The discrete homotopy hypothesis for directed graphs

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 ${\mathsf{DGra}}_{\infty }$. Our main result shows that ${\mathsf{DGra}}_{\infty }$ is equivalent to the $\infty$-category of spaces.