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Thomason model structures in all strict higher dimensions

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Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

The paper states

Theorem 1.1

In Thomason Model Structures in Every Strict Higher Dimension

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

For every n\in\{1,2,3,\ldots\}\cup\{\infty\}, the classes (C_n,W_n,F_n) in (1.1) form a proper combinatorial model structure on \nCat. It is cofibrantly generated by (L_nI,L_nJ), where

I=\{\partial\Delta[m]\hookrightarrow\Delta[m]:m\ge0\},\qquad J=\{\Lambda^k[m]\hookrightarrow\Delta[m]:m\ge1,\ 0\le k\le m\}.

Moreover,

L_n:\sSet_{\KQ}\rightleftarrows(\nCat,C_n,W_n,F_n):R_n

is a Quillen equivalence.

Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
Papers and explanations
One current paper, readable here as a page beside its PDF.

Papers

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Thomason Model Structures in Every Strict Higher Dimension

Undated, current edition.

Read it herePDF

We prove the higher-dimensional Thomason model-structure conjecture of Ara and Maltsiniotis. For every \(1\le n\le\infty\), the category of small strict globular n-categories admits a proper combinatorial model structure that is Quillen equivalent to simplicial sets. Its weak equivalences and fibrations are detected by the twice-extended Street nerve \(\mathrm{Ex}^2N_n\), and the Quillen equivalence is given by \(c_n\mathrm{Sd}^2\dashv\mathrm{Ex}^2N_n\). Thus strict higher categories model the homotopy theory of spaces in every positive finite dimension and in dimension ω.

One Lean statement in this family

Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

ThomasonModelStructures

The paper states

Theorem 1.1

In Thomason Model Structures in Every Strict Higher Dimension

For every n\in\{1,2,3,\ldots\}\cup\{\infty\}, the classes (C_n,W_n,F_n) in (1.1) form a proper combinatorial model structure on \nCat. It is cofibrantly generated by (L_nI,L_nJ), where

I=\{\partial\Delta[m]\hookrightarrow\Delta[m]:m\ge0\},\qquad J=\{\Lambda^k[m]\hookrightarrow\Delta[m]:m\ge1,\ 0\le k\le m\}.

Moreover,

L_n:\sSet_{\KQ}\rightleftarrows(\nCat,C_n,W_n,F_n):R_n

is a Quillen equivalence.

The words they share: combinatorial, model, proper, quillen, structure.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Thomason model structures and nerve detection in all positive strict dimensions

ThomasonModelStructures.lean, the release lists it as a supporting statement.

theorem Thomason.main : OAI.Thomason.FullMain.{u} := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

The file states one more theorem: nerveDetection_allDimensions.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: Thomason.main : FullMain asserts, for strict ω-categories and for every n ≥ 1 for n-categories (NCategory n), an ExactModelEndpoint for the adjunction (Sd² ⋙ categorification) ⊣ (Street nerve ⋙ Ex ⋙ Ex): a model structure with weak equivalences and fibrations created by the right adjoint, cofibrantly generated by boundary/horn images, complete, cocomplete, locally presentable, left and right proper, with the Quillen-equivalence criterion against simplicial sets (plus the Kan-Quillen model on SSet). nerveDetection_allDimensions identifies the weak equivalences with Street-nerve detection for ω and every n, matching the summary and docs.

Definitions the group asks readers to compare with the paper: structure OmegaCategory (cells with source/target : ℕ → Cell → Cell, partial comp k, interchange, finite_dimension : ∃ k, source k x = x) and NCategory n := {C // ∀ x, C.source n x = x} are hand-built strict globular ω-/n-categories (no Mathlib counterpart); Classical.choose only picks cell dimensions in the presentation colimit; ExactModelEndpoint bundles the model structure, its classes and the Quillen-equivalence criterion by hand (quillenEquivalence : ∀ K Y fibrant, ∀ f : L K ⟶ Y, weq f ↔ sweq (adj.homEquiv f)); simplicialWeakEquivalences = weak homotopy equivalences of geometric realizations (π₀ bijection and π_n for all n ≥ 1 at all basepoints)

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The formalization proves the higher-dimensional Thomason model-structure theorem for small strict globular $n$-categories in every positive finite dimension and in dimension $\omega$. Each category has the stated proper combinatorial model structure, Quillen equivalent to simplicial sets through the twice-subdivided categorification and twice-extended Street nerve adjunction.

The selected statements also identify weak equivalences by the Street nerve in all these dimensions. Thus the constructions model the homotopy theory of spaces throughout the stated range.

AI-drafted comparisons, with no recorded human review

  • Thomason model structures and nerve detection in all positive strict dimensions ThomasonModelStructures

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: proper combinatorial Thomason model structures on strict n-categories for every n ≥ 1 and n = ω, Quillen equivalent to simplicial sets, with weak equivalences detected by the Street nerve.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Thomason model structures and nerve detection in all positive strict dimensions ThomasonModelStructures

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 317 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 317 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form