Result 317, Topology

Thomason model structures in all strict higher dimensions

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.

Lean formalization Proof

The bigger picture

Why it matters

Spaces can be studied through systems of objects, arrows and higher arrows, rather than through geometric coordinates. This manuscript claims that strict higher categories provide such a model in every positive finite dimension and with infinitely many layers.

What changes?

The unreviewed manuscript reports proper combinatorial Thomason model structures on small strict globular n-categories for every positive finite n and for n = omega, allowing infinitely many layers. These categories have arrows between objects, higher arrows between parallel arrows, and composition laws holding exactly. A model structure specifies weak equivalences, fibrations and cofibrations for doing homotopy theory. Here weak equivalences and fibrations are detected by applying the Street nerve, which converts categories into simplicial sets, followed by the extension operation twice.

What does that help mathematicians do?

The claimed Quillen equivalence means these categorical models recover the same homotopy theory as simplicial sets, which encode spaces using vertices, edges, triangles and higher simplices. Crucially, the categorical dimension does not bound the homotopy types represented: even a fixed positive finite dimension models the homotopy theory of all spaces. The reported comparison uses subdivision twice followed by categorification in one direction, and the twice-extended Street nerve in the other.

Are there practical applications?

The immediate value is foundational: researchers gain a framework for translating homotopy-theoretic questions between spaces and strict higher categories. Properness supports compatibility with gluing and pullback constructions under the model-structure hypotheses. The combinatorial property supplies set-based generation and presentability, making the framework amenable to general model-category methods. The supplied abstract claims no direct technological application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Thomason Model Structures in Every Strict Higher Dimension

September 25, 2026 35 pages

We prove the higher-dimensional Thomason model-structure conjecture of Ara and Maltsiniotis. For every 1≤n≤∞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 Ex2Nn\mathrm{Ex}^2N_n, and the Quillen equivalence is given by cnSd2⊣Ex2Nnc_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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026,
  author = {{OpenAI}},
  title = {{Thomason Model Structures in Every Strict Higher Dimension}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026/paper.pdf}{OAI:Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/317.md.

Thomason model structures in all strict higher dimensions

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves the higher-dimensional Thomason model-structure theorem for small strict globular nn-categories in every positive finite dimension and in dimension ω\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.

Comparator links

Result Comparator statement
Thomason model structures and nerve detection in all positive strict dimensions ThomasonModelStructures.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.