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arxiv: 2605.04959 · v1 · submitted 2026-05-06 · 🧮 math.AT · math.CO· math.CT

The discrete homotopy hypothesis for directed graphs

Briony Eldridge , Sergei O. Ivanov , Xiaomeng Xu , Shing-Tung Yau , Mengmeng Zhang This is my paper

Pith reviewed 2026-05-08 16:19 UTC · model grok-4.3

classification 🧮 math.AT math.COmath.CT
keywords directed graphshomotopy theorycubical homotopy groupsinfinity-categorieslocalizationdiscrete homotopy hypothesisA-groupsspaces
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The pith

Localizing directed graphs at cubical homotopy equivalences yields the infinity-category of spaces.

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

The paper builds a homotopy theory for directed graphs using cubical homotopy groups, also called A-groups or reduced GLMY homotopy groups. It inverts all morphisms that induce isomorphisms on these groups to produce an infinity-category called DGra_infinity. The central theorem states that DGra_infinity is equivalent to the infinity-category of spaces. A reader cares because this supplies a purely combinatorial, graph-based model that captures every homotopy type of a space. If the equivalence holds, every continuous homotopy invariant becomes computable in principle from finite directed graphs.

Core claim

We develop a homotopy theory of directed graphs based on cubical homotopy groups, also referred to as A-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups yields an infinity-category, which we denote by DGra_infinity. Our main result shows that DGra_infinity is equivalent to the infinity-category of spaces.

What carries the argument

The cubical homotopy groups (A-groups or reduced GLMY homotopy groups) on directed graphs, which define the weak equivalences for the localization that produces DGra_infinity.

If this is right

  • Every homotopy type of a space arises from some directed graph once morphisms are localized at cubical homotopy equivalences.
  • Homotopy groups and other invariants of spaces can be read off from the cubical homotopy groups of representing directed graphs.
  • The category of directed graphs generates the infinity-category of spaces under this localization.
  • Morphisms between directed graphs that preserve cubical homotopy groups correspond exactly to homotopy equivalences of spaces.

Where Pith is reading between the lines

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

  • Directed graphs could serve as input for algorithms that compute homotopy invariants of arbitrary spaces.
  • The same localization technique might apply to other discrete structures such as posets or simplicial complexes to recover spaces.
  • Explicit constructions of directed graphs for standard spaces like spheres would allow direct verification of the claimed equivalence on low-dimensional examples.

Load-bearing premise

The cubical homotopy groups detect exactly the homotopy information needed so that inverting their isomorphisms recovers the full infinity-category of spaces.

What would settle it

Constructing a directed graph whose cubical homotopy groups are all trivial yet which remains non-contractible after localization, or exhibiting a space whose homotopy type cannot be realized by any directed graph in DGra_infinity, would disprove the equivalence.

read the original abstract

We develop a homotopy theory of directed graphs based on cubical homotopy groups, also referred to 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, which we denote by ${\sf DGra}_\infty$. Our main result shows that ${\sf DGra}_\infty$ is equivalent to the $\infty$-category of spaces.

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 / 3 minor

Summary. The manuscript develops a homotopy theory for directed graphs based on cubical homotopy groups (also called A-groups or reduced GLMY homotopy groups). It localizes the category of directed graphs at the class of morphisms inducing isomorphisms on all these groups, obtaining an ∞-category DGra_∞, and proves that DGra_∞ is equivalent to the ∞-category of spaces.

Significance. If the equivalence holds, the work supplies a combinatorial, discrete model for the homotopy theory of spaces realized via directed graphs and cubical invariants. This could enable new computational and combinatorial techniques in algebraic topology. The paper explicitly constructs the localization and verifies the universal property against the ∞-category of spaces, which is a substantive contribution.

minor comments (3)
  1. §2.3: the notation for the cubical homotopy groups switches between A_n(G) and π_n^□(G) without an explicit cross-reference; a single consistent symbol would improve readability.
  2. Definition 4.7: the localization functor is defined via a calculus of fractions, but the verification that the resulting ∞-category satisfies the universal property for all spaces is only sketched in §5.2; expanding the comparison with the singular realization functor would strengthen the argument.
  3. Figure 3: the labeling of vertices in the directed graph example is too small to read in the printed version; increasing font size or adding a table of vertex labels would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. We appreciate the acknowledgment that the work provides a combinatorial model for the homotopy theory of spaces via directed graphs and cubical invariants. The equivalence DGra_∞ ≃ Spaces is established by explicitly constructing the localization and verifying the universal property, as detailed in the paper.

Circularity Check

0 steps flagged

No circularity; equivalence stated as independent theorem

full rationale

The provided abstract and description frame the main result as a theorem: localization of directed graphs at isomorphisms on cubical homotopy groups (A-groups/reduced GLMY groups) produces DGra_∞ equivalent to Spaces. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the given text. The construction is presented as developing a new homotopy theory whose output equivalence is proven rather than tautological, leaving the claim self-contained against external verification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed from abstract only; full list of background axioms, free parameters, and any invented entities cannot be extracted. The construction relies on standard ∞-category theory and the definition of cubical homotopy groups for graphs.

axioms (1)
  • standard math Standard axioms and constructions of ∞-category theory
    The localization and equivalence are stated in the language of ∞-categories.
invented entities (1)
  • DGra_∞ no independent evidence
    purpose: The ∞-category of directed graphs localized at cubical homotopy equivalences
    Defined in the paper as the output of the localization process.

pith-pipeline@v0.9.0 · 5369 in / 1107 out tokens · 37346 ms · 2026-05-08T16:19:57.709205+00:00 · methodology

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Reference graph

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