{"id":"b3cf9f7b-427b-4d5e-bbdf-58bb7da50d72","arxiv_id":"2506.21867","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stable infinity-categories with left and right complete t-structures have deformation functors that are formal E2-moduli problems determined by their Hochschild cohomology.","lead":"This paper proves that for a stable category with a complete t-structure, Hochschild cohomology exactly controls all deformations that respect the t-structure. The result turns a previously known obstruction into a working deformation theory, with applications to schemes and D-modules.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Formal-moduli step of Theorem 4.9 is delegated to unpublished [13]; the central claim's pullback property is unverified.","rationale":"The reader's weakest_assumption locates the risk in the symmetric monoidality of the left completion functor bL, which is indeed used in Propositions 3.16 and 3.18. That assumption is cited to Lurie's DAG-VIII and is probably standard, though not proved in detail in the preprint. However, the more urgent unverified point is Theorem 4.9: the proof of the formal moduli property is delegated to an unpublished note and to a 'small modification' of DAG-X. Theorem 4.5 alone gives only R-wise equivalences; the natural transformation and the pullback preservation needed for the formal moduli statement are not demonstrated in the text. Since the headline statement of Theorem 1.1 literally asserts that Deform_E is a formal E2-moduli problem, this omitted verification is the most load-bearing concern. It does not amount to a demonstrated falsehood, so the appropriate evaluation is the reader's CONDITIONAL verdict, not rejection.","tokens_in":29310,"tokens_out":16622,"duration_ms":181912,"concrete_test":"Obtain [13, Section 4.4] and check that its formal-moduli criterion applies to the left fibration r: RLin^∧(k)^†_{/(E,k)} -> Alg^+_2(Mod_k^{≤0}) of Definition 4.8, with the natural transformation whose R-component is the inverse of P_R from Theorem 4.5. Independently, take a nontrivial pullback square of Artin E2-algebras with surjective H^0 maps, compute Deform_E on it directly from Definition 4.8 using the maps induced by bL ∘ Φ_R, and verify that the resulting square of spaces is homotopy Cartesian. If this verification cannot be carried out from the paper alone, the proof of Theorem 4.9 is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central assertion of Theorem 1.1 is not merely the pointwise formula in Theorem 4.5, but also that Deform_E is a formal E2-moduli problem. Theorem 4.5 gives, for each Artin R, an equivalence of spaces Deform_E(R) ≃ Map_{Alg2(Modk)}(D2(R), HH•(E/k)). A formal moduli problem requires more: a natural transformation Deform_E -> F^(2)_{k⊕HH•(E/k)} compatible with the pullback squares in Art2, and verification of the Schlessinger-type pullback condition. The proof of Theorem 4.9 in Section 4.2 says: 'The construction is done in [19, X, Construction 5.3.18] up to a small modification... Besides, we can also apply the axiomatic formulation in [13, Section 4.4].' Reference [13] is an unpublished note, and the adaptation to PrL_{t±} with left-complete t-structures is not carried out. If the axiomatic criterion in [13] does not apply to the left fibration of Definition 4.8, or if the 'small modification' changes the behavior of the t-structures, Theorem 1.1 fails at exactly this step. The later corollaries, including Corollary 4.10 and the obstruction theory of Theorem 4.12, inherit this gap. This is a missing proof, not an internal contradiction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims (Theorem 1.1, proved as Theorem 4.5 and Theorem 4.9) that for a k-linear presentable stable infinity-category E equipped with a left and right complete accessible t-structure, the deformation functor Deform_E, which classifies deformations of E together with its t-structure to Artin E2-algebras, is the formal E2-moduli problem F^(2)_{k⊕HH•(E/k)} associated with the augmented E2-algebra k⊕HH•(E/k)→k. The proof proceeds through a categorified Koszul duality: for an Artin E2-algebra R, a module over the Koszul dual D2(R) is converted, via a left-completion adjunction, into an R-linear category; Theorem 4.5 establishes the pointwise equivalence Deform_E(R) ≃ Map(D2(R), HH•(E/k)), and Theorem 4.9 asserts that this equivalence is compatible with pullback squares, making Deform_E a formal moduli problem. Applications include a corollary for commutative bases and an obstruction theory.","tokens_in":29517,"tokens_out":6126,"duration_ms":61065,"significance":"The intended result is significant: it gives a positive answer, under t-structure completeness hypotheses, to the problem of when a categorical deformation functor is governed by Hochschild cohomology, and it is consistent with the known failure of the naive functors to satisfy Schlessinger conditions. Strengths of the manuscript include a precise and intrinsic definition of HH•(E/k) as an E2-algebra of endomorphisms of the identity endofunctor, a detailed categorical framework for left completion, and explicit examples and applications. The pointwise theorem is supported by a real argument, not by a formal manipulation. The full theorem, however, is currently conditional on the deferred formal-moduli construction in Theorem 4.9.","major_comments":[{"comment":"The proof of Theorem 4.9 does not establish the formal-moduli property of Deform_E. Theorem 4.5 gives, for each R in Art2, an equivalence of spaces Deform_E(R) ≃ Map_{Alg2(Modk)}(D2(R), HH•(E/k)), but this is weaker than the statement that Deform_E is the formal moduli problem F^(2)_{k⊕HH•(E/k)}: one must also construct a natural transformation Deform_E → F^(2)_{k⊕HH•(E/k)} compatible with the pullback squares in Art2 and verify the Schlessinger-type pullback condition. The proof instead says 'The construction is done in [19, X, Construction 5.3.18] up to a small modification... Besides, we can also apply the axiomatic formulation in [13, Section 4.4].' Reference [13] is an unpublished note, and the 'small modification' is not carried out; in particular it is not checked that the left/right module convention change and the use of PrL_{t±} preserve the pullback properties. Since Corollary 4.10 and Theorem 4.12 depend on Theorem 4.9, this is a load-bearing gap in the proof of the main theorem as stated.","section":"Section 4.2, Theorem 4.9"},{"comment":"The proof of Proposition 3.16 relies on assertions that are not proved in the text: that bL is symmetric monoidal and preserves small colimits in the way needed to commute with the relative tensor products appearing in the chain of equivalences, and that the left completion of LMod_{D2(R)} is identified with End^l_R(Modk). These identifications are exactly what makes the unit of the adjunction an equivalence after left completion, and they are also used in the proof of Theorem 4.5. The sketch 'by the construction... This proves our assertion' does not allow the reader to verify the several hidden coherence conditions; a lemma stating the required properties of bL and proving the identification would be needed.","section":"Section 3.3, Proposition 3.16"},{"comment":"Lemma 3.14, which supplies the left completion functor RMod_{k⊗D2(R)k} → LMod_R and the rank-one freeness used in Proposition 3.18, is only partially proved. The assertion that Ind(LCoh(R)) → LMod_R is a left completion functor is dispatched by 'the argument similar to the proof of Lemma 3.10' plus a reference to [6, Proposition 1.3.4] for the commutative case, and the rank-one freeness proof contains a long informally described module action whose coherence is not written out. Since Proposition 3.18 is used in Theorem 4.5 to show that the counit is an equivalence after left completion, this is another load-bearing point that should be expanded.","section":"Section 3.4, Lemma 3.14"}],"minor_comments":[{"comment":"There are numerous typographical errors ('Exmaple 2.4', 't-strucutre', 'argumented', 'catgories', 'B-mdoule') that should be corrected in revision.","section":"Throughout"},{"comment":"The cross-reference 'Section refMOA' is broken; it should point to the relevant part of Section 2.2.","section":"Section 2.2, Construction 2.15"},{"comment":"Corollary 3.17 says 'Use Lemma 3.2', but no Lemma 3.2 appears in the paper; the intended reference appears to be Proposition 3.2.","section":"Section 3.1, Corollary 3.17"},{"comment":"Example 4.6 writes 'D2(k ⊕ k)' where the square-zero extension is presumably 'k ⊕ k[n]'; the displayed formula should be stated consistently.","section":"Section 4.1, Example 4.6"}],"recommendation":"major_revision","confidential_remarks":"The key issue for the editor is whether the author can supply the missing formal-moduli argument. Since [13] is an unpublished note by the same author, the published version of this paper should either include a complete proof of Theorem 4.9 or replace that reference by a published account. The rest of the paper is promising, but the current version should not be accepted without that step."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main takeaway: this paper proves the right version of Lurie's old conjecture—once you carry a left and right complete t-structure and deform it along with the category, the deformation functor is a formal E2-moduli problem governed by HH•. That is a genuine advance, and the main construction is coherent.\n\nWhat is new: Theorem 4.5 gives a pointwise equivalence between complete deformations and maps from the E2-Koszul dual into HH•. Theorem 4.9 upgrades this to a formal moduli problem, and the applications (E∞-restriction, obstruction theory) follow formally. The conceptual contribution—left completion as categorified Koszul duality—is attractive and the paper does a good job motivating it. The pointwise proof, via the adjunction between left LMod_R modules and left LMod_{D2(R)} modules in t-structure categories, is a solid argument even though it is written tersely.\n\nThe soft spots are real. Theorem 4.9 is the load-bearing step that turns pointwise equivalences into a formal moduli problem, and its entire proof is a reference: 'up to a small modification' of Lurie's DAG-X Construction 5.3.18, or the axiomatic formulation in the author's unpublished [13]. The compatibility of the natural transformation with pullback squares—the Schlessinger condition—is not checked in the text. If the unpublished axiomatic framework does not apply in the PrL_{t±} setting, the theorem fails at that step. I agree with the stress-test note here. Also, Propositions 3.16 and 3.18 depend on dense arguments (Lemma 3.14 in particular) and would benefit from expansion. Minor: Corollary 3.17 cites a 'Lemma 3.2' that does not exist; likely a typo.\n\nThis deserves a serious referee. The result is important, the strategy is identifiable, and the gaps are fixable in principle. But the paper as posted is not yet citable as a black box for the formal-moduli statement. I would send it to review and ask for a concrete proof of Theorem 4.9 and public versions of [12] and [13]. Probably a conversation-starter for a higher algebra reading group.","headline":"A significant theorem with a real gap: the pointwise equivalence is proven, but the formal-moduli step is delegated to unpublished notes.","tokens_in":30148,"tokens_out":4308,"would_cite":false,"duration_ms":40906,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For stable ∞-categories with left- and right-complete t-structures, all deformations are controlled by the Hochschild cohomology $E_2$-algebra.","keywords":["Hochschild cohomology","stable ∞-categories","t-structures","deformation theory","formal moduli problems","E2-algebras","Koszul duality","left completion"],"falsifier":"Compute the claimed equivalence in a concrete instance, e.g. $E = \\mathrm{LMod}_A$ for a connective $E_2$-algebra $A$ over $k$ and $R = k \\oplus k[n]$, where the theorem predicts $\\pi_0(\\mathrm{Deform}_E(R)) \\cong \\mathrm{HH}^{n+2}(E/k)$; finding a deformation to $k \\oplus k[n]$ that does not correspond to a class in $\\mathrm{HH}^{n+2}$, or an extra class, would refute it. Alternatively, exhibit a pair of right-complete t-structures for which $bL$ fails to be symmetric monoidal or fails to commute with a relative tensor product, since the unit and counit equivalences rest on that premise.","tokens_in":29030,"feed_emoji":"🔄","tokens_out":14773,"duration_ms":139589,"temperature":0.7,"pith_summary":"This paper proves that the deformation theory of a stable ∞-category equipped with a left- and right-complete t-structure is entirely controlled by its Hochschild cohomology. For a $k$-linear presentable stable ∞-category $E$ with an accessible, left- and right-complete t-structure, the functor sending an Artin $E_2$-algebra $R$ to the space of complete deformations of $(E, E_{\\le 0})$ to $R$ is a formal $E_2$-moduli problem, equivalent to the space of $E_2$-algebra maps from the $E_2$-Koszul dual $D_2(R)$ into the Hochschild cohomology $E_2$-algebra $\\mathrm{HH}^\\bullet(E/k)$. This makes deformation questions cohomological: deformations to $k \\oplus k[n]$ are classified by $\\mathrm{HH}^{n+2}(E/k)$, and lifting along elementary extensions is governed by obstructions in $\\mathrm{HH}^{i+2}(E/k) \\otimes V$. Many categories of geometric origin carry such t-structures, and the earlier naive deformation functor was known to fail the Schlessinger conditions, so the theorem identifies the correct refinement: keep the t-structure and require completeness.","feed_headline":"Hochschild cohomology classifies deformations of stable categories","feed_subtitle":"Every deformation of a left- and right-complete t-structured category is a map into its Hochschild cohomology.","key_machinery":"The load-bearing object is the left completion functor $bL : \\mathrm{Pr}^L_{t+} \\to \\mathrm{Pr}^L_{t\\pm}$, which sends a right-complete accessible t-structure $(C, C_{\\le 0})$ to its left completion $\\lim_{n \\to -\\infty} C_{\\ge n}$, making it both left- and right-complete. The proof works through categorified Koszul duality: an augmented $E_2$-algebra $A$ is viewed through its module category $\\mathrm{LMod}_A$ with its canonical t-structure, and an $A$-linear stable ∞-category with t-structure is a module over $\\mathrm{LMod}_A$. The decisive technical points are that $bL$ is symmetric monoidal and colimit-preserving, that $bL(\\mathrm{LMod}_{D_2(R)})$ is identified with the endomorphism algebra of $\\mathrm{Mod}_k$ as an $R$-module, and that after left completion the unit and counit of the adjunction between $\\mathrm{LMod}_R$-modules and $\\mathrm{LMod}_{D_2(R)}$-modules become equivalences (Propositions 3.16 and 3.18). Those identifications produce the equivalence between the deformation space and the $E_2$-algebra mapping space.","core_discovery":"The paper's central claim is Theorem 1.1: for $E$ a $k$-linear presentable stable ∞-category equipped with a left- and right-complete accessible t-structure $(E_{\\le 0}, E_{\\ge 0})$, the deformation functor $\\mathrm{Deform}_E : \\mathrm{Art}_2 \\to \\widehat{\\mathcal{S}}$ is a formal $E_2$-moduli problem, namely $\\mathrm{F}^{(2)}_{k \\oplus \\mathrm{HH}^\\bullet(E/k)}$, the formal moduli problem associated to the augmented $E_2$-algebra $k \\oplus \\mathrm{HH}^\\bullet(E/k) \\to k$. In particular there is a canonical equivalence $\\mathrm{Deform}_E(R) \\simeq \\mathrm{Map}_{\\mathrm{Alg}_2(\\mathrm{Mod}_k)}(D_2(R), \\mathrm{HH}^\\bullet(E/k))$ for every Artin $E_2$-algebra $R$, where $D_2$ is $E_2$-Koszul duality. Restricting to commutative Artin bases gives a formal $E_\\infty$-moduli problem governed by the dg Lie algebra $\\mathrm{HH}^\\bullet(E/k)[1]$, with an explicit $\\mathrm{F}^{(\\infty)}$ description in characteristic zero. The theorem also yields the classification for square-zero extensions and an obstruction theory.","pith_inferences":["A natural extension, which the paper explicitly flags as likely, is to $E_n$-monoidal stable ∞-categories with a compatible left-complete t-structure, where deformations should be governed by the $E_{n+1}$-Hochschild cochain complex in a formal $E_{n+2}$-moduli problem.","The same formalism suggests that the heart of the t-structure deforms along with the category: a complete deformation $D$ of $E$ induces an $H^0(R)$-linear deformation of the abelian heart $E^\\heartsuit$, giving Hochschild cohomology control over deformations of abelian categories as well.","A computable test is to take a smooth proper derived scheme $X$: deformations of $\\mathrm{QCoh}(X)$ to square-zero extensions should match ordinary deformations of $X$ when the standard comparison between Hochschild cohomology and polyvector fields holds, and any mismatch would reveal a missing hypothesis.","The result also suggests that completeness, rather than compact generation, is the structural condition making categorical deformation theory algebraic; reformulating other deformation problems in terms of left completion may yield similar classification theorems."],"forward_implications":["Every complete deformation of $(E, E_{\\le 0})$ to an Artin $E_2$-algebra $R$ is classified by an $E_2$-algebra map $D_2(R) \\to \\mathrm{HH}^\\bullet(E/k)$; for $R = k \\oplus k[n]$ the isomorphism classes of deformations are exactly the elements of $\\mathrm{HH}^{n+2}(E/k)$.","For commutative Artin bases, the deformation functor is a formal $E_\\infty$-moduli problem with underlying dg Lie algebra $\\mathrm{HH}^\\bullet(E/k)[1]$, so Koszul duality over $E_\\infty$ algebras computes these deformations.","A deformation over $R$ lifts along an elementary extension $R' \\to R$ with kernel $V[i]$ precisely when an obstruction in $\\mathrm{HH}^{i+2}(E/k) \\otimes V$ vanishes; because every surjection of Artin local algebras factors into elementary extensions, this gives an iterative obstruction theory.","For quasi-coherent sheaves on quasi-compact separated derived schemes, the theorem applies verbatim, so deformations of $\\mathrm{QCoh}(X)$ with its standard t-structure are governed by $\\mathrm{HH}^\\bullet(\\mathrm{QCoh}(X)/k)$.","The theorem identifies the right deformation problem: deformations must remember the t-structure and require left and right completeness, repairing the failure of the naive deformation functor to satisfy the Schlessinger conditions."],"supporting_citations":[{"why":"Supplies the higher-algebra framework: $E_2$-algebras, module categories, endomorphism algebras, Dunn additivity, and Koszul duals as centralizers.","marker":"[18]"},{"why":"Supplies the core theorems on t-structures, left/right completion, formal $E_2$-moduli problems, Artin $E_2$-algebras, Koszul duality $D_2$, and the universal map $\\mathrm{Def}_C \\to \\mathrm{F}^{(2)}$.","marker":"[19]"},{"why":"Supplies the method for proving that the functor $\\mathrm{Ind}(\\mathrm{RCoh}(k \\otimes_A k)) \\to \\mathrm{RMod}_{k \\otimes_A k}$ is a left completion, used in Lemma 3.10.","marker":"[6]"},{"why":"Supplies the axiomatic formulation of formal moduli problems used to build the map $\\mathrm{Deform}_E \\to \\mathrm{F}^{(2)}_{k \\oplus \\mathrm{HH}^\\bullet(E/k)}$ in Theorem 4.9.","marker":"[13]"},{"why":"Supplies the obstruction-theory setup of elementary extensions and pullback squares used in Theorem 4.12.","marker":"[4]"}],"fun_headline_variants":["Stable category deformations: classified by Hochschild cohomology","Hochschild cohomology is the moduli of stable deformations","Deformation theory of stable categories: a Hochschild moduli","Every stable deformation arises from Hochschild cohomology","Stable t-structure deformations: governed by Hochschild cohomology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the left completion functor $bL$ being symmetric monoidal and preserving small colimits when applied to right-complete t-structures, so that it commutes with relative tensor products and with module categories; if this failed, the unit and counit maps linking $\\mathrm{LMod}_R$-modules and $\\mathrm{LMod}_{D_2(R)}$-modules would not be equivalences and the main theorem would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Stable category deformations: classified by Hochschild cohomology","Hochschild cohomology is the moduli of stable deformations","Deformation theory of stable categories: a Hochschild moduli","Every stable deformation arises from Hochschild cohomology","Stable t-structure deformations: governed by Hochschild cohomology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001561,"raw_usage":{"total_tokens":6194,"prompt_tokens":865,"completion_tokens":5329,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":5239}},"tokens_in":481,"tokens_out":5329,"duration_ms":42505,"temperature":1.0,"reasoning_tokens":5239,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:16:54.235939+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the claimed equivalence in a concrete instance, e.g. $E = \\mathrm{LMod}_A$ for a connective $E_2$-algebra $A$ over $k$ and $R = k \\oplus k[n]$, where the theorem predicts $\\pi_0(\\mathrm{Deform}_E(R)) \\cong \\mathrm{HH}^{n+2}(E/k)$; finding a deformation to $k \\oplus k[n]$ that does not correspond to a class in $\\mathrm{HH}^{n+2}$, or an extra class, would refute it. Alternatively, exhibit a pair of right-complete t-structures for which $bL$ fails to be symmetric monoidal or fails to commute with a relative tensor product, since the unit and counit equivalences rest on that premise.","supporting_citations":[{"cited_title":"Lurie, Higher Algebra, Draft 2017","cited_arxiv_id":null,"evidence_quote":"Supplies the higher-algebra framework: $E_2$-algebras, module categories, endomorphism algebras, Dunn additivity, and Koszul duals as centralizers."},{"cited_title":"Lurie, Derived Algebraic Geometry Series, preprint","cited_arxiv_id":null,"evidence_quote":"Supplies the core theorems on t-structures, left/right completion, formal $E_2$-moduli problems, Artin $E_2$-algebras, Koszul duality $D_2$, and the universal map $\\mathrm{Def}_C \\to \\mathrm{F}^{(2)}$."},{"cited_title":"Gaitsgory, Ind-coherent sheaves, Mosc","cited_arxiv_id":null,"evidence_quote":"Supplies the method for proving that the functor $\\mathrm{Ind}(\\mathrm{RCoh}(k \\otimes_A k)) \\to \\mathrm{RMod}_{k \\otimes_A k}$ is a left completion, used in Lemma 3.10."},{"cited_title":"Iwanari, Moduli theory associated to Hochschild pairs, available at the author’s webpage","cited_arxiv_id":null,"evidence_quote":"Supplies the axiomatic formulation of formal moduli problems used to build the map $\\mathrm{Deform}_E \\to \\mathrm{F}^{(2)}_{k \\oplus \\mathrm{HH}^\\bullet(E/k)}$ in Theorem 4.9."},{"cited_title":"Deformations and Lifts of Calabi-Yau Varieties in Characteristic $p$","cited_arxiv_id":"2407.09256","evidence_quote":"Supplies the obstruction-theory setup of elementary extensions and pullback squares used in Theorem 4.12."}],"review_version":1}