REVIEW 3 major objections 5 minor 11 references
Open Condensed Subgroups and Mackey's Formula
T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read The paper proves a condensed analogue of Mackey's formula for open condensed subgroups, decomposing the restriction of an induced module over double cosets.
desk verdict Correct and honestly-scoped formal paper: Mackey's formula for condensed groups holds, and the stress-test counterexample miscounts orbits. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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 $$\mathrm{Res}^G_K \mathrm{Ind}^G_H M \;\cong\; \bigoplus_{g \in H \backslash G / K(\ast)} \mathrm{Ind}^{K}_{K\cap $g^{{-1}}$Hg} \mathrm{Res}^{$g^{{-1}}$Hg}_{K\cap $g^{{-1}}$Hg}\bigl(M \otimes_{R[H]} R[Hg]\bigr).$$ The 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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [Definition 2.2] 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.
- [Propositions 4.5 and 4.7] 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.
- [Lemma 6.1] 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.
minor comments (5)
- [Definition 2.1] 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.
- [Abstract and Introduction] There are typos in the abstract and introduction, such as 'W e' and 'ab out' on page 1.
- [Corollary 5.3] 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.
- [Example 4.2(ii)] 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.
- [References] 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.
Circularity Check
No significant circularity: the Mackey formula proof is a formal derivation from an independent openness condition, with only background self-citations.
full rationale
The central result, Theorem 5.5, is derived by decomposing G as a disjoint union of double cosets (Proposition 4.7), which in turn rests on the orbit decomposition of Proposition 4.5. The openness condition in Definition 4.1 is not defined in terms of Mackey's formula or in terms of the double-coset index set; it is defined by representability of certain pullbacks by open subsets. Proposition 4.5 then proves, rather than assumes, the equivalence between openness and discreteness of the quotient. The proof of Theorem 5.5 uses no fitted parameters and no data-derived quantities; the isomorphism is algebraic and formal once the decompositions are available. The paper's self-citations to [8] are to standard condensed-module facts, such as the Hom-tensor adjunction and projective generators, which are background tools rather than the load-bearing content of the Mackey formula. The cited definition of open subgroup follows Scholze [1], an external source. The skeptic's concern about Definition 2.2 — that the pointwise orbit set may not satisfy the sheaf condition — is a potential mathematical defect in the paper's construction, but it is not a circularity: even if the pointwise quotient must be sheafified, that is a repair to the definition, not a case of the theorem being equivalent to its own input. Therefore no circular step can be exhibited with the required specificity, and the appropriate score is 0.
Assumptions & free parameters
assumptions (5)
- standard math Condensed groups, sets, and modules are sheaves on CHED with finite jointly surjective covers (Definition 2.1).
- standard math The condensation functor from topological spaces to condensed sets preserves limits and colimits as stated in Section 2.
- standard math The Hom-tensor adjunction and the projective generators Z[S] ⊗ R[G] of the module category (cited from [8, A.21, A.15]).
- domain assumption Solid modules and solidification preserve colimits and are monoidal (cited from [2, Theorems 5.8, 6.2]).
- standard math The abstract Mackey formula for group modules is known and can be applied to the presheaf level (Lemma 5.4 proof).
Cite this review
Pith. "Pith review of Open Condensed Subgroups and Mackey's Formula." pith.science (2026). https://pith.science/paper/BFKEO3OA
@misc{pith2026241211968,
author = {Pith},
title = {Pith review of: Open Condensed Subgroups and Mackey's Formula},
year = {2026},
howpublished = {\url{https://pith.science/paper/BFKEO3OA}},
note = {Machine review of arXiv:2412.11968}
}
read the original abstract
We define what it means for a condensed group action to be open (following Scholze) and show that for open subgroups, many elementary results about abstract modules hold for condensed modules, such as the existence of Mackey's Formula for condensed groups. We also indicate how these results can be "solidified" to obtain their solid versions.
Reference graph
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Reviewed August 11, 2026 · model on record in the stance chip above.
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