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Important Classes of Morphisms and the Relative Cotangent Sequence in Tangent Categories

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.07874 v3 pith:CZHN576F submitted 2025-06-09 math.CT math.AGmath.DG

Jean-Simon Pacaud Lemay , Geoff Vooys This is my paper
classification math.CTmath.AGmath.DG MSC 18F4013N9914B1053B9953C9957R99
keywords tangentcategoryhorizontaldescentrelativecotangentsequencedifferentialbundleimmersionsubmersionunramifiedmorphismétale
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Section 7.2, Theorem 7.2.2] 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.
  2. [Section 7.1, Definition 7.1.1] 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.
minor comments (4)
  1. [Section 4.2, proof of Theorem 4.2.1] 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.
  2. [Section 7.2, Corollary 7.2.3] 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.
  3. [Sections 2.2, 3.2, 8.3] 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.
  4. [Throughout] There are numerous typographical errors (e.g., 'Rosciský', 'manfiodls', 'sumbersion', 'isomoprhism', and the title header 'IMPOR T ANT'). A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: Theorem 4.2.1 is proved from the tangent-category axioms and the stated carrability hypotheses, and the example translations rely on published, proof-carrying prior work.

full rationale

Theorem 4.2.1, the paper's central claim, is derived from the tangent category axioms together with the two stated pullback hypotheses, not from its own conclusion. The relative tangent bundle T(X/Y) is introduced in Definition 4.1.1 as the T-pullback of the zero section 0_Y along Tf, whereas the theorem exhibits T(X/Y) as the equalizer, in DBun(X), of the parallel pair (theta_f, f*(0_Y)∘p_X); these are different universal properties, and the proof of Theorem 4.2.1 genuinely reduces the equalizer condition to the pullback condition (Tf∘k = 0_Y∘f∘q for a bundle map k), using p-carrability to know that f*(TY) is a differential bundle and 0-carrability to know that T(X/Y) and its lift lambda_(X/Y) exist and are T-preserved. Neither carrability assumption follows from the conclusion, and the paper checks them separately in each example (e.g., Example 4.1.7 for SMan). I also checked the surrounding theorems: Theorem 6.2.3 combines Theorem 4.2.1 with the standard Ab-enriched characterization of Lemma 6.2.2, and Proposition 9.1.2 is definitional in the harmless sense that a pullback is a weak-pullback-plus-prepullback. I find no fitted input renamed as a prediction, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in via citation: the horizontal descent theta_f = <p_X, Tf> (Definition 3.2.2) is exhibited from the pullback universal property on the page, and Section 3.1 spells out the differential bundle structure on f*(E) rather than importing it as a black box. The self-citations ([CL23] for the equivalences DBun(A) ≃ A-Mod and DBun(X) ≃ QCoh(X)^op used to translate the abstract theorem into schemes, and [CL24] for the carrable and T-etale definitions) are published, proof-carrying results, so under the review rules they are real evidence and do not raise the circularity score. One item should be flagged for completeness rather than circularity: Theorem 7.2.2 is imported from [CL25a], 'Private Communication, 2025', and the paper says 'we strongly suggest that the reader see the upcoming work of Cruttwell and Lanfranchi for details' (Section 7.2); this leaves the T-submersion/theta_f-T-epic equivalence unproved in the present text, but it is not a self-citation by the present authors and it does not support the central equalizer theorem. A minor typo in the proof of Theorem 4.2.1, where the displayed identity q∘0_X = Tq∘lambda should read 0_X∘q = Tq∘lambda, is a writing slip rather than a circular step.

Assumptions & free parameters 0 free parameters · 6 assumptions · 3 invented entities

The paper introduces no numerical free parameters. Its central claim rests on tangent category axioms plus explicit pullback-existence hypotheses, on a key unproved external theorem about submersions, and on previously established equivalences between differential bundles and modules or quasi-coherent sheaves.

assumptions (6)
  • domain assumption C is a Cockett-Cruttwell tangent category
    The paper works throughout with the tangent category axioms from Definition 2.1.2 and assumes the reader is familiar with them.
  • ad hoc to paper The morphism f is p-carrable and 0-carrable
    These are explicit hypotheses in Theorem 4.2.1 and most later results; they require existence and T-preservation of specific pullbacks.
  • standard math DBun(X) is CMon-enriched with finite biproducts
    Used in Section 4.2 to interpret the relative cotangent sequence as an exact sequence; cited to Lucyshyn-Wright [LW18].
  • domain assumption Rosicky tangent categories have Ab-enriched DBun(X) categories
    Used in Theorem 6.2.3 to identify T-immersions with T-unramified maps in the presence of negatives.
  • ad hoc to paper Theorem 7.2.2 is valid
    The paper states without proof that for p- and 0-carrable maps, being a T-submersion is equivalent to theta_f being T-epic, attributing the result to private communication [CL25a].
  • standard math Equivalences DBun(A) approximating A-Mod and DBun(X) approximating QCoh(X)
    Used to translate tangent-categorical characterizations into module and sheaf statements in examples; cited to [CL23].
invented entities (3)
  • T-immersion
    purpose: Defines immersion-like maps in a tangent category via T-prepullbacks of the projection naturality square.
    A new definition introduced in Section 6; its evidence is internal consistency and verification in SMan, schemes, and CDCs, not an external falsifiable prediction.
  • T-unramified
    purpose: Captures maps whose tangent map has trivial kernel using a T-pullback of the zero naturality square.
    New in Section 4.3; in examples it matches formally unramified maps, but this is checked within the paper.
  • Horizontal descent theta_f
    purpose: A canonical map from TX to f*(TY) used to define submersions, immersions, etale maps, and the relative cotangent sequence.
    It is an abstract redefinition of a classical construction in differential geometry; the paper's examples identify it with known maps, but it has no empirical content outside the theory.

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Cite this review

Pith. "Pith review of Important Classes of Morphisms and the Relative Cotangent Sequence in Tangent Categories." pith.science (2026). https://pith.science/paper/CZHN576F

@misc{pith2026250607874,
  author       = {Pith},
  title        = {Pith review of: Important Classes of Morphisms and the Relative Cotangent Sequence in Tangent Categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CZHN576F}},
  note         = {Machine review of arXiv:2506.07874}
}
read the original abstract

In this paper we provide a deep and systematic study of what it means to be an immersion, a submersion, a local diffeomorphism, and unramified in a tangent category. We also give a systematic study of the ways in which these classes of morphisms interact, their properties, and give very explicit and concrete characterizations of how each class appears in algebraic geometry, differential geometry, algebra, and in Cartesian differential categories. Additionally, we discuss the notion of being carrable with respect to the tangent bundle projection, then use this to define the notion of horizontal descent in a tangent category, which we then use as a key tool to study the aforementioned classes of morphisms. In particular, we use this to define a de Rham relative cotangent complex in an arbitrary tangent category.

Figures

Figures reproduced from arXiv: 2506.07874 by the authors.

Figure 1
Figure 1. The corresponding classes of morphism in each given (family) of tangent categories. T-submersions, T-monic, and T-´etale. In particular, the place where we can really see where working with the horizontal descent directly provides a powerful perspective is when working with T-submersions. While the general definition of a T-submersion is stated in terms of weak pullbacks, we can use the horizontal descent to simply … view at source ↗

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