The discrete homotopy hypothesis for directed graphs

TL;DR

This paper establishes a homotopy theory for directed graphs using cubical homotopy groups, showing an equivalence between a localized category of directed graphs and the category of spaces.

Contribution

It introduces a homotopy framework for directed graphs and proves an equivalence with the category of spaces via localization at specific morphisms.

Findings

The category of directed graphs can be localized to form an $$-category.

The localized category ${ m DGra}_$ is equivalent to the $$-category of spaces.

Homotopy groups for directed graphs are developed and used to establish this equivalence.

Abstract

We develop a homotopy theory of directed graphs based on cubical homotopy groups, also referred to as A-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups yields an $\infty$-category, which we denote by ${\sf DGra}_\infty$. Our main result shows that ${\sf DGra}_\infty$ is equivalent to the $\infty$-category of spaces.