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Infinitary combinatorics in condensed math and strong homology

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Pith's one-line read The paper shows that one nonvanishing derived limit, $\lim^1 A_{\omega,\omega_1}$, simultaneously produces failures of fullness in condensed derived categories, of additivity in strong homology, and, in higher degrees, of product…

desk verdict Solid paper with a real result; the main soft spot is the under-specified ∞-categorical reduction in §2.2, not the higher-degree dictionary the reader flagged. read the letter →

arxiv 2412.19605 v2 pith:7EOREWF3 submitted 2024-12-27 math.AT math.CTmath.LO

Jeffrey Bergfalk , Chris Lambie-Hanson This is my paper · ORCID
classification math.ATmath.CTmath.LO MSC 18F1018G8003E0503E3503E7513D05
keywords condensedmathematicsanimacompactprojectivesderivedlimitsn-coherencestronghomologyBanach-Smithdualityinfinitarycombinatorics
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 establishes that a single family of inverse systems — the groups $A_{\kappa,\lambda}[H]$ indexed by finite-support functions from $\kappa$ to $[\lambda]^{<\omega}$ — carries a cluster of structural failures in condensed mathematics and in strong homology. Its central ZFC computation is $\lim^1 A_{\omega,\omega_1}\neq 0$, a nonvanishing higher derived limit produced from a nontrivially coherent family of finite-to-one functions. From that computation the authors derive that the natural functor from pro-derived abelian groups to derived condensed abelian groups is not full, that strong homology fails to be additive on a countable sum of compact spaces, and that the derived limit functors $\lim^n$ are not additive for $n=1,2$. In higher degrees, the same machinery shows that products of compact projective condensed anima need not be compact, because constant sheaves on products of large extremally disconnected spaces can have infinite injective dimension. The paper also shows that consistently vanishing higher limits repair these failures and restore a derived Banach–Smith duality.

What carries the argument

The central objects are the inverse systems $A_{\kappa,\lambda}[H]$ with terms $\bigoplus_{X(f)}H$ and projection maps, indexed by functions $f:\kappa\to[\lambda]^{<\omega}$ ordered by inclusion of their graphs $X(f)$. Their higher derived limits are governed by coherence: by Lemma 2.9 and its generalization Lemma 2.18, $\lim^n A_{\kappa,\lambda}[H]=0$ exactly when every $n$-coherent family of functions $X(\vec f)\to H$ is trivial. The proof of the ZFC nonvanishing in degree one builds a nontrivially coherent family from a classical ladder of finite-to-one functions $e_\alpha:\alpha\to\omega$; the key claim is that no single function on $\omega\times\omega_1$ can trivialize all the induced restrictions. In higher degrees the mechanism is a classical vanishing theorem for derived limits above the cofinality of the indexing order, together with transfinite recursion and a pressing-down argument to produce nontrivial $n$-coherent families and to rule out their trivializations.

What would settle it

Take the coherent family $\Phi=\langle\varphi_f:X(f)\to\mathbb{Z}\rangle$ constructed in the proof of Theorem 2.10(5) from a ladder of finite-to-one functions $e_\alpha:\alpha\to\omega$ such that consecutive restrictions agree modulo finite sets, and check whether a global $\psi:\omega\times\omega_1\to\mathbb{Z}$ exists whose restriction to each $X(f)$ agrees with $\varphi_f$ modulo finite sets. Claim 2.12 asserts that no such $\psi$ exists; producing one would force $\lim^1 A_{\omega,\omega_1}=0$ and would collapse Theorems A and C. The higher-degree analogue is to search for an $(n-1)$-trivialization of an $n$-coherent family built in Theorem 4.11; finding one would make the corresponding $\mathrm{H}^n(U;\bigoplus_\kappa K)$ vanish.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the higher derived limits of the systems $A_{\kappa,\lambda}[H]=\bigoplus_{X(f)}H$, with $f:\kappa\to[\lambda]^{<\omega}$ and $X(f)=\{(i,j)\mid j\in f(i)\}$, are the common combinatorial heart of several apparently unrelated questions. Theorem 2.10(5) proves $\lim^1 A_{\omega,\omega_1}\neq 0$ in ZFC; equivalently, there is a nontrivially coherent family of functions indexed by $\omega([\omega_1]^{<\omega})$. The immediate corollaries are that the natural functor $\mathrm{Pro}(D(\mathrm{Ab}))^b\to D(\mathrm{Cond}(\mathrm{Ab}))$ is not full, that $\mathrm{H}^{n-1}(\coprod_\omega Y^{n,\omega_1})\not\cong\bigoplus_\omega \mathrm{H}^{n-1}(Y^{n,\omega_1})$ where $Y^{n,\omega_1}$ is the compact one-point compactification of a coproduct of $\omega_1$ open $n$-balls, and that $\lim^n$ is not additive for $n=1,2$. Under the consistent hypothesis that $\lim^n A_{\aleph_0,\aleph_0}[H]=0$ for all $n>0$ and all $H$, the paper derives degree-zero concentration of certain $\mathrm{RHom}$ expressions, commutation of $\mathrm{Ext}$ with countable limits, and a derived Banach–Smith duality for separable solid $\mathbb{Q}_p$-Banach spaces. With additional set-theoretic hypotheses such as the axiom of constructibility, the nonvanishing extends to $\lim^{n+1} A_{\aleph_n,\aleph_{n+1}}\neq 0$. Finally, Theorem 4.11 constructs open subsets $U\subseteq\beta X$ with $\mathrm{H}^n(U;\bigoplus_\kappa K)\neq 0$ whenever $|X|,\kappa\geq\aleph_\omega$, yielding Theorem D: the constant sheaf on $S\times T$ has infinite injective dimension for large extremally disconnected $S,T$, and products of compact projective condensed anima are not in general compact.

Load-bearing premise

The bridge from vanishing higher derived limits to triviality of all $n$-coherent families is stated for every degree $n$, but for $n>1$ its proof is delegated to an earlier source rather than given here; the higher-dimensional claims of the paper stand on that dictionary.

Editorial extensions

If this is right

  • The functor $\mathrm{Pro}(D(\mathrm{Ab}))^b\to D(\mathrm{Cond}(\mathrm{Ab}))$ is not full, so the condensed derived category contains homomorphisms that no pro-object morphism can see (Corollary 2.11).
  • Strong homology is not additive in ZFC on a countable sum of compact spaces: $\mathrm{H}^{n-1}(\coprod_\omega Y^{n,\omega_1})$ is not isomorphic to $\bigoplus_\omega \mathrm{H}^{n-1}(Y^{n,\omega_1})$ (Theorem C, Corollary 3.4).
  • The derived limit functors $\lim^1$ and $\lim^2$ on pro-abelian groups are not additive in ZFC (Corollary 3.5).
  • If all higher limits $\lim^n A_{\aleph_0,\aleph_0}[H]$ vanish, then the $\mathrm{RHom}$ computations in $D(\mathrm{Cond}(\mathrm{Ab}))$ concentrate in degree zero and the classical Banach–Smith duality extends to a derived duality for separable solid $\mathbb{Q}_p$-Banach spaces (Theorems 2.13 and 2.14).
  • For extremally disconnected $S,T$ of cardinality at least $\aleph_\omega$, the constant sheaf on $S\times T$ has infinite injective dimension, and products of compact projective condensed anima need not be compact (Theorems 4.3 and 4.4).

Reading between the lines

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

  • If the degree-by-degree pattern of the paper's higher nonvanishing theorem persists, the question of whether $\lim^3$ is consistently additive is likely independent of ZFC and tied to the behaviour of $\mathrm{H}^2(\omega_2;\mathbb{Z})$; the paper itself leaves this as an open question.
  • The $\aleph_\omega$ threshold in Theorem D suggests a testable dichotomy: either products of extremally disconnected spaces below $\aleph_\omega$ are compact in every model of ZFC, or the threshold itself fluctuates with set-theoretic hypotheses such as the strong limit status of $\aleph_\omega$.
  • Since the authors note the pyknotic translation is straightforward, translating Theorems A–D to pyknotic categories would confirm that these failures are not an artifact of the condensed-site conventions.
  • The same $n$-coherent families may have functional-analytic shadows: nonvanishing derived limits could appear as nonzero $\mathrm{Ext}$ classes in Banach–Smith duality outside the separable case, giving a concrete place to look for a failure of derived duality.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper develops a package of infinitary combinatorial tools—multidimensional coherent families, the inverse systems A_{κ,λ}, and their higher derived limits lim^n A_{κ,λ}[H]—and applies them to four main theorems. Theorem A asserts that the natural functor Pro(D(Ab))^b → D(Cond(Ab)) is not full. Theorem B gives consistency results, conditional on lim^n A[H] = 0, concerning Ext groups in Cond(Ab), the structure of RHom against products of free abelian groups, and solid Q_p-Banach duality. Theorem C gives a ZFC counterexample to additivity of strong homology, witnessed by a countable sum of compact spaces Y^{n,ℵ_1}. Theorem D states that constant sheaves on extremally disconnected spaces are injective for finite fields, that certain products of Čech–Stone compactifications have infinite injective dimension, and that products of compact projective condensed anima need not be compact. The combinatorial core includes the ZFC computation lim^1 A_{ω,ω_1} ≠ 0 (Theorem 2.10(5)), the consistent higher-degree nonvanishing of Theorem 2.20, and the construction of nontrivial n-coherent families in Theorem 4.11.

Significance. If the ∞-categorical transfer in §2.2 is made fully rigorous, Theorems A, B, and D are substantial and will be influential. The strong homology part of the paper, Theorem C together with Theorem 3.3 and Corollary 3.4, is largely independent of that transfer and appears sound; it provides a notably simpler, compact ZFC counterexample to additivity than Prasolov's. The explicit combinatorial constructions, especially the proof of Theorem 2.10(5) and the recursive construction in Theorem 4.11, are a genuine strength: they are detailed, checkable, and likely to be reusable. The paper also performs a useful service by organizing several Clausen–Scholze questions and connecting them to a substantial set-theoretic literature. The main weakness is that the reduction from fullness of the natural functor to equality (4), and hence to vanishing of the derived limits, is asserted through a 'relaxed or naive' ∞-categorical reading and is not proved with the necessary hypotheses.

major comments (3)
  1. [§2.2 (pp. 8–10), Proposition 2.5, Corollary 2.11] The reduction from full faithfulness of the natural functor (3) to equality (4), and then to vanishing of lim^n A_{I,J}[H], is the load-bearing step for Theorem A and for Theorem 2.13 (hence Theorem B), but it is not established in the manuscript. The paper explicitly says it works with a 'relaxed or naive' reading of the ∞-categorical manipulations, and the specific claims 'RHom commutes with all finite limits and colimits' and the 'standard dévissage' reduction to free groups concentrated in degree zero are not proved; the cited references [53, 4.4.2.7], [52, Prop. 15.4.2], [77, Thm. 5.8], and [39] are not accompanied by the boundedness, size, and projectivity checks needed here. In particular, the identification of the right-hand side of (4) with Rlim A_{I,J}[H] depends on the projectivity of ∏_i Z^{f(i)} in Solid and on slenderness, and the reader cannot verify that these apply for arbitrary I and J. Since Corollary 2.11 is literally 'immediate from item (5) of Theorem 2.10, together with Proposition 2.5', the non-fullness theorem is only as solid as this reduction. Please either prove Proposition 2.5 directly for the specific objects G_i = ⊕_J Z and H = ⊕_K Z, which would avoid most of the problematic reduction, or supply a complete ∞-categorical proof with all hypotheses verified.
  2. [Lemma 2.9 (p. 12)] The n = 1 case of Lemma 2.9 is proved in the text, but for n > 1 the proof is delegated to [18, Section 2.1] with a brief assurance that the argument is 'close in spirit' and 'only a bit more tedious'. This lemma is the dictionary used to translate nonvanishing of lim^n into nontrivial n-coherent families in Theorem 2.10, Theorem 2.20, and Theorem 4.11. Because those theorems are among the paper's central new contributions, the higher-degree case should either be proved in full or the precise statement from [18] should be quoted with the hypotheses verified for the systems A_{κ,λ}[H]. In particular, the alternating conventions and the use of equation (7) need to be checked explicitly.
  3. [Theorem A (p. 2) and Corollary 2.11 (p. 13)] The proof establishes, at most, that the natural functor is not fully faithful in the ∞-categorical sense: nonvanishing lim^1 contradicts the implication 'fully faithful ⇒ vanishing' of Proposition 2.5. The theorem, however, states that the functor is 'not full'. For ordinary categories, 'full' and 'fully faithful' are not interchangeable unless faithfulness is known separately, and the paper does not prove faithfulness of the functor (3). Please either change the statement and abstract to 'not fully faithful', or add an argument showing that the induced map on Hom sets is not surjective (or that the functor is faithful), so that 'not full' follows.
minor comments (3)
  1. [§2.2 (p. 9)] The sentence 'RHom commutes with all finite limits and colimits' should be made precise. In the relevant stable ∞-categorical setting, mapping spectra preserve finite limits in each variable, but they do not preserve arbitrary colimits; the colimits over I([J]<ω) used later are not finite, so the intended statement needs qualification.
  2. [§4.1, proof of Theorem 4.11 (pp. 38–40)] The final nontriviality argument in the induction step is very compressed. In particular, the step from the failure of equation (20) to the claim that the associated family Ψ is trivial, and the later notation 'supp(φ_α)' for the fixed tuple α obtained from Fodor's lemma, should be expanded for the reader to check the contradiction.
  3. [References] Several key references are unpublished or online lecture notes ([76], [77], [27]) and personal communication ([78]). This is common in the field, but for the journal version the authors should give the most stable available references or precise pointers (theorem numbers, dates, or published versions) for the claims cited from them.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims rest on a directly proved ZFC computation of lim^1 A_{ω,ω1} and a stated, non-circular reduction; self-citations are to prior proven theorems rather than to the paper's own conclusions.

full rationale

The main derivation chain is not circular. Theorem 2.10(5), the ZFC nonvanishing of lim^1 A_{ω,ω1}, is proved in the paper by constructing a nontrivially coherent family from a classical nontrivially coherent family E of finite-to-one functions; the proof is self-contained and does not assume the desired conclusion. Corollary 2.11 then applies Proposition 2.5, which states the forward direction: full faithfulness would force all lim^n A_{I,J}[H] to vanish. Nonvanishing of lim^1 therefore yields non-fullness by contrapositive, and the reduction in Section 2.2 is an argued chain of equivalences, not a restatement of the conclusion. Theorem C similarly follows from the same directly proved lim^1 computation via the strong-homology computation in Theorem 3.3, which generalizes classical arguments rather than importing the target result. The sheaf-theoretic Theorem D uses a dictionary between higher derived limits and n-coherent families; the higher-n case of Lemma 2.9 is delegated to the authors' prior paper [18], but that is a cited prior theorem with a stated proof, not an unverified premise identical to the paper's conclusion, and the n=1 case used for the central non-fullness and strong-homology claims is proved in the paper itself. The paper also openly attributes the forms of Theorems A, B, and D to Clausen and Scholze and identifies its own contribution as the derived-limit analyses. No fitted parameter is relabeled as a prediction, and no equation is used to prove itself. The skeptical concern about the ∞-categorical reduction in Section 2.2 is a question of external rigor and completeness, not of circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical parameters are fitted to data; the paper's cardinals and abelian groups are variables in theorems, not fitted constants. The mathematical constructions, such as the systems A_kappa_lambda and n-coherent families, are defined explicitly and are not speculative entities. The set-theoretic hypotheses are declared as assumptions, not disguised as derived results.

assumptions (6)
  • standard math ZFC axioms
    All theorems are proven within ZFC or under additional set-theoretic hypotheses that are stated explicitly.
  • standard math Lemma 2.9 for higher n, delegated to [18, Section 2.1]
    The dictionary converting lim^n A_kappa_lambda = 0 into triviality of n-coherent families is stated but not proved for n > 1; the reader is referred to prior work by the same authors.
  • standard math Goblot's theorem (Theorem 2.15)
    Used repeatedly to obtain trivializations at limit stages in Theorem 2.20 and Theorem 4.11, and to prove Lemma 4.13.
  • domain assumption Hypothesis of Theorem B and Theorem 2.14: lim^n A[H] = 0 for all n > 0 and all abelian groups H
    This is a consistency hypothesis known to hold in Cohen models (by [16] and [5]), not a ZFC theorem; the Banach-Smith duality conclusion is conditional on it.
  • domain assumption Theorem 2.20 hypotheses: stationary S in I[aleph_{n+1}] and, in Case 2, diamond on S and a nontrivial n-coherent family
    These are additional set-theoretic axioms that hold in L and in certain canonical inner models; the nonvanishing result is not claimed in ZFC.
  • standard math Stone duality correspondence between open subsets of beta X and ideals on X
    Used in equations (16)-(19) in Section 4.1 to convert sheaf cohomology on open subsets of Stone-Cech compactifications into derived limits over ideals.

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Cite this review

Pith. "Pith review of Infinitary combinatorics in condensed math and strong homology." pith.science (2026). https://pith.science/paper/7EOREWF3

@misc{pith2026241219605,
  author       = {Pith},
  title        = {Pith review of: Infinitary combinatorics in condensed math and strong homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7EOREWF3}},
  note         = {Machine review of arXiv:2412.19605}
}
read the original abstract

Recent advances in our understanding of higher derived limits carry multiple implications in the fields of condensed and pyknotic mathematics, as well as for the study of strong homology. These implications are thematically diverse, pertaining, for example, to the sheaf theory of extremally disconnected spaces, to Banach--Smith duality, to the productivity of compact projective condensed anima, and to the structure of the derived category of condensed abelian groups. Underlying each of these implications are the combinatorics of multidimensionally coherent families of functions of small infinite cardinal height, and it is for this reason that we convene accounts of them together herein.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Higher limits of wider systems

    math.LO 2025-07 unverdicted novelty 8.0 of 10

    Under GCH plus diamond principles, and in Gödel's constructible universe, the higher derived limits lim^n A_λ are nonzero for every cardinal λ where Goblot's vanishing theorem does not force them to zero.

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