Thomason model structures in all strict higher dimensions

1 · The essentials

Spaces can be studied through mathematical descriptions that organize objects and the connections between them. This result concerns descriptions with multiple levels of connections, all obeying exact rules. The manuscript shows that they can represent the homotopy theory of spaces—the study of spaces through continuous deformation. It does so for every positive finite number of levels, and also for a setting with arbitrarily many finite levels. The claim applies to the particular descriptions called small strict globular higher categories, not to every kind of higher-dimensional description. It resolves the conjecture attributed in the source to Ara and Maltsiniotis. The main result has a Lean formalization, although the supplied material does not specify its detailed coverage.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine describing the same building with two different systems of plans. The result supplies a mathematically controlled translation between two systems for describing spaces; it does not say that the individual plans are identical.

Why it matters

The result establishes that strict higher categories model the homotopy theory of spaces in every dimension stated in the source. The comparison is a Quillen equivalence, so it connects entire homotopy theories rather than merely matching selected examples. Its uniform scope includes both every positive finite dimension and dimension $ω$ .

2 · Going further

The precise problem is to equip the category of small strict globular $n$ -categories with a proper combinatorial Thomason model structure, for every positive finite $n$ and for $n = ω$ , and to compare it with simplicial sets by a Quillen equivalence. The manuscript identifies this as the higher-dimensional Thomason model-structure conjecture of Ara and Maltsiniotis. Its abstract uses the notation $1 \leq n \leq \infty$ , while the accompanying description identifies the infinite-dimensional case as dimension $ω$ .

The central construction described in the abstract uses the Street nerve $N_{n}$ and the twice-applied extension operation $Ex^{2}$ . Weak equivalences and fibrations are detected by $Ex^{2} N_{n}$ : a map of strict $n$ -categories belongs to either class exactly when its image belongs to the corresponding class of simplicial-set maps. The stated Quillen equivalence is the adjunction $c_{n} Sd^{2} ⊣ Ex^{2} N_{n}$ , with twice-applied subdivision on the left and twice-applied extension on the right.

This gives the construction's central mechanism, but the supplied abstract does not describe the argument establishing the model-structure axioms, properness, combinatoriality, or the Quillen equivalence. Only the main manuscript is supplied, so no relationship to companion manuscripts can be reported. The main result has a Lean formalization; the supplied material does not identify which additional claims or construction details are included in its scope.

Useful notions

Homotopy and continuous deformation
Categories and higher categories
Simplicial sets
Model structures
Adjunctions and Quillen equivalences

What was known before / what is new

Before

The source identifies the target statement as a conjecture of Ara and Maltsiniotis. It gives no account of earlier dimension-specific results or partial progress, so it does not support a more detailed history.

New

The manuscript shows the existence of proper combinatorial Thomason model structures uniformly in every positive finite strict globular dimension and in dimension $ω$ . It specifies weak equivalences and fibrations through $Ex^{2} N_{n}$ and supplies the Quillen equivalence $c_{n} Sd^{2} ⊣ Ex^{2} N_{n}$ . This is the stated resolution of the Ara–Maltsiniotis conjecture.

Limits and assumptions

The hypotheses concern small strict globular $n$ -categories, with $n \geq 1$ finite or $n = ω$ ; the statement does not cover arbitrary higher-category frameworks or dimension zero. The comparison is an equivalence of homotopy theories, not an assertion that strict higher categories and simplicial sets are identical. The main result has a Lean formalization, but its detailed scope is not specified in the supplied material, so verification of every associated claim cannot be asserted. The abstract supplies no proof details and identifies no remaining open problems.

Mini glossary

Strict globular higher category: A structure of objects, arrows, and higher arrows between parallel lower-dimensional arrows, whose composition laws hold exactly.
Simplicial set: A mathematical structure built from abstract simplices linked by face and degeneracy operations.
Model structure: A framework that designates weak equivalences, fibrations, and cofibrations to organize a homotopy theory.
Quillen equivalence: An adjunction compatible with model structures that expresses an equivalence of their homotopy theories.
Proper: A property of a model structure requiring weak equivalences to be preserved under pushout along cofibrations and pullback along fibrations.
Combinatorial: A model structure is combinatorial when its underlying category is locally presentable and the structure is cofibrantly generated.

Keywords : Strict higher categories · Thomason model structures · Quillen equivalence

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/317.md

Manuscripts in this family

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Thomason Model Structures in Every Strict Higher Dimension

25/09/2026

To be verified

We prove the higher-dimensional Thomason model-structure conjecture of Ara and Maltsiniotis. For every $1 \leq n \leq \infty$ , 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 $Ex^{2} N_{n}$ , and the Quillen equivalence is given by $c_{n} Sd^{2} ⊣ Ex^{2} N_{n}$ . Thus strict higher categories model the homotopy theory of spaces in every positive finite dimension and in dimension ω.

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