Connections on a principal Lie groupoid bundle and representations up to homotopy

Naga Arjun S J; Saikat Chatterjee

REVIEW 4 major objections 5 minor 44 references

Connections on a principal Lie groupoid bundle and representations up to homotopy

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read For a principal bundle whose structure object is a Lie groupoid, choosing an Ehresmann connection on the groupoid produces an Atiyah-like short exact sequence of diffeological groupoids, and a suitably equivariant Lie-algebroid-valued…

desk verdict New Atiyah-sequence construction for Lie groupoid bundles, worth refereeing, but the unproved well-definedness of the quotient relations is load-bearing. read the letter →

arxiv 2502.02284 v4 pith:3C5M2FPS submitted 2025-02-04 math.DG math.CT

Saikat Chatterjee , Naga Arjun S J This is my paper · ORCID

classification math.DGmath.CT MSC 53C0522A2255R9918F15

keywords LiegroupoidprincipalbundleEhresmannconnectionrepresentationuptohomotopyAtiyahsequencediffeologicaladjoint1-form

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 proposes a working notion of a connection on a principal bundle whose structure object is a Lie groupoid rather than a Lie group, a setting where no natural adjoint action exists. Its strategy is to choose an auxiliary Ehresmann connection on the Lie groupoid itself, which produces actions up to homotopy on the relevant graded vector bundles. From these actions the paper builds two diffeological groupoids, Ad(P) and At(P), over the base manifold, and proves that they fit into a short exact sequence that mirrors the classical Atiyah sequence of a principal G-bundle. It then defines a connection 1-form on the bundle as an equivariant A-valued form and shows that such a form splits the sequence; when the Lie groupoid connection is Cartan, splitting and connection become equivalent. If the construction is sound, it gives Lie groupoid bundles a genuine differential-geometric infrastructure, with the classical theory recovered when the groupoid is a Lie group.

What carries the argument

The load-bearing object is a representation up to homotopy of a Lie groupoid on a graded vector bundle, together with the basic curvature K^bas_σ(g,h) that measures the failure of the induced quasi actions to compose functorially. The paper takes the quasi actions on A ⊕ a*TX0 and TP ⊕ a*TX0, generated by a chosen Ehresmann connection on X, and kills their homotopy errors by explicit equivalence relations on the fibres. The resulting quotients are the diffeological groupoids Ad(P) and At(P), and the well-definedness of their structure maps and of the fibre-wise graded vector-space structures is what makes the Atiyah sequence in Theorem 1 meaningful.

What would settle it

Take a Lie groupoid with nonzero basic curvature and compute the relation (46) for its unit bundle X1 → X0; exhibit two finite error sums that give the same class under (46) but different classes after applying the target map or after addition via (53). Such a pair would make the target map or the vector-space structure on Conn($π^{{-1}}$(m)) ill-defined and would collapse the exactness of (101).

Watch

Extended reading notes

Core claim

Fix a principal [X : X1 ⇒ X0]-bundle π : P → M and an Ehresmann connection H ⊂ TX1 on the Lie groupoid X. The paper's central discovery is that H turns the fundamental vector-field map dδ and the tangent projection dπ into morphisms of quasi-equivariant bundles up to homotopy, so that after quotienting by the homotopy errors one obtains a length-three sequence of diffeological groupoids over M, 0 → Ad(P) → At(P) → Act(π*TM) → 0, whose fibre-wise connected components form graded vector spaces and form a short exact sequence for each m ∈ M (Theorem 1). The paper calls this the Atiyah sequence of the X-bundle. A connection 1-form ω : TP → P ×_{X0} A satisfying equivariance (111) and normalization (112) defines a splitting of this sequence, and conversely a splitting defines such a form whenever the Lie groupoid connection is Cartan (Proposition 21).

Load-bearing premise

The whole construction rests on the equivalence relations generated by the quasi-action errors being compatible with the groupoid structure maps and the fibre-wise vector-space operations; the paper asserts this with a verification rather than proving it, and if the compatibility fails the exact sequence in Theorem 1 is not well-defined.

Editorial extensions

If this is right

For any principal X-bundle and any Lie groupoid connection H, Theorem 1 provides an Atiyah sequence of diffeological groupoids, so every such bundle carries a canonical long-exact object over M.
Every connection 1-form in the sense of Definition 20 splits this sequence, so connections on Lie groupoid bundles can be studied through splittings just as in classical principal bundle theory.
When the Lie groupoid connection is Cartan, splittings and connections are in one-to-one correspondence, so the classical bijection between connections and horizontal distributions survives in the Cartan case.
For a Lie group G viewed as the groupoid G ⇒ ∗, the construction reproduces the classical adjoint bundle, Atiyah bundle, and Atiyah sequence, so the classical theory is a special case.
Any two connection pairs (ω, H) and (ω′, H′) are isomorphic objects in the category CONNECTION(X, P → M), meaning the choice of Lie groupoid connection does not create inequivalent connection theories.

Reading between the lines

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

An implicit consequence is a route to characteristic classes for Lie groupoid bundles: once a connection splits the Atiyah sequence, Chern–Weil-type constructions become available, something the paper mentions as motivation but does not carry out.
A natural test is whether parallel transport defined by such a connection is well-defined on homotopy classes of X-arrows, which would connect the construction to holonomy and monodromy for foliations and differentiable stacks.
The groupoids Ad(P) and At(P) may serve as stacky quotients; a testable extension is to show that for Morita-equivalent Lie groupoids and equivalent bundles, the resulting Atiyah sequences are Morita-equivalent, which would let the construction descend to differentiable stacks.
Since a connection's horizontal distribution is stable under the quasi action, the framework likely specializes to partial connections along foliations of M, matching the F-partial connections of [30].

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: this paper actually does something new—it builds an Atiyah-like short exact sequence of diffeological groupoids for a principal Lie groupoid bundle, using Abad–Crainic's representations up to homotopy. The architecture is plausible and the goal is worthwhile. But the construction depends on a quotient by quasi-action errors, and the paper never verifies that the equivalence relation is compatible with the structure maps or the fiberwise addition. That is a load-bearing missing proof, not a typo.

What deserves credit: the definitions of Ad(P), At(P), and the category CONNECTION are genuinely new as far as I can tell from the cited literature. The idea of using an Ehresmann connection on the Lie groupoid to induce quasi-actions on P ×_X0 A and TP, then killing the errors by quotienting, is natural and well motivated. The appendix contains real work—Proposition 9's compatibility condition is checked in detail, and Proposition 14's equivariance is proved at length. The authors are also careful to state that the splitting–connection correspondence is one-way unless the Lie groupoid connection is Cartan (Prop 21), so there is no overclaiming there.

The soft spot is exactly what the stress-test note says. In Section 5.1.2 the groupoid structure on Ad(P) is defined after quotienting by (46), and the text says 'one verifies' that the structure maps are well-defined. The addition on connected components (53) is likewise asserted. Section 5.2.2 and 5.2.4 do the same for At(P). These checks are not cosmetic. The target map uses the quasi-action λ_{g^{-1}} on representatives, so it only descends if the relation is closed under that action; addition only descends if the error subspace is a subgroup/submodule. If either fails, Theorem 1's sequence isn't even defined. The Cartan case avoids the issue because the errors vanish, but Theorem 1 is stated for an arbitrary Ehresmann connection.

I don't see circularity. The paper uses [1] as an external tool and doesn't feed its own conclusions back into assumptions. The self-citations are to adjacent settings (principal 2-bundles, Atiyah sequences for Lie group bundles), not to this result.

Who should read it: people working on Lie groupoid geometry, differentiable stacks, and foliations. A careful referee could either fill the missing verifications or find a counterexample; either way referee time is justified. I would not cite it in my own work until the well-definedness is written out, but I would send it to a serious referee.

Referee Report

4 major / 5 minor

Summary. The paper develops a connection theory for principal bundles over a Lie groupoid [X: X1 ⇒ X0]. Fixing an Ehresmann connection H on the groupoid, the authors use representations up to homotopy to define quasi actions of X on P×_{X0}A⊕a^*TX0 and on TP⊕a^*TX0. They then quotient by the errors of these quasi actions to construct diffeological groupoids Ad(P), At(P), and Act(π^*TM), and prove an Atiyah-type short exact sequence (Theorem 1, eqs. (100)–(101)). A connection 1-form ω on P is defined by equivariance and normalization conditions (Definition 20), shown to be equivalent to a horizontal subbundle (Theorem 2), and shown to split the Atiyah sequence, with a converse in the Cartan case (Proposition 21). The paper also introduces a category of connection pairs and proves any two such pairs are isomorphic (Proposition 20).

Significance. If the constructions are made fully rigorous, the paper would provide a natural Atiyah-sequence analogue for principal Lie groupoid bundles, connecting the classical Atiyah sequence to representations up to homotopy and to diffeological groupoids. The framework is novel and potentially useful for differentiable stacks, foliations, and higher gauge theory. The exposition is detailed, with many examples, explicit formulas, and an appendix of algebraic computations; the paper also correctly identifies the need to work with diffeological rather than Lie groupoids. However, the main claims currently rest on several unproved well-definedness and exactness assertions, so the significance is conditional on completing those verifications.

major comments (4)

[§5.1.2 and §5.2.2, eqs. (46), (48), (51), (53), (72)] The equivalence relation (46) is defined by requiring α−α′ and v−v′ to be finite sums of quasi-action errors, and the analogous relation is used for At(P). The source/target maps (48), composition, and the fiberwise vector-space operations (51), (53), (72) are asserted to be well-defined with only 'one verifies' or 'not difficult to verify'. This is not a cosmetic gap. For example, for the target map to descend one must show that replacing (α,v) by an equivalent pair changes (λ_{g^{-1}}α, λ_{g^{-1}}v) by error sums based at pg; this depends on identities (26)–(29) and on the structure of the relation. Similarly, the additions in (53) and (72) must be independent of the connecting arrow g and of representatives. If any of these compatibility checks fail, Ad(P) and At(P) are not groupoids, the connected components do not carry the claimed vector-space structure, and Theorem 1 is not defined. These checks are load-bearing and must be supplied in full.
[§5.4, Theorem 1, eq. (101)] The exactness of the sequence of linear maps on connected components is introduced with 'A direct verification confirms' before Theorem 1. The proof must establish injectivity of the induced ̅dδ, equality im(̅dδ)=ker(̅dπ), surjectivity of ̅dπ, and the linearity of ̅dπ—the last of which the text explicitly skips ('We skip the verification of the linearity of the map ̅dπ'). These are not routine formalities: kernels in quotient groupoids depend on the relations (46) and (70), and surjectivity of dπ only gives a set-level statement. Since exactness in (101) is the central claim of Theorem 1, this verification must appear rather than be left to the reader.
[Appendix A, eqs. (135)–(138)] In the verification of the compatibility condition (69) for Proposition 9, the first displayed computation is labelled τ^1_k Υ(g,h)(v). But τ^1_k acts on a^*TX0 while Υ(g,h)(v) is a tangent vector in T_{pgh}P; the manipulations displayed are those of the tangent quasi action τ^0_k. As written, eq. (135) is not well-formed, and the proof that the four terms in (69) cancel is therefore incomplete. Please correct the label and spell out explicitly how the terms combine with identity (29) to give zero.
[§6.2, Definition 17 and Proposition 21] The passage from a connection form ω to a quasi-connection F requires proving that F0([p,u,v])=[p,ω_p(u),v] is well-defined on the quotient (70). When u is replaced by u plus an error term, one must use Proposition 18 (eq. (113)) to identify the image in Ad(P); the text only says 'It is not difficult to verify'. Conversely, in Proposition 21 the Cartan hypothesis makes the object spaces vector bundles, but one still needs to prove that F0 is additive and scalar-linear on tangent fibres, not merely equivariant and normalized on fundamental vector fields. The current proof does not supply these linearity checks. Since Definition 17 and Proposition 21 are the bridge between the connection theory and the Atiyah sequence, these arguments must be written out.

minor comments (5)

[§5.2.2, just after eq. (70)] The relation defining At(P) is written as 'u−v and u′−v′ can be written as the sum of errors'; it should read 'u−u′ and v−v′ can be written as finite sums of errors'. As typeset, the relation compares different components.
[§5.2.2, eq. (70)] The notation Obj(At(P))=TP/∼ is slightly abusive: the relation is defined on TP⊕a^*TX0, so the object space should be denoted (TP⊕a^*TX0)/∼.
[§6, Definition 17(1)] The condition 'F∘̅dδ = Id_{At(P)}' has mismatched types: F:At(P)→Ad(P) and ̅dδ:Ad(P)→At(P), so the identity should be on Ad(P). The later text and Proposition 21 confirm that the intended condition is F∘̅dδ = Id_{Ad(P)}.
[§5.3.3 and §5.4] The degree grading on the 'graded vector spaces' in (101) is never made explicit for Ad(P) and At(P); the reader must infer the degrees from the representation-up-to-homotopy convention in Proposition 4. A sentence fixing the degrees of A and TX0 would remove ambiguity.
[Throughout] The phrases 'one verifies', 'not difficult to verify', and 'direct verification' occur at several load-bearing points (§5.1.4, §5.2.4, §5.4, §6.2). Even if the major comments are addressed, it would improve the paper to collect the quotient-compatibility and linearity checks in a systematic lemma or appendix.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the construction is self-contained modulo external results from Abad-Crainic, and the main sequence is not fed back into its own assumptions.

full rationale

The paper's central derivation chain is not circular. The quasi-actions and the representation up to homotopy are imported from Abad and Crainic [1], an external source, not from the authors' own prior work. The authors' self-citations [8], [12], and [14] appear as context, comparisons, or examples (e.g., Example 11) and do not carry the proof of Theorem 1. The Atiyah sequence is obtained by first constructing the diffeological groupoids Ad(P) and At(P) from explicit quotient relations (46) and (70), then verifying that dδ and dπ descend to functors; the exactness on connected components follows from the fiberwise injectivity and surjectivity of the fundamental vector-field map and the projection, not from a parameter fitted to the sequence. The connection 1-form in Definition 20 is not defined as a splitting of the sequence; Proposition 21 proves the equivalence with quasi-connections under the Cartan assumption by an argument rather than by renaming. The main weakness is a proof gap: the well-definedness of the quotient relations and of the vector-space operations on connected components is asserted with phrases such as 'one verifies' and 'not difficult to verify' rather than proved. That is a correctness risk, not circularity, because the compatibility conditions in equations (26) through (29) are stated independently and are not assumed by defining the groupoid. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical free parameters or invented physical entities. The framework rests on existing Lie groupoid technology, chiefly Abad-Crainic representations up to homotopy and Moerdijk-Mrcun results on Lie groupoids. The main new premise is the error-quotient compatibility used to build Ad(P) and At(P).

assumptions (4)

domain assumption Every Lie groupoid admits an Ehresmann connection (Lemma 2.9 of [1]).
Invoked in Remark 8 and used throughout to choose H on X; without this existence, the quasi-actions and the constructions of Ad(P) and At(P) are undefined.
standard math The Abad-Crainic characterization of representations up to homotopy via operators (R0, R1, R2), including the basic curvature identities in Proposition 2.15 of [1].
Used in Sections 3.2 and 5 to build the adjoint representation on A⊕TX0 and to control the error terms of the quasi-actions.
domain assumption The identity d(1∘t)_{1x}(α) − (di)_{1x}(α) − α = 0 for α ∈ ker(ds)_{1x} (Lemma 1), proved using the theorem of [30] that s^{-1}(x)∩t^{-1}(x) is a Lie group.
Used in Proposition 14 to establish equivariance of the fundamental vector field map dδ; if this identity fails, the functor \bar{dδ} is not well-defined.
ad hoc to paper The quotient by the relation in eq. (46) is compatible with the groupoid structure maps and the fiberwise operations in eqs. (47)-(53), (70)-(72).
This is asserted with 'one verifies' rather than proved. The exactness of Theorem 1 and the vector-space structure on connected components depend directly on this compatibility.

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Pith. "Pith review of Connections on a principal Lie groupoid bundle and representations up to homotopy." pith.science (2026). https://pith.science/paper/3C5M2FPS

@misc{pith2026250202284,
  author       = {Pith},
  title        = {Pith review of: Connections on a principal Lie groupoid bundle and representations up to homotopy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3C5M2FPS}},
  note         = {Machine review of arXiv:2502.02284}
}

abstract

A Lie groupoid principal $\mbbX$ bundle is a surjective submersion $\pi\colon P\to M$ with an action of $\mathbb{X}$ on $P$ with certain additional conditions. This paper offers a suitable definition for the notion of a connection on such bundles. Although every Lie groupoid $\mathbb{X}$ has its associated Lie algebroid $A:=1^*\ker ds\to X_0$, it does not admit a natural action on its Lie algebroid. There is no natural action of $\mathbb{X}$ on $TP$ either. Choosing a connection $\mathbb{H}\subset TX_1$ on the Lie groupoid $\mathbb{X},$ and considering its induced action up to homotopy of $\mathbb{X}$ on graded vector bundle $TX_0\oplus A,$ we prove the existence of a short exact sequence of diffeological groupoids over the discrete category $M$ (with appropriate vector space structures on the fibres) for the $\mbbX$ bundle $\pi\colon P\to M.$ We introduce a notion of connection on $\mbbX$ bundle $\pi\colon P\to M,$ and show that such a connection $\omega$ splits the sequence. Finally, we show that a connection pair $(\omega, \mathbb{H})$ on $\mbbX$ bundle $\pi\colon P\to M$ is isomorphic to any other connection pair.}

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