The discrete homotopy hypothesis for directed graphs

Abstract

We develop a homotopy theory of directed graphs based on cubical homotopy groups, also known 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, denoted by ${\mathsf{DGra}}_{\infty }$. We prove that ${\mathsf{DGra}}_{\infty }$ is equivalent to the $\infty$-category of spaces, establishing a directed version of the discrete homotopy hypothesis of Carranza and Kapulkin.