arxiv: 2604.22394 · v2 · submitted 2026-04-24 · 🧮 math.DG · math-ph· math.MP· math.SG

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Multiplicative Ehresmann connections for Lie groupoid fibrations

Matthijs Lau , Ioan M\u{a}rcu\c{t}

Authors on Pith no claims yet

Pith reviewed 2026-05-08 09:50 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MPmath.SG

keywords Lie groupoidsEhresmann connectionsmultiplicative connectionsgroupoid fibrationslocal trivialitysource-proper groupoidsMorita submersionskernel bundle

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The pith

For families of source-proper Lie groupoids, local triviality is equivalent to the existence of complete multiplicative Ehresmann connections.

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

The paper defines multiplicative Ehresmann connections on surjective submersions of Lie groupoids as a structure that extends classical connections on fiber bundles and multiplicative connections on groupoid extensions. It shows that such connections need not exist for arbitrary proper Lie groupoids, but they do exist for Morita submersions, uniform fibrations, locally trivial families, and proper families. The central theorem states that a family of source-proper Lie groupoids is locally trivial if and only if it carries a complete multiplicative Ehresmann connection. Completeness of the connection is controlled by the induced connection on the kernel bundle and, under connectivity assumptions, by the base connection. This equivalence supplies a practical criterion for detecting when a family of groupoids can be locally trivialized.

Core claim

For families of source-proper Lie groupoids the family is locally trivial precisely when it admits a complete multiplicative Ehresmann connection; completeness of any such connection is determined by the connection induced on the kernel bundle together with the base connection once connectivity assumptions are in place.

What carries the argument

Multiplicative Ehresmann connection on a Lie groupoid fibration: a connection on the surjective submersion that is compatible with the groupoid multiplication and preserves the source and target maps.

If this is right

Local triviality of source-proper families can be established by constructing or verifying a single complete multiplicative connection.
Existence of multiplicative Ehresmann connections is guaranteed for Morita submersions and for proper families of Lie groupoids.
Completeness of the connection reduces to checking the kernel-bundle connection and the base connection separately.
General proper Lie groupoids may admit no multiplicative Ehresmann connection at all.

Where Pith is reading between the lines

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

The equivalence supplies a concrete test for local triviality that could be applied to concrete examples such as orbit spaces or moduli spaces of geometric structures.
One could attempt to build explicit trivializations by integrating the complete connection along paths in the base.
The same completeness criterion might extend to other classes of groupoid fibrations once suitable properness or connectivity replacements are identified.

Load-bearing premise

The Lie groupoid fibrations are source-proper or satisfy connectivity conditions that let completeness be read off from the kernel and base connections alone.

What would settle it

An explicit family of source-proper Lie groupoids that is locally trivial yet carries no complete multiplicative Ehresmann connection, or that admits a complete multiplicative connection while failing to be locally trivial.

read the original abstract

We introduce multiplicative Ehresmann connections on surjective submersions of Lie groupoids, extending both the classical notion of Ehresmann connections on fibre bundles and the more recent notion of multiplicative connections on Lie groupoid extensions. We investigate the existence of such connections, showing that, in general, they may fail to exist even for proper Lie groupoids. In contrast, positive results hold for Morita submersions, uniform Lie groupoid fibrations, locally trivial families of Lie groupoids, and proper families of Lie groupoids. Our main results concern completeness. For Lie groupoid fibrations, we prove that the completeness of a multiplicative connection is governed by the induced connection on the kernel bundle and, under connectivity assumptions, by the base connection. For families of source-proper Lie groupoids, we prove the equivalence between local triviality and the existence of complete multiplicative Ehresmann connections.

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.

Desk Editor's Note private letter to a colleague

The paper defines multiplicative Ehresmann connections on Lie groupoid submersions and proves that local triviality of source-proper families is equivalent to the existence of a complete one.

read the letter

The main takeaway is that for families of source-proper Lie groupoids, local triviality is equivalent to admitting a complete multiplicative Ehresmann connection. They introduce the definition by requiring the horizontal distribution on a groupoid submersion to be both Ehresmann (complementary to the vertical bundle) and multiplicative (compatible with the groupoid multiplication). Existence holds for Morita submersions, uniform fibrations, locally trivial families, and proper families, while it can fail for general proper groupoids. Completeness of the connection reduces to the induced kernel connection and, under connectivity, to the base connection, with source-properness ensuring the horizontal flows are globally defined without escape in finite time. The equivalence follows by constructing the trivialization from the complete horizontal lifts in one direction and the product connection from local triviality in the other. This is solid work because it cleanly extends both classical Ehresmann theory and recent multiplicative connections, and the equivalence gives a practical criterion that ties geometric triviality directly to global horizontal structure. The arguments use standard Lie groupoid properties without circularity. The main limitation is the restriction to source-proper families for the equivalence, which is necessary for controlling the flows but narrows the scope. General non-existence is acknowledged, so the positive results are appropriately conditional rather than overstated. This is for specialists in Lie groupoids and differential geometry who work with fibrations, foliations, or related structures in Poisson geometry. A reader already familiar with groupoid connections will pick up usable new tools and criteria. It deserves a serious referee because the results are new, the setup is careful, and the claims rest on verifiable geometric arguments.

Referee Report

2 major / 2 minor

Summary. The paper introduces multiplicative Ehresmann connections on surjective submersions of Lie groupoids, extending classical Ehresmann connections on fibre bundles and multiplicative connections on Lie groupoid extensions. It investigates existence, showing failure in general for proper Lie groupoids but positive results for Morita submersions, uniform Lie groupoid fibrations, locally trivial families, and proper families. Main results address completeness: for Lie groupoid fibrations, completeness of a multiplicative connection is governed by the induced connection on the kernel bundle and (under connectivity assumptions) by the base connection. For families of source-proper Lie groupoids, local triviality is equivalent to the existence of complete multiplicative Ehresmann connections.

Significance. If the results hold, the work supplies a coherent extension of connection theory to Lie groupoid fibrations with direct implications for local triviality questions. The equivalence theorem for source-proper families is the strongest contribution, as it ties a geometric triviality property to the existence of globally defined multiplicative horizontal lifts. The reduction of completeness to kernel and base connections is a useful structural insight. The manuscript draws on standard Lie groupoid properties without introducing free parameters or ad-hoc axioms, and the source-properness hypothesis is explicitly used to control compactness of source fibres and global flow existence.

major comments (2)

[Main results on completeness and equivalence] The equivalence between local triviality and existence of complete multiplicative Ehresmann connections (stated for families of source-proper Lie groupoids) is load-bearing; the argument that a complete multiplicative connection induces a local trivialization via horizontal lifts, while local triviality supplies a product connection that is automatically complete, requires explicit verification that the horizontal flows remain globally defined precisely when source fibres are compact.
[Completeness analysis for Lie groupoid fibrations] The statement that completeness is governed by the induced connection on the kernel bundle (and by the base connection under connectivity assumptions) needs a precise formulation of the connectivity hypotheses; without them the reduction may not hold in full generality for arbitrary Lie groupoid fibrations.

minor comments (2)

Ensure that the distinction between existence results (for Morita submersions, uniform fibrations, etc.) and the completeness results is clearly signposted in the introduction and in the statements of the theorems.
Define the multiplicative property of the connection form and the horizontal distribution in a single preliminary section so that later references to multiplicativity are unambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major points below and will revise the paper accordingly to strengthen the exposition and precision of the main results.

read point-by-point responses

Referee: [Main results on completeness and equivalence] The equivalence between local triviality and existence of complete multiplicative Ehresmann connections (stated for families of source-proper Lie groupoids) is load-bearing; the argument that a complete multiplicative connection induces a local trivialization via horizontal lifts, while local triviality supplies a product connection that is automatically complete, requires explicit verification that the horizontal flows remain globally defined precisely when source fibres are compact.

Authors: We agree that the equivalence theorem is central and that the proof benefits from more explicit verification of global flow existence. In the revised version we will insert a dedicated lemma (in the section on source-proper families) showing that, because the source map is proper, the source fibres are compact; this compactness, together with the smoothness of the horizontal distribution, guarantees that the horizontal vector fields are complete. The local trivialization is then obtained by integrating these complete flows from a local section. The converse direction (local triviality yields a complete product connection) is already direct, but we will add a short remark confirming that the product structure preserves completeness. These additions will be placed immediately before the statement of the equivalence theorem. revision: yes
Referee: [Completeness analysis for Lie groupoid fibrations] The statement that completeness is governed by the induced connection on the kernel bundle (and by the base connection under connectivity assumptions) needs a precise formulation of the connectivity hypotheses; without them the reduction may not hold in full generality for arbitrary Lie groupoid fibrations.

Authors: We thank the referee for this observation. The manuscript already qualifies the reduction to the base connection by “connectivity assumptions,” but we will make the hypotheses fully explicit in the revised statement of the theorem: the base manifold is connected and each fibre of the groupoid fibration is connected. Under these conditions the horizontal lifts on the kernel bundle determine the base connection up to the usual identification, and completeness reduces accordingly. We will also add a brief counter-example remark illustrating why the reduction can fail without connectivity. The revised theorem will appear in the section on completeness for general Lie groupoid fibrations. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation introduces multiplicative Ehresmann connections via new definitions extending classical Ehresmann and multiplicative connections, then proves the central equivalence for source-proper families by explicit constructions: complete connections induce local trivializations via horizontal flows, while local triviality yields product connections that are complete and multiplicative under source-properness. These steps rely on standard Lie groupoid properties (compact source fibers controlling flows) and direct verification rather than self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. The completeness governance by kernel/base connections is an auxiliary result, not the core claim. The paper is self-contained against external benchmarks with no quoted reductions to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard axioms of Lie groupoid theory and differential geometry; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)

standard math Standard properties of Lie groupoids, submersions, and connections from classical differential geometry
The paper extends prior notions without introducing new foundational axioms.

pith-pipeline@v0.9.0 · 5461 in / 1122 out tokens · 53403 ms · 2026-05-08T09:50:15.211672+00:00 · methodology

discussion (0)

Reference graph

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