{"id":"dc553671-7c0b-468a-81e9-64c14eb60ff2","arxiv_id":"2506.07874","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In any tangent category satisfying the p-carrable and 0-carrable conditions, the relative tangent bundle is the kernel of the horizontal descent, yielding a relative cotangent sequence that recovers the classical algebraic and differential geometric sequences.","lead":"The paper builds a tangent-category dictionary for morphism classes such as immersions, submersions, unramified maps, and etale maps, and proves that from one common construction, the horizontal descent, a relative cotangent sequence exists in any tangent category with the required pullbacks. The dictionary specializes to familiar notions in smooth manifolds, schemes, commutative algebras, and Cartesian differential categories.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 4.2.1 appears correct; its carrability hypotheses are genuine but are stated and checked, and the proof has only minor typographical issues.","rationale":"The reader's conditional verdict is driven by external dependencies—the unproved theorem from private communication [CL25a] and deferred scheme computations—rather than by an identified flaw in the central equalizer theorem. My stress-test of Theorem 4.2.1 found no internal error. The theorem is precisely conditioned on p-carrability and 0-carrability, which are legitimate hypotheses verified in the main examples. The proof is long but structurally sound; the only blemish is a type-error typo in the linearity computation, easily corrected. Therefore I do not see a reason to alter the reader's verdict.","tokens_in":59412,"tokens_out":12265,"duration_ms":144191,"concrete_test":"Independently verify that the horizontal descent θf is a linear morphism of differential bundles over X and re-derive the linearity step of Theorem 4.2.1 using the corrected identity 0X∘q = Tq∘λ (not q∘0X = Tq∘λ). If θf failed to be linear, the equalizer statement in DBun(X) would be ill-posed; this verification confirms the proof's coherence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no substantive flaw in Theorem 4.2.1. The proof correctly uses the two carrability hypotheses: p-carrability supplies the differential bundle f*(TY) and the horizontal descent θf; 0-carrability supplies TX/Y with its differential bundle structure and the T-preserved pullback needed for λX/Y. The linearity argument for the induced map ρ is valid up to a minor typo: the displayed identity q∘0X = Tq∘λ should read 0X∘q = Tq∘λ; the intended equation follows from the additive-bundle axiom and naturality of 0. The equalizer universal property is proved directly in DBun(X), and the proof's use of the pullback property of T(TX/Y) is sound. The remaining unproved private-communication theorem [CL25a] affects Section 7, not this central statement. Thus the only genuinely load-bearing premise is the existence of the two pullbacks, which is a stated hypothesis, not a hidden assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops tangent-categorical analogues of immersions, submersions, unramified morphisms, local diffeomorphisms, and related classes, chiefly through the horizontal descent θ_f of a p-carrable morphism f. The central technical result is Theorem 4.2.1: if f is both 0-carrable and p-carrable, then the relative tangent bundle TX/Y becomes the equalizer of θ_f and f^*(0_Y)∘p_X in DBun(X), yielding a tangent-categorical relative cotangent sequence. The paper also introduces T-immersions, T-submersions, split T-submersions, and T-étale maps, with classifications in smooth manifolds, affine schemes, schemes, and Cartesian differential categories.","tokens_in":59621,"tokens_out":7319,"duration_ms":91227,"significance":"If the main theorem stands, the paper gives a genuinely general formulation of the relative cotangent sequence that specializes to the classical exact sequences in algebraic and differential geometry. The proof of Theorem 4.2.1 is detailed and, as far as I can check, correct; the carrability hypotheses are explicit existence assumptions and are checked separately in each main example. The systematic study of submersions, however, rests on Theorem 7.2.2, which is not proved in the manuscript and is attributed to private communication, so the full scope of the paper's claims is conditional on that missing argument. The examples connecting the abstract notions to schemes and CDCs are valuable and mostly well grounded in the published equivalence DBun(X) ≃ QCoh(X)^op.","major_comments":[{"comment":"Theorem 7.2.2 is a load-bearing result for the paper's treatment of T-submersions, but it is stated without proof and attributed to the private communication [CL25a]. It is used essentially in Proposition 7.2.5, Corollary 7.2.6, Corollary 7.2.8, and Example 7.1.8, where the equivalence between f being a T-submersion and θ_f being T-epic is needed. As the manuscript stands, these systematic claims about submersions are not independently checkable. The authors should either provide a full proof of Theorem 7.2.2 or replace the reference with a publicly available, verifiable source.","section":"Section 7.2, Theorem 7.2.2"},{"comment":"The definition of T-submersion states that θ_f is a 'T-coequalizer in C', but a coequalizer requires a specified parallel pair and none is given. The surrounding arguments treat the condition as being that θ_f is a regular epimorphism preserved by all powers of T, as in Lemma 7.2.1 and Theorem 7.2.2. The definition should be made precise, for example by defining T-submersions via the appropriate T-coequalizer of the kernel pair of θ_f, or by explicitly defining what 'T-coequalizer' means for a single morphism here.","section":"Section 7.1, Definition 7.1.1"}],"minor_comments":[{"comment":"In the proof of Theorem 4.2.1, the displayed identity 'q∘0_X = Tq∘λ' should read '0_X∘q = Tq∘λ'; the intended equation follows from the additive-bundle morphism axiom and naturality of 0, and the rest of the proof is unaffected.","section":"Section 4.2, proof of Theorem 4.2.1"},{"comment":"The displayed sequence in Corollary 7.2.3 is garbled: it reads 'X TX/Y T X f^*(T Y) X pr0 θf' without clear arrows. Please rewrite the sequence with explicit arrows so that the claimed exactness is unambiguous.","section":"Section 7.2, Corollary 7.2.3"},{"comment":"Several scheme-theoretic constructions, in particular the tangent category structure on Sch/S, are deferred to the forthcoming work [Voo25]; for example, Example 2.2.5 states that full details are in [Voo25]. Since the paper's classifications in algebraic geometry depend on these details, the authors should either include the needed definitions in the present paper or clearly mark those examples as conditional on the forthcoming reference.","section":"Sections 2.2, 3.2, 8.3"},{"comment":"There are numerous typographical errors (e.g., 'Rosciský', 'manfiodls', 'sumbersion', 'isomoprhism', and the title header 'IMPOR T ANT'). A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central equalizer theorem appears sound, and I do not see a circularity problem. My main concern is that Section 7's theory of submersions hinges on an unproved theorem attributed to private communication; that is a correctness-risk issue rather than a mere presentation issue. I would suggest asking the authors to supply a proof or a public reference for Theorem 7.2.2, and to tighten Definition 7.1.1, before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a serious, mostly sound paper. Theorem 4.2.1—that for a 0-carrable and p-carrable map f, the relative tangent bundle T_{X/Y} sits as the kernel of the horizontal descent θ_f in DBun(X)—is the real contribution, and the proof checks out. The distinction between T-immersions and T-unramified maps is genuinely new and pays off: they coincide in Rosický categories but not in general, with CMon providing a concrete counterexample.\n\nWhat the paper does well: it builds a coherent framework—carrability, horizontal descent, the relative cotangent sequence—and then computes what each class of morphisms is in smooth manifolds, affine schemes, schemes, commutative algebras, and Cartesian differential categories. The Figure 1 table is useful. The authors are honest about what depends on work not included here: Theorem 7.2.2 (submersion iff horizontal descent is T-epic) is attributed to Cruttwell and Lanfranchi and not proved in this paper, and a few scheme calculations point to forthcoming work. That is a genuine soft spot for the submersion section, but it is not load-bearing for the main equalizer theorem.\n\nThe carrability hypotheses are not hidden: p-carrable and 0-carrable are exactly the pullback existence conditions needed to define the objects in the statement, and the paper checks them in each example. The proof of 4.2.1 is detailed; the only issue I see is a small typo in the linearity argument (q∘0_X should be 0_X∘q), which does not affect the argument.\n\nWho is this for: anyone working in tangent categories, categorical differential geometry, or the interface between algebraic and differential geometry. It will likely become the standard reference for these morphism classes. The submersion part should be tightened—either include the proof of 7.2.2 or clearly mark it as dependent on [CL25a]—but the paper as a whole deserves a serious referee. I would send it to review.","headline":"A genuine contribution to tangent categories: the relative cotangent sequence theorem is correct and well proved, though the submersion section leans on an unproved private communication.","tokens_in":60109,"tokens_out":2442,"would_cite":true,"duration_ms":28782,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18F40","13N99","14B10","53B99","53C99","57R99"],"pacs":[],"model":"deepseek-v4-flash","headline":"In a tangent category, whenever the needed pullbacks exist, a map's relative tangent bundle is the kernel of its horizontal descent—the relative cotangent sequence holds in full generality.","keywords":["tangent category","horizontal descent","relative cotangent sequence","differential bundle","immersion","submersion","unramified morphism","étale morphism"],"falsifier":"In the tangent category of commutative R-algebras, take any map $f:A\\to B$ and compute the equalizer of $\\theta_f(a+a'\\epsilon)=(a,f(a'))$ and $f^*(0_Y)\\circ p_X$ in $\\mathrm{DBun}(A)$; Theorem 4.2.1 says the result is $A\\ltimes \\mathrm{Ker}(f)$. Finding that this equalizer differs from $A\\ltimes \\mathrm{Ker}(f)$ for some rig $R$, or finding any 0- and p-carrable map in any tangent category whose $TX/Y$ is not the kernel of $\\theta_f$, would falsify the central claim.","tokens_in":59217,"feed_emoji":"📐","tokens_out":16903,"duration_ms":176089,"temperature":0.7,"pith_summary":"The paper's aim is to give tangent categories—categories equipped with an abstract tangent bundle functor—a uniform way to say what an immersion, a submersion, a local diffeomorphism, and an unramified morphism are, and to show that in smooth manifolds, schemes, commutative algebras, and Cartesian differential categories (categories with a derivative combinator) these tangent-categorical notions agree with the classical ones. Its central result is a relative cotangent sequence theorem: if a morphism $f:X\\to Y$ admits the two pullbacks needed to form $TX/Y$ and $f^*(TY)$, and all powers of the tangent functor preserve them, then the relative tangent bundle $TX/Y$ is the kernel of the horizontal descent $\\theta_f$ inside the category $\\mathrm{DBun}(X)$ of differential bundles (the abstract analogue of vector bundles) over $X$. This gives every tangent category a de Rham relative cotangent complex, recovering the scheme-theoretic sequence $f^*\\Omega^1_{Y/S}\\to\\Omega^1_{X/S}\\to\\Omega^1_{X/Y}\\to 0$ and the manifold vertical-bundle sequence as special cases. A further structural finding is that being $T$-unramified is weaker than being a $T$-immersion in general, with equality forcing a tangent category with negatives.","feed_headline":"The relative tangent bundle is always the kernel of horizontal descent","feed_subtitle":"Immersions, submersions, unramified maps, and étale maps get one categorical classification matching geometry and algebra.","key_machinery":"The load-bearing object is the horizontal descent $\\theta_f=\\langle p_X,Tf\\rangle:TX\\to f^*(TY)$, the unique map induced by the pullback that defines the horizontal bundle $f^*(TY)$ of a p-carrable map $f$. The relative tangent bundle $TX/Y$ is defined as the $T$-pullback of the zero section $0_Y$ along $Tf$, which requires 0-carrability. The proof that $TX/Y$ equalizes $\\theta_f$ and $f^*(0_Y)\\circ p_X$ in $\\mathrm{DBun}(X)$ uses the fact that zero sections are monic, the universality of the vertical lift, and the biproduct structure of $\\mathrm{DBun}(X)$; those three facts carry the homological content of the equalizer statement.","core_discovery":"The central discovery is Theorem 4.2.1: for a 0-carrable and p-carrable morphism $f:X\\to Y$ in any tangent category, the diagram $TX/Y\\xrightarrow{\\pi_0} TX\\xrightarrow{\\theta_f} f^*(TY)$ is an equalizer in $\\mathrm{DBun}(X)$, where $\\theta_f=\\langle p_X,Tf\\rangle$ is the horizontal descent and the second map of the parallel pair is $f^*(0_Y)\\circ p_X$. Equivalently, $TX/Y$ is the kernel of the horizontal descent, so the relative cotangent sequence $X\\to TX/Y\\to TX\\to f^*(TY)$ is exact in the sense of differential bundles. The paper uses this to define a de Rham relative cotangent complex in an arbitrary tangent category, and then builds on the horizontal descent to characterize $T$-unramified morphisms, $T$-immersions, $T$-submersions, split $T$-submersions, and $T$-étale morphisms, matching the classical classes in each main example.","pith_inferences":["Editorial extension: the equalizer form of Theorem 4.2.1 suggests defining a derived relative cotangent complex by replacing the kernel $TX/Y$ with a chain object in any tangent category with enough exactness, whereas the paper itself builds the exact, degree-zero sequence.","Editorial extension: the separation of $T$-unramified from $T$-immersion in tangent categories without negatives makes monoid-based categories such as $\\mathrm{CMon}$ the natural place to look for ramification phenomena that rings and manifolds cannot exhibit.","Editorial extension: since carrability is checked example by example rather than derived, a testable criterion would be to show that any map whose tangent bundle projection is a display morphism is automatically 0- and p-carrable, which would let Theorem 4.2.1 apply without per-map hypotheses.","Editorial extension: the paper's announced Zariski-topology project would naturally take monic $T$-étale maps as its open immersions, and Proposition 9.2.4, which classifies monic $T$-étale maps as $T$-monic split $T$-submersions, is exactly the classification such a topology would build on."],"forward_implications":["In the tangent category of schemes over a base $S$, Theorem 4.2.1 reproduces the classical relative cotangent sequence $f^*\\Omega^1_{Y/S}\\to\\Omega^1_{X/S}\\to\\Omega^1_{X/Y}\\to 0$ through the equivalence $\\mathrm{DBun}(X)^{\\mathrm{op}}\\simeq \\mathrm{QCoh}(X)$.","In smooth manifolds, the sequence becomes the fibre-wise exact sequence $0\\to \\mathrm{Ker}(D[f](x))\\to T_xX\\to T_{f(x)}Y$, so the relative tangent bundle is the vertical bundle of the horizontal descent.","In a tangent category with negatives, a p-carrable map is a $T$-immersion if and only if it is $T$-unramified; the two notions separate only without negatives, as the paper shows in $\\mathrm{CMon}$.","A p-carrable and 0-carrable map is a $T$-submersion exactly when its horizontal descent is $T$-epic, and a split $T$-submersion exactly when the horizontal descent has a section; in tangent categories with negatives the section can be chosen linear.","$T$-étale maps are exactly the maps that are both $T$-immersions and split $T$-submersions, and for p-carrable maps this is equivalent to the horizontal descent being an isomorphism."],"supporting_citations":[{"why":"This supplies abstract tangent functors and the equalizer description of the relative tangent bundle.","marker":"[Ros84]"},{"why":"This defines tangent structure and the pullback lemma that makes carrability meaningful.","marker":"[CC14]"},{"why":"This defines differential bundles and the universality of the vertical lift used in Theorem 4.2.1.","marker":"[CC18]"},{"why":"This identifies differential bundles over smooth manifolds with vector bundles, anchoring the manifold examples.","marker":"[Mac21]"},{"why":"This supplies the equivalences $\\mathrm{DBun}(A)\\simeq A\\text{-Mod}$ and $\\mathrm{DBun}(X)^{\\mathrm{op}}\\simeq \\mathrm{QCoh}(X)$ that convert the theorem into scheme statements.","marker":"[CL23]"},{"why":"This is the source of the relative cotangent sequence and tangent scheme in algebraic geometry, which the theorem recovers.","marker":"[GD67]"},{"why":"This supplies the tangent display morphisms and the $T$-étale definitions the paper builds on.","marker":"[CL25b]"}],"fun_headline_variants":["Horizontal descent kernel yields exact relative cotangent","Tangent categories classify morphisms via descent kernel","Relative tangent bundle: kernel of horizontal descent","Unified categorical view of immersions and submersions","Exact relative cotangent sequence from horizontal descent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the map $f$ is both p-carrable and 0-carrable: the pullbacks $f^*(TY)$ and $TX/Y$ must exist and be preserved by all powers of the tangent functor, since without them the horizontal descent and the relative tangent bundle are undefined and the theorem has no content.","fun_headline_variants_meta":{"raw":{"variants":["Horizontal descent kernel yields exact relative cotangent","Tangent categories classify morphisms via descent kernel","Relative tangent bundle: kernel of horizontal descent","Unified categorical view of immersions and submersions","Exact relative cotangent sequence from horizontal descent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000156,"raw_usage":{"total_tokens":1199,"prompt_tokens":907,"completion_tokens":292,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":220}},"tokens_in":523,"tokens_out":292,"duration_ms":4377,"temperature":1.0,"reasoning_tokens":220,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:24:01.100873+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the tangent category of commutative R-algebras, take any map $f:A\\to B$ and compute the equalizer of $\\theta_f(a+a'\\epsilon)=(a,f(a'))$ and $f^*(0_Y)\\circ p_X$ in $\\mathrm{DBun}(A)$; Theorem 4.2.1 says the result is $A\\ltimes \\mathrm{Ker}(f)$. Finding that this equalizer differs from $A\\ltimes \\mathrm{Ker}(f)$ for some rig $R$, or finding any 0- and p-carrable map in any tangent category whose $TX/Y$ is not the kernel of $\\theta_f$, would falsify the central claim.","supporting_citations":[],"review_version":1}