pith. sign in

arxiv: 2606.28693 · v2 · pith:DFDAVRKYnew · submitted 2026-06-27 · 🧮 math.AT · math.CO

Discrete homotopy groups of cubical sets

Daisuke Kishimoto , Yichen Tong This is my paper

Pith reviewed 2026-07-02 21:05 UTC · model grok-4.3

classification 🧮 math.AT math.CO
keywords cubical setsdiscrete homotopy groupsquasisymmetric cubical setsgeometric realizationadjunctionweak equivalenceHurewicz theorem
0
0 comments X

The pith

The discrete homotopy groups of quasisymmetric cubical sets coincide with the homotopy groups of their geometric realizations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends discrete homotopy groups from graphs to arbitrary cubical sets. It shows that these groups are naturally isomorphic to the homotopy groups of the geometric realization precisely when the cubical set carries quasisymmetric structure, meaning coordinate permutations compatible with faces and degeneracies. The proof proceeds by constructing the left adjoint to the forgetful functor from quasisymmetric cubical sets to ordinary cubical sets and proving that the unit of this adjunction is an objectwise weak equivalence. The resulting isomorphism then transfers to all cubical sets, supplying a purely combinatorial description of their homotopy groups and establishing the Hurewicz theorem for the quasisymmetric case.

Core claim

Discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to the homotopy groups of their geometric realizations. This isomorphism is obtained from a purely combinatorial left adjoint to the forgetful functor whose unit is an objectwise weak equivalence; the same construction yields a combinatorial description of homotopy groups for arbitrary cubical sets and the Hurewicz theorem in the quasisymmetric setting.

What carries the argument

The left adjoint to the forgetful functor from the category of quasisymmetric cubical sets to the category of cubical sets, together with the proof that its unit is an objectwise weak equivalence.

If this is right

  • A purely combinatorial description of the homotopy groups of the geometric realizations of arbitrary cubical sets follows directly from the isomorphism.
  • The Hurewicz theorem holds for the discrete homotopy groups of quasisymmetric cubical sets.
  • Discrete homotopy groups supply a combinatorial model for the homotopy groups of any cubical set.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The combinatorial description may allow homotopy groups to be computed directly from the face and degeneracy data without constructing a geometric realization.
  • The adjunction technique could be examined for transfer to other combinatorial models such as simplicial sets.

Load-bearing premise

The unit of the adjunction from quasisymmetric cubical sets to cubical sets is an objectwise weak equivalence.

What would settle it

A counterexample would be any quasisymmetric cubical set in which at least one discrete homotopy group differs from the corresponding homotopy group of its geometric realization.

read the original 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 an objectwise 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 Hurewicz theorem for the discrete homotopy groups of quasisymmetric cubical sets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper extends discrete homotopy groups from graphs to arbitrary cubical sets. For quasisymmetric cubical sets (cubical sets with compatible coordinate permutation symmetries), it proves that the discrete homotopy groups are naturally isomorphic to the homotopy groups of the geometric realizations. It constructs a purely combinatorial left adjoint to the forgetful functor from quasisymmetric cubical sets to cubical sets and proves that the unit of this adjunction is an objectwise weak equivalence. This yields a combinatorial description of homotopy groups for all cubical sets and establishes the Hurewicz theorem in the quasisymmetric case.

Significance. If the central claims hold, the work supplies an explicit combinatorial model for homotopy groups of cubical sets via discrete homotopy groups, which is a notable contribution to algebraic topology. The combinatorial left adjoint and the proof that its unit is an objectwise weak equivalence are strengths, as they enable transfer of results without additional parameters and support direct computation. The Hurewicz theorem application further demonstrates utility.

minor comments (2)
  1. The definition of quasisymmetric cubical sets (in the section introducing the category) should explicitly state whether the symmetries commute with all face and degeneracy maps or only a subset; the current phrasing leaves the compatibility conditions slightly ambiguous.
  2. In the statement of the main isomorphism theorem, the naturality in the cubical set variable is asserted but the verification that the isomorphism respects morphisms in the category of cubical sets is only sketched; a short diagram chase or reference to the adjunction unit would clarify this.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. There are no major comments to address.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claims rest on an explicit combinatorial construction of the left adjoint to the forgetful functor from quasisymmetric cubical sets and a direct proof that the unit of this adjunction is an objectwise weak equivalence. These are presented as self-contained combinatorial arguments that establish the isomorphism for the quasisymmetric case without definitional reduction, fitted inputs renamed as predictions, or load-bearing self-citations. The extension to arbitrary cubical sets follows as a consequence of the adjunction, and the Hurewicz theorem application is likewise derived from the established isomorphism. No step in the provided abstract or described derivation chain reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard category-theoretic and topological background; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of cubical sets, face and degeneracy maps, and geometric realization functors
    Invoked throughout the statements about quasisymmetric structures and weak equivalences.
  • standard math Existence and basic properties of homotopy groups and the Hurewicz theorem in classical topology
    Used as the target of the isomorphism and for the application.

pith-pipeline@v0.9.1-grok · 5665 in / 1273 out tokens · 27974 ms · 2026-07-02T21:05:18.919940+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

26 extracted references · 4 canonical work pages · 2 internal anchors

  1. [1]

    Antolini, Geometric realisations of cubical sets with connections, and classifying spaces of categories, Applied Cat

    R. Antolini, Geometric realisations of cubical sets with connections, and classifying spaces of categories, Applied Cat. Str.10(2002), 481-494. 7, 25

    2002

  2. [2]

    Atkin, An algebra for patterns on a complex, I, Internat

    R.H. Atkin, An algebra for patterns on a complex, I, Internat. J. Man-Mach. Stud.6(1974), 285-307. 1

    1974

  3. [3]

    Atkin, An algebra for patterns on a complex, II, Internat

    R.H. Atkin, An algebra for patterns on a complex, II, Internat. J. Man-Mach. Stud.8(1976), no. 5, 483-498. 1

    1976

  4. [4]

    Babson, H

    E. Babson, H. Barcelo, M. de Longueville, and R. Laubenbacher, Homotopy theory of graphs, J. Algebraic Combin.24(2006), no. 1, 31-44. 1, 42

    2006

  5. [5]

    Barcelo, C

    H. Barcelo, C. Greene, A.S. Jarrah, and V. Welker, Homology groups of cubical sets with connections, Appl. Categ. Struct.29(2021), no. 3, 415-429. 7

    2021

  6. [6]

    Barcelo, X

    H. Barcelo, X. Kramer, R. Laubenbacher, and C. Weaver, Foundations of a connectivity theory for simplicial complexes, Adv. Appl. Math.26(2001), no. 2, 97-128. 1, 42

    2001

  7. [7]

    Barcelo and R

    H. Barcelo and R. Laubenbacher, Perspectives on𝐴-homotopy theory and its applications, Discrete Math. 298(2005), no. 1-3, 39-61. 1

    2005

  8. [8]

    Brown and P

    R. Brown and P. J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra21(1981), no. 3, 233-260. 4

    1981

  9. [9]

    Brown, P

    R. Brown, P. J. Higgins, R. Sivera,Nonabelian Algebraic Topology, EMS Tracts Math.15, European Mathematical Society (EMS), Z¨ urich, 2011. 10, 19

    2011

  10. [10]

    Carranza and C

    D. Carranza and C. Kapulkin, Homotopy groups of cubical sets, Expo. Math.41(2023), no. 4, 125518. 1, 8, 9

    2023

  11. [11]

    Carranza and C

    D. Carranza and C. Kapulkin, Cubical setting for discrete homotopy theory, revisited, Compositio Math. 160(2024), no. 12, 2856-2903. 1, 3, 42, 43, 47, 48

    2024

  12. [12]

    Carranza, C

    D. Carranza, C. Kapulkin, and A. Tonks, The Hurewicz theorem for cubical homology, Math. Z.305 (2023), article number 61. 41

    2023

  13. [13]

    Cisinski, Les pr ´efaisceaux comme mod `eles des types d’homotopie, Ast ´erisque308(2006), xxiv+390

    D.-C. Cisinski, Les pr ´efaisceaux comme mod `eles des types d’homotopie, Ast ´erisque308(2006), xxiv+390. 6, 7

    2006

  14. [14]

    Doherty, K

    B. Doherty, K. Kapulkin, Z. Lindsey, and C. Sattler, Cubical models of(∞, 𝑛)-categories, 2020. to appear in Mem. Amer. Math. Soc. 6

    2020

  15. [15]

    The discrete homotopy hypothesis for directed graphs

    B. Eldridge, S. O. Ivanov, X. Xu, S.-T. Yau, M. Zhang, The discrete homotopy hypothesis for directed graphs, arXiv:2605.04959. 47

    work page internal anchor Pith review Pith/arXiv arXiv

  16. [16]

    Federer, Lectures in Algebraic Topology, Brown University, Providence, R.I

    H. Federer, Lectures in Algebraic Topology, Brown University, Providence, R.I. (1962) 7

    1962

  17. [17]

    Grandis and L

    M. Grandis and L. Mauri, Cubical sets and their site, Theory Appl. Categ.11(2003), no. 8, 185-211. 2, 4, 13

    2003

  18. [18]

    Homologies of path complexes and digraphs

    A. Grigor’yan, Y. Lin, Y. Muranov, and S.T. Yau, Homologies of path complexes and digraphs, arXiv:1207.2834v4. 3

    work page internal anchor Pith review Pith/arXiv arXiv

  19. [19]

    Grigor’yan, Y

    A. Grigor’yan, Y. Lin, Y. Muranov, and S.-T. Yau, Homotopy theory for digraphs, Pure Appl. Math. Q. 10(2014), no. 4, 619-674. 3, 44 DISCRETE HOMOTOPY GROUPS OF CUBICAL SETS 49

    2014

  20. [20]

    Grigor’yan and Y

    A. Grigor’yan and Y. Muranov, On homology theories of cubical digraphs, Pacific J. Math.322(2023), no. 1, 39-58. 3, 42, 43

    2023

  21. [21]

    Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs99, American Mathematical Society, Providence, RI, 2003 24, 25

    P.S. Hirschhorn, Model Categories and Their Localizations, Mathematical Surveys and Monographs99, American Mathematical Society, Providence, RI, 2003 24, 25

    2003

  22. [22]

    Isaacson, Symmetric cubical sets, J

    S.B. Isaacson, Symmetric cubical sets, J. Pure Appl. Algebra216(2011), no. 6, 1146-1173. 2, 14

    2011

  23. [23]

    Kan, Abstract homotopy

    D.M. Kan, Abstract homotopy. III, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 419–421. 35

    1956

  24. [24]

    Kishimoto and Y

    D. Kishimoto and Y. Tong, The fundamental group and the magnitude-path spectral sequence of a directed graph, J. Lond. Math. Soc. (2)112(2025), no. 1, Paper No. e70210. 3, 11, 44

    2025

  25. [25]

    J. Li, J. Wu, S.-T. Yau, M. Zhang, Homotopy groups and Puppe sequence of digraphs, arXiv:2402.01110. 3, 42, 44, 45

    work page arXiv

  26. [26]

    Theriault, J

    S. Theriault, J. Wu, S.-T. Yau, M. Zhang, Calculating Higher Digraph Homotopy Groups, arXiv:2504.04119. 3, 46 Faculty of Mathematics, Kyushu University , Fukuoka 819-0395, Japan Email address:kishimoto@math.kyushu-u.ac.jp Institute for Theoretical Sciences, Westlake Institute of Advanced Study , Westlake University , Hangzhou 310030, China Email address:t...

    work page arXiv