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REVIEW 3 major objections 5 minor 34 references

El mar que envuelve a las piedras: espacios de Stone, profinitos y su papel en la aritm\'etica contempor\'anea

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper develops Stone duality, profinite spaces, and Stone–Čech compactification from scratch, then argues that profinite spaces are the generators of condensed mathematics, giving arithmetic objects like p-adic integers and Galois grou

desk verdict A readable Spanish-language survey of Stone duality and profinite spaces; the classical core is fine, but the closing bridge to condensed mathematics is built on a wrong site definition and a false proposition about regular open sets. read the letter →

arxiv 2511.21128 v2 pith:Z2PBHRWH submitted 2025-11-26 math.GN math.CTmath.NT

J.R. Pérez-Buendía This is my paper
classification math.GNmath.CTmath.NT MSC 06E1554B2554D3554G0518F20
keywords StonedualityBooleanalgebrasprofinitespacesStone–Čechcompactificationcondensedmathematicsp-adicintegersabsoluteGaloisgroupsextremallydisconnected
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

The paper builds up Stone duality, profinite spaces, and Stone–Čech compactification from Boolean algebras and ultrafilters, and then makes a larger claim: these classical spaces are the basic building blocks of condensed mathematics. The intended payoff is that arithmetic objects such as the p-adic integers and absolute Galois groups—which are profinite spaces—can be treated as sheaves on a site, so that they live in an abelian category where ordinary homological algebra applies. A sympathetic reader can see the paper as a didactic bridge from elementary topology to the machinery used in contemporary arithmetic geometry, and as a statement that the old Boolean perspective is still the structural core.

What carries the argument

The machinery is the Stone-space construction: for a Boolean algebra B, the set X_B of ultrafilters, topologized by the clopens â={U∈X_B:a∈U}. This one construction proves the representation theorem, produces inverse limits of finite spaces for profinite spaces, realizes the Stone–Čech compactification βX as the Stone space of the power set P(X), and in the last section supplies the profinite probes S↦Cont(S,T) that define condensed sets. The entire argument rides on the fact that clopens in a Stone space are exactly the Boolean-algebra elements, so discrete Boolean logic and compact totally disconnected geometry are the same data.

What would settle it

Compare the paper's site with the standard one on the two-point profinite space: the canonical cover by its two singleton clopen subsets (each proper, but jointly covering) is a cover in the latter but not under Definition 11.1. Exhibiting a presheaf on profinite spaces that satisfies the equalizer condition for every cover in the paper's sense but fails gluing for that two-singleton cover would settle the mismatch.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the single chain Bool ≃ Stone^op ≃ ProFin^op. The author proves the representation theorem of Stone—every Boolean algebra is the algebra of clopen subsets of the space of its ultrafilters—and the converse recovery of a Stone space from its clopens, then identifies this class with inverse limits of finite discrete spaces. The final move is to put profinite spaces at the base of condensed mathematics: every compact Hausdorff space T is assigned the presheaf T^♢(S)=Cont(S,T) on profinite probes, and the paper claims that profinite spaces generate the resulting topos, so that Z_p and Galois groups become sheaves whose homological algebra is stan

Load-bearing premise

The bridge to condensed mathematics rests on the profinite-site definition in Section 11.1, where a cover is required to be a finite family of maps each already surjective onto the whole space; the standard condensed site allows finite families whose images jointly cover, and the sheaf-theoretic consequences are different under the definition as written.

Editorial extensions

If this is right

  • Condensed abelian groups, as described in the paper, form a Grothendieck abelian category, so cohomology of profinite groups and p-adic cohomology can be defined by standard derived functors.
  • The condensation functor from compact Hausdorff spaces to condensed sets is faithful, meaning continuous maps between compact Hausdorff spaces are detected entirely by their behaviour on profinite probes.
  • Because profinite spaces generate the category, every condensed set is a colimit of representable profinite probes; morphisms and isomorphisms of condensed objects can be checked on profinite spaces such as Z_p and Galois groups.
  • The Stone–Čech compactification βX, realized as the Stone space of P(X), is extremally disconnected, and Gleason's theorem makes such spaces precisely the projective objects in compact Hausdorff spaces; βN is thus a projective object carrying all the Boolean information about N.
  • The duality Bool≃ProFin^op gives a working dictionary: finite inverse systems, clopen partitions, and Boolean subalgebras are the same data, which organizes the arithmetic examples systematically.

Reading between the lines

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

  • The paper leaves implicit a gap in its site theory: the cover condition in Definition 11.1 (each map individually surjective onto the whole space) is stronger than the usual finite jointly-surjective condition, and the sheaf-theoretic consequences differ; replacing that definition while keeping the narrative would make the bridge to the condensed framework cleaner.
  • The 'granule' approximation operator in Section 10 suggests a finite-combinatorial approximation theory for regular open sets and Gleason covers; refining finite Boolean subalgebras should yield explicit approximants, though the paper only defines the operator.
  • The same generation-by-probes pattern could be tested on pro-étale or pro-finite sites in arithmetic geometry: if profinite spaces are the correct generators for condensed sets, then analogous sheaf categories should admit a presentation by profinite-like probes.
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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 / 5 minor

Summary. The paper is an expository survey, in Spanish, aimed at advanced undergraduates and beginning graduate students. It develops the classical theory of Boolean algebras, Stone duality, profinite spaces as inverse limits of finite spaces, and the Stone–Čech compactification, with an emphasis on arithmetic examples (Z_p, absolute Galois groups, profinite completions). It then adds a section on extremely disconnected spaces and Gleason's theorem, and concludes with a chapter presenting Clausen–Scholze's condensed mathematics, claiming that profinite/Stone spaces are the fundamental generators of the category of condensed sets. The stated goal is pedagogical: to reorganize known material so as to bridge classical duality theory and modern condensed mathematics.

Significance. If the exposition were fully correct, the paper would be a useful teaching text: the classical parts are mostly standard and many proofs (e.g., Stone's representation theorem, the ultrafilter construction of βX, and the ED–projective equivalence) are presented in a readable way. The paper claims no new theorems and is clear that it is a review. Its main value is as a self-contained route from Boolean algebras to profinite spaces and, potentially, to condensed mathematics. However, the condensed-math bridge contains a load-bearing error: the profinite site is defined with a topology that is strictly coarser than the Clausen–Scholze site, so the sheaf category described as Cond is not the usual category of condensed sets. In addition, a central proposition in the section on complete Boolean algebras is stated with a false formula. These issues are fixable within the scope of a revision, but they currently undermine the claimed modern framework.

major comments (3)
  1. [§11.1, Definition 11.1] The definition of the profinite site is not the Clausen–Scholze site. Clause (i) requires each map f_i:S_i→S to be an epimorphism, i.e. f_i(S_i)=S, which makes clause (ii) redundant. In the standard condensed site a cover is a finite family whose images merely cover S jointly; individual maps need not be surjective. For example, the two inclusions {0}→{0,1} and {1}→{0,1} form a cover in the standard site but not under Definition 11.1. Consequently the sheaf category defined in Definition 11.4 is strictly larger than the usual condensed sets, and Proposition 11.7, which only sketches sheafness for the paper's weaker topology, does not establish the bridge to Clausen–Scholze's category. The central claim of §11 — that profinite spaces generate Cond in the Clausen–Scholze sense — is therefore unsupported as written.
  2. [§10.1, Proposition 10.3, Eq. (22)] The statement that arbitrary intersections of regular open sets give the infimum in RO(X) is false. The proof begins with the assertion that an arbitrary intersection of open sets is open, which is not true. A concrete counterexample in R is U_n = (-1/n,1/n) ∪ (10,11); each U_n is regular open, but ⋂_n U_n = {0} ∪ (10,11) is not regular open and is not open. The correct infimum in RO(X) is int(cl(⋂ U_i)), not ⋂ U_i. This error affects the proof that RO(X) is complete, which is used in the proof of Theorem 10.6. The theorem itself can be repaired, but Eq. (22) and its proof must be corrected.
  3. [§7.2, Theorem 7.7, continuity part] The proof of continuity of the extension ar f:βX→K contains an invalid step. The paper claims that if U ∈ \widehat{f^{-1}(W)} then ar f(U)∈W. This does not follow from ultrafilter convergence: an ultrafilter on a compact Hausdorff space may contain an open set that does not contain its limit. For instance, on K=[0,1], an ultrafilter generated by the open set (0,1] and all neighborhoods of 0 converges to 0 while containing (0,1], which does not contain 0. The theorem is true, but the proof should use closed sets or a different compactness argument.
minor comments (5)
  1. [§10.1, Lemma 10.2] The proof of the complement law for U∨U* says "U∪U* = X", which is false for a regular open set such as U=(0,1) in R. The correct statement is int(cl(U∪U*)) = X. The result is true but the justification needs to be rewritten.
  2. [§10.1, Proposition 10.3 proof] In the proof of the infimum part, the line "siempre se tiene V⊆int(V)" is not correct for an arbitrary set V. If V is open then V=int(V), but V has not been shown to be open. This is part of the same issue as Major Comment 2.
  3. [§11.1, Definition 11.4] There is a typo: "Denotamos por Conda la categoría" should read "Denotamos por Cond la categoría".
  4. [§7.2, Lemma 7.6] The letter U is used both for an ultrafilter and for an open set in the proof. This makes the proof harder to follow and should be clarified.
  5. [§11.3, Theorem 11.15] The proof is only an "Esbozo" and relies on cited notes [16] and [1]. For an expository paper this is acceptable, but since the theorem is central to the final section, the authors should either give a more complete proof or state more explicitly that it is quoted from the literature.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is an expository review; the main concern in the condensed-math section is a site-definition fidelity issue, not a circular reduction.

full rationale

The paper's classical chapters (Stone duality, profinite spaces, Stone–Čech compactification, ED spaces, Gleason projectivity) are self-contained derivations from definitions, occasionally citing standard external monographs for routine details. There are no fitted parameters, no empirical predictions, and no author self-citations that carry load. The condensed-math epilogue invokes external sources (Clausen–Scholze, Bhatt) for the generator theorem, and its own Definition 11.1 gives a nonstandard notion of profinite cover — each map is required to be surjective, making condition (ii) redundant — which is not the Clausen–Scholze finite jointly-surjective site. As a result, the bridge to Clausen–Scholze's condensed mathematics is not supported as written; Proposition 11.7 and Theorem 11.15 are at best established for the paper's weaker site. However, this is a correctness/fidelity problem, not circularity: the paper does not define its conclusion in terms of itself, and the external references are not self-citations. No load-bearing step reduces to its own input by construction.

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

No free parameters or invented entities appear. The axioms are standard background from order theory, topology, and the cited condensed-math literature; the main caveat is the paper's incorrect statement of the regular-open meet (Prop. 10.3) and the nonstandard cover definition (Def. 11.1), which are flagged separately.

assumptions (6)
  • standard math Ultrafilter lemma / Zorn's lemma: every proper filter is contained in an ultrafilter.
    Used in Obs. 3.4 and Thm. 4.4 to extend the filter generated by a∧¬b to an ultrafilter.
  • standard math Tychonoff's theorem: arbitrary products of compact spaces are compact.
    Used in Thm. 4.3 to prove the Stone space X_B is compact as a closed subspace of 2^B.
  • standard math A compact Hausdorff space is 0-dimensional iff totally disconnected (clopens form a base).
    Used throughout, esp. Lem. 5.4 and Prop. 6.6.
  • standard math The Boolean algebra RO(X) of regular open sets is complete, with meet int(⋂ U) and join int(closure(⋃ U)).
    The paper states a false version (Prop. 10.3); the corrected result is needed for Thm. 10.6.
  • standard math Gleason's theorem: projective objects in CHaus are exactly extremally disconnected spaces.
    Invoked as Thm. 10.15 without proof.
  • domain assumption Condensed mathematics (Clausen–Scholze): Cond is a topos; profinite spaces generate Cond; T^♢ is a sheaf for T∈CHaus.
    Used in Section 11; the paper only sketches these results and cites [16], [1].

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Pith. "Pith review of El mar que envuelve a las piedras: espacios de Stone, profinitos y su papel en la aritm\'etica contempor\'anea." pith.science (2026). https://pith.science/paper/Z2PBHRWH

@misc{pith2026251121128,
  author       = {Pith},
  title        = {Pith review of: El mar que envuelve a las piedras: espacios de Stone, profinitos y su papel en la aritm\'etica contempor\'anea},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2PBHRWH}},
  note         = {Machine review of arXiv:2511.21128}
}
abstract

Stone duality establishes a contravariant equivalence between the category of Boolean algebras and the category of compact, Hausdorff, totally disconnected topological spaces (Stone spaces). These spaces are precisely the profinite spaces and form the natural environment for arithmetic objects such as the $p$-adic integers, absolute Galois groups, and profinite completions. In this article we give a rigorous and self-contained development of the construction of the Stone space associated to a Boolean algebra, prove Stone's representation theorem, and describe profinite spaces as inverse limits of finite systems. From there we introduce the Stone--\v{C}ech compactification via its universal property and discuss central arithmetic examples (such as $\beta\mathbb{N}$ and profinite completions), emphasizing their structural role in arithmetic geometry and in the framework of Condensed Mathematics of Clausen--Scholze. The text is aimed at advanced undergraduate and beginning graduate students, as well as readers interested in the interface between topology, number theory, and new categorical formulations of topology.

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Reference graph

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