Models and van Kampen theorems for directed homotopy theory

TL;DR

This paper develops models and van Kampen theorems for the fundamental category in directed homotopy theory, extending classical concepts to non-reversible paths and providing new tools for analyzing directed spaces.

Contribution

It introduces models of the fundamental category, such as the fundamental bipartite graph and extremal models, and proves van Kampen theorems for these structures.

Findings

Defined the fundamental category for directed spaces

Introduced models like the fundamental bipartite graph

Proved van Kampen theorems for subcategories and models

Abstract

We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. We define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, we prove van Kampen theorems for subcategories, retracts, and models of the fundamental category.