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PreprintCC0arXiv

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 ∞-category, denoted by sf DGra∞. We prove that sf DGra∞ is equivalent to the ∞-category of spaces, establishing a directed version of the discrete homotopy hypothesis of Carranza and Kapulkin.

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Cite

BibTeXRIS
BibTeX
@misc{eldridge2026discrete,
  title = {{The discrete homotopy hypothesis for directed graphs}},
  author = {Briony Eldridge and Sergei O. Ivanov and Xiaomeng Xu and Shing-Tung Yau and Mengmeng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.04959},
  eprint = {2605.04959},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.04959},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - The discrete homotopy hypothesis for directed graphs
AU  - Briony Eldridge
AU  - Sergei O. Ivanov
AU  - Xiaomeng Xu
AU  - Shing-Tung Yau
AU  - Mengmeng Zhang
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2605.04959
UR  - https://arxiv.org/abs/2605.04959
AB  - 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 ∞-category, denoted by sf DGra_∞. We prove that sf DGra_∞ is equivalent to the ∞-category of spaces, establishing a directed version of the discrete homotopy hypothesis of Carranza and Kapulkin.
ER  -