Smith, it is locally presentable.