{"id":"d2787c7c-a338-4d3c-b8d4-f885e2380e03","arxiv_id":"2607.16953","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An E_n-algebra is (n+1)-dualizable in the higher Morita category exactly when it is dualizable as a module over each sphere-shaped factorization homology, confirming conjectures of Lurie and Brochier–Jordan–Safranov–Snyder.","lead":"This mathematics paper proves a conjecture of Jacob Lurie: an algebra with n compatible multiplications can power an (n+1)-dimensional topological field theory exactly when it is 'dualizable' over certain sphere-shaped invariants. The proof required a new technical framework — pointless factorization algebras — that removes a structural obstruction in earlier models.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.0.1 depends on unproven identification of uMor_n(C) with the Morita category: additivity (Prop 2.4.42) and composition-as-relative-tensor-product (Prop 2.4.43) are deferred to [SSS], and no comparison with Haugseng's model (Remark 3.1.1) is provided.","rationale":"The reader identified the model identification as the weakest assumption, and I agree. Theorem 3.0.1 is not self-contained: two statements essential for interpreting uMor_n(C) as the higher Morita category are deferred to an unpublished follow-up, and the comparison with Haugseng's established model is explicitly left open. The paper's visible components—localization of disk operads, the Segal condition, and regular bimodules—are internally consistent as far as they go, but the central theorem's status is conditional on the deferred results. I found no clear internal contradiction in the visible text. The n=1 reduction and the geometric sketch are sensible. Since the reader already marked the paper CONDITIONAL, my read does not change the verdict. The main concern is not that the theorem is false, but that it is currently unverified at the load-bearing joints; this supports keeping the conditional assessment rather than accepting the theorem as proven.","tokens_in":67681,"tokens_out":5403,"duration_ms":58662,"concrete_test":"Prove Proposition 2.4.43 in the lowest nontrivial case n=2 by an explicit computation: take two composable 1-morphisms in uMor_2(C) given by pointless constructible factorization algebras on the stratified square ◻^2_{[2]}, push forward along the collapse-rescale map ϱ(α):◻^2_{[2]}→◻^2_{[1]}, and compare the resulting bimodule with the relative tensor product of the two corresponding bimodules in BMod(Alg(C)). If the two agree up to canonical equivalence, the reduction step survives this case; if not, the proof of Theorem 3.0.1 has a concrete gap. An independent proof for general n would settle the model-identification concern completely.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 3.0.1: an E_n-algebra in uMor_n(C) is (n+1)-dualizable iff it is left dualizable as a left module over the stated factorization homologies. The proof's first pillar is the lifting-of-adjoints theorem (Theorem 3.1.3), which reduces adjointability of an n-morphism to adjointability of an underlying 1-morphism in uMor_1(C). That reduction is valid only if uMor_n(C)'s k-morphisms are genuinely iterated bimodules in E_{n-k}-algebras and if composition is the relative tensor product. These are exactly Proposition 2.4.42 (pointless additivity) and Proposition 2.4.43 (composition as relative tensor product), and both are explicitly deferred: \"A proof will appear in upcoming joint work of the first two authors with Anja Švraka [SSS].\" They are used in essential places: quasi-unitality (Prop 2.4.44), the Cartesian fibration structure (proof of Prop 2.4.61), regular bimodules and their adjointability (Prop 2.4.64(3) and Cor 2.4.69), and finally the application of the known 1-dimensional bimodule adjoint criterion [Lur17, Prop 4.6.2.13]. If either proposition fails, the dualizability data constructed in Section 3 may produce adjoints in a category whose morphisms are not bimodules or whose composition is not Morita composition. Additionally, Remark 3.1.1 concedes that no comparison to Haugseng's combinatorial Morita category is proved. Since Lurie's conjecture is about the Morita category of E_n-algebras, a dualizability theorem for an uncompared new model does not settle the conjecture as stated. This is a gap in support, not a visible internal contradiction; the surrounding localization results and the n=1 sanity check are coherent. The verdict should remain conditional.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a new model uMor_n(C) for the higher Morita category of E_n-algebras, based on 'pointless' constructible factorization algebras on marked stratified cubes. Its main result, Theorem 3.0.1, asserts that an E_n-algebra A in this category is (n+1)-dualizable if and only if, for every 0 ≤ k ≤ n, A is left dualizable as a left module over the factorization homology ∫_{S^{k-1} × R^{n-k+1}} A. From this it derives the invertibility criterion of Corollary 3.5.1 and a relative dualizability theorem, Theorem 4.1.3, aimed at relative/twisted field theories. The proof combines a lifting-of-adjoints argument (Section 3.1) with a Morse-theoretic computation (Section 3.3), building on previous work of the first author and collaborators.","tokens_in":68090,"tokens_out":5567,"duration_ms":62409,"significance":"If the central proof is completed, the paper would prove a long-standing conjecture of Lurie (Remark 4.1.27 of [Lur09b]) and the Brochier–Jordan–Safranov–Snyder invertibility conjecture, and would supply a broad framework for fully extended framed TFTs with Morita-type targets. The paper also contains several auxiliary results that are valuable in their own right: the operadic localization theorem 2.3.15, the Segal condition for the pointless factorization model (Proposition 2.4.40), and the identification of pointless constructible factorization algebras on (R,0) with bimodules (Corollary 2.3.18). The n=1 case of the main criterion matches Lurie's known bimodule adjoint criterion [Lur17, Prop. 4.6.2.13]. However, the significance is conditional: the category uMor_n is not yet shown to be the accepted higher Morita category of E_n-algebras, and several load-bearing structural results are deferred to unpublished work.","major_comments":[{"comment":"The central construction and proof depend on two results that are explicitly deferred to [SSS]: Pointless additivity (Prop. 2.4.42) and the identification of composition with the relative tensor product (Prop. 2.4.43). These are not optional refinements: they are used to prove quasi-unitality (Prop. 2.4.44), the Cartesian fibration structure of morphism categories (proof of Prop. 2.4.61), the properties of regular bimodules (Prop. 2.4.64 and Cor. 2.4.69), and ultimately the lifting-of-adjoints theorem 3.1.3 and Theorem 3.0.1. Without these propositions, Theorem A is a dualizability statement for a newly assembled category whose k-morphisms have not been identified with iterated bimodules and whose composition has not been identified with Morita composition. This is a load-bearing gap. The same concern applies to the relative Theorem B. The authors should either include full proofs of Pro","section":"§2.4.5, Propositions 2.4.42 and 2.4.43"},{"comment":"The paper concedes that a precise comparison between uMor_n(C) and Haugseng's combinatorial higher Morita category is beyond its scope. Since the conjecture of Lurie that the paper claims to prove is about the Morita category of E_n-algebras, a dualizability theorem for a model that is not compared to an established model does not by itself settle the conjecture. If the intended contribution is to prove the conjecture, the equivalence with Haugseng's model (or with another accepted model) must be established in this paper or replaced by a precise reference with a complete proof. Otherwise the claim 'we prove a conjecture of Lurie' in the abstract is premature.","section":"Remark 3.1.1"}],"minor_comments":[{"comment":"The title contains apparent spelling errors: 'INVER TIBILLITY' and 'CA TEGOR Y' should be corrected.","section":"Title"},{"comment":"The text contains many corrupted or overlapping Unicode arrow symbols such as '/leftr⫯g⊸tl⫯ne→' and '⫯ne→'. These should be cleaned up before publication.","section":"Throughout"},{"comment":"The introduction states 'Theorem 4.1.3 and Corollary 4.2.7' and adds 'even = oplax case'. The relationship between the main relative theorem and the lax/oplax terminology could be clarified, especially whether the lax case is also proven.","section":"Introduction, Theorem B"},{"comment":"Since the theorem is stated for both Mor_n(C) and uMor_n(C), it may help to add a short reminder that univalent completion does not affect dualizability; Remark 2.4.52 already explains this, but a sentence in the main theorem would reduce potential confusion.","section":"Theorem 3.0.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is ambitious and contains several solid auxiliary results, but the main theorem is not self-contained: it relies on unpublished work for Pointless additivity and composition-as-relative-tensor-product, and it explicitly defers the comparison with Haugseng's model. This is not a presentation issue but a genuine gap in the proof of the paper's central claim as stated. I would only accept after these results are either proved in the paper or replaced by a precise and available reference with complete statements and proofs. If the journal requires submissions to be fully self-contained, a reject may be warranted; otherwise major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is that this is a serious preprint with a genuinely new main theorem, but the proof is conditional on two deferred results that identify the new model with the Morita category. The n=1,2 and Pr cases were known; the general En statement and the invertibility corollary are first here. The core idea is well-motivated: pointless constructible factorization algebras remove exactly the pointings that blocked higher dualizability in earlier factorization models.\n\nThe auxiliary results visible in the text are coherent and clean. The localization theorem for disk operads (2.3.15), the identification of pointless (R,0)-algebras with bimodules (2.3.18), and the Segal condition (2.4.40) are all proven in the paper. The n=1 reduction of the relative theorem matching Lurie's known criterion is a useful sanity check. The paper is also honest about what it does not prove.\n\nThe soft spots are real and load-bearing. Proposition 2.4.42 (pointless additivity) and Proposition 2.4.43 (composition as relative tensor product) are explicitly deferred to [SSS]. These propositions are needed to know that k-morphisms really are iterated bimodules and that composition is relative tensor product; they are then used in quasi-unitality, the Cartesian fibration structure, regular bimodules, and adjointability. Without them, Theorem 3.0.1 is a dualizability statement about a new category uMor_n, not necessarily the higher Morita category in Lurie's conjecture. Remark 3.1.1 concedes that no comparison with Haugseng's model is proved. The visible copy of §3.1 also cuts off mid-proof of the lifting-of-adjoints theorem. These are gaps in support, not internal contradictions: the framework is credible, and the deferrals are openly flagged.\n\nI would send this to peer review. Referees should push on the dependence on [SSS]—ideally the authors include those proofs or state clearly which results survive without them. The paper deserves a serious referee despite its conditional status; if the foundational identification goes through, this resolves a long-open conjecture. I would cite it as the current best evidence for the conjecture, with the caveat noted.","headline":"A serious, technically advanced preprint that plausibly proves Lurie's (n+1)-dualizability conjecture and the BJSS invertibility corollary, but the central proof is conditional on unpublished results identifying the new pointless factorization model with the Morita category.","tokens_in":68790,"tokens_out":2149,"would_cite":true,"duration_ms":25135,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18N70","57K16"],"pacs":[],"model":"deepseek-v4-flash","headline":"An E_n-algebra in the Morita category is (n+1)-dualizable exactly when it is left dualizable as a module over each sphere-shaped factorization homology.","keywords":["higher Morita categories","dualizability","E_n-algebras","factorization homology","factorization algebras","topological field theories","adjointability","relative field theories"],"falsifier":"Take an E_n-algebra A whose factorization-homology modules over S^{k-1} × R^{n-k+1} are all left dualizable and compute its (n+1)-dualizability in an independent, established model of the higher Morita category; if the dual fails there, the criterion is false as a statement about the standard Morita category. A low-dimensional check is the n=1 or n=2 case, where the criterion should reproduce the known classifications of 2-dualizable algebras and 3-dualizable tensor categories; a mismatch, or a failure of the model's composition to be the relative tensor product, would falsify the paper's cent","tokens_in":67340,"feed_emoji":"🌀","tokens_out":7613,"duration_ms":86042,"temperature":0.7,"pith_summary":"The paper proves a long-standing conjecture characterizing when an algebra with n compatible multiplications is (n+1)-dualizable in the higher Morita category: this happens precisely when the algebra is dualizable as a left module over each of n+1 explicitly defined sphere-shaped factorization homologies. Because of the Cobordism Hypothesis, such dualizable objects are exactly the inputs for fully extended framed topological field theories, so the result converts a finite set of module-duality conditions into a large family of field theories. The proof works by rebuilding the higher Morita category from 'pointless' factorization algebras, which remove unwanted basepoint data that had blocked earlier approaches, and then using a critical-point geometric argument to identify the bimodules witnessing dualizability with the required factorization homologies. A corollary gives a criterion for invertibility in the Morita category, and a relative version produces lax and oplax relative field theories.","feed_headline":"A sphere-module duality test decides when algebras yield full TFTs","feed_subtitle":"Higher Morita dualizability reduces to finitely many sphere-module dualities—unlocking fully extended TFTs.","key_machinery":"The load-bearing new object is the pointless higher Morita category, built from constructible 'pointless' factorization algebras on stratified cubes: objects are E_n-algebras, k-morphisms for k < n are iterated bimodules, and n-morphisms are bimodules without chosen elements. The absence of pointings is what makes (n+1)-dualizability possible, since previously the pointings forced all sufficiently dualizable objects to be trivial. Two supporting mechanisms carry the proof: a lifting-of-adjoints theorem that reduces adjointability of an n-morphism to adjointability of an underlying ordinary bimodule, and a critical-point geometric computation identifying the bimodules that witness n-dualizabi","core_discovery":"The central claim is Theorem A: in the Morita (∞, n+1)-category of a presentably symmetric monoidal ∞-category, an E_n-algebra A is (n+1)-dualizable if and only if for every 0 ≤ k ≤ n, A is left dualizable as a left module over the factorization homology of A over S^{k-1} × R^{n-k+1}. This is the missing characterization of high dualizability: the right-to-left direction says that finitely many module-dualizability conditions over explicit geometric spaces suffice to produce all the higher adjoints that dualizability requires. In particular, by the Cobordism Hypothesis, such an A gives a fully extended framed (n+1)-dimensional topological field theory that assigns A to the framed point.","pith_inferences":["The paper defers the proof that composition in the pointless model is the relative tensor product and leaves the comparison with the established combinatorial model open; if that identification fails, Theorem A is a dualizability statement about a new category rather than the Morita category in the original conjecture.","The criterion is finite and explicit, so it could be used as a test for dualizability in concrete algebraic examples (e.g., braided tensor categories and fusion 2-categories), potentially revealing which familiar objects admit fully extended framed TFTs.","The success of 'unpointing' suggests a general principle: basepoint data in factorization models of higher categories obstruct dualizability, and marking-preserving inclusions may be a broadly applicable remedy in other higher-categorical settings.","The relative criterion likely produces more twisted field theories than the two boundary cases named here; any morphism whose half-sphere factorization-homology modules are dualizable should give an interface between the absolute theories of its source and target."],"forward_implications":["Every (n+1)-dualizable E_n-algebra in a nice symmetric monoidal ∞-category yields, via the Cobordism Hypothesis, a fully extended framed (n+1)-dimensional topological field theory valued in the Morita category.","An E_n-algebra is invertible in the Morita category exactly when it is (n+1)-dualizable and each sphere factorization homology maps canonically to the corresponding E_{n-k}-center as an equivalence.","The relative theorem gives a criterion for a morphism between E_n-algebras to be n-times right adjointable, in terms of dualizability over certain stratified half-sphere factorization homologies, producing oplax relative fully extended field theories.","The two standard kinds of relative boundary theories arise uniformly: the regular module from the unit always gives an oplax relative theory, while the reverse module gives one exactly when it is n-dualizable in the lower Morita category.","The dualizability and invertibility results extend to Morita categories valued in higher (∞, d)-categories, because the relevant properties are detected in the underlying (∞, n+1)-category after truncation."],"fun_headline_variants":["Sphere-module dualities decide full TFTs","Finite sphere checks ensure higher dualizability","Dualizability reduces to sphere-module conditions","New theorem unlocks fully extended TFTs","Morita dualizability pinned by sphere modules"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the newly built pointless category is genuinely the higher Morita category of E_n-algebras—that its k-morphisms are iterated bimodules and its composition is the relative tensor product; the proofs of these identifications are deferred to later work, and the equivalence with the established combinatorial model is not proved here.","fun_headline_variants_meta":{"raw":{"variants":["Sphere-module dualities decide full TFTs","Finite sphere checks ensure higher dualizability","Dualizability reduces to sphere-module conditions","New theorem unlocks fully extended TFTs","Morita dualizability pinned by sphere modules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000579,"raw_usage":{"total_tokens":2525,"prompt_tokens":662,"completion_tokens":1863,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":406,"completion_tokens_details":{"reasoning_tokens":1808}},"tokens_in":406,"tokens_out":1863,"duration_ms":14888,"temperature":1.0,"reasoning_tokens":1808,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T19:28:59.506994+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an E_n-algebra A whose factorization-homology modules over S^{k-1} × R^{n-k+1} are all left dualizable and compute its (n+1)-dualizability in an independent, established model of the higher Morita category; if the dual fails there, the criterion is false as a statement about the standard Morita category. A low-dimensional check is the n=1 or n=2 case, where the criterion should reproduce the known classifications of 2-dualizable algebras and 3-dualizable tensor categories; a mismatch, or a failure of the model's composition to be the relative tensor product, would falsify the paper's cent","supporting_citations":[],"review_version":1}