{"id":"3be14cb7-f65a-4abc-a133-2d80b166c885","arxiv_id":"2507.00146","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every weak (∞,n)-category embeds into a semi-strict algebraic model via an acyclic cofibration, forming the derived unit of a Quillen equivalence between weak model categories.","lead":"The paper proves that every weak (∞,n)-category embeds into a semi-strict algebraic model through an acyclic cofibration, giving the first equivalence between a geometric-style and an algebraic-style model for all n. It also provides the first semi-strict model of classical homotopy types with algebraic units and composition.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central Quillen equivalence is asserted rather than demonstrated in the available text: Section 4.4, where the derived-unit condition of Proposition 2.21 must be verified via merge-inflate distributivity, is absent. Until that proof is supplied, the main theorem remains unverified.","rationale":"I read the available portions of the paper in good faith. Sections 1 through 4.3 form a coherent development: the combinatorial framework of regular directed complexes, subdivisions, cylindrical collapses, the weak model structures M_n, the characterisation of fibrant objects as (∞,n)-categories, and the construction of the merge-complex model are internally consistent as far as I can verify. I found no explicit mathematical contradiction or circularity in these parts. The paper also honestly flags that comparison with standard geometric models for n>0 remains conjectural (Conjecture 3.82), but that is not load-bearing for the internal equivalence claimed in Theorem 4.43. The single most load-bearing point is therefore the missing proof of Theorem 4.43 itself. The theorem promises the Quillen equivalence of M_n and M_{M,n}, plus the semi-strictification square with sigma an acyclic cofibration. Theorem 4.21 provides only the Quillen adjunction; the equivalence conditions of Proposition 2.21 are not established in the visible text. The reader's verdict of CONDITIONAL is appropriate: the claim is plausible, the preparation is substantial, but the decisive final verification is absent from the reviewed text. I do not see a reason to move the verdict, but I also cannot upgrade it to acceptance without the missing proof. My concern agrees partially with the reader's weakest assumption: the reader points to Proposition 1.30 and Proposition 1.36 as the combinatorial core, and I agree that these are exactly where a hidden failure would surface, but the immediate unverified step is the derived-unit/counit check in Section 4.4 that would use those propositions.","tokens_in":63723,"tokens_out":11139,"duration_ms":137460,"concrete_test":"Obtain the complete proof of Theorem 4.43 and check the two conditions of Proposition 2.21 for the adjunction F^M_m -| U^M_m. Specifically: (i) for every J_n-fibrant marked directed complex (X,A), identify the map X -> U^M_m(F^M_m X, A_inv) with the derived unit, construct its inverse in Ho(M_n) using the merge-inflate distributive law, and confirm compatibility with marked-equivalence classes; (ii) verify the counit condition on a fibrant marked merge-complex by showing that U^M_m reflects equivalences onto M_n. If (i) cannot be derived from Propositions 1.30 and 1.36 without adding a new coherence axiom, the central theorem is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the visible text, Theorem 4.21 establishes only a Quillen adjunction between M_n and M_{M,n}; Theorem 4.43 additionally claims that this adjunction is a Quillen equivalence and that the canonical maps sigma_{(X,A)}: (X,A) -> U^M_m(F_M X, A_inv) are acyclic cofibrations. By Proposition 2.21, this requires checking, for every fibrant (X,A), that the transpose of a fibrant replacement of F^M_m(X,A) is an equivalence in M_n, and dually for the counit. That verification is the content of the missing Section 4.4. The proof is said to rely on a merge-inflate distributive law obtained from the ternary factorisation system (S,C,L) of Proposition 1.36 and from the free-generation statement for cylindrical collapses, Proposition 1.30; these are the same combinatorial premises the reader identifies as the weakest assumptions. If the distributive law fails, or if Proposition 1.30 hides non-trivial higher-dimensional relations among collapses, the maps sigma would not lift to semi-strict functors and the equivalence would fail. Thus the central claim is plausible and well-prepared but not confirmed by the text available for review.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":63930,"tokens_out":5607,"duration_ms":64680,"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":[{"comment":"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.","section":"§4.4 / Theorem 4.43"},{"comment":"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.","section":"§1.3, Proposition 1.36"},{"comment":"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.","section":"§1.2, Proposition 1.30"}],"minor_comments":[{"comment":"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.","section":"§4.3, last paragraph"},{"comment":"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.","section":"Introduction, Theorem statement"},{"comment":"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.","section":"Various lemmas (e.g., Lemma 2.40, Lemma 2.74)"},{"comment":"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.","section":"Title page"}],"recommendation":"major_revision","confidential_remarks":"The provided text is truncated before Section 4.4. If the missing section is present in the submitted version, this report should be read as conditional on it, and I would be willing to re-review the complete file. The paper relies heavily on prior work of the same authors ([Had24], [CH24a-c]) and on references of the form 'same proof as ...'; this is acceptable in a series of papers, but the editor may wish to ensure that referees have access to those documents."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—here's my read on Chanavat–Hadzihasanovic.\n\nThe 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.\n\nThe 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).\n\nNow 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.\n\nThere 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.\n\nWho 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.","headline":"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.","tokens_in":64509,"tokens_out":2327,"would_cite":false,"duration_ms":27309,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18N45"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["(∞,n)-categories","semi-strictification","weak model categories","regular directed complexes","merge-complexes","inflate-complexes","weak units conjecture","homotopy hypothesis"],"falsifier":"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.","tokens_in":63473,"feed_emoji":"🔗","tokens_out":11774,"duration_ms":122538,"temperature":0.7,"pith_summary":"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.","feed_headline":"Every weak (∞,n)-category embeds into a semi-strict one","feed_subtitle":"The embedding is an acyclic cofibration, functors lift, and weak and semi-strict model categories become Quillen equivalent.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the combinatorial theory of regular directed complexes, including maps, comaps, subdivisions, collapses, and tensor products on which the whole construction is built.","marker":"[Had24]"},{"why":"Proves the homotopy hypothesis for a partially algebraic model of $\\infty$-groupoids with round composition but no algebraic units; this is the starting point extended here to all $(\\infty,n)$-categories.","marker":"[Hen18]"},{"why":"Introduces weak model categories and the framework used in the paper to put model structures on marked directed and merge-complexes and to define Quillen equivalences.","marker":"[Hen20]"},{"why":"Pioneering work on directed complexes and the duality between composition and subdivision that motivates the definition of merge-complexes.","marker":"[Ste93]"},{"why":"Develops the preceding diagrammatic model of $(\\infty,n)$-categories whose marked structures and weak model are adapted and modified here.","marker":"[CH24c]"},{"why":"The classical strictification of bicategories is the two-dimensional template whose proof structure the semi-strictification generalises to arbitrary dimension.","marker":"[ML63]"},{"why":"Establishes that fully strict models cannot present all homotopy types, setting the target of semi-strictification with only unitality weakened.","marker":"[Sim09]"}],"fun_headline_variants":["First equivalence: weak and semi-strict (∞,n)-categories","Semi-strictification: every weak (∞,n)-category embeds","Quillen equivalence between weak and semi-strict (∞,n)-categories","First semi-strict model of homotopy types with algebraic units","Explicit semi-strictification via acyclic cofibrations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["First equivalence: weak and semi-strict (∞,n)-categories","Semi-strictification: every weak (∞,n)-category embeds","Quillen equivalence between weak and semi-strict (∞,n)-categories","First semi-strict model of homotopy types with algebraic units","Explicit semi-strictification via acyclic cofibrations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000919,"raw_usage":{"total_tokens":4004,"prompt_tokens":1070,"completion_tokens":2934,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":686,"completion_tokens_details":{"reasoning_tokens":2836}},"tokens_in":686,"tokens_out":2934,"duration_ms":23486,"temperature":1.0,"reasoning_tokens":2836,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:22:40.468838+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}