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arxiv: 2602.02420 · v2 · submitted 2026-02-02 · 🧮 math.DG

The functor between two categories of mathbb{Z}-graded manifolds

Martha Valentina Guarin Escudero , Alexei Kotov This is my paper

Pith reviewed 2026-05-16 08:21 UTC · model grok-4.3

classification 🧮 math.DG
keywords Z-graded manifoldsBatchelor-Gawedzki theoremBorel-Whitney theoremhomogeneity morphismsgraded vector bundlesfunctorformal neighborhoodsdifferential geometry
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The pith

A functor from Z-graded vector bundles to Z-graded manifolds is full and surjective on objects.

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

The paper shows that every finite-dimensional Z-graded manifold arises noncanonically as the formal neighborhood of the zero section in a graded vector bundle equipped with a homogeneity structure. It extends Kotov-Salnikov's graded Borel lemma to a Borel-Whitney theorem that lifts homogeneity morphisms between formal neighborhoods to smooth maps on the bundles. These two results together make the natural functor from the category of graded bundles to the category of graded manifolds surjective on objects and full on morphisms. A reader cares because the construction reduces questions about graded manifolds to questions about graded vector bundles with explicit polynomial filtrations.

Core claim

Every Z-graded manifold over base M is noncanonically isomorphic to the formal neighborhood of the zero section of its canonical Batchelor-Gawedzki bundle with the induced homogeneity structure. The graded Borel-Whitney theorem states that any homogeneity morphism between such formal neighborhoods lifts to a smooth homogeneity map between the underlying graded bundles. Therefore the functor F from the category B_Z of Z-graded vector bundles with homogeneity morphisms to the category Man_Z of Z-graded manifolds is full and surjective on objects.

What carries the argument

The functor F that assigns to each finite-dimensional Z-graded vector bundle the formal neighborhood of its zero section with the induced homogeneity structure.

If this is right

  • Every finite-dimensional Z-graded manifold is locally equivalent to the formal neighborhood of a graded vector bundle.
  • Morphisms between graded manifolds are determined by their restrictions to formal neighborhoods and lift to bundle morphisms.
  • The two natural polynomial filtrations on local models of graded manifolds become equivalent in finite dimensions.
  • Graded manifolds can be constructed explicitly from graded bundles without additional choices beyond the Batchelor-Gawedzki bundle.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Geometric structures defined on graded bundles can be transferred to the corresponding graded manifolds via the functor.
  • The result suggests that classification problems for graded manifolds reduce to classification of graded vector bundles up to homogeneity morphisms.
  • The same lifting technique might extend to infinite-dimensional graded manifolds if the Borel-Whitney statement generalizes.

Load-bearing premise

The graded Borel-Whitney theorem holds in the stated form, so that homogeneity morphisms between formal neighborhoods lift to smooth homogeneity maps on the Batchelor-Gawedzki bundles.

What would settle it

A concrete finite-dimensional Z-graded manifold together with a homogeneity morphism on its formal neighborhood that cannot be lifted to a smooth homogeneity map on the associated graded bundle, or a graded manifold not isomorphic to any formal neighborhood of a graded bundle.

read the original abstract

This paper examines $\mathbb{Z}$-graded manifolds as semiformal homogeneity structures, comparing two polynomial filtrations from their local models. In finite dimensions, these are componentwise equivalent, yielding isomorphic graded completions; generally, one induces a finer topology. By the Batchelor-Gawedzki-type theorem (Kotov--Salnikov), every $\mathbb{Z}$-graded manifold over base $M$ is noncanonically isomorphic to one associated with its canonical $\mathbb{Z}$-graded bundle (Batchelor-Gawedzki bundle). In finite dimensions, this is the formal neighborhood of the zero section with the induced homogeneity structure. Kotov-Salnikov's graded Borel lemma extends weight-$k$ functions from the formal neighborhood to smooth ones of the same weight. Here, this generalizes to a Borel--Whitney theorem: homogeneity morphisms of formal neighborhoods lift to smooth homogeneity maps between Batchelor-Gawedzki bundles. Categorically, let $\mathsf{B}_{\mathbb{Z}}$ be the category of finite-dimensional $\mathbb{Z}$-graded vector bundles with homogeneity morphisms, and $\mathsf{Man}_{\mathbb{Z}}$ the category of finite-dimensional $\mathbb{Z}$-graded manifolds. The functor $\mathsf{F}\colon \mathsf{B}_{\mathbb{Z}} \to \mathsf{Man}_{\mathbb{Z}}$ sends bundles to formal neighborhoods of their zero sections. The graded Batchelor-Gawedzki and Borel-Whitney theorems imply $\mathsf{F}$ is full and surjective on objects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines a functor F from the category B_Z of finite-dimensional Z-graded vector bundles equipped with homogeneity morphisms to the category Man_Z of finite-dimensional Z-graded manifolds. The functor sends each bundle to the formal neighborhood of its zero section with the induced homogeneity structure. Using a graded Batchelor-Gawedzki theorem (from Kotov-Salnikov) for surjectivity on objects and a new Borel-Whitney theorem generalizing the graded Borel lemma for fullness, the paper concludes that F is full and surjective on objects. The two polynomial filtrations on local models are shown to be equivalent in finite dimensions, yielding isomorphic graded completions.

Significance. If the Borel-Whitney lifting holds, the result provides a concrete categorical bridge between Z-graded bundles and Z-graded manifolds, allowing morphisms between graded manifolds to be realized via lifts from formal neighborhoods. This strengthens the Batchelor-Gawedzki correspondence in the Z-graded setting and may simplify computations involving homogeneity structures by reducing them to bundle data, with potential applications in graded differential geometry.

major comments (2)
  1. [Borel-Whitney theorem statement and proof sketch] The Borel-Whitney theorem (generalizing Kotov-Salnikov's graded Borel lemma) is the load-bearing step for fullness of F. The manuscript states that homogeneity morphisms of formal neighborhoods lift to smooth homogeneity maps on Batchelor-Gawedzki bundles, but provides no explicit construction, filtration-preservation argument, or verification that the lift exists when weights fail to align with polynomial filtration degrees; without this, surjectivity on Hom sets cannot be confirmed.
  2. [Functor definition and properties] The claim that F is full relies on the non-canonical isomorphism from the graded Batchelor-Gawedzki theorem. Section discussing the functor properties should clarify how the choice of isomorphism affects the induced maps on morphisms, as different choices could alter whether a given homogeneity morphism in Man_Z lifts uniquely or at all.
minor comments (2)
  1. [Introduction] The abstract and introduction use 'semiformal homogeneity structures' without a precise definition or reference to the local model; adding a short paragraph defining the filtrations explicitly would improve readability.
  2. [Throughout] Notation for the categories is introduced as B_Z and Man_Z but the script font is used inconsistently in later sections; standardize to match the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each major comment below and will incorporate revisions to strengthen the presentation of the Borel-Whitney theorem and the functor properties.

read point-by-point responses

  1. Referee: [Borel-Whitney theorem statement and proof sketch] The Borel-Whitney theorem (generalizing Kotov-Salnikov's graded Borel lemma) is the load-bearing step for fullness of F. The manuscript states that homogeneity morphisms of formal neighborhoods lift to smooth homogeneity maps on Batchelor-Gawedzki bundles, but provides no explicit construction, filtration-preservation argument, or verification that the lift exists when weights fail to align with polynomial filtration degrees; without this, surjectivity on Hom sets cannot be confirmed.

    Authors: We agree that the manuscript currently states the Borel-Whitney theorem without a full explicit construction or detailed verification for all weight alignments. In the revised version we will expand the relevant section to include a complete proof sketch: we first recall the graded Borel lemma of Kotov-Salnikov, then use the equivalence of the two polynomial filtrations (established earlier in the paper for finite dimensions) to construct the lift explicitly via a Whitney-type extension that preserves homogeneity degrees. The argument proceeds by inducting on the filtration levels and verifying that the lift commutes with the homogeneity morphisms even when weights do not coincide with the polynomial degrees, by decomposing into homogeneous components and applying the finite-dimensional equivalence of completions. revision: yes

  2. Referee: [Functor definition and properties] The claim that F is full relies on the non-canonical isomorphism from the graded Batchelor-Gawedzki theorem. Section discussing the functor properties should clarify how the choice of isomorphism affects the induced maps on morphisms, as different choices could alter whether a given homogeneity morphism in Man_Z lifts uniquely or at all.

    Authors: We acknowledge the need for clarification on the non-canonical isomorphism. In the revision we will add a dedicated paragraph in the functor-properties section explaining that, although the Batchelor-Gawedzki isomorphism is not unique, any two such isomorphisms differ by a homogeneity-preserving automorphism of the bundle. Because the Borel-Whitney theorem guarantees a lift for every homogeneity morphism between formal neighborhoods, the existence of the lift in Man_Z is independent of the choice; the induced map on morphisms is well-defined up to this automorphism, which does not affect fullness of F. We will also note that uniqueness of the lift is not claimed, only existence. revision: yes

Circularity Check

0 steps flagged

Minor self-citation on prior Kotov-Salnikov theorems; central implication remains independent

full rationale

The paper defines categories B_Z and Man_Z, constructs functor F sending bundles to formal neighborhoods, and concludes F is full and surjective on objects by invoking the graded Batchelor-Gawedzki theorem (cited as Kotov-Salnikov) together with a new generalization of their graded Borel lemma to a Borel-Whitney lifting theorem for homogeneity morphisms. No equation or definition reduces the target statement to its inputs by construction, no parameter is fitted and renamed as a prediction, and the non-canonical isomorphism is standard rather than smuggled. The self-citation is present but not load-bearing for the logical implication step, which supplies independent categorical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim depends on the Batchelor-Gawedzki-type theorem and the graded Borel lemma from Kotov-Salnikov as background assumptions, plus the new generalization to Borel-Whitney; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Every Z-graded manifold over base M is noncanonically isomorphic to the graded manifold associated with its canonical Z-graded bundle
    Invoked as the Batchelor-Gawedzki-type theorem (Kotov-Salnikov) to identify graded manifolds with formal neighborhoods of zero sections.
  • domain assumption Kotov-Salnikov graded Borel lemma: weight-k functions on the formal neighborhood extend to smooth weight-k functions
    Used as the starting point for the generalization to the Borel-Whitney theorem on morphisms.

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