Discrete homotopy groups of cubical sets

Abstract

We extend the notion of discrete homotopy groups of graphs to arbitrary cubical sets, and show that the discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to the homotopy groups of their geometric realizations. Here, quasisymmetric cubical sets are cubical sets equipped with coordinate permutation symmetries that are compatible with faces and degeneracies, but not necessarily with connections. We give a purely combinatorial construction of the left adjoint of the forgetful functor from the category of quasisymmetric cubical sets to the category of cubical sets, and prove that the unit of this adjunction is a natural weak equivalence. As a consequence, we obtain a purely combinatorial description of the homotopy groups of the geometric realizations of arbitrary cubical sets. As an application, we establish the discrete Hurewicz map from the discrete homotopy groups of a cubical set to the homology groups, and prove a Hurewicz theorem with respect to the discrete Hurewicz map whenever the cubical set is quasisymmetric. Applied to the 1-nerve of a directed graph, it also yields a Hurewicz theorem relating the homotopy groups of Li, Wu, Yau, and Zhang to singular cubical homology.

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BibTeX RIS

BibTeX

@misc{kishimoto2026discrete,
  title = {{Discrete homotopy groups of cubical sets}},
  author = {Daisuke Kishimoto and Yichen Tong},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2606.28693},
  eprint = {2606.28693},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.28693},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Discrete homotopy groups of cubical sets
AU  - Daisuke Kishimoto
AU  - Yichen Tong
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2606.28693
UR  - https://arxiv.org/abs/2606.28693
AB  - We extend the notion of discrete homotopy groups of graphs to arbitrary cubical sets, and show that the discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to the homotopy groups of their geometric realizations. Here, quasisymmetric cubical sets are cubical sets equipped with coordinate permutation symmetries that are compatible with faces and degeneracies, but not necessarily with connections. We give a purely combinatorial construction of the left adjoint of the forgetful functor from the category of quasisymmetric cubical sets to the category of cubical sets, and prove that the unit of this adjunction is a natural weak equivalence. As a consequence, we obtain a purely combinatorial description of the homotopy groups of the geometric realizations of arbitrary cubical sets. As an application, we establish the discrete Hurewicz map from the discrete homotopy groups of a cubical set to the homology groups, and prove a Hurewicz theorem with respect to the discrete Hurewicz map whenever the cubical set is quasisymmetric. Applied to the 1-nerve of a directed graph, it also yields a Hurewicz theorem relating the homotopy groups of Li, Wu, Yau, and Zhang to singular cubical homology.
ER  -

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