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arxiv: 1907.05670 · v1 · pith:BODEOC4Nnew · submitted 2019-07-12 · 🧮 math.DG · math.RT· math.SG

Deformations of Vector Bundles over Lie Groupoids

Pier Paolo La Pastina , Luca Vitagliano This is my paper

Pith reviewed 2026-05-24 22:32 UTC · model grok-4.3

classification 🧮 math.DG math.RTmath.SG
keywords VB-groupoidsdeformationscochain complexMorita invariancevan Est theoremLie groupoidsvector bundlesLie algebroids
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The pith

Every VB-groupoid comes equipped with a cochain complex whose cohomology classes parametrize its deformations.

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

The paper establishes a cochain complex attached to any VB-groupoid that governs how the structure can be deformed while remaining a VB-groupoid. This applies uniformly to objects such as Lie group representations and 2-vector spaces, which arise as special cases. The construction is shown to be invariant under Morita equivalence and to satisfy a van Est theorem that relates it to the underlying Lie algebroid. Readers interested in deformation problems in Lie theory would gain a single algebraic tool that replaces case-by-case geometric arguments.

Core claim

To every VB-groupoid the authors attach a cochain complex that controls its deformations. The complex is Morita invariant, meaning equivalent groupoids produce the same deformation theory up to isomorphism, and it admits a van Est map whose image relates the deformations to those of the associated Lie algebroid. The paper illustrates the complex on several families of examples and derives basic properties such as its behavior under direct sums and duals.

What carries the argument

The cochain complex attached to a VB-groupoid, whose cohomology classes correspond to infinitesimal deformations of the vector bundle structure over the groupoid.

If this is right

  • Deformations of VB-groupoids are classified by the cohomology of the attached complex.
  • Morita equivalent VB-groupoids have isomorphic deformation theories.
  • A van Est theorem relates the deformation cohomology of the VB-groupoid to that of its Lie algebroid.
  • The complex behaves functorially under standard operations such as duals and direct sums of VB-groupoids.

Where Pith is reading between the lines

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

  • The same complex may be used to compare deformation problems across different presentations of the same geometric object.
  • Computations in cases where the groupoid is transitive could reduce to ordinary Lie algebra cohomology via the van Est map.
  • The framework supplies a candidate for studying deformations of higher categorical structures that generalize VB-groupoids.

Load-bearing premise

That the cohomology of this single cochain complex classifies all actual deformations of the VB-groupoid rather than only formal or infinitesimal ones.

What would settle it

An explicit VB-groupoid together with a deformation that cannot be recovered from any cohomology class in the attached complex.

read the original abstract

VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids. We attach to every VB-groupoid a cochain complex controlling its deformations and discuss its fundamental features, such as Morita invariance and a van Est theorem. Several examples and applications are given.

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

0 major / 3 minor

Summary. The paper associates to each VB-groupoid (a vector bundle in the category of Lie groupoids) a cochain complex whose cohomology is claimed to control the deformations of the VB-groupoid. It proves that this complex is Morita invariant and satisfies a van Est theorem relating it to the deformation complex of the underlying Lie groupoid. The construction is illustrated with examples including Lie group representations and 2-vector spaces, and applications to deformation theory are discussed.

Significance. If the central construction is correct, the result supplies a deformation-theoretic tool for VB-groupoids that unifies several classical objects and respects Morita equivalence, a key equivalence relation in the category of Lie groupoids. The van Est theorem provides a bridge to ordinary Lie algebroid cohomology, which is likely to be useful for explicit computations. The explicit examples and applications strengthen the case for the framework's utility.

minor comments (3)
  1. §2.3: the definition of the cochain complex is given in terms of the tangent complex of the VB-groupoid; it would help to include a short diagram or explicit formula for the differential in the case of a trivial VB-groupoid to make the construction more immediately verifiable.
  2. §4.2, Theorem 4.5 (van Est): the statement assumes the base manifold is compact; clarify whether the result extends to non-compact bases or whether an additional properness hypothesis is needed.
  3. Notation: the symbol for the VB-groupoid is sometimes overloaded with its underlying groupoid; a consistent distinction (e.g., via boldface or subscript) would improve readability across sections 3 and 5.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of our contributions on the deformation cochain complex for VB-groupoids, its Morita invariance and van Est theorem, and the recommendation of minor revision. We are pleased that the significance for unifying classical objects and providing computational bridges is recognized.

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper defines a cochain complex attached to each VB-groupoid and proves its properties (Morita invariance, van Est theorem) as standard results in Lie groupoid deformation theory. No equations reduce a claimed prediction or theorem to a fitted parameter or prior self-citation by construction. The central claim is a direct construction whose validity rests on explicit differential-geometric definitions rather than on any input that is redefined as output. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be identified from the text.

pith-pipeline@v0.9.0 · 5592 in / 1006 out tokens · 43336 ms · 2026-05-24T22:32:47.178646+00:00 · methodology

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Reference graph

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