Semi-strictification of $(\infty, n)$-categories

Amar Hadzihasanovic; Cl\'emence Chanavat

REVIEW 3 major objections 4 minor 16 references

Semi-strictification of $(\infty, n)$-categories

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves the first equivalence between a weak non-algebraic model and a semi-strict algebraic model of $(\infty,n)$-categories, via an explicit combinatorial semi-strictification.

desk verdict A serious, detailed candidate proof of semi-strictification for (∞,n)-categories, whose central Quillen equivalence still rests on a section (4.4) that is missing from the posted text. read the letter →

arxiv 2507.00146 v1 pith:HNITO6I5 submitted 2025-06-30 math.CT math.ATmath.CO

Clémence Chanavat , Amar Hadzihasanovic This is my paper · ORCID

classification math.CTmath.ATmath.CO MSC 18N45

keywords (∞n)-categoriessemi-strictificationweakmodelcategoriesregulardirectedcomplexesmerge-complexesinflate-complexesunitsconjecturehomotopyhypothesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves the first equivalence between a weak, non-algebraic model of $(\infty,n)$-categories and a semi-strict algebraic model. The main theorem constructs, for every functor between weak $(\infty,n)$-categories, a square in which the weak categories are embedded by acyclic cofibrations into semi-strict merge-$n$-categories and the functor is replaced by a semi-strict functor. This square is the derived unit of a Quillen equivalence between a weak model structure on marked directed complexes and one on marked merge-complexes. The result gives a rigorous, fully explicit combinatorial form to the conjecture that only unitality needs to be weak: associativity and interchange can hold strictly for round pasting diagrams, while algebraic units are supplied independently. If the paper is right, the traditional divide between weak, geometric models and algebraic, semi-strict models of higher categories is bridged for the first time in all dimensions.

What carries the argument

The load-bearing object is a regular directed complex: a combinatorial shape built from atoms with oriented input and output boundaries, which supports both strict $\omega$-categorical pasting and a topological cell decomposition. Three classes of morphisms between these shapes carry the argument: subdivisions, which generate composition; cylindrical collapses, which generate units and degeneracies; and local embeddings. Proposition 1.36 assembles them into a ternary factorisation system $(S,C,L)$ on local subdivision-collapses, and Proposition 1.30 states that cylindrical collapses are freely generated by codimension-1 collapses. From this the paper defines an inflate monad, adding algebraic units, and a merge monad, adding round composition, and shows that they interact through a distributive law: a unit on a composite is a composite of units. The Quillen equivalence runs between the weak model structure $\mathcal{M}_n$ on marked directed complexes and $\mathcal{M}_{M,n}$ on marked merge-complexes, both built from a functorial cylinder given by a tensor product of shapes ('Gray product') and from generating cofibrations and anodyne extensions.

What would settle it

An explicit collapse between atoms of dimension at least 4 with two different decompositions into codimension-1 cylindrical collapses would contradict Proposition 1.30; equally, a pair consisting of a subdivision and a collapse that fails the ternary factorisation of Proposition 1.36 would break the construction of algebraic units in Theorem 3.15 and with it the Quillen equivalence. A reader could search for such a counterexample in the combinatorics of regular directed complexes at the first dimension where strict 4-category pasting is known to diverge.

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Extended reading notes

Core claim

The paper's central claim is that the category of marked directed complexes with the weak model structure $\mathcal{M}_n$, whose fibrant objects are its $(\infty,n)$-categories, is Quillen equivalent to the category of marked merge-complexes with the weak model structure $\mathcal{M}_{M,n}$, whose fibrant objects are, up to acyclic fibration, its semi-strict merge-$n$-categories. The equivalence is exhibited by a semi-strictification functor: given any functor $f:(X,A)\to (Y,B)$ of $(\infty,n)$-categories, there is a diagram in which $F_M f$ is a semi-strict functor of merge-$n$-categories and the vertical maps are equivalences of $(\infty,n)$-categories, in fact acyclic cofibrations. In the semi-strict model, round pasting diagrams compose with strict associativity and interchange, units are algebraic, and globular composition is recovered from units together with round composition; semi-strict functors preserve round composition strictly but units only weakly. The authors present this as the first equivalence between a weak non-algebraic and a semi-strict algebraic model and, at $n=0$, the first semi-strict model of classical homotopy types with algebraic units and composition.

Load-bearing premise

The proof stands or falls on a combinatorial fact about collapsing and subdividing cells: every cylindrical collapse can be decomposed uniquely into a chain of the simplest one-step collapses, and subdivisions and collapses mesh in a clean three-way factorisation; if hidden collapse relations or factorisation failures appear in high dimensions, the algebraic units on which the semi-strictification is built would no longer be definable.

Editorial extensions

If this is right

Every functor between weak $(\infty,n)$-categories can be replaced, without changing source and target up to equivalence, by a semi-strict functor of merge-$n$-categories.
The weak model structure on marked directed complexes and the semi-strict model structure on marked merge-complexes are Quillen equivalent for every $n\in\mathbb{N}\cup\{\infty\}$.
The semi-strict model has algebraic units and strict associativity and interchange for round pasting diagrams, while globular composition appears as a derived operation combining units with round composition.
At $n=0$, the equivalence yields a semi-strict algebraic model of classical homotopy types with both algebraic units and composition, the first such model.
Semi-strict functors preserve round composition strictly and preserve units only weakly, matching the expected shape of the weak units conjecture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the Quillen equivalence extends to the comparison with the complicial model conjectured in the paper, the semi-strict model would inherit equivalences to the standard geometric models for all $n$, connecting the algebraic and geometric clusters of higher-category models.
The explicit, combinatorial nature of the construction makes it a plausible foundation for computational higher-dimensional rewriting, with the derived globular operations giving a concrete bridge between round and globular presentations.
The paper's discussion of dimension 4 suggests a sharp test: proving or refuting the expected equivalence at $n=4$ would reveal whether the semi-strict model captures topologically sound pasting relations that the algebra of strict 4-categories misses.
Because semi-strict functors preserve units only weakly, the theorem can be read as a precise formulation of the idea that unitality is the single axiom that must remain weak in a general higher-categorical model.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—here's my read on Chanavat–Hadzihasanovic.

The headline result is exactly what the abstract says: a Quillen equivalence between their weak model M_n and a semi-strict algebraic model M_{M,n}, with the derived unit giving a semi-strictification of weak functors. If the proof goes through, this is the first bridge between a geometric-style weak model and a semi-strict algebraic model for all n, and for n=0 it gives a semi-strict model of homotopy types with algebraic units and composition. That would be a genuinely important result.

The paper does a lot well before the missing step. The construction of marked directed complexes, the explicit inflate monad, the saturations, the characterisation of fibrant objects as (∞,n)-categories (Theorem 3.50), and the homotopy hypothesis for n=0 are all laid out in considerable detail. The combinatorial backbone—cylindrical collapses, the ternary factorisation system, the merge monad—is hard and mostly self-contained, and the authors are honest about what is conjecture (e.g. Conjecture 3.82 comparing to complicial models for n>0).

Now the soft spots. The biggest one is exactly where the reader's stress-test lands: the proof of the Quillen equivalence is in Section 4.4, and the version I have stops before it. Theorem 4.21 proves a Quillen adjunction; Theorem 4.43 claims the equivalence, and the verification of the derived-unit condition via Proposition 2.21 is not in the visible text. The authors say it uses the merge-inflate distributive law obtained from the ternary factorisation system and the free generation of cylindrical collapses. Those premises are at least stated and partially proved earlier (Propositions 1.30 and 1.36), so I don't see circularity. But the central claim is, at this stage, asserted rather than demonstrated. This is not a minor gap: without Section 4.4, the main theorem is unverified.

There are also smaller issues. The paper leans heavily on the authors' own prior work ([Had24], [CH24a-c], [Cha25]) for many technical lemmas; some of those citations are necessary, but a referee will want those results checked or at least precisely referenced. The comparison with standard geometric models for n>0 remains conjectural, which the authors acknowledge.

Who is this for? Anyone working on models of higher categories, diagrammatic rewriting, or coherence. It deserves a serious referee: the machinery is original, the claimed result is important, and the paper is written with unusual care. But I would not cite it as a proof until Section 4.4 is supplied. My recommendation: send it to peer review, and insist that the missing proof be included or made available in full before acceptance.

Referee Report

3 major / 4 minor

Summary. The paper develops two explicit combinatorial models for (∞,n)-categories: a weak model M_n on marked directed complexes (with fibrant objects characterised as (∞,n)-categories), and a semi-strict model M_{M,n} on marked merge-complexes. The announced main theorem (Theorem 4.43, to be proved in Section 4.4) states that every functor of (∞,n)-categories embeds into a semi-strict functor of merge-n-categories via equivalences σ that are acyclic cofibrations, and that the free-forgetful adjunction between marked directed complexes and marked merge-complexes is a Quillen equivalence. Sections 1–3 and 4.1–4.3 build the necessary combinatorics (subdivisions, cylindrical collapses, a ternary factorisation system, inflate and merge monads) and prove supporting results, including the homotopy hypothesis for n=0. The text provided for review ends in Section 4.3, before the proof of Theorem 4.43.

Significance. If the claims are correct, the paper achieves the first equivalence between a weak non-algebraic model and a semi-strict algebraic model of (∞,n)-categories, and at n=0 provides a semi-strict model of classical homotopy types with algebraic units and composition; this is a substantial step toward Simpson's weak units conjecture and is potentially very useful for diagrammatic algebra and rewriting. The paper is commendable for the explicitness of its constructions, the care with which the main theorem is stated, and the fact that the weak and semi-strict models are defined independently before the Quillen equivalence is attempted; the homotopy hypothesis for n=0 is proved in the text rather than assumed. Its main limitation in the available text is that the central semi-strictification proof is absent, and several load-bearing combinatorial premises are justified only tersely or by reference to prior work of the same authors.

major comments (3)

[§4.4 / Theorem 4.43] The proof of the main theorem is not available in the text provided for review: Section 4.4, where the Quillen equivalence between M_n and M_{M,n} and the acyclicity of the maps σ are to be established, is absent, and the visible text ends in Section 4.3. By Proposition 2.21, the Quillen equivalence requires checking the derived-unit and derived-counit conditions; the text indicates that this is done via the merge-inflate distributive law obtained from the ternary factorisation system (S,C,L) of Proposition 1.36 and the free generation of cylindrical collapses (Proposition 1.30), but the verification itself is not shown. This is load-bearing: until Section 4.4 is supplied, Theorem 4.43 is unsubstantiated. I recommend major revision rather than rejection because the surrounding development is detailed and the missing part is clearly identified.
[§1.3, Proposition 1.36] The ternary factorisation system (S,C,L) is asserted with a proof that consists of the sentence 'The fact that (S,CL) is an orthogonal factorisation systems holds essentially by construction, and we conclude by Proposition 1.27.' Since this factorisation system is used in the definition of the merge monad and is a key ingredient of the missing merge-inflate distributivity in Section 4.4, the construction and uniqueness of the (S,CL) factorisation need to be spelled out, or an exact reference to a published proof must be provided. As written, the assertion is too terse for a load-bearing combinatorial premise.
[§1.2, Proposition 1.30] The claim that cylindrical collapses of atoms are freely generated by codimension-1 collapses is load-bearing: it underlies the construction of algebraic units in Theorem 3.15 and the expected absence of hidden higher-dimensional relations in the merge-inflate distributive law. The proof of uniqueness in the case m>1 uses a characterisation of two faces described as 'independent of the factorisation', but I do not see in the visible text a complete argument that rules out non-trivial higher-dimensional relations among collapses, beyond the claimed induction. Please expand the proof or provide a precise reference that includes this freeness statement.

minor comments (4)

[§4.3, last paragraph] The paragraph on globular composition asserts that the operations −∗_k− satisfy associativity and unitality up to marked-equivalence, with a reference to '[CH24b, Section 5]' and the phrase 'we can show'. Since these operations are advertised as the link to traditional algebraic models, please state the precise theorem and either prove it or give a complete proof sketch.
[Introduction, Theorem statement] The display of Theorem 4.43 in the introduction is missing arrow symbols in the text as provided, so the commutative square is hard to read, especially the labels σ_(X,A), σ_(Y,B), and U^M_m F^M f. Please typeset the square with clearly labelled arrows.
[Various lemmas (e.g., Lemma 2.40, Lemma 2.74)] Several technical lemmas are justified by 'Same as [CH24b, Lemma ...]' or by an analogous statement without a full proof. For a journal submission, please include the statements at least, and for lemmas that are transferred to a new setting (marked directed complexes without algebraic units) explain the transfer explicitly.
[Title page] The header says 'Current version: 25th September 2025' while the arXiv line reads 'arXiv:2507.00146v1 [math.CT] 30 Jun 2025'. Please align the version information.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the weak and semi-strict models are constructed independently, and the claimed Quillen equivalence is not derived from its conclusion in the visible text.

full rationale

No circular step is visible in the manuscript under review. The weak model M_n is built by a Cisinski–Olschok transfer on marked directed complexes (Theorem 2.78), while the semi-strict model M_{M,n} is defined independently on marked merge-complexes, with fibrations created by the forgetful functor to M_n (Theorem 4.21(2)); this is a definitional transfer, not a hidden assumption of the main theorem. Theorem 4.43, the Quillen equivalence, is stated in the introduction and is said to be proved in Section 4.4, which is absent from the supplied text; that is a missing proof and a correctness risk, not a circularity. The paper's heavy reliance on self-citations ([Had24], [CH24a-c]) concerns combinatorial background that is either re-proved on the spot (e.g., Propositions 1.30 and 1.36, whose proofs are given using factorisation systems and free-generation arguments) or consists of prior independent results about regular directed complexes and diagrammatic sets; none of these citations asserts the target equivalence, and the target theorem is not used in their proofs. No parameter is fitted to data and then renamed a prediction: the semi-strictification maps and fibrant-object characterizations are constructed, not posited as consequences of the conclusion. Consequently, the derivation chain visible in the paper does not reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces many definitions but no free parameters or invented entities analogous to new physical objects. The axioms listed are the background theories cited from prior work, which the paper relies on without re-proving.

assumptions (4)

domain assumption The theory of regular directed complexes, maps, comaps, and pastings as in [Had24].
Section 1.1 summarizes but relies on [Had24] for fundamental results, e.g. Proposition 6.2.30 and 6.3.13, used throughout.
domain assumption Henry's theory of weak model categories and the Cisinski-Olschok method [Hen20].
Section 2.1 summarizes this theory and uses it to construct the model structures Mn and MM,n.
domain assumption The Kan-Quillen model structure on semi-simplicial sets presents classical homotopy types [Hen20, Theorem 5.2.1].
Used in Section 3.4 to prove the homotopy hypothesis for n = 0.
standard math Standard set-theoretic foundations with choice and categorical universal algebra (locally presentable categories, monads, adjunctions).
The paper uses locally presentable presheaf categories, reflective subcategories, and the small object argument without further justification.

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Cite this review

Pith. "Pith review of Semi-strictification of $(\infty, n)$-categories." pith.science (2026). https://pith.science/paper/HNITO6I5

@misc{pith2026250700146,
  author       = {Pith},
  title        = {Pith review of: Semi-strictification of $(\infty, n)$-categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HNITO6I5}},
  note         = {Machine review of arXiv:2507.00146}
}

abstract

We prove the first equivalence between a weak non-algebraic model and a semi-strict algebraic model of $(\infty, n)$-categories. This takes the form of a natural semi-strictification, whereby a weak $(\infty, n)$-category is embedded into a semi-strict one through an acyclic cofibration, in such a way that weak functors lift to semi-strict functors; this constitutes the derived unit of a Quillen equivalence between weak model categories whose fibrant objects are, respectively, the weak $(\infty, n)$-categories and (up to an acyclic fibration) the semi-strict ones. The semi-strict model has algebraic units and composition of round pasting diagrams, satisfying a strict form of associativity and interchange as in Henry's regular version of Simpson's weak units conjecture; semi-strict functors strictly preserve round composition, but only weakly preserve units. Globular composition operations are obtained from a combination of units and round composition. Since the models satisfy the homotopy hypothesis in the case $n = 0$, this result also exhibits the first semi-strict model of the classical homotopy types that has algebraic units and composition. The constructions are based on the combinatorics of regular directed complexes and are entirely explicit and combinatorial, in the spirit of Mac Lane's strictification of bicategories.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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Model structures for diagrammatic $(\infty, n)$-categories

[CH24c] C. Chanavat and A. Hadzihasanovic. Model structures for diagrammatic (∞,n )-categories. Online preprint arXiv:2410.19053,

work page Pith review arXiv

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Gray products of diagrammatic $(\infty, n)$-categories

[Cha25] C. Chanavat. Gray products of diagrammatic (∞,n )-categories. Online preprint arXiv:2505.01387,

work page Pith review arXiv

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[Had24] A. Hadzihasanovic. Combinatorics of higher-categorical diagrams. Online preprint arXiv:2404.07273v2,

work page Pith review arXiv

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To appear in London Mathematical Society Lecture Note Series. [Hen16] S. Henry. Algebraic models of homotopy types and the homotopy hypo- thesis. Online preprint arXiv:1609.04622,

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[Sch25] L. Schelstraete. Rewriting modulo in diagrammatic algebras and applica- tion to categorification. Online preprint arXiv:2502.03028,

arXiv

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[Ver08] D.R.B. Verity. Weak complicial sets I. Basic homotopy theory. Advances in Mathematics, 219(4):1081–1149, 2008

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Benjamin and I

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work page 2006