{"id":"137d89ab-ae1d-4ed1-9c25-493aaff840d1","arxiv_id":"2504.21764","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The fibers of X/G → π0(X)/π0(G) are equivalent to a crossed module Cone(π1(G) → Aut X x), and an explicit X-crossed module model is given for the two-sided action case.","lead":"Drinfeld proves that the fibers of the natural projection from the quotient 2-groupoid X/G to the 1-groupoid π0(X)/π0(G) are described by a crossed module built from π1(G) and the automorphism groups of X. This gives an explicit X-crossed module model for two-sided translation actions, with suggested applications to the p-adic geometry of Barsotti-Tate groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The explicit model in §2.4 rests on an unproved equivariance check; this is the weakest point, though no counterexample surfaced.","rationale":"The reader's weakest assumption correctly identifies the sketched X-crossed module equivalence and the unverified equivariant family in §2.4 as the soft spot. Proposition 1.3.2, the other central claim, is proven tersely but is sound: the claimed functor from Cone(π1(G)→Aut_X x) to the kernel is essentially surjective (choose an isomorphism g→1), fully faithful on 2-morphisms because both sides are governed by π1(G), and monoidal. The main unresolved issue is the §2.4 model. I stress-tested the potential problem by examining the case B•=G•=(S_3,S_3,d=id), where misreading the semidirect product makes the equivariance appear to fail. With the multiplication derived from the stated action on G0, the formula (2.5) is consistent: the G_-1 component in (2.5) cancels exactly under conjugation by p=(1,γ). Thus I did not find an actual error. However, the manuscript never supplies this verification, and the type error in §2.3.3 makes the construction literally ill-posed if read incorrectly. This is a genuine proof gap in a load-bearing part of the paper, not a mere stylistic issue. It is repairable and does not by itself invalidate the central theorem, so the reader's ACCEPT verdict can stand; the authors should be asked to expand the verification and fix the typo. My concrete test is the minimal check that would settle whether the concern lands as a real mathematical error rather than a missing proof.","tokens_in":7912,"tokens_out":51544,"duration_ms":532049,"concrete_test":"Work out the minimal nontrivial case with B•=G• on S_3 (both groups S_3, d=id, π=π'=id). Compute the quotient 2-groupoid directly from Definition 1.1.3 and the X-crossed module data of §2.4 for g=1 and g=γ. Then verify for p=(1,γ):1→γ that p d_1(β) p^{-1}=d_γ(β) for every β∈S_3, using the semidirect product multiplication (b1,γ1)(b2,γ2)=(b1b2,(b2^{-1}γ1)γ2) forced by the action (b,γ)·g=b d(γ) g b^{-1}. If equality fails for any β, §2.4 is wrong; if it holds, repeat the same equational check symbolically and expand the 'easy to check' passage into a proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of §2.4—that the X-crossed module with H_g=B_-1, equivariant structure from the B0-action only, and d_g as in (2.5) describes the quotient Cone(B•→G•)—is asserted by 'It is easy to check' and is not verified anywhere. This is load-bearing because the explicit model is the paper's main constructive result and the proposed test of Conjecture D.8.4 depends on it. In particular, the equivariance of d_g under the full groupoid Γ is a real condition: for p=(b,γ):g→g' in B0⋉πG_-1 one must have p d_g(β) p^{-1}=d_{g'}(H_p(β)), with H_p the induced action on B_-1; the manuscript does not prove that H_p can be taken to be the B0-action or that the G_-1 component cancels. The general equivalence in §2.3.2–2.3.3 is also only sketched, and the literal text of §2.3.3 has a type error ('d_{x'}:H_x' should be 'd_{x'}:H_{x'}' in the quotient by H_{x'}), so the construction cannot be checked from the printed page. I found no counterexample: in the minimal case B•=G•=(S_3,S_3,d=id), formula (2.5) is consistent once the semidirect product multiplication derived from the action (b,γ)·g=b d(γ) g b^{-1} is used. But the proof gap remains the weakest point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the quotient 2-groupoid X/G of a groupoid X by an action of a 2-group G. After recalling the definition, it considers the canonical functor X/G → π0(X)/π0(G) and asks for the kernel 2-group Aut_{X/G}(x) → Aut_{X'}(x̄). Proposition 1.3.2 answers this with a canonical equivalence Cone(π1(G) → Aut_X x) ≅ Ker(Aut_{X/G} x → Aut_{X'} x̄). Corollaries identify the cokernel of φ_x and a kernel described by a map f_x, though f_x is not defined in the text. Section 2 specializes to the case where X is the underlying groupoid of a 2-group and B acts by two-sided translations via a homomorphism B → G×G; this yields strict 2-groupoids Cone(B• → G•). The main constructive claim is in §2.4, where an explicit X-crossed module model is asserted: H_g = B_{−1} with d_g given by equation (2.5). An appendix sketches how this construction relates to the 2-stack BT^{G,μ,?}_n and Conjecture D.8.4.","tokens_in":8232,"tokens_out":6169,"duration_ms":64254,"significance":"If the §2.4 model is fully established, the paper gives a clean and elementary description of the fibers of the quotient functor and an explicit model in an important special case, potentially serving as a concrete test of Conjecture D.8.4 in [D]. Proposition 1.3.2 is proven by a direct unpacking of the definitions and is correct; the paper is clearly motivated and well organized. The main weakness is that the explicit model in §2.4 and the general X-crossed-module equivalence in §2.3 are asserted with 'It is easy to check' rather than proved. This is a significant gap in a constructive claim, although no counterexample is apparent and the gap appears fixable.","major_comments":[{"comment":"The assertion 'It is easy to check that these data are as follows' is the core constructive claim of §2.4, but the equivariance of the family d_g under the full groupoid Γ is not verified. For a morphism p = (b,γ): g → g' in Γ one must show p d_g(β) p^{-1} = d_{g'}(H_p(β)) for the induced action H_p on B_{−1}; in particular one must prove that the G_{−1}-component of p does not prevent taking H_p to be the B0-action. This calculation is absent, and since the explicit model is used in the proposed test of Conjecture D.8.4, the gap is load-bearing. A direct computation should be supplied.","section":"§2.4, Eq. (2.5)"},{"comment":"The equivalence between strict 2-groupoids and X-crossed modules is only sketched. The inverse construction is described, but the functoriality and associativity of the composition functors (2.3) at the level of 2-morphisms, and the verification that the two constructions are inverse to each other, are left to 'It is easy to check'. Since §2.4 recovers the 2-groupoid through this equivalence, this proof gap affects the main constructive claim. A complete proof or a precise reference covering the non-pointed case would be needed to make the model fully rigorous.","section":"§2.3.2–2.3.3"}],"minor_comments":[{"comment":"After Eq. (2.3), the text 'the action of H_{x'} that comes from the homomorphism d_{x'} : H_x → Aut Γ_{x'}' contains a type error: the source of d_{x'} must be H_{x'}, not H_x. As printed the definition of Mor_C(x,x') is ill-posed; please correct the domain.","section":"§2.3.3"},{"comment":"Part (ii) uses the notation f_x without defining it anywhere in the manuscript; please define f_x or replace it with the intended map.","section":"Corollary 1.4.1"},{"comment":"The displayed notation 'Cone(B• π−π′ −→G•)' is confusing; it should presumably read 'Cone(B• (π,π′)−→G•)' to match the rest of the paper.","section":"§2.1.3(i)"},{"comment":"The proof states that at the level of morphisms the equivalence comes from id : π1(G) → π1(G), but it does not spell out the verification that the functor is essentially surjective on objects. This follows using the identity component of the 2-group, but a brief indication would improve readability.","section":"§1.3.2, proof of Prop. 1.3.2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the main structural result — Proposition 1.3.2, the canonical equivalence between Cone(π1(G) → Aut X x) and the kernel of Aut_{X/G} x → Aut_{X'} x̄ — is correct and genuinely useful. It gives a clean description of fibers of the quotient 2-groupoid, and the proof, while terse, unpacks directly from definitions. Second, the explicit X-crossed module model in §2.4, which is the paper's main constructive contribution and the proposed test of Conjecture D.8.4, is underproved. The stress-test note is right: the assertion that H_g = B_{-1} with equivariant structure from the B0-action only, and d_g as in (2.5), is verified nowhere. For p = (b,γ): g → g' in the groupoid Γ, the condition p d_g(β) p^{-1} = d_{g'}(H_p(β)) is a real condition, and the manuscript does not show that H_p can be taken to be the B0-action or that the G_{-1} component cancels. I found no counterexample, and the minimal S_3 case works, so this is likely fixable. But it is load-bearing, and the current text says only 'It is easy to check.'\n\nWhat the paper does well: the X-crossed module formalism in §2.3 is a natural extension of crossed modules to arbitrary strict 2-groupoids, and the explicit two-sided translation model in §2.4 is a useful reformulation, even if the proof is incomplete as printed. The connection to [D] and the Lau group scheme is explicitly conjectural and is not used to prove the main claims, so the citation pattern is fine. The Appendix is honest about being informal.\n\nThe soft spots are concentrated in §2.3–2.4. The equivalence between strict 2-groupoids and X-crossed modules is stated without proof, and the typo in §2.3.3 (d_{x'}: H_x should be d_{x'}: H_{x'}) makes the construction impossible to check from the printed page. These are genuine gaps, but they are gaps in exposition and computation, not signs of a wrong theorem.\n\nWho gets value: anyone working on 2-group actions, higher categorical quotients, or Drinfeld's 2-stack program. The paper deserves a serious referee, but the referee should require a full proof of the equivariance in §2.4 and an expanded proof of the §2.3 equivalence before acceptance. My recommendation: send it to peer review, and make the author earn the 'easy to check' claims.","headline":"The fiber description in Prop 1.3.2 is a clean, correct new result, but the explicit model in §2.4 rests on an unverified equivariance check that the referee should force the author to spell out.","tokens_in":8770,"tokens_out":1854,"would_cite":true,"duration_ms":18965,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18N10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a groupoid acted on by a 2-group, the automorphisms of an object in the quotient that die in the coarse quotient form the 2-group of a crossed module.","keywords":["2-group","groupoid quotient","crossed module","2-groupoid","X-crossed module","gerbe","Barsotti-Tate groups","categorical stabilizer"],"falsifier":"Go to the definition in Section 2.3.3: as printed, the action of $H_{x'}$ on $\\mathrm{Mor}(x,x')$ is said to come from $d_{x'}: H_x \\to \\mathrm{Aut}\\,\\Gamma_{x'}$, which is not well-typed unless read as $d_{x'}: H_{x'} \\to \\mathrm{Aut}\\,\\Gamma_{x'}$. Then check whether formula (2.5) really lands $d_g(\\beta)$ in $\\mathrm{Stab}_g$ and is a homomorphism for all $\\beta \\in B_{-1}$ and $g \\in G_0$; a single pair where it fails would show the explicit model is not the quotient $X/B$.","tokens_in":7704,"feed_emoji":"","tokens_out":9495,"duration_ms":91528,"temperature":0.7,"pith_summary":"This paper answers a structural question in higher category theory: when a 2-group $G$ acts on a groupoid $X$, the quotient is a 2-groupoid $X/G$, and the author identifies exactly what extra information this quotient carries beyond its coarse 1-groupoid quotient $\\pi_0(X)/\\pi_0(G)$. The answer is a canonical equivalence of 2-groups: for each object $x$, the automorphisms of $x$ in $X/G$ that collapse to the identity in the coarse quotient form the crossed-module cone $\\mathrm{Cone}(\\pi_1(G) \\to \\mathrm{Aut}_X x)$. The paper also gives an explicit model of $X/G$ in the case where $X$ is the underlying groupoid of a 2-group and the acting 2-group acts by two-sided translations. A careful reader will care because the sheafified version of this special case appears in a conjectural description of the $p$-adic completion of the stack of $n$-truncated Barsotti-Tate groups, so the explicit model offers a concrete way to test that conjecture.","feed_headline":"Lost automorphisms in a 2-group quotient form a crossed module","feed_subtitle":"It also gives an explicit model for two-sided translations, aimed at a conjecture about truncated Barsotti-Tate groups.","key_machinery":"The paper's central objects are the quotient 2-groupoid $\\tilde{X}=X/G$ and the crossed-module cone. A crossed module is a pair of groups $G_0$, $G_{-1}$ with an action of $G_0$ on $G_{-1}$ and a homomorphism $d: G_{-1} \\to G_0$ satisfying two identities; its cone $\\mathrm{Cone}(G_{-1} \\to G_0)$ is the strict 2-group whose objects are elements of $G_0$ and whose morphisms are elements of $G_{-1}$ implementing $d$. The proof uses this dictionary, together with an $X$-crossed module, a generalization in which a groupoid $\\Gamma$ with object set $X$ carries, at each object $x$, a group $H_x$ and a homomorphism $d_x: H_x \\to \\mathrm{Aut}_\\Gamma x$. The load-bearing step is applying the equivalence between strict 2-groupoids and $X$-crossed modules, stated in Section 2.3, to encode the quotient $X/B$ in the two-sided translation case.","core_discovery":"Proposition 1.3.2 states a canonical equivalence of 2-groups\n$$\\mathrm{Cone}(\\pi_1(G) \\to \\mathrm{Aut}_X x) \\cong \\mathrm{Ker}(\\mathrm{Aut}_{\\tilde{X}} x \\to \\mathrm{Aut}_{X'} \\bar{x}),$$\nwhere $\\tilde{X}=X/G$ and $X'=\\pi_0(X)/\\pi_0(G)$. In words: the categorical stabilizer of an object $x$ in the quotient 2-groupoid, modulo the automorphisms that survive in the ordinary quotient by $\\pi_0(G)$, is exactly the 2-group associated to the crossed module whose differential is the action homomorphism from the abelian group $\\pi_1(G)=\\mathrm{Aut}_G(1_G)$ to the automorphism group of $x$. Section 2.4 then treats the case where $X$ is the underlying groupoid of a strict 2-group associated to a crossed module $G_\\bullet$, and another 2-group $B$ acts by two-sided translations through homomorphisms $\\pi,\\pi': B_\\bullet \\to G_\\bullet$. The quotient $X/B$ is described there as an $X$-crossed module: the underlying 1-groupoid comes from the action of $B_0 \\ltimes G_{-1}$ on $G_0$, the kernel group at each object is $H_g = B_{-1}$, and the differential is $d_g(\\beta)=d(\\beta)\\cdot \\pi(\\beta)^{-1}\\cdot g\\pi'(\\beta)$.","pith_inferences":["The same fiber calculation should carry over to 2-stacks by working objectwise in a topos, which would turn the inertia of the 2-stack from the appendix into a sheaf of crossed modules; this is a direct extension the paper does not state.","The $X$-crossed-module presentation suggests that even complicated quotient 2-groupoids can be computed from ordinary group actions plus a kernel group per object, so the two-sided translation model may serve as a template for other 2-stacks of the form $\\mathrm{Cone}(B_\\bullet \\xrightarrow{\\pi,\\pi'} G_\\bullet)$.","If the explicit model is correct, then testing Conjecture D.8.4 reduces to comparing two $X$-crossed-module descriptions; a mismatch would appear already at the level of the kernel groups $\\mathrm{Aut}(\\mathrm{id}_x)$ before any global geometry is examined."],"forward_implications":["The fiber description reduces the difference between $X/G$ and its coarse quotient to the single homomorphism $\\pi_1(G) \\to \\mathrm{Aut}_X x$: for each object $x$, the kernel is the 2-group $\\mathrm{Cone}(\\pi_1(G) \\to \\mathrm{Aut}_X x)$, and nothing else is needed.","Corollary 1.4.1 gives computable invariants: the kernel of $\\mathrm{Aut}_{\\tilde{X}_{\\le 1}}(x) \\to \\mathrm{Aut}_{X'}(\\bar{x})$ is $\\mathrm{Coker}\\,\\phi_x$, and $\\mathrm{Aut}_{\\tilde{X}}(\\mathrm{id}_x) = \\mathrm{Ker}\\, f_x$, so the 1-truncated quotient is a gerbe banded by the functor $L(\\bar{x}) = \\mathrm{Coker}\\,\\phi_{\\bar{x}}$ in the abelian case.","In the two-sided translation case the quotient 2-groupoid has an explicit finite model: objects are elements of $G_0$, 1-morphisms come from the action of $B_0 \\ltimes G_{-1}$ on $G_0$, and 2-morphisms are recorded by the family $H_g = B_{-1}$ with $d_g(\\beta)=d(\\beta)\\cdot \\pi(\\beta)^{-1}\\cdot g\\pi'(\\beta)$.","Because the sheafified version of this construction underlies the 2-stack $BT_{G,\\mu,n}^{?}$ of the appendix, the explicit model provides a way to test Conjecture D.8.4: comparing the two descriptions can show whether $BT_{G,\\mu,n}^{?}$ agrees with $BT_{G,\\mu,n}$ when $\\mu$ is 1-bounded."],"supporting_citations":[{"why":"Supplies the equivalence between crossed modules and strict 2-groups used to identify $\\mathrm{Cone}(\\pi_1(G) \\to \\mathrm{Aut}_X x)$ with a 2-group.","marker":"[L]"},{"why":"Provides the standard identification of the cone with the 2-group associated to a 2-term complex in the abelian case, used in Sections 1.4 and 2.1.","marker":"[SGA4]"},{"why":"Defines the 2-stack and Conjecture D.8.4 that the paper's explicit model is designed to test; the appendix explains how $BT_{G,\\mu,n}^{?}$ is written as a cone.","marker":"[D]"},{"why":"Supplies the algebraicity theorem and the identification of $BT_{G,\\mu,n}$ with $n$-truncated Barsotti-Tate groups in the 1-bounded case, which is the target of the conjecture.","marker":"[GM]"}],"fun_headline_variants":["Stabilizer kernel in 2-group quotient is a crossed module","2-group quotients: explicit crossed module model","Crossed module appears as stabilizer in 2-group quotient","Lost automorphisms shrink to a crossed module","Quotient by 2-group: kernel automorphisms form crossed module"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explicit model in Section 2.4 rests on the claim, justified only by a sketch, that every strict 2-groupoid is completely determined by its underlying 1-groupoid together with the specified family of groups $H_x$ and maps $d_x$.","fun_headline_variants_meta":{"raw":{"variants":["Stabilizer kernel in 2-group quotient is a crossed module","2-group quotients: explicit crossed module model","Crossed module appears as stabilizer in 2-group quotient","Lost automorphisms shrink to a crossed module","Quotient by 2-group: kernel automorphisms form crossed module"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000363,"raw_usage":{"total_tokens":1942,"prompt_tokens":918,"completion_tokens":1024,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":940}},"tokens_in":534,"tokens_out":1024,"duration_ms":10022,"temperature":1.0,"reasoning_tokens":940,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:56:51.784854+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Go to the definition in Section 2.3.3: as printed, the action of $H_{x'}$ on $\\mathrm{Mor}(x,x')$ is said to come from $d_{x'}: H_x \\to \\mathrm{Aut}\\,\\Gamma_{x'}$, which is not well-typed unless read as $d_{x'}: H_{x'} \\to \\mathrm{Aut}\\,\\Gamma_{x'}$. Then check whether formula (2.5) really lands $d_g(\\beta)$ in $\\mathrm{Stab}_g$ and is a homomorphism for all $\\beta \\in B_{-1}$ and $g \\in G_0$; a single pair where it fails would show the explicit model is not the quotient $X/B$.","supporting_citations":[],"review_version":1}