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A simplicial set is d-Segal exactly when it is (d+1)-coskeletal and satisfies the d-Segal conditions in the two lowest relevant dimensions.

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A simplicial set is upper/lower d-Segal iff it is (d+1)-coskeletal and satisfies the d-Segal conditions in dimensions d+1 and d+2 (with the proved 'if' direction using (d+2)-coskeletality).

T0 review reviewed 2026-07-31 challenge →

load-bearing objection Clean uniform recognition theorem for d-Segal sets; the abstract/body gap the reader flagged is not real. the 3 major comments →

arxiv 2607.24476 v1 pith:5KYQJ4XG submitted 2026-07-27 math.AT math.CT

Coskeletality and the higher Segal conditions

Philip Hackney This is my paper
classification math.AT math.CT MSC 18N5055U1018G30
keywords higher Segal conditionscoskeletal simplicial setsdécalagepath space criteriongapped subsets2-Segal setssimplicial sets
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Higher Segal conditions ask that certain cubes built from a simplicial set be cartesian; they generalize the classical Segal condition that models categories. Checking them in every dimension looks infinite. This paper proves that for ordinary simplicial sets the infinite list collapses: the set is upper or lower d-Segal if and only if it is (d+1)-coskeletal and the conditions hold in just the two lowest non-vacuous dimensions (called d-critical). The converse direction also shows every d-Segal set is automatically (d+1)-coskeletal. The reduction turns an infinite exactness check into a finite one, and it is sharp: the analogous statement fails for simplicial spaces.

Core claim

For a simplicial set X and d ≥ 0, X is upper (respectively lower) d-Segal if and only if it is (d+1)-coskeletal and upper (respectively lower) d-critical—that is, it satisfies the corresponding d-Segal cube conditions in simplicial dimensions d+1 and d+2. Equivalently, (d+2)-coskeletal plus d-critical already forces full d-Segality, while every d-Segal set is at least (d+1)-coskeletal.

What carries the argument

The path-space criterion relating Segality of X to Segality of its upper and lower décalages, together with the Wiggle Lemma that propagates cartesianness of one gapped cube to all others in the same dimension, and the comparison theorems that transfer coskeletality between X and its décalages. These close the inductive step from the two critical dimensions to all higher ones.

Load-bearing premise

The argument assumes that unique fillers for the two lowest gapped cubes, once coskeletality is known, can be propagated to every higher dimension by wiggling between adjacent gapped sets and by the décalage comparison; if that transfer failed for any parity, the induction would stop.

What would settle it

Exhibit a simplicial set that is (d+1)-coskeletal and satisfies the two critical d-Segal conditions yet fails a higher-dimensional gapped-cube condition, or a d-Segal simplicial set whose (d+1)-spheres do not fill uniquely.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • d-Segality of a finite simplicial set (finitely many nondegenerate simplices) becomes a finite, effective check once d is large enough relative to skeletal dimension.
  • The hierarchy of higher Segal conditions is strict: familiar examples such as Δ^n/∂Δ^n are 2n-Segal but not lower (2n−1)-Segal.
  • Every upper or lower d-Segal simplicial set is automatically (d+1)-coskeletal, recovering the classical facts for nerves of categories (d=1) and for 2-Segal sets.
  • The corresponding reduction fails for simplicial spaces, so the result is special to discrete simplicial sets.

Where Pith is reading between the lines

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

  • The same coskeletal-plus-critical pattern may supply practical recognition algorithms for higher Segal structures arising in Hall algebras or incidence coalgebras once those structures are presented by finite generators.
  • Because n-skeletal sets are automatically (2n−1)-coskeletal, the criterion immediately classifies the Segal height of many quotients and suspensions already studied in algebraic topology.
  • A computer-checkable test for d-criticality on truncated simplicial sets could turn the theorem into a decision procedure for small examples.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proves a coskeletality criterion for the higher Segal conditions on simplicial sets: X is upper (resp. lower) d-Segal if and only if it is (d+1)-coskeletal and satisfies the upper (resp. lower) d-Segal conditions in the lowest two non-vacuous dimensions, d+1 and d+2 ("d-critical"). Theorem 25 states the slightly sharper body version: (d+2)-coskeletal plus d-critical implies d-Segal (Theorems 15 and 24), and d-Segal implies (d+1)-coskeletal (Proposition 12 and Theorem 22). The lower-odd case is proved directly by an inductive filler argument (Lemmas 16–17, Proposition 18) built on the Wiggle Lemma; the remaining three parities are reduced to it via Poguntke's path-space criterion and a careful comparison of coskeletality under décalage (Proposition 20, Lemma 21). Remark 26 shows the analogue for simplicial spaces fails, via a constant simplicial space on a space with infinitely many nontrivial homotopy groups.

Significance. The result resolves a question attributed to Walker Stern and unifies two classical recognition principles — the nerve criterion (2-coskeletal plus unique inner-horn fillers in dimensions 2 and 3) and the Bergner–Osorno–Ozornova–Rovelli–Scheimbauer / Stern criterion for 2-Segal sets (3-coskeletal plus square and pentagon conditions) — into a single statement valid for all d. The practical content is real: combined with the Kennett–Riehl–Roy–Zaks result that n-skeletal simplicial sets are (2n−1)-coskeletal, the criterion reduces d-Segality of a finite simplicial set (for d large enough) to a finite check in dimensions d+1 and d+2, and the author has already used this to exhibit the strictness of the higher Segal hierarchy on the examples Δn/∂Δn. The proofs are self-contained and combinatorial, with the four parity cases handled explicitly, the degenerate k = 0 cases quarantined by a correct hand computation (Lemma 19), and the failure for simplicial spaces delimited by a correct counterexample (Remark 26) that prevents overgeneralization. There are no fitted parameters or external black boxes beyond standard, cited results (Walde's theorem, Poguntke's path-space criterion).

major comments (3)
  1. [Abstract / Theorem 25] I have no load-bearing objections. I specifically checked the one place where the abstract appears stronger than the body: Theorem 25's reverse direction assumes (d+2)-coskeletality, while the abstract asserts the iff with (d+1)-coskeletal. By Definition 3, n-coskeletality quantifies over all m-spheres with m > n, so (d+1)-coskeletal implies (d+2)-coskeletal and Theorem 25 applies verbatim; the forward direction is Theorem 22 exactly. Footnotes 2 and 4 describe precisely this slack in the classical cases, and the line 'See Theorem 25 for a slightly stronger statement' is accurate. The abstract is therefore supported by the body.
  2. [§3, Lemmas 16-17, Proposition 18] The induction in Theorem 15 rests on Lemma 16 and Lemma 17 producing a full compatible collection from the I-indexed data, with Proposition 18 then giving existence and uniqueness. The gap-filling for J = {1,3,...,2k+1} (Lemma 17(2)) is the most intricate step: the verification that L_{j,j'} is gapped in S_{j,j'} via the isomorphism to the even elements of [0,2k] in [0,n-2], and the induction on N = (j'-j)/2, are correct as written. I confirmed the N=1 base case is fully handled by the even-element argument and that Lemma 2 is applied with consistent parameter choices throughout. No correction needed, but this is the argument a reader must trust, and it holds up.
  3. [§4, Proposition 20, Lemma 21, Theorems 22 and 24] The parity reduction via décalage (Propositions 9 and 20, Lemma 21) is consistent in all four cases. In particular: Theorem 22 (upper odd) correctly chains Theorem 8 with Proposition 12 applied to dec^T X and Lemma 21; Theorem 24(3) invokes item (1) rather than Theorem 15, which is legitimate since (1) is established (via (2) and duality) earlier in the same proof — there is no circularity. Lemma 21's two sharpened clauses match exactly the hypotheses available in each parity (the 'in particular' sentence is correct: ue^k_{2k+1}/le^k_{2k+1} hold for 2k-critical, uo^k_{2k+2} for (2k+1)-critical). The k=0 exceptional cases are quarantined by Lemma 19 before the general argument, which is necessary (Remark 14 shows Proposition 12 fails at k=0).
minor comments (6)
  1. [Proposition 9] The proof is outsourced entirely to [10, Proposition 3.13] ('As in the proof of...'). Since this proposition is used in every parity reduction in §4, one or two sentences indicating the idea (or at least a more precise pointer to the relevant steps in [10]) would make the paper more self-contained.
  2. [Lemma 19] Only the upper 1-critical case is treated, with 'the other cases being similar.' The lower 0-critical case is not literally identical (it uses le^0_1 rather than uo^0_2), so a sentence indicating what changes would be welcome, especially since this lemma is the sole guard for the k=0 exceptions.
  3. [References, footnote 3] Reference [14] is an nLab page (with a revision date); for the classical fact that nerves of categories are 2-coskeletal, a stable reference (e.g., a Kerodon tag, which is already cited as [13], or a textbook source) would be preferable.
  4. [Remark 10] The claim that the cyclic-polytope formulation of d-critical is equivalent to Definition 7 is supported only by 'it's not so arduous to prove this directly' plus a reading of Walde's Theorem 7.2.1. A slightly more detailed sketch, or an explicit statement of which steps of [18] are needed, would help readers who want to use the geometric form.
  5. [Introduction, paragraph on effectiveness] The effectiveness claim in the introduction ('allows for an effective, finite check of d-Segality... for d suitably large') would benefit from stating the explicit bound: an n-skeletal simplicial set is (2n-1)-coskeletal by [12, Theorem 3.19], so the criterion applies once d+1 >= 2n-1, i.e. d >= 2n-2. Giving this inequality explicitly would make the remark immediately usable.
  6. [§1, notation; Abstract] The notation S_i = S \ {i} and e_i is convenient but collides visually with simplicial face notation; the remark after the definition helps, but a one-line reminder at its first use in §2 (Definition 4 context) would prevent confusion. Also, in the abstract line of the arXiv text, spacing appears to be lost in 'isd-Segal'/'is(d+1)-coskeletal' — presumably a rendering artifact, but worth checking in the final version.

Circularity Check

0 steps flagged

No circularity: self-contained combinatorial proof relating coskeletality to higher Segal conditions

full rationale

The paper proves a clean if-and-only-if between d-Segality of a simplicial set and the conjunction of (d+1)-coskeletality with the two lowest-dimensional d-Segal checks (d-criticality). All load-bearing steps are direct arguments about unique fillers of compatible collections and cubes (Prop. 12, Lemmas 16–17, Prop. 18, Thm. 15; décalage transfer in Prop. 20 and Lemma 21; path-space reduction via Thm. 8/Prop. 9; Wiggle Lemma 11). Definitions of coskeletality, higher Segal conditions, and décalage are taken from the external literature (Dyckerhoff–Kapranov, Poguntke, Walde, Kennett–Riehl–Roy–Zaks) and used as inputs; the new content is the implication chain among them. Author self-citations ([8], [9], [10]) appear only for context, presentation conventions, or non-essential remarks and are not required to close the main theorems. There are no fitted parameters, no empirical predictions, and no uniqueness claims imported solely by self-citation. Remark 26 correctly delimits the result to simplicial sets. The abstract’s sharper (d+1)-coskeletal formulation is an immediate corollary of Theorem 25 once monotonicity of coskeletality is noted. Score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper works entirely inside standard simplicial-set theory and the existing higher-Segal literature. No free parameters. Load-bearing external inputs are the definitions of higher Segal conditions, Poguntke's path-space criterion, Walde's equivalence with cyclic-polytope triangulations, and the Kennett–Riehl–Roy–Zaks coskeletality bound for skeletal simplicial sets. No new physical or combinatorial entities are postulated.

axioms (5)
  • standard math Standard simplicial-set theory: face/degeneracy identities, coskeletality as unique filling of m-spheres (or right Kan extension along Δ≤n ↪ Δ).
    Used throughout §§1–4 as the ambient category; Definition 3 and the equivalent formulations cited from Kerodon.
  • domain assumption Higher Segal conditions via cartesianness of cubes indexed by gapped subsets (Definition 4), equivalent to unique fillers for compatible collections on simplicial sets (Remark 5).
    Taken from Dyckerhoff–Kapranov / Walde / Hackney–Lynd presentation; the whole paper is a recognition theorem relative to these definitions.
  • domain assumption Poguntke path-space criterion: upper/lower 2k-Segal and (2k+1)-Segal reduce to lower (2k−1)-Segal of décalages (Theorem 8, Proposition 9).
    Invoked to transfer the lower-odd case to all remaining parities in Theorems 22–24.
  • domain assumption Walde's theorem equating cube-cartesianness formulations with cyclic-polytope triangulation locality (used to justify 'critical' via geometric conditions in Remark 10).
    Cited to equate Definition 7 with the geometric critical conditions; not re-proved.
  • domain assumption n-skeletal simplicial sets (n>1) are automatically (2n−1)-coskeletal (Kennett–Riehl–Roy–Zaks, Thm. 3.19).
    Used only for the effectiveness corollary in the introduction (finite check for large d); not needed for Theorem 25 itself.

reviewed 2026-07-31 · how reviews work

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Cite this review

Pith. "Pith review of Coskeletality and the higher Segal conditions." pith.science (2026). https://pith.science/paper/5KYQJ4XG

@misc{pith2026260724476,
  author       = {Pith},
  title        = {Pith review of: Coskeletality and the higher Segal conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KYQJ4XG}},
  note         = {Machine review of arXiv:2607.24476}
}
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read the original abstract

A simplicial set is $d$-Segal if and only if it is $(d{+}1)$-coskeletal and satisfies the $d$-Segal condition in the two lowest relevant simplicial dimensions.

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Reference graph

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This paper was first reviewed by grok-4.5 on July 31, 2026.