Multiplicative Ehresmann Connection

• Multiplicative Ehresmann connection is a geometric structure on Lie groupoids where a horizontal distribution complements the vertical bundle and is compatible with groupoid multiplication.
• It establishes connection 1-forms, curvature 2-forms, and a Bianchi identity analogously to principal bundles, ensuring that horizontal lifts and inverses preserve structure.
• Applications span gauge theory and Poisson geometry, linking Lie algebroid frameworks with obstruction theory, Morita invariance, and generalized Yang–Mills models.

A multiplicative Ehresmann connection is an Ehresmann connection on a Lie-groupoid-level submersion that is compatible with the groupoid multiplication. In the foundational setting, one considers a surjective submersion of Lie groupoids $\Phi:G\to H$ covering the identity on the object manifold $M$, or more generally a bundle of ideals $k\subset A$ in the Lie algebroid $A$ of $G\rightrightarrows M$. The connection is given by a horizontal distribution complementary to the relevant vertical directions, and multiplicativity requires that this horizontal distribution be a wide Lie subgroupoid of the tangent groupoid $TG\rightrightarrows TM$. The resulting theory generalizes principal bundle connections, admits connection $1$-forms, curvature $2$-forms, and a Bianchi identity, and has infinitesimal counterparts on Lie algebroids. It was developed systematically for Lie groupoid submersions covering the identity in (Fernandes et al., 2022), and later extended to representation-valued Bott–Shulman–Stasheff and Weil complexes (Grad, 11 Mar 2025), to Lie groupoid fibrations (Lau et al., 24 Apr 2026), and to a generalized Yang–Mills framework (Grad, 10 Jul 2025).

1. Foundational setting and basic definition

Let $G\rightrightarrows M$ be a Lie groupoid with source and target maps $s,t:G\to M$, multiplication $M$0, unit $M$1, and inverse $M$2. Its Lie algebroid is $M$3 with anchor $M$4. A Lie groupoid morphism covering the identity is a smooth functor $M$5 such that $M$6, and the basic theory assumes that $M$7 is a surjective submersion. The vertical subbundle is $M$8, and the kernel subgroupoid is

$M$9

a bundle of Lie groups over $k\subset A$0 (Fernandes et al., 2022).

An Ehresmann connection for $k\subset A$1 is a horizontal distribution $k\subset A$2 such that

$k\subset A$3

and $k\subset A$4 is a vector bundle isomorphism. The connection is multiplicative when $k\subset A$5 is a Lie subgroupoid of the tangent groupoid $k\subset A$6. Equivalently, products of horizontal tangent arrows are horizontal and inverses preserve horizontality (Fernandes et al., 2022).

The same construction admits a more intrinsic formulation in terms of a bundle of ideals $k\subset A$7, where $k\subset A$8 and $k\subset A$9 is invariant under the conjugation representation

$A$0

Left and right translations of $A$1 define the associated “smearing”

$A$2

A bundle of ideals is partially split if there exists a wide Lie subgroupoid $A$3 such that

$A$4

In this language, a multiplicative Ehresmann connection is precisely such a splitting (Fernandes et al., 2022).

Later work generalized the notion from submersions covering the identity to Lie groupoid fibrations $A$5. In that setting, a multiplicative Ehresmann connection is equivalently a horizontal VB-subgroupoid $A$6 complementary to $A$7, a multiplicative horizontal lift

$A$8

or a multiplicative vertical projection $A$9 (Lau et al., 24 Apr 2026).

2. Equivalent formulations and connection forms

For a partially split bundle of ideals $G\rightrightarrows M$0, a multiplicative Ehresmann connection $G\rightrightarrows M$1 is encoded by a $G\rightrightarrows M$2-valued multiplicative connection $G\rightrightarrows M$3-form

$G\rightrightarrows M$4

It is characterized by three properties: its simplicial differential vanishes, $G\rightrightarrows M$5; it annihilates horizontal vectors, $G\rightrightarrows M$6; and it reproduces verticals, $G\rightrightarrows M$7 (Fernandes et al., 2022).

The same datum admits several equivalent descriptions. Complementary VB subgroupoids $G\rightrightarrows M$8 with $G\rightrightarrows M$9 correspond to VB-groupoid morphisms $TG\rightrightarrows TM$0 splitting the natural projection, to $TG\rightrightarrows TM$1-valued multiplicative $TG\rightrightarrows TM$2-forms $TG\rightrightarrows TM$3 with $TG\rightrightarrows TM$4, and to linear, closed, multiplicative $TG\rightrightarrows TM$5-forms on $TG\rightrightarrows TM$6 whose restriction to $TG\rightrightarrows TM$7 is canonical symplectic (Fernandes et al., 2022). This places multiplicative Ehresmann connections inside the broader calculus of VB-groupoids and multiplicative forms.

A frequently used equivalent formula, written for a connection form $TG\rightrightarrows TM$8, is

$TG\rightrightarrows TM$9

for composable tangent vectors $1$0, together with the normalization $1$1 (Grad, 11 Mar 2025). This formulation makes the analogy with principal bundle connections explicit: multiplicativity replaces ordinary equivariance by a groupoid cocycle condition.

The principal-bundle case appears through gauge groupoids. For a principal $1$2-bundle $1$3 with connection $1$4, the gauge groupoid $1$5 has anchor $1$6, and principal bundle connections $1$7 are in $1$8-to-$1$9 correspondence with multiplicative Ehresmann connections $2$0, via

$2$1

This is one of the central reasons the theory is regarded as a genuine generalization of classical connection theory (Fernandes et al., 2022).

3. Curvature, structure equation, and Bianchi identity

A multiplicative Ehresmann connection determines a horizontal projector $2$2 and a linear connection $2$3 on the Lie algebra bundle $2$4. Using the pullback connection $2$5 on $2$6, one defines the horizontal exterior covariant derivative

$2$7

The curvature of the connection is then

$2$8

(Fernandes et al., 2022).

This curvature is multiplicative. For horizontal vector fields $2$9, it satisfies the explicit bracket-defect formula

$G\rightrightarrows M$0

Thus curvature measures precisely the vertical component of the failure of the horizontal distribution to be involutive (Fernandes et al., 2022). In the formulation of (Grad, 11 Mar 2025), $G\rightrightarrows M$1 if and only if the horizontal distribution is involutive.

The theory admits direct analogues of the Cartan structure equation and the Bianchi identity: $G\rightrightarrows M$2 These are groupoid-level analogues of the familiar principal-bundle formulas and are derived from multiplicativity together with the connection identities of the theory (Fernandes et al., 2022).

A major later development is that the same operator $G\rightrightarrows M$3 extends naturally to representation-valued Bott–Shulman–Stasheff and Weil complexes. In that context, the horizontal projection $G\rightrightarrows M$4 and the induced connection define a vertical operator $G\rightrightarrows M$5 that commutes with the simplicial differential $G\rightrightarrows M$6, yielding a curved double complex. This avoids the restrictive invariance conditions required by the naive covariant derivative $G\rightrightarrows M$7 (Grad, 11 Mar 2025).

4. Infinitesimal theory and Lie-algebroid description

For a surjective Lie algebroid morphism $G\rightrightarrows M$8 covering $G\rightrightarrows M$9, with kernel $s,t:G\to M$0 in the short exact sequence

$s,t:G\to M$1

the infinitesimal counterpart of a multiplicative Ehresmann connection is an IM Ehresmann connection. It is a VB-subalgebroid $s,t:G\to M$2 such that

$s,t:G\to M$3

(Fernandes et al., 2022).

Equivalently, an IM Ehresmann connection is a $s,t:G\to M$4-valued IM $s,t:G\to M$5-form, or Spencer operator, $s,t:G\to M$6, whose symbol $s,t:G\to M$7 satisfies $s,t:G\to M$8 and whose operator $s,t:G\to M$9 obeys the IM compatibility equations. One of these is the symbol equation

$M$00

supplemented by bracket and anchor compatibilities (Fernandes et al., 2022).

If $M$01 is target $M$02-connected with Lie algebroid $M$03, multiplicative Ehresmann connections on $M$04 and IM Ehresmann connections on $M$05 correspond $M$06-to-$M$07. Under this correspondence, the groupoid-level connection form $M$08 differentiates to $M$09, and curvature also has an infinitesimal counterpart as an IM $M$10-form (Fernandes et al., 2022).

A useful algebraic encoding is given by coupling data $M$11. Fixing a splitting $M$12 via $M$13, one defines

$M$14

These satisfy structure equations expressing bracket preservation on $M$15, the relation between curvature and the adjoint action of $M$16, and a mixed cocycle identity. Conversely, any coupling $M$17 satisfying those equations determines a Lie algebroid structure on $M$18 (Fernandes et al., 2022).

Related work on multiplicative linear connections in the tangent bundle developed a parallel Lie theory. There, a linear connection $M$19 on $M$20 is multiplicative if and only if its horizontal distribution is a multiplicative subgroupoid of $M$21, equivalently if its torsion is a multiplicative tensor and its geodesic spray is a multiplicative vector field. In the source simply connected case, these multiplicative connections are in bijection with IM connections on $M$22, and their global obstruction is an Atiyah cocycle in a deformation complex (Pugliese et al., 2020). Although this is a distinct framework, it is closely aligned with the infinitesimal philosophy of multiplicative Ehresmann connections.

5. Existence, obstructions, Morita invariance, and completeness

The existence problem has both geometric and cohomological aspects. If $M$23 is partially split, then $M$24 is a locally trivial bundle of Lie algebras, and each isotropy Lie algebra $M$25 splits as a direct sum of ideals,

$M$26

At the groupoid level, for a surjective submersion $M$27 covering $M$28, a multiplicative Ehresmann connection is complete if and only if the kernel $M$29 is a locally trivial bundle of groups (Fernandes et al., 2022).

The theory is Morita invariant. If two Lie groupoids are Morita equivalent, bundles of ideals correspond $M$30-to-$M$31 and partial splittings are preserved. As a consequence, every bundle of ideals in a proper Lie groupoid is partially split. The existence proof uses local slice models, averaging on proper action groupoids, and a $M$32-invariant partition of unity to glue local multiplicative $M$33-forms while preserving the equation $M$34 (Fernandes et al., 2022).

When the kernel $M$35 is abelian, cohomological obstructions appear. The extension $M$36 of $M$37 by $M$38 is classified by a $M$39-cocycle $M$40 with class $M$41, and a kernel-flat partial splitting exists if and only if there is a flat connection $M$42 on $M$43 with $M$44 and $M$45 lies in the image of $M$46. In rank one, the theory gives complete criteria for totally flat, leafwise flat, kernel flat, and principal-type IM Ehresmann connections in terms of $M$47 and $M$48 (Fernandes et al., 2022).

A common misconception is that properness alone guarantees existence in every generalized setting. For extensions or morphisms covering the identity, properness indeed implies partial splitting (Fernandes et al., 2022). For general Lie groupoid fibrations, however, existence may fail even for proper Lie groupoids. The action-morphism examples of (Lau et al., 24 Apr 2026) show that a connected Lie group $M$49 acting nontrivially on $M$50 yields a proper Lie groupoid fibration $M$51 with no multiplicative Ehresmann connection unless the action is trivial. Positive results do hold for Morita fibrations, uniform Lie groupoid fibrations, locally trivial families of Lie groupoids, and proper families of Lie groupoids (Lau et al., 24 Apr 2026).

Completeness in the fibration setting is governed by path lifting. For a multiplicative horizontal lift $M$52, the horizontal lift $M$53 of a path $M$54 is defined by

$M$55

The connection is complete when these lifts exist globally. For Lie groupoid fibrations, completeness is equivalent to completeness of the induced connection on the kernel bundle, and if the kernel is source-connected, it is further equivalent to completeness of the base connection on $M$56. For families of source-proper Lie groupoids, local triviality is equivalent to the existence of complete multiplicative Ehresmann connections (Lau et al., 24 Apr 2026).

6. Examples, analogies, and applications

The theory encompasses a wide range of standard constructions. For a connected Lie group $M$57 with ideal $M$58, a partial splitting exists if and only if

$M$59

as ideals, and the multiplicative Ehresmann connection is $M$60. For a product groupoid $M$61 with $M$62, there is a complete multiplicative connection $M$63. For a bundle of Lie groups $M$64, a multiplicative Ehresmann connection is equivalent to a Cartan connection on $M$65, and for connected fibers existence is equivalent to local triviality. Every transitive groupoid has partially split $M$66, and the gauge-groupoid construction produces the connection explicitly (Fernandes et al., 2022).

Action groupoids and gauge groupoids provide especially transparent models. For an action groupoid $M$67, a $M$68-equivariant splitting $M$69 implies partial splitting, and for proper actions such a splitting can be obtained by averaging (Fernandes et al., 2022). By contrast, in the more general fibration framework, action morphisms illustrate genuine nonexistence phenomena (Lau et al., 24 Apr 2026).

The relation with symplectic and Poisson geometry is structural. A closed multiplicative $M$70-form $M$71 with $M$72 defines a bundle of ideals $M$73, and partial splitting is crucial in coisotropic embeddings and local models in Poisson geometry. In particular, multiplicative Ehresmann connections are necessary in the construction of local models around Poisson submanifolds and in linearization results (Fernandes et al., 2022).

The theory also interacts with higher and representation-valued constructions. The horizontal covariant derivative $M$74 supplies curved double complexes on representation-valued Bott–Shulman–Stasheff and Weil complexes, and on multiplicative forms it is compatible with the van Est map (Grad, 11 Mar 2025). In a further extension, principal bundle Yang–Mills theory is reformulated by replacing principal connections with multiplicative Ehresmann connections; in that setting, the classical Yang–Mills equation is upgraded to a gauge-invariant pair of equations describing longitudinal and transversal dynamics relative to the orbit foliation, with $M$75-bundle gerbes as an example (Grad, 10 Jul 2025).

A plausible implication is that multiplicative Ehresmann connections occupy a unifying position between principal-bundle gauge theory, multiplicative differential geometry on Lie groupoids, and the deformation-theoretic Lie theory of Lie algebroids. The existing literature supports that interpretation through the simultaneous presence of connection forms, curvature identities, Morita invariance, obstruction classes, and integration theorems across these settings (Fernandes et al., 2022).