{"id":"2cffaaf6-3d22-48f4-a9cc-dba0e2b3fb4a","arxiv_id":"2601.17677","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Tate cohomology is defined and studied in any three-functor formalism, unifying local-system and quasicoherent-sheaf examples with the same norm map and universal property.","lead":"This paper builds a general framework for Tate cohomology, a construction that mixes homology and cohomology, and shows the same definitions work for local systems on spaces and for quasicoherent sheaves on stacks. It matters because it unifies a tool used in topological cyclic homology, chromatic homotopy theory, and algebraic geometry, so future proofs can be done once and reused everywhere.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.8.7 inherits all its force from unverified lemmas in [13]; if [13, Lemmas 4.5.5–4.5.8 or D.2.8] fail, the universal property and the examples are unsupported.","rationale":"I read the entire paper and checked the proof of Theorem 1.8.7, Proposition 1.8.10, Proposition 1.8.15, and the examples. Internally, the arguments are coherent: the localization construction in Prop. 1.8.10 is standard, the use of monadicity to generate D(T) from i_♯ is correct (modulo the i^*/i_* typo), and Prop. 1.8.15 correctly factors the norm map through fi using Prop. 1.7.9. The only serious risk is external: the paper explicitly imports the entire (∞,2)-categorical machinery of norm maps, adjunction criteria, and descent from Heyer–Mann [13], a 2024 preprint. The reader's weakest assumption matches mine, so I recommend no change to the CONDITIONAL verdict. A focused verification of the imported lemmas would settle whether the central claim stands.","tokens_in":4,"tokens_out":39186,"duration_ms":448115,"concrete_test":"Extract and fully verify [13, Lemma 4.5.7] and [13, Prop. D.2.8] from the definitions in [13, Appendices C–D]. In particular, check that the equivalence in Prop. 1.2.11 does not secretly assume the mapping ∞-categories are stable, and that the locality of primness under D-∗-covers holds without additional finiteness hypotheses. If either cannot be reproduced, Theorem 1.8.7 and the cover arguments in §2 would need to be reworked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central theorem (1.8.7) is not self-contained. It rests on the Heyer–Mann framework [13] at three load-bearing points: (1) Proposition 1.2.11 = [13, Prop. D.2.8] converts weak adjoints into honest adjoints, which is how primness/suaveness are detected; (2) Propositions 1.6.9 and 1.7.11 = [13, Lemmas 4.5.7–4.5.8] give locality results needed to produce D-Tate covers in Examples 2.1.11 and 2.2.11; (3) Proposition 2.2.10 = [13, Thm 3.4.11] supplies the extension of QCoh to prestacks. A gap in any of these recent-preprint results would invalidate the claimed universal property or the examples; the present paper does not re-prove them. (The statement of Definition 1.8.5(2) also has a typo: conservativity of i_* should be conservativity of i^*, as required by the proof of Theorem 1.8.7 via Lemma 1.8.14.)","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a general definition of norm maps and Tate cohomology in the setting of Heyer–Mann three-functor formalisms. For a D-!-able map f:T→S, using weak adjoints in the kernel (∞,2)-category K_D(S), it defines the ∗-norm and ♯-norm maps, then the norm map Nm_f:f_♯(-⊗ν_f)→f_*, whose cofiber is Tate cohomology f^t. It establishes base change and composition formulas for these norms, proves a Browder-style theorem for left-divisible A2-monoids (Thm 1.5.6), and, assuming D is stable and presentable, proves a universal property of Tate cohomology (Thm 1.8.7) together with induced lax symmetric monoidal structures and functoriality (Thm 1.9.3). It then instantiates the formalism in local systems valued in a stable presentable symmetric monoidal ∞-category and in quasicoherent sheaves on prestacks, recovering classical Tate cohomology and giving examples such as S^1 with K(1)-local coefficients.","tokens_in":1456,"tokens_out":2644,"duration_ms":184951,"significance":"If the imported Heyer–Mann results are correct, this is a useful unifying framework: it gives a clean categorical mechanism for norm maps, recovers Nikolaus–Scholze's universal property, and supplies new examples beyond spectra (chromatic local systems and QCoh of classifying stacks). The paper is clearly organized and carefully attributes each step; Proposition 1.8.10 is a nice self-contained categorical localization result that is likely of independent use. Its main limitation is external dependence: the central theorems are conditional on specific lemmas in the recent preprint [13].","major_comments":[{"comment":"The proof of the main theorem and the examples depend on imported results from [13]: Proposition 1.2.11 (D.2.8), Propositions 1.6.9 and 1.7.11 (Lemmas 4.5.7–4.5.8), and Proposition 2.2.10 (Theorem 3.4.11). These are load-bearing for the existence of D-Tate covers and hence for the universal property and monoidality/functoriality statements. As [13] is a recent unreviewed preprint, the paper is conditional on external results. Please either re-prove the imported statements or explicitly state the dependency and list the exact lemmas, making the theorem explicitly conditional.","section":"§1.8, Theorem 1.8.7; §2.1, Example 2.1.11; §2.2, Example 2.2.11"},{"comment":"With the paper's convention that f_* denotes the right adjoint of f^*, condition (2) of Definition 1.8.5 and Lemma 1.8.13 should read i^*:D(T)→D(V) (the pullback functor) is conservative, not i_*. This is the hypothesis used in the proof of Theorem 1.8.7 via Lemma 1.8.14. The notation error should be corrected.","section":"§1.8, Definition 1.8.5 and Lemma 1.8.13"}],"minor_comments":[{"comment":"In part (2), the displayed definition after 'δ_f :=' should be ω_f, not δ_f.","section":"§1.5, Proposition 1.5.3(2)"},{"comment":"The step saying 'if fib(γ) preserves colimits, then γ is initial' is compressed; please spell out that fib(γ) is left Kan extended from D(T)_{i-nil}, so it is the localization.","section":"§1.8, proof of Theorem 1.8.7"},{"comment":"The assertion that the localization q of Ind(B^κ) is compatible with the symmetric monoidal structure is stated in one 'hence'; please supply a sentence explaining why the tensor ideal hypothesis suffices.","section":"§1.8, Proposition 1.8.10(3)"},{"comment":"Typo: 'isomorphim' should be 'isomorphism'.","section":"§2.2, Example 2.2.9"}],"recommendation":"major_revision","confidential_remarks":"The paper is careful and likely useful if the Heyer–Mann framework is accepted, but its main theorems are currently conditional on an unreviewed preprint. Make the dependency explicit and fix the small notation errors."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a well-executed synthesis, not a blockbuster theorem. The author says outright in §0.3 that it is largely “reviewing and lightly mixing ideas” from Heyer–Mann [13] and Nikolaus–Scholze [25], and that is accurate. What is new and worth having: Tate cohomology is defined purely from weak adjoints in the kernel (∞,2)-categories of a three-functor formalism; Theorem 1.5.6 gives a Browder-style Poincaré duality for H-spaces in that generality; Theorem 1.8.7 packages the Nikolaus–Scholze universal property (with lax symmetric monoidal structure) in this language; and the examples are real—particularly the S1-functoriality for Tate cohomology used in [9] and the QCoh treatment of stably dualizable Hopf algebras (§2.2). The writing is clear and the categorical bookkeeping is honest: each step is traced to a lemma of [13] or [25].\n\nThe soft spots are proportionate. First, the dependence on [13] is not cosmetic: Propositions 1.2.11, 1.6.9, 1.7.11, and 2.2.10 are imported and are all load-bearing for the universal property and the QCoh examples. [13] is a 2024 preprint that has not been independently vetted. The author is transparent about this, and it is normal for an expository paper to build on a framework, but it means the present paper's central theorem has an external epistemic risk. A referee should check those specific citations rather than the whole framework. Second, Definition 1.8.5(2) says i_* is conservative; it has to be i^* (the proof of Theorem 1.8.7 uses Lemmas 1.8.13–1.8.14 exactly that way). That is a typed slip, not a mathematical flaw, but it should be fixed before publication. I did not find other issues: the arguments I checked are sound, the examples are consistent, and the citation pattern is honest—self-citations appear only in examples or companion work.\n\nBottom line: this paper is for people who want a reusable home for Tate cohomology beyond local systems on spaces—chromatic homotopy, quasicoherent sheaves, THH of Z. It deserves a serious referee. I would send it out, ask for the typo fix and a brief note on the status of [13], and accept after those.","headline":"Clean, honestly-labeled axiomatization of Tate cohomology via three-functor formalisms; genuinely useful examples, but the whole edifice rests on the un-vetted Heyer–Mann preprint and a definition has a typo.","tokens_in":43950,"tokens_out":4088,"would_cite":true,"duration_ms":43988,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Tate cohomology can be defined in any three-functor formalism, with a universal property that yields its monoidality and functoriality.","keywords":["Tate cohomology","three functor formalism","norm map","Poincaré duality","lax symmetric monoidal","universal property","local systems","quasicoherent sheaves"],"falsifier":"A concrete refutation would be an example of a stable presentable three-functor formalism with a D-Tate map f and Tate cover i in which the canonical map f_*→f^t is not initial among accessible functors vanishing on i-nilpotent objects—or, more basically, a failure of the cited local-descent lemma that would invalidate the existence of such covers in the examples.","tokens_in":42986,"feed_emoji":"","tokens_out":8424,"duration_ms":82320,"temperature":0.7,"pith_summary":"This paper establishes that Tate cohomology—the amalgam of homology and cohomology originally built for finite groups—can be defined in the very general setting of three-functor formalisms, which axiomatize how homology and cohomology behave for spaces, stacks, and other geometric objects. The definition is a cofiber of a norm map constructed from weak adjoints in a category of kernels, and the main theorem gives this construction a universal property: the canonical map from cohomology to Tate cohomology is initial among accessible functors that vanish on a specified class of nilpotent objects. From this universal property, the paper derives that Tate cohomology is automatically lax symmetric monoidal and functorial, generalizing what was previously known for local systems of spectra on spaces. Two families of examples show the scope: C-valued local systems on spaces, recovering and extending known group-theoretic Tate cohomology, and quasicoherent sheaves on prestacks, covering dualizable Hopf algebras and stably dualizable groups. If correct, this provides a uniform mechanism for producing Tate cohomology and its ring structures in any context with a six-functor-like calculus.","feed_headline":"Tate cohomology generalizes to every three-functor formalism","feed_subtitle":"The cofiber of the norm map gets monoidality and functoriality for free from a single universal property.","key_machinery":"The paper's machinery is the three-functor formalism D: a formal package assigning to each space (or stack) a symmetric monoidal ∞-category of coefficient systems, with ∗-pullback, !-pushforward, and ♯-pushforward functors subject to base-change and projection formulas. The key objects are the (∞,2)-category of kernels KD(S), in which weak adjoints (rather than honest adjoints) are enough to construct a norm map Nm_f:f_♯(-⊗ν_f)→f_* for any D-Tate map f. Tate cohomology f^t is the cofiber of this norm map. The universal property (Theorem 1.8.7) is proved using the left Kan extension / localization machinery relative to the thick tensor ideal of i-nilpotent objects, for i a D-Tate cover.","core_discovery":"The paper defines, for every D-Tate map f:T→S in a three-functor formalism D, a Tate cohomology functor f^t:D(T)→D(S) as the cofiber of a norm map Nm_f:f_♯(-⊗ν_f)→f_*, where ν_f is a twisting object assembled from the diagonal of f. The central theorem is that, when a D-Tate cover i:V→T exists, the canonical map can:f_*→f^t is initial among accessible functors from D(T) to D(S) that vanish on i-nilpotent objects; moreover can and f^t carry canonical lax symmetric monoidal structures making can initial among such maps of lax symmetric monoidal functors. From this universal property the paper derives that Tate cohomology is functorial under maps that are D-Poincaré, and that norm maps are comp","pith_inferences":["If the underlying three-functor formalism is accepted, the same construction would supply Tate cohomology in other six-functor settings (for example p-adic or rigid-analytic geometries), where no group or space-level description is available, giving candidates for new equivariant invariants.","The universal property suggests interpreting Tate cohomology as a colocalization or completion of cohomology that kills the homology part; this could be formalized in a general stable ∞-categorical context, yielding a notion of 'Tate objects' independent of any geometric input.","The paper's results in the local-systems and quasicoherent-sheaf settings coincide where the two frameworks overlap (for dualizable group spaces), hinting that the 'correct' domain for Tate cohomology is the abstract three-functor formalism itself rather than either example."],"forward_implications":["In any stable presentable three-functor formalism with a Tate cover, the Tate cohomology functor is automatically lax symmetric monoidal and the canonical map from cohomology respects this structure, so coefficient ring structures pass to Tate cohomology.","For C-valued local systems, the universal property endows Tate cohomology of a compact group space with lax symmetric monoidal structures, and yields functoriality of Tate cohomology under finite covering maps, and under the multiplication-by-n maps on the circle when C is 1-semiadditive.","For quasicoherent sheaves on prestacks, the same results give Tate cohomology for classifying stacks of dualizable Hopf algebras, providing a systematic account that includes the theory of stably dualizable groups.","The norm maps and Tate cohomology are compatible with base change and composition, so the classes of Poincaré, smooth, and terse maps form a genuine six-functor calculus.","The universal property gives a practical criterion: to construct a lax symmetric monoidal map out of Tate cohomology, it suffices to construct one out of cohomology and check vanishing on i-nilpotent objects."],"fun_headline_variants":["Tate cohomology: one universal property for any three-functor setup","Tate cohomology generalizes via universal property to all 3-functor settings","Three-functor Tate cohomology: norm cofiber gets monoidal structure","Tate cohomology for all three-functor formalisms: from a single norm map","Universal Tate cohomology: monoidality and functoriality for free"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole edifice rests on the correctness of the recently introduced three-functor formalism, including the (∞,2)-categorical machinery of kernels and the cited lemmas establishing norm maps, Poincaré duality criteria, and descent; if any of those imported results is incomplete or wrong, the definitions and theorems here inherit the flaw.","fun_headline_variants_meta":{"raw":{"variants":["Tate cohomology: one universal property for any three-functor setup","Tate cohomology generalizes via universal property to all 3-functor settings","Three-functor Tate cohomology: norm cofiber gets monoidal structure","Tate cohomology for all three-functor formalisms: from a single norm map","Universal Tate cohomology: monoidality and functoriality for free"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001208,"raw_usage":{"total_tokens":4756,"prompt_tokens":629,"completion_tokens":4127,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":373,"completion_tokens_details":{"reasoning_tokens":4015}},"tokens_in":373,"tokens_out":4127,"duration_ms":23373,"temperature":1.0,"reasoning_tokens":4015,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T08:11:12.496198+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete refutation would be an example of a stable presentable three-functor formalism with a D-Tate map f and Tate cover i in which the canonical map f_*→f^t is not initial among accessible functors vanishing on i-nilpotent objects—or, more basically, a failure of the cited local-descent lemma that would invalidate the existence of such covers in the examples.","supporting_citations":[],"review_version":1}