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arXiv · 2605.04959

The discrete homotopy hypothesis for directed graphs

Briony Eldridge·Sergei O. Ivanov·Xiaomeng Xu·Shing-Tung Yau·Mengmeng Zhang
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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 ${\sf DGra}_\infty$. We prove that ${\sf DGra}_\infty$ is equivalent to the $\infty$-category of spaces, establishing a directed version of the discrete homotopy hypothesis of Carranza and Kapulkin.

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BibTeXRIS

Briony Eldridge, Sergei O. Ivanov, Xiaomeng Xu, Shing-Tung Yau, Mengmeng Zhang. 2026-05-06. The discrete homotopy hypothesis for directed graphs. https://arxiv.org/abs/2605.04959

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