{"id":"bb27ac73-5260-4e09-80ce-4d15fed3c3cf","arxiv_id":"2412.11968","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Condensed analogues of Mackey's formula and related restriction-induction results hold for open subgroups of condensed groups.","lead":"This paper defines a notion of open actions for condensed groups and proves a Mackey formula for induced modules along open subgroups. It shows that several elementary results from abstract group cohomology carry over to the condensed setting.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 2.2's pointwise orbit set is not a sheaf in general, and Proposition 4.5, hence Theorem 5.5, relies on that invalid object.","rationale":"The paper's central claim is Theorem 5.5, Mackey's formula for condensed groups with an open subgroup. The proof route is: define open actions via the orbit set, prove in Proposition 4.5 that open actions are exactly those with discrete orbit space, use this to decompose G into orbits and double cosets, and then assemble the Mackey decomposition. The reader identified the equivalence in Proposition 4.5 as the weakest assumption. My stress-test locates a more basic problem underneath that equivalence: the orbit set G\\X is introduced in Definition 2.2 as a pointwise quotient and asserted to be a sheaf, but this assertion has a concrete counterexample even for a finite group acting on a finite discrete constant sheaf. If G\\X is not a sheaf, then the pullback diagram used in Definition 4.1 to define an orbit is not a diagram in CondSet, and the proof of Proposition 4.5, which explicitly treats G\\X as a sheaf and compares it with the constant sheaf on its underlying set, is not valid. This is an internal inconsistency rather than a disagreement with current consensus, because the definition itself is false. The error is likely repairable: the standard construction of quotients in CondSet is the sheafified coequalizer, and with that replacement the intended results may survive. Therefore the verdict should remain CONDITIONAL rather than REJECT, but the condition should now include fixing the quotient definition and rewriting Proposition 4.5 accordingly. I agree only partially with the reader's weakest-assumption analysis: the reader pointed to the openness/discreteness equivalence, which is indeed pivotal, but the specific mechanism I identify is the non-sheafness of the pointwise quotient that the equivalence presupposes.","tokens_in":10151,"tokens_out":28072,"duration_ms":290456,"concrete_test":"Compute the pointwise quotient Q(S) = C(S,{0,1})/Z/2 for the action of Z/2 swapping the two points, for S = {0,1} and for the cover {0} -> {0,1}, {1} -> {0,1}. Observe that the two global sections of Q(S) restrict to the same compatible pair, so Q is not a sheaf. Then replace G\\X in Definition 4.1 and Proposition 4.5 by the sheafified colimit quotient and check whether the proof of Proposition 4.5 still yields discreteness of the orbit space; in particular, determine whether the argument that a map S -> G\\X locally factors through a constant map survives without pointwise surjectivity of global sections.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 2.2 claims that the pointwise quotient (G\\X)(S) = G(S)\\X(S) defines a sheaf. This is false. Let G be the constant condensed group Z/2 and let X be the constant sheaf on {0,1}, with G acting by swapping the two points. For S = {0,1}, the set X(S) has four elements and G(S) has two orbits on it, so Q(S) has two elements. For S = {0} and S = {1}, Q is terminal. The sheaf condition for the cover {0} -> S, {1} -> S would force Q(S) to have exactly one element, but it has two. Thus Q is not even separated. Since Definition 4.1 and Proposition 4.5 use this orbit set G\\X as a condensed set, and Proposition 4.7 uses the double quotient H\\G/K as the index set for the double-coset decomposition underlying Theorem 5.5, the central Mackey formula is not established as written. The likely repair is to define G\\X as the sheafified quotient (the coequalizer in CondSet) and then reprove Proposition 4.5 for that object; this replacement is not supplied in the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a notion of open action of a condensed group on a condensed set, following Scholze, and claims that for open subgroups the usual orbit decomposition holds. It then derives structural results about induction and restriction of condensed modules, culminating in a condensed version of Mackey's formula (Theorem 5.5). A final section sketches analogous statements for solid modules, including a projective-restriction result for profinite subgroups.","tokens_in":10342,"tokens_out":13486,"duration_ms":130368,"significance":"If corrected, the paper would provide a natural transfer of elementary module-theoretic facts to condensed groups with open subgroups. The main theorem is a formal consequence of the orbit decomposition; the value lies in identifying the correct openness condition and showing that it suffices. The paper is candid about the limits of the approach, credits related work in [3], and offers a potentially useful framework for condensed representation theory. The flaws identified below are significant but appear repairable rather than fatal to the central claim.","major_comments":[{"comment":"The assertion that the pointwise quotient (G\\X)(S)=G(S)\\X(S) defines a sheaf is false. Let G be the constant condensed group Z/2 and X the constant sheaf on {0,1} with G acting by swapping. For S={0,1} in CHED, the pointwise quotient has two elements, while the quotient on each singleton is terminal; the sheaf condition for the jointly surjective cover {0}->S, {1}->S is violated. Thus G\\X is not even a separated presheaf. Since G\\X is used as a condensed set in Definition 4.1, Propositions 4.5 and 4.7, and the index set of Theorem 5.5, this error is load-bearing. The repair is to define G\\X as the sheafified coequalizer in CondSet and to redevelop the orbit decomposition for that object; the paper does not supply this replacement.","section":"Definition 2.2"},{"comment":"The proof of Proposition 4.5 conflates the pointwise quotient presheaf with its sheafification. It first treats G\\X as a pointwise-defined object and then invokes sheafification via the injection of presheaves Δ0(G\\X) into G\\X without explicitly separating the two. More importantly, the open-action condition in Definition 4.1 refers to pullbacks along x:*->G\\X, which is not a well-defined condensed-set map until G\\X is known to be a sheaf. A viable repair is to define orbits as images of the action morphism G->X for each x in X(*), prove the decomposition X = ∐ Gx directly, and then show the quotient sheaf is the constant sheaf on G\\X(*). This is more than a notational change and needs to be written out for the central theorem to be established.","section":"Propositions 4.5 and 4.7"},{"comment":"The proof of Lemma 6.1 assumes a non-canonical H-set decomposition G = H × G/H for profinite groups. A continuous section of the quotient map G->G/H does not always exist; surjective maps of compact Hausdorff spaces, even Stone spaces, need not split. The paper's remark that one can choose a preimage under G(X) -> (G/H)(X) is not justified because an epimorphism of sheaves need not be surjective on sections. This result is not needed for Theorem 5.5, but it is a stated claim in the solid section and should be corrected or its proof revised.","section":"Lemma 6.1"}],"minor_comments":[{"comment":"The condition given for a condensed set — involving only T(∅)=* and T(S1⊔S2)≅T(S1)×T(S2) — is not equivalent to the sheaf condition on CHED with finite jointly surjective families. The full sheaf condition for arbitrary finite covers is needed.","section":"Definition 2.1"},{"comment":"There are typos in the abstract and introduction, such as 'W e' and 'ab out' on page 1.","section":"Abstract and Introduction"},{"comment":"The verification that the isomorphism Ind = Coind is canonically an R[G]-module isomorphism is only sketched; a more explicit display of the R[G]-actions would improve clarity.","section":"Corollary 5.3"},{"comment":"It would be helpful to explicitly state that an open (and hence clopen) subset of an extremally disconnected space is again extremally disconnected, so that the pullback P is indeed in CHED.","section":"Example 4.2(ii)"},{"comment":"Reference [1] is a MathOverflow answer; consider citing a published source if one exists, as this is a foundational reference for the notion of openness.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is a short formal note whose main theorem is plausible, but the current version contains a foundational error in the definition of the orbit set that undermines the proof of the central result. The repair — replacing pointwise quotients by sheafified coequalizers — is natural and likely within the author's reach, so I do not recommend rejection. The solid-section Lemma 6.1 also needs reworking; it is independent of the main theorem but should be corrected. The editor may wish to ask the author to consult the existing literature on condensed quotients and to provide a rigorous treatment of the orbit decomposition for the sheafified quotient."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nTang's paper is a clean, modest formal contribution. It generalizes Scholze's open-subgroup notion to group actions, proves orbit and double-coset decompositions, and derives Mackey's formula for condensed modules over arbitrary condensed rings. The proofs are formal, and the author says so himself. Corollaries 5.1 and 5.3 were already in Zou for Z, but Theorem 5.5 for arbitrary condensed rings appears new. The central argument looks sound to me.\n\nThe best part is the honesty: prior work is credited, the results are not oversold, and the double-coset decomposition (Prop. 4.7) is the right mechanism.\n\nNow to the stress-test note: I do not think it lands. For S={0,1}, G(S)=Z/2×Z/2 and X(S) has four elements; the pointwise action is transitive, so Q(S) has one element, not two. The alleged sheaf violation disappears. More broadly, the site CHED is selected because extremally disconnected spaces are projective in CondSet, which is exactly what makes the pointwise quotient in Definition 2.2 a sheaf. The paper should still justify that claim with a line or two, but it is not a hidden flaw.\n\nSoft spots, in proportion: Definition 2.2's \"does define a sheaf\" is asserted without proof—worth adding a sentence or a citation. Lemma 6.1 assumes, for profinite H≤G, that G ≅ H×G/H as H-spaces; a section exists, but the proof is only sketched in the remark, and the solid section is clearly not the main event. The proof of Proposition 4.5 is also a bit terse around the comparison of presheaves and sheaves, but the idea is correct.\n\nWho this is for: anyone doing induction, restriction, or cohomology in condensed settings. It is not groundbreaking, but it fills a real gap and appears correct. I would send it to a serious referee, with the small tightenings above as the main requests.\n\nRecommendation: deserves peer review, conditional on minor revisions.","headline":"Correct and honestly-scoped formal paper: Mackey's formula for condensed groups holds, and the stress-test counterexample miscounts orbits.","tokens_in":10922,"tokens_out":30859,"would_cite":true,"duration_ms":323534,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18F20","20J06"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a condensed analogue of Mackey's formula for open condensed subgroups, decomposing the restriction of an induced module over double cosets.","keywords":["condensed mathematics","condensed groups","open subgroups","Mackey formula","restriction and induction","solid modules","profinite groups","orbit decompositions"],"falsifier":"Exhibit a condensed group action satisfying Definition 4.1 whose orbit space is not isomorphic to the constant condensed set on its underlying points. Proposition 4.5 says this cannot happen; such an example would falsify Theorem 5.5, whose proof depends on that equivalence.","tokens_in":9893,"feed_emoji":"🧮","tokens_out":11133,"duration_ms":110452,"temperature":0.7,"pith_summary":"Condensed mathematics treats topological algebraic objects as sheaves, and this paper asks whether the basic module-theoretic machinery of group theory survives in that setting. The answer is that it does, provided one of the subgroups is open in a precise sheaf-theoretic sense. The main theorem is a condensed Mackey formula: for open $H \\leq G$ and any condensed ring $R$, restricting the module induced from $H$ to $G$ back down to a subgroup $K$ is isomorphic to a direct sum, indexed by double cosets $H \\backslash G / K(\\ast)$, of modules induced from the intersections $K \\cap g^{-1}Hg$. This matters because restriction does not preserve projectives for arbitrary condensed subgroups, so the ordinary coset arguments fail; openness restores a canonical discrete orbit decomposition that makes those arguments go through. The paper also solidifies the formula, recovering the known profinite case, and suggests that the main content is the openness notion itself rather than the formal corollaries.","feed_headline":"Mackey's formula survives in condensed groups","feed_subtitle":"When one subgroup is open, induced condensed modules restrict as double-coset direct sums, as in abstract groups.","key_machinery":"The machinery is the paper's openness condition (Definition 4.1): a condensed group $G$ acting on a condensed set $X$ is open if, for every test object $S$ in the site of compact Hausdorff extremally disconnected spaces and every map $S \\to X$, pulling back any orbit $Gx$ gives a subobject of $S$ representable by an open subset of the topological space $S$. Proposition 4.5 shows that this is equivalent to the orbit space $G \\backslash X$ being discrete, and hence gives the canonical decomposition $X \\cong \\coprod_{x \\in G\\backslash X(\\ast)} Gx$. The same statement for bisets (Proposition 4.7) decomposes $G$ as $\\coprod_{g \\in H\\backslash G/K(\\ast)} HgK$, turning the condensed group ring $R[G]$ into a direct sum of $R[H]$-$R[K]$-bimodules. Lemma 5.4 then identifies each summand $R[HgK]$ as the tensor product of $R[Hg]$ with $R[K]$ along the intersection $K \\cap g^{-1}Hg$, which is exactly the assembly needed for Mackey's formula.","core_discovery":"The paper's central claim is Theorem 5.5: for condensed groups $H, K \\leq G$ with $H$ open (or $K$ open), and for any condensed ring $R$ and right $R[H]$-module $M$, there is an $R[K]$-module isomorphism\n$$\\mathrm{Res}^G_K \\mathrm{Ind}^G_H M \\;\\cong\\; \\bigoplus_{g \\in H \\backslash G / K(\\ast)} \\mathrm{Ind}^{K}_{K\\cap $g^{{-1}}$Hg} \\mathrm{Res}^{$g^{{-1}}$Hg}_{K\\cap $g^{{-1}}$Hg}\\bigl(M \\otimes_{R[H]} R[Hg]\\bigr).$$\nThe direct sum is indexed by the underlying point set of the condensed double-coset space, and the paper's convention is that every equality sign denotes an isomorphism. The result is not a new analytic artifact: it reproduces the abstract-group Mackey formula once the subgroup is open, and it specializes to the known profinite version after applying solidification.","pith_inferences":["A reader could use Theorem 5.5 as a transfer principle: any group-theoretic construction expressed through induction, restriction, and double cosets should work verbatim for condensed modules once the relevant subgroup is open, since the proof is formal after the orbit decomposition.","The paper does not develop cohomological consequences, but since induction is exact and restriction has a well-behaved derived version in projective settings, Theorem 5.5 should yield a Mackey decomposition for condensed group cohomology in the same way as the classical statement.","Lemma 6.1 suggests that in the solid category the natural finiteness condition may be representability of the quotient as a profinite space rather than openness, and that a broader class of subgroups might satisfy version of these results after solidification."],"forward_implications":["For an open subgroup $H \\leq G$, restriction from $G$ to $H$ sends projective $R[G]$-modules to projective $R[H]$-modules, because the canonical coset decomposition makes $R[G]$ a direct sum of copies of $R[H]$.","If $H$ has finite index in $G$, induction and coinduction from $H$ to $G$ are canonically isomorphic, matching the abstract-group situation.","The Mackey formula gives a complete double-coset decomposition of $\\mathrm{Res}^G_K \\mathrm{Ind}^G_H M$ whenever either subgroup is open.","Applying the solidification functor to Theorem 5.5 yields the solid version of Mackey's formula, and condensation of profinite data preserves all the operations involved, so the known Mackey formula for profinite groups is recovered.","For profinite groups, restriction of solid modules preserves projectives even without openness, a statement stronger than the open-subgroup version in the solid setting."],"supporting_citations":[{"why":"This reference supplies the original notion of open subgroup in condensed groups that Definition 4.1 generalises.","marker":"[1]"},{"why":"This reference provides the condensed-mathematics foundations, including solidification functors whose preservation properties are used in Section 6.","marker":"[2]"},{"why":"This reference contains the earlier proof that restriction to an open subgroup preserves projectives for condensed abelian groups, which Corollary 5.1 extends to all condensed rings.","marker":"[3]"},{"why":"This reference gives the sheaf-theoretic identification used in Proposition 4.5 to show openness is equivalent to discreteness of the orbit space.","marker":"[5]"},{"why":"This reference is the classical Mackey formula for profinite groups that Theorem 5.5 generalises after condensation and solidification.","marker":"[7]"},{"why":"This reference supplies the tensor-Hom adjunction, enriched Hom, projective generators, and solid module facts used throughout the module arguments.","marker":"[8]"},{"why":"This reference provides the lemma that non-totally-disconnected compact groups have non-projective condensed group rings, motivating why openness is needed.","marker":"[10]"}],"fun_headline_variants":["Condensed groups keep Mackey's formula for open subgroups","Open subgroups bring Mackey's formula to condensed modules","Mackey's formula generalizes via open subgroups in condensed groups","Open condensed subgroups enable Mackey's double-coset decomposition","Condensed Mackey's formula holds for open subgroups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that openness of a condensed group action is equivalent to discreteness of its orbit space (Proposition 4.5), since the coset decompositions used to prove Mackey's formula are drawn from that equivalence.","fun_headline_variants_meta":{"raw":{"variants":["Condensed groups keep Mackey's formula for open subgroups","Open subgroups bring Mackey's formula to condensed modules","Mackey's formula generalizes via open subgroups in condensed groups","Open condensed subgroups enable Mackey's double-coset decomposition","Condensed Mackey's formula holds for open subgroups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000337,"raw_usage":{"total_tokens":1789,"prompt_tokens":795,"completion_tokens":994,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":913}},"tokens_in":411,"tokens_out":994,"duration_ms":7701,"temperature":1.0,"reasoning_tokens":913,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:25:41.304467+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a condensed group action satisfying Definition 4.1 whose orbit space is not isomorphic to the constant condensed set on its underlying points. Proposition 4.5 says this cannot happen; such an example would falsify Theorem 5.5, whose proof depends on that equivalence.","supporting_citations":[{"cited_title":"Structure of a profinite group as a conden sed set with an action of an open subgroup","cited_arxiv_id":null,"evidence_quote":"This reference supplies the original notion of open subgroup in condensed groups that Definition 4.1 generalises."},{"cited_title":"Condensed mathematics","cited_arxiv_id":null,"evidence_quote":"This reference provides the condensed-mathematics foundations, including solidification functors whose preservation properties are used in Section 6."},{"cited_title":"Sheaves in geometry and logic: A first introduction to topos t heory","cited_arxiv_id":null,"evidence_quote":"This reference gives the sheaf-theoretic identification used in Proposition 4.5 to show openness is equivalent to discreteness of the orbit space."},{"cited_title":"Profinite groups","cited_arxiv_id":null,"evidence_quote":"This reference is the classical Mackey formula for profinite groups that Theorem 5.5 generalises after condensation and solidification."},{"cited_title":"Condensed group cohomology","cited_arxiv_id":null,"evidence_quote":"This reference provides the lemma that non-totally-disconnected compact groups have non-projective condensed group rings, motivating why openness is needed."}],"review_version":1}