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Thomason Model Structures in Every Strict Higher Dimension
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Abstract
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.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope note for family 317: Thomason model structures in all strict higher dimensions (lean/docs/317.md, 2026). The release's own scope note says the formalization covers the proper combinatorial model structures and Quillen equivalences for every finite n ≥ 1 and for n = ω, and that the Street nerve detects weak equivalences; it links the comparator statements. [family-level]
- OpenAI math release, ThomasonModelStructures.lean (comparator challenge) (openai/math, 2026). This file holds the formal challenge statement linked from the scope note, defining strict ω-categories, n-truncation, augmented directed complexes and normalized chains of posets.
Possibly relevant literature (found by the agent, not checked)
- Dimitri Ara and Georges Maltsiniotis, Comparison of the n-categorical nerves (arXiv:2010.00266, Algebr. Geom. Topol. 22 (2022)). It shows that the Street, cellular and multi-simplicial nerves define the same Thomason equivalences, which bears on the class W_n defined via the Street nerve in Theorem 1.1.
- Philip Hackney, Viktoriya Ozornova, Emily Riehl and Martina Rovelli, Pushouts of Dwyer maps are (∞,1)-categorical (arXiv:2205.02353, Algebr. Geom. Topol. 24 (2024)). It refines Thomason's comparison between pushouts of nerves and nerves of pushouts along Dwyer maps for n = 1, which the paper's attachment comparison generalizes to strict n-categories.
- Dimitri Ara, Model category structures à la Thomason on 2-Cat (arXiv:1607.03644, Cahiers Topol. Géom. Différ. Catég. 56 (2015)). It constructs Thomason-type model structures on 2-Cat for almost every basic localizer, which places the paper's n = 2 case alongside other localizers.
Digests and expositions
- Latent Seminar, Thomason structures: strict higher categories and stationary prisms (blog exposition, 2026-10-07). Explains the manuscript's stationary Gray cylinders, arbitrary-target pushout argument, sieve-cover gluing and model-structure transfer, describing its coverage as selected proof mechanisms rather than independent verification. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 317
Independent sites; their content is not reviewed here.
Cite
@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 provides Lean files for this family: lean/docs/317.md. The Lean statements have not been checked by people against the paper's claims.
Family 317: 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.
This is the only paper in this family.