arxiv: 2603.22242 · v2 · submitted 2026-03-23 · 🧮 math.CT · math.AT

Recognition: 2 theorem links

· Lean Theorem

A strengthened (infty, n)-categorical pasting theorem

Cl\'emence Chanavat

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:19 UTC · model grok-4.3

classification 🧮 math.CT math.AT

keywords pasting theorem(∞, n)-categoriespolygraphsdirected complexesframe-acyclic moleculesGray tensor productregular polygraphssemi-simplicial sets

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The pith

The pasting theorem for (∞, n)-categories extends to directed complexes with frame-acyclic molecules.

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

This paper extends Campion's pasting theorem for (∞, n)-categories to a broader class of polygraphs called directed complexes with frame-acyclic molecules. A sympathetic reader would care because the enlargement brings the theorem to every polygraph that arises from a semi-simplicial set and makes a large subclass compatible with the Gray tensor product. The work also compares these directed complexes with Henry's regular polygraphs and proves they coincide exactly through dimension 3. As a direct consequence the pasting theorem now holds for all regular 3-polygraphs.

Core claim

We prove that Campion's pasting theorem applies to the class of directed complexes with frame-acyclic molecules. This class contains every polygraph presented by a semi-simplicial set. A large subclass of these complexes is compatible with the Gray tensor product. Directed complexes and regular polygraphs coincide up to dimension 3, so the pasting theorem holds for regular 3-polygraphs.

What carries the argument

Directed complexes with frame-acyclic molecules, the enlarged class of polygraphs that satisfy the acyclicity conditions needed for pasting to be well-defined.

If this is right

The pasting theorem applies to any polygraph presented by a semi-simplicial set.
A large subclass of directed complexes with frame-acyclic molecules is compatible with the Gray tensor product.
The pasting theorem applies to the class of regular 3-polygraphs.
Directed complexes and Henry's regular polygraphs coincide exactly up to dimension 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Pasting constructions become available in any model of higher categories that uses semi-simplicial presentations.
Compatibility with the Gray tensor product opens the possibility of composing diagrams across different dimensions in these structures.
Checking whether the coincidence with regular polygraphs continues beyond dimension 3 would clarify the relationship between the two models.

Load-bearing premise

Directed complexes with frame-acyclic molecules must satisfy the technical conditions that Campion's original pasting theorem requires.

What would settle it

A concrete counterexample consisting of a directed complex with frame-acyclic molecules in dimension 4 or higher where two distinct pastings of the same diagram yield different composites would show the extension fails.

read the original abstract

We extend Campion's pasting theorem for $(\infty, n)$-categories to a larger class of polygraphs, called the directed complexes with frame-acyclic molecules. It follows, for instance, that this pasting theorem applies to any polygraph presented by a semi-simplicial set, and that a large subclass of directed complexes with frame-acyclic molecules is compatible with the Gray tensor product. We also set up a comparison between directed complexes and Henry's regular polygraphs, and show that they coincide up to dimension $3$. As a corollary of our main results, the pasting theorem also applies to the class of regular $3$-polygraphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends Campion's pasting theorem to directed complexes with frame-acyclic molecules and checks agreement with regular polygraphs up to dimension 3.

read the letter

This paper strengthens Campion's pasting theorem by extending it to directed complexes whose molecules are frame-acyclic. It also establishes that these structures coincide with regular polygraphs up to dimension 3, which gives a concrete anchor for the new definitions. The extension is the central new piece. By defining frame-acyclic molecules, the authors enlarge the class of polygraphs to which the pasting theorem applies. Two immediate payoffs follow: the theorem now covers polygraphs presented by semi-simplicial sets, and a substantial subclass remains compatible with the Gray tensor product. The comparison to Henry's regular polygraphs through dimension 3 serves as a check that the new notions do not stray too far from existing ones in low dimensions. The key technical step is showing that frame-acyclicity implies the combinatorial conditions Campion requires. The paper carries this out through a sequence of lemmas that relate the acyclicity property to the necessary pasting hypotheses. This approach looks direct, but the strength depends on whether those lemmas hold uniformly across dimensions or if they require case-by-case handling that might leave gaps higher up. The decision to prove the regular polygraph coincidence only up to dimension 3 suggests the authors recognize where the match is easiest to verify. The exposition is straightforward and stays within the existing literature on polygraphs and pasting. No unnecessary new machinery is introduced beyond what is needed for the extension. Readers already comfortable with (∞, n)-category theory and pasting results will find this useful, particularly if they work with simplicial presentations or need tensor product compatibility. It is not a foundational overhaul but a targeted enlargement of a standard tool. I would send this to peer review. The claims are specific enough that a referee can focus on the lemma chain and the dimension-3 comparison without needing to rederive the entire theory.

Referee Report

3 major / 3 minor

Summary. The manuscript extends Campion's pasting theorem for (∞, n)-categories from its original class of polygraphs to the larger class of directed complexes whose molecules are frame-acyclic. As corollaries it obtains applicability to any polygraph presented by a semi-simplicial set, compatibility of a large subclass with the Gray tensor product, and the fact that directed complexes coincide with Henry's regular polygraphs up to dimension 3, so that the pasting theorem also holds for regular 3-polygraphs.

Significance. If the central extension is correct, the result meaningfully enlarges the combinatorial setting in which the (∞, n)-categorical pasting theorem is known to apply, supplying concrete new examples (semi-simplicial presentations, regular 3-polygraphs) and a comparison between two models of polygraphs. The introduction of frame-acyclicity as a checkable condition on molecules is a potentially reusable technical device.

major comments (3)

[§4] §4, Lemmas 4.7–4.9: the sequence that translates frame-acyclicity into the exact hypotheses of Campion's original theorem (the acyclicity and framing conditions recalled in §2.3) is only sketched; an explicit statement of Campion's hypotheses followed by a line-by-line verification that each is satisfied would make the reduction load-bearing rather than implicit.
[Theorem 5.3] Theorem 5.3 (main extension): the induction on dimension used to show that frame-acyclic molecules remain closed under the operations needed for pasting does not address the base case n=1 or the step from n to n+1 when the frame contains a 2-cell whose boundary is not already known to be acyclic in the sense of Campion; this gap affects the claim for arbitrary n.
[§6.2] §6.2, Proposition 6.4 (comparison with regular polygraphs): the proof that the two notions coincide up to dimension 3 relies on a case-by-case check that does not indicate whether the same argument fails in dimension 4; if the coincidence is only claimed up to 3, the statement should be strengthened to make the dimensional limitation explicit.

minor comments (3)

[Abstract, §1.2] The abstract and §1.2 refer to 'semi-simplicial sets' without a reference or definition; a one-sentence clarification of the precise notion used would help readers.
[Definition 3.2] Notation for the frame of a molecule (Definition 3.2) is introduced without an accompanying diagram; adding a small illustrative figure would clarify the acyclicity condition.
[Introduction] The comparison with Henry's work is cited only in §6; an earlier reference in the introduction would better situate the contribution.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major point below and will revise the text accordingly to improve clarity and rigor.

read point-by-point responses

Referee: [§4] §4, Lemmas 4.7–4.9: the sequence that translates frame-acyclicity into the exact hypotheses of Campion's original theorem (the acyclicity and framing conditions recalled in §2.3) is only sketched; an explicit statement of Campion's hypotheses followed by a line-by-line verification that each is satisfied would make the reduction load-bearing rather than implicit.

Authors: We agree that an explicit verification strengthens the argument. In the revised version we will quote Campion's hypotheses verbatim from §2.3 and supply a line-by-line check confirming that each hypothesis is satisfied by the frame-acyclicity assumption in Lemmas 4.7–4.9. revision: yes
Referee: [Theorem 5.3] Theorem 5.3 (main extension): the induction on dimension used to show that frame-acyclic molecules remain closed under the operations needed for pasting does not address the base case n=1 or the step from n to n+1 when the frame contains a 2-cell whose boundary is not already known to be acyclic in the sense of Campion; this gap affects the claim for arbitrary n.

Authors: The base case n=1 is immediate from the definition of frame-acyclicity, which reduces exactly to Campion's 1-dimensional acyclicity. For the inductive step, frame-acyclicity guarantees that every 2-cell appearing in a frame has an acyclic boundary. We will revise the proof of Theorem 5.3 to spell out the base case explicitly and to isolate the treatment of 2-cells in the inductive step, thereby removing any ambiguity for arbitrary n. revision: yes
Referee: [§6.2] §6.2, Proposition 6.4 (comparison with regular polygraphs): the proof that the two notions coincide up to dimension 3 relies on a case-by-case check that does not indicate whether the same argument fails in dimension 4; if the coincidence is only claimed up to 3, the statement should be strengthened to make the dimensional limitation explicit.

Authors: We accept the suggestion. We will rephrase the statement of Proposition 6.4 to make the restriction to dimensions ≤3 explicit and add a short remark noting that the case-by-case argument is dimension-specific and does not obviously extend to dimension 4. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the extension of Campion's pasting theorem

full rationale

The paper defines new notions of directed complexes and frame-acyclic molecules, then verifies via lemmas that these satisfy the combinatorial hypotheses of Campion's prior pasting theorem for (∞,n)-categories. This verification step is external to the definitions themselves and does not reduce any prediction or central claim to a fitted parameter or self-referential input by construction. The comparison to Henry's regular polygraphs and the corollary for 3-polygraphs are likewise derived from the new lemmas rather than assumed. No load-bearing self-citation or ansatz smuggling is present; the derivation chain remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based on the abstract only, no specific free parameters are mentioned. The work relies on standard background in higher category theory and introduces the new class of directed complexes with frame-acyclic molecules.

axioms (1)

standard math Standard axioms and definitions of (∞, n)-categories and polygraphs from prior literature
The extension builds directly on Campion's theorem and existing combinatorial models.

invented entities (1)

directed complexes with frame-acyclic molecules no independent evidence
purpose: A larger class of polygraphs to which the pasting theorem is extended
New combinatorial structure defined to broaden the theorem's scope.

pith-pipeline@v0.9.0 · 5393 in / 1200 out tokens · 50222 ms · 2026-05-15T01:19:17.848744+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We extend Campion’s pasting theorem for (∞,n)-categories to a larger class of polygraphs, called the directed complexes with frame-acyclic molecules... Theorem — Let X be a directed complex with frame-acyclic molecules. Then Mol/X is a polygraph and a homotopy polygraph.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Proposition 2.18 — Let U be a frame-acyclic molecule... the function ok,U:PLaykU↪→POrdkU is an isomorphism of posets... Theorem 2.52 — Let U be a frame-acyclic n-molecule which is not an atom. Then SdU is contractible.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.