{"id":"2f3d510f-cfcb-40cc-9165-879b6db9b82f","arxiv_id":"2510.24629","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Approximate fibrations, previously defined only for topological spaces, are reformulated for higher topoi and shown to coincide with the classical notion for proper maps of locally compact ANRs.","lead":"The paper introduces a version of approximate fibrations, a relaxed notion of fiber bundle, for higher topoi and proves it matches the classical notion for maps between nice spaces called ANRs. It gives shape theorists and geometric topologists a new dictionary for translating questions about approximate fibrations into the language of higher topos theory.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central equivalence in Theorem 4.27 depends on unpublished Martini–Wolf properness characterization; the risk is real but already priced into the reader's moderate-confidence ACCEPT.","rationale":"The reader identified the same weakest assumption: the paper's main theorems depend on the correctness of the unpublished Martini–Wolf preprints [MW23] and [MW25]. I find no internal contradiction or circular reasoning in the manuscript itself; the proofs are coherent once those external inputs are granted. The central claim is therefore plausible but not fully self-contained, which supports the reader's moderate confidence rather than high confidence. Because the concern is about an external dependency that the reader already flagged and weighed, I do not see a reason to change the verdict. The appropriate response is for the community to verify the specific Martini–Wolf theorems used, rather than to reject or conditionally accept the paper on the basis of a visible internal flaw.","tokens_in":19442,"tokens_out":25169,"duration_ms":241395,"concrete_test":"Isolate the exact [MW23] statement used in Theorem 3.11 and Remark 4.14 (properness implies preservation of internally filtered colimits for the associated Y-functors) and ask the authors or an independent referee to provide a fully expanded proof, ideally formalized in a proof assistant. As a minimal concrete cross-check, instantiate the statement for Y = Shv(S^1) and f : Shv(S^1 × S^1) → Shv(S^1) the projection, computing whether f_*f^* preserves internally filtered colimits according to the external criterion in Proposition 2.4; if the two criteria disagree, the bridge between the internal and external notions breaks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central Theorem 4.27 (and Theorem 3.11 and Remark 4.14) rests on two unpublished inputs: [MW23], which characterizes proper geometric morphisms as those whose associated Y-functors preserve internally filtered colimits, and [MW25, Theorem 3.2.5.1], which identifies internal Y-topoi with Top/Y. Concretely, in Theorem 3.11 the proof that hereditary shape equivalence implies relative shape equivalence needs q_*q^* and p_*p^* to preserve internally filtered colimits, so that the transformation (3.12) can be checked on internally compact objects. In Remark 4.14, the claim that a proper approximate fibration has locally constant compact corepresenting object uses the same [MW23] theorem. If this theorem is false or has a hidden hypothesis, then the internal definition of approximate fibration in Definition 4.11 is not linked to properness, and the equivalence with shape fibrations in Theorem 4.27 loses its foundation. Similarly, Section 4.2 repeatedly uses Top(Y) ≃ Top/Y to identify the relevant Y-functors. This is a genuine correctness risk rather than an internal inconsistency: the arguments in the manuscript are coherent conditional on these inputs. The preprints are not machine-checked and are not included with the submission, so the load-bearing step is precisely the correctness of these external statements.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a topos-theoretic counterpart of the classical theory of approximate fibrations. It defines an approximate fibration of ∞-topoi as a proper geometric morphism f:X→Y for which the Y-functor F_*F^* is internally corepresentable, then proves that, for locally contractible X and Y with enough points in the hypercompletion of Y, this notion is equivalent to being a shape fibration and to being a shape quasi-fibration (Theorem 1.2 and Corollary 4.30). It also proves a cell-like characterization (Theorem 1.1 and Corollary 3.13): cell-like proper geometric morphisms are exactly hereditary shape equivalences and, under local contractibility and hypercompleteness hypotheses, exactly maps whose point fibers have trivial shape. Finally, Corollary 1.3 identifies the new notion with the classical Coram–Duvall approximate fibrations for proper maps of locally compact ANRs, via the shape-fibration reformulation of [HTW90]. The proofs are long reductions in internal higher category theory, relying on the Martini–Wolf framework.","tokens_in":19605,"tokens_out":8102,"duration_ms":75091,"significance":"The results, if correct, would be a meaningful unification: they provide an internal, categorical definition of approximate fibration that is checked externally against the classical notion, they generalize Lurie's shape-theoretic characterization of cell-like maps, and they offer a route to import higher topos theory into geometric topology. The manuscript is careful with hypotheses and does not fit parameters or rely on circular reasoning. The main source of uncertainty is external: the central equivalence arguments depend on unpublished preprints by Martini and Wolf, especially the characterization of proper geometric morphisms via preservation of internally filtered colimits. Conditional on those inputs, the paper's internal logic is coherent.","major_comments":[{"comment":"The proof that proper geometric morphisms preserve internally filtered colimits and that Top(Y) is equivalent to Top/Y is imported from [MW23] and [MW25, Theorem 3.2.5.1] without stating the precise theorems or their hypotheses. These results are load-bearing for the equivalence between approximate fibrations and shape fibrations and for the cell-like Theorem 3.11; if either statement has a hidden hypothesis, Theorem 4.27 and Corollary 1.3 lose their foundation. The authors should either restate these results as explicit assumptions or appendix material, or include a stable version of the preprints with the submission, and give exact references for each use.","section":"§4.2 (Theorem 4.27), §3 (Theorem 3.11), Remark 4.14"},{"comment":"The reduction from invertibility of (4.20) on internally compact objects to checking the global-sections transformation (4.28) is compressed. In particular, the passage from (4.29) for each U to the conclusion is justified by an unstated descent/internal-Yoneda argument, and the earlier identification in Remark 4.21 that composes η^* and η_* is only sketched. Since this is the core of the proof that every shape fibration is an approximate fibration, I ask that this reduction be expanded with the precise internal category-theoretic statements, including why section-wise invertibility at each U is sufficient for an equivalence of Y-functors.","section":"§4.2, proof of Theorem 4.27, (2)⇒(1)"}],"minor_comments":[{"comment":"The sentence 'f♯ factors through a functor a functor η♯' contains a duplicated phrase; it should read 'factors through a functor η♯'.","section":"§2.7"},{"comment":"Using the notation f♯(1_X) for the corepresenting object may be confused with the left adjoint f♯ from Definition 2.9, since an approximate fibration is not assumed to be locally contractible. A neutral notation, or an explicit warning that f♯ here is auxiliary, would improve readability.","section":"Definition 4.11"},{"comment":"The reference [MW23] is listed as 'arXiv preprint, 2023' without an arXiv identifier; because that preprint is central to the paper, the authors should provide a stable public identifier or a versioned reference.","section":"References"},{"comment":"The phrase 'copresented by lim←' should be 'corepresented by' (or 'computed as a limit'), since the relevant Y-functor is represented by a pro-object.","section":"Corollary A.6"}],"recommendation":"major_revision","confidential_remarks":"I see no evidence of circularity or fitted parameters. The main risk is the unpublished status and possible instability of [MW23] and [MW25]; this may be acceptable for a specialist journal, but it should be resolved before archival publication. I would not reject over the dependence alone, provided the authors make the imported results precise and confirm the preprints are stable. The paper fits the scope of a topology/category theory journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real contribution, not a translation exercise. Kremer and Volpe define approximate fibrations for geometric morphisms of ∞-topoi using internal corepresentability, prove they match shape fibrations and shape quasi-fibrations under local contractibility and points assumptions, and recover Coram-Duvall's classical theory for proper maps of locally compact ANRs. They also give a topos-theoretic proof of Lurie's cell-like characterization that generalizes it to hereditary shape equivalences and pointwise trivial shape. That is more than repackaging: the internal definition is new, and the equivalences in Theorems 1.2 and 4.27 are substantive.\n\nThe organization is good. Hypotheses are stated precisely, proofs are structured as long reductions, and the paper is honest about what is imported. There is no circularity: the new notion is tested against the classical Coram-Duvall notion via [HTW90], an external benchmark. The only self-reference, [Vol25, Prop. A.6] in Lemma 4.8, is a published compactness argument and is legitimate.\n\nThe soft spot is exactly the one the stress-test flags, and it is not minor: the proof of Theorem 4.27 (and Theorem 3.11, Remark 4.14) leans on unpublished Martini-Wolf results, specifically [MW23]'s characterization of proper geometric morphisms as those whose associated Y-functors preserve internally filtered colimits, and [MW25]'s identification Top(Y) ≃ Top/Y. If either has a hidden hypothesis, the internal definition of approximate fibration (Definition 4.11) is not linked to properness and the main equivalence loses its foundation. The authors state these dependencies clearly, and the arguments are coherent conditional on them, but this is a real correctness risk that a referee needs to check. It is priced into the reader's moderate confidence, and I agree with that calibration.\n\nMinor softness: Theorem 4.33 imports [HTW90, Theorems 12.13 and 12.15] as a black box. That is fine for a paper whose point is topos theory, not classical geometric topology, but it means the bridge to the classical theory is only as solid as the cited theorem.\n\nWho this is for: people working in higher topos theory, shape theory, or geometric topology who want the categorical packaging of approximate fibrations and cell-like maps. The paper deserves a serious referee. The risk is identifiable and checkable; it is not a vague gap. Recommend.","headline":"A genuinely new internal ∞-topos definition of approximate fibrations; the main risk is the unpublished Martini-Wolf inputs it leans on, but the paper is coherent and deserves serious refereeing.","tokens_in":20217,"tokens_out":2718,"would_cite":true,"duration_ms":23653,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18N60","55P55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines approximate fibrations for geometric morphisms of ∞-topoi as internal corepresentability of the Y-shape functor, and proves this agrees with shape fibrations and, classically, with approximate fibrations of locally…","keywords":["approximate fibrations","∞-topos theory","shape theory","cell-like maps","geometric morphisms","internal higher category theory","hereditary shape equivalence","proper maps"],"falsifier":"A concrete counterexample would be a proper geometric morphism f:X→Y of locally contractible ∞-topoi with Y^hyp having enough points that is a shape quasi-fibration but not a shape fibration; Theorems 1.2 and 4.10 assert the two coincide. A second decisive check is to find a proper map of locally compact ANRs that is an approximate fibration in the classical sense defined by homotopy lifting up to open covers but whose associated sheaf geometric morphism fails internal corepresentability, contradicting Corollary 1.3.","tokens_in":19143,"feed_emoji":"🌐","tokens_out":11475,"duration_ms":96213,"temperature":0.7,"pith_summary":"This paper aims to transfer the classical theory of approximate fibrations—maps that lift homotopies only up to an open cover—into higher topos theory, where continuous maps are replaced by geometric morphisms of ∞-topoi. It proposes that a proper geometric morphism f:X→Y is an approximate fibration precisely when its Y-shape functor F_*F^* is internally corepresentable, and it proves that under local contractibility and a mild point condition this is equivalent to being a shape fibration and also to being a shape quasi-fibration. A sympathetic reader should care because this is not a parallel new definition: Corollary 1.3 shows that for proper maps of locally compact ANRs the new notion is exactly the classical approximate fibration notion for spaces, so topos-theoretic tools now point at manifold topology. The paper also upgrades the earlier topos-theoretic characterization of cell-like maps to a purely topos-theoretic theorem identifying them with hereditary shape equivalences and, under the same hypotheses, with maps whose point-fibres have trivial shape.","feed_headline":"Shape conditions capture approximate fibrations in ∞-topoi","feed_subtitle":"A new internal definition recovers classical manifold maps and re-proves the cell-like criterion.","key_machinery":"The central object is the Y-shape of a geometric morphism f:X→Y: the Y-functor F_*F^*:Ω_Y→Ω_Y, defined sectionwise by V↦f_*f^*V×_{f_*f^*U}U over U, which packages all base-changed shape information. Approximate fibration is defined as internal corepresentability of this functor, i.e. F_*F^* ≃ Map_{Ω_Y}(f_♯(1_X),−) for some object f_♯(1_X) in Y. The proof machinery is the framework of internal higher category theory the paper cites as [MW25, MW23]: the equivalence between internal Y-topoi and the over-category Top/Y, the characterization of proper geometric morphisms as those whose Y-functors preserve internally filtered colimits, and the identification of internally compact objects in Ω_Y with locally constant objects with compact values. These allow the authors to reduce statements about all objects to statements about locally constant compact objects and then to points of Y.","core_discovery":"The central discovery is that both cell-like maps and approximate fibrations can be recognized by shape-theoretic data internal to the base topos. For a geometric morphism f:X→Y, the paper defines an approximate fibration as internal corepresentability of the Y-functor F_*F^*:Ω_Y→Ω_Y, the Y-shape of f, whose sections record the shape of X pulled back over each object of Y. Theorem 1.2 proves that when X and Y are locally contractible and the hypercompletion of Y has enough points, this internal condition is equivalent to f being a shape fibration—each square sh(X/f^*U)→sh(X) over sh(Y/U)→sh(Y) is a pullback in Pro(An)—and equivalent as well to f being a shape quasi-fibration, where only the fibres over points of Y need the same pullback property. Theorem 1.1 proves that a proper geometric morphism is cell-like, meaning f_* is fully faithful, exactly when it is a hereditary shape equivalence, and under the same point assumption exactly when every fibre over a point has trivial shape; this generalizes the earlier characterization and gives a purely topos-theoretic proof of the classical hereditary-shape criterion for cell-like maps. Finally, Corollary 1.3 feeds the classical pullback characterization of approximate fibrations of locally compact ANRs back into the new definition: a proper map of locally compact ANRs is an approximate fibration in the original sense exactly when its associated geometric morphism of sheaf ∞-topoi is an approximate fibration.","pith_inferences":["If the equivalence with shape quasi-fibrations extends beyond locally contractible topoi, pointwise fibre checks would become the standard detection method for approximate fibrations, replacing explicit ε-lifting arguments in geometric topology.","The internal corepresentability formulation could be imported into any setting with a workable internal category theory—sites, stacks, or derived geometry—where no metric or open-cover notion of closeness exists.","The paper leaves the connection to fibering obstructions implicit; a natural test is whether Farrell-type K-theoretic obstructions can be recast as the failure of the Y-shape functor to be internally corepresentable.","One testable extension is whether Corollary 1.3 survives for paracompact spaces of finite covering dimension that are not ANRs, where the classical theory is less developed."],"forward_implications":["Cell-like geometric morphisms are exactly hereditary shape equivalences; under local contractibility and enough points, they are also exactly the maps whose point-fibres have trivial shape.","Under the same hypotheses, approximate fibrations, shape fibrations, and shape quasi-fibrations are three equivalent descriptions of a single class of proper geometric morphisms.","Every proper approximate fibration of ∞-topoi has a well-defined internal corepresenting object f_♯(1_X) that is locally constant with compact values, encoding the shape of the fibres.","For proper maps of locally compact ANRs, the classical approximate fibrations are exactly the new topos-theoretic approximate fibrations, giving a dictionary between manifold topology and ∞-topos theory.","The notion is compatible with étale base change: the corepresenting object for f/U is η_∗ sh(f/U)_! 1_{X/f^*U}, so approximate fibration is inherited by restriction to slices."],"supporting_citations":[{"why":"Supplies the internal higher-category theory: the equivalence Top(Y)≃Top/Y, internal compactness, and Yoneda ingredients used throughout Section 4.2.","marker":"[MW25]"},{"why":"Provides the characterization of proper geometric morphisms as those whose Y-functors preserve internally filtered colimits, used in Theorems 3.11 and 4.27.","marker":"[MW23]"},{"why":"Introduces the classical approximate fibration notion for maps of spaces that Corollary 1.3 must match.","marker":"[CPFD77]"},{"why":"Gives the pullback characterization of approximate fibrations of locally compact ANRs invoked in Theorem 4.33.","marker":"[HTW90]"},{"why":"Supplies the foundation of ∞-topos theory and the earlier cell-like-map characterization that Theorem 1.1 generalizes.","marker":"[Lur09]"},{"why":"States the classical hereditary-shape criterion for cell-like maps that Theorem 1.1 re-proves and extends.","marker":"[Lac69]"},{"why":"Provides facts about locally constant objects, hypercompletion, and the shape of ANRs used in Examples and Lemma 2.14.","marker":"[Lur17]"},{"why":"Gives the Yoneda lemma for internal higher categories used in Construction 4.15 and Remark 4.24.","marker":"[Mar21]"}],"fun_headline_variants":["Internal shape condition pins approximate fibrations in ∞-topoi","Shape theory internalizes approximate fibrations for ∞-topoi","Cell-like maps get hereditary shape test in ∞-topoi","Approximate fibrations: shape-theoretic characterization in ∞-topoi"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the correctness of the unpublished internal higher-category theory used here—especially the characterization of proper geometric morphisms by preservation of internally filtered colimits and the equivalence between internal Y-topoi and the over-category Top/Y—so a flaw in those manuscripts would force repairs in the main theorems.","fun_headline_variants_meta":{"raw":{"variants":["Internal shape condition pins approximate fibrations in ∞-topoi","Shape theory internalizes approximate fibrations for ∞-topoi","Cell-like maps get hereditary shape test in ∞-topoi","Approximate fibrations: shape-theoretic characterization in ∞-topoi"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000883,"raw_usage":{"total_tokens":3809,"prompt_tokens":938,"completion_tokens":2871,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":2810}},"tokens_in":554,"tokens_out":2871,"duration_ms":18527,"temperature":1.0,"reasoning_tokens":2810,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:40:09.813238+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete counterexample would be a proper geometric morphism f:X→Y of locally contractible ∞-topoi with Y^hyp having enough points that is a shape quasi-fibration but not a shape fibration; Theorems 1.2 and 4.10 assert the two coincide. A second decisive check is to find a proper map of locally compact ANRs that is an approximate fibration in the classical sense defined by homotopy lifting up to open covers but whose associated sheaf geometric morphism fails internal corepresentability, contradicting Corollary 1.3.","supporting_citations":[],"review_version":2}