The discrete homotopy hypothesis for directed graphs

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 $\mathrm{\infty }$-category, which we denote by ${\mathsf{D}\mathsf{G}\mathsf{r}\mathsf{a}}_{\mathrm{\infty }}$. Our main result shows that ${\mathsf{D}\mathsf{G}\mathsf{r}\mathsf{a}}_{\mathrm{\infty }}$ is equivalent to the $\mathrm{\infty }$-category of spaces.