Weighted VB-Groupoids

• Weighted VB-groupoids are Lie groupoids equipped with a graded homogeneity structure that allows for nonlinear, weight-preserving maps, extending the classical VB-groupoid to higher degrees.
• They integrate both graded bundle and filtration-based approaches, ensuring all structure maps remain compatible with multiplicative weightings and yield a natural layered tower.
• Applications include modeling higher tangent bundles, multiplicative Poisson–Lie groupoids, and advanced cohomology theories, thereby unifying graded geometry with Lie theory.

Searching arXiv for papers on weighted VB-groupoids, weighted Lie groupoids, and related structures. arxiv_search(query="weighted VB-groupoids Lie groupoids graded bundles VB-groupoids", max_results=10, sort_by="relevance") arxiv_search(query="\"VB-structures and generalizations\" (Grabowski et al., 2021) weighted Lie groupoids", max_results=10, sort_by="relevance") Weighted VB-groupoids are Lie groupoid structures endowed with a compatible notion of weight. In the graded-bundle formulation, they arise by replacing the degree-$1$ Euler homogeneity of a vector bundle with a degree-$k$ homogeneity structure and requiring every groupoid structure map to respect that action; the classical VB-groupoid is recovered when $k=1$ (Grabowski et al., 2021). In a second, filtration-based line of work, the same expression is used for VB-groupoids equipped with compatible multiplicative weightings along subgroupoids, placing vector-bundle and groupoid data inside the theory of weighted manifolds and weighted deformation spaces (Hudson, 14 Aug 2025). Across these formulations, the central theme is the same: compatibility between linear or graded fiberwise structure and multiplicative groupoid geometry.

1. Terminological scope and basic definition

In the graded-bundle literature, a weighted Lie groupoid of degree $k$ is a Lie groupoid $\Gamma_k \rightrightarrows B_k$ equipped with a multiplicative homogeneity structure $h_t$ of degree $k$, and VB-groupoids are precisely the degree-$1$ case (Bruce et al., 2015). The paper "VB-structures and generalizations" formulates the same idea as follows: a weighted groupoid of degree $k \ge 1$ is a graded bundle $(F,h)$ of degree $k$0 equipped with a Lie groupoid structure $k$1 such that, for all $k$2, the maps $k$3 are Lie groupoid morphisms (Grabowski et al., 2021). In this sense, a weighted VB-groupoid is a Lie groupoid object in the category of graded bundles, and the familiar VB-groupoid is its regular degree-$k$4 specialization.

Classically, a VB-groupoid is a Lie groupoid object in the category of vector bundles: the arrow and object manifolds are vector bundles over the corresponding base groupoid, and source, target, multiplication, inversion, and unit are linear maps, with multiplication bilinear on composable arrows (Bruce et al., 2015). Bursztyn–Cabrera–del Hoyo reformulated this by observing that a VB-groupoid is equivalently a Lie groupoid equipped with a compatible regular homogeneity structure of degree $k$5; the weighted viewpoint extends this by admitting arbitrary non-negative degrees and polynomial graded transition laws instead of strict linearity (Bruce et al., 2015).

A distinct but related usage appears in the theory of multiplicative weightings along subgroupoids. There, one begins with a weighting of a manifold along a submanifold, understood as a filtration of the sheaf of smooth functions, and then imposes compatibility of that filtration with the groupoid structure maps. In that framework, a “weighted VB-groupoid” means a VB-groupoid together with a compatible linear weighting on the vector-bundle side and a multiplicative weighting on the groupoid side (Hudson, 14 Aug 2025). The two usages are not identical: one is based on homogeneity structures on graded bundles, the other on filtrations and weighted deformation spaces.

2. Homogeneity structures, graded coordinates, and local polynomiality

A homogeneity structure on a manifold $k$6 is a smooth action $k$7 of the multiplicative monoid $k$8 such that $k$9, $k=1$0, and $k=1$1 is a smooth submanifold $k=1$2 serving as the base of a graded bundle structure (Grabowski et al., 2021). Locally there exist homogeneous coordinates $k=1$3 adapted to the projection $k=1$4 with $k=1$5, $k=1$6, and

$k=1$7

The associated complete weight vector field is

$k=1$8

A smooth function is homogeneous of weight $k=1$9 precisely when $k$0, equivalently $k$1 for $k$2 (Grabowski et al., 2021).

This local model governs the geometry of weighted VB-groupoids. In homogeneous coordinates on the arrow manifold $k$3, the weight-$k$4 coordinates are unchanged by $k$5, while each positive-weight coordinate scales by its assigned degree. The graded-bundle transition maps are polynomial in the positive-weight coordinates and respect weights, so higher-degree weighted VB-groupoids allow nonlinear but still weight-preserving structure maps (Grabowski et al., 2021). This is the key difference from the classical VB case, where every positive-weight coordinate has degree $k$6 and all admissible coordinate changes are linear.

The same weight mechanism induces a grading on functions, tensors, cochains, and differential forms. For vector fields, $k$7 is equivalent to $k$8; for differential forms, $k$9 (Grabowski et al., 2021). In cohomological applications, this produces $\Gamma_k \rightrightarrows B_k$0-homogeneous subcomplexes on both the groupoid and algebroid side, a feature exploited by the Van Est theory for homogeneous cochains (Cabrera et al., 2016).

3. Compatibility with the groupoid structure

The defining compatibility condition for a weighted groupoid is that the homogeneity action be multiplicative. Writing the groupoid structure maps as

$\Gamma_k \rightrightarrows B_k$1

the requirement is

$\Gamma_k \rightrightarrows B_k$2

$\Gamma_k \rightrightarrows B_k$3

$\Gamma_k \rightrightarrows B_k$4

for all $\Gamma_k \rightrightarrows B_k$5 (Grabowski et al., 2021). These identities express that each $\Gamma_k \rightrightarrows B_k$6 is a Lie groupoid morphism, and they are the direct graded analogue of linearity in a classical VB-groupoid.

In the degree-$\Gamma_k \rightrightarrows B_k$7 case, this condition says exactly that source, target, inversion, and units are fiberwise linear and multiplication is bilinear on composable arrows. In local coordinates $\Gamma_k \rightrightarrows B_k$8 of total degree $\Gamma_k \rightrightarrows B_k$9, the structure maps have fiberwise linear or bilinear form, with coefficients smooth in the base variables (Bruce et al., 2015). For higher degree, the structure maps remain graded morphisms but may be polynomial in higher-weight fiber coordinates.

Several immediate consequences follow. The base $h_t$0 is itself a graded subbundle, and $h_t$1 is a Lie subgroupoid of $h_t$2. Moreover, the projection $h_t$3 is a Lie groupoid morphism (Grabowski et al., 2021). The degree-$h_t$4 truncation thus separates the underlying ungraded groupoid from the higher-weight directions. This underlies the “tower of levels” viewpoint: projecting away coordinates of weight $h_t$5 yields lower-degree weighted groupoids, and at $h_t$6 one recovers a VB-groupoid (Bruce et al., 2015).

The filtration-based theory of multiplicative weightings recasts compatibility differently. A weighting of a Lie groupoid $h_t$7 along a subgroupoid $h_t$8 is multiplicative if $h_t$9 is a weighted submanifold, $k$0 are weighted submersions, $k$1 is a weighted morphism, and $k$2 is a weighted morphism (Hudson, 14 Aug 2025). This criterion is equivalent to a graph characterization involving the graph of multiplication and to the existence of a weighted deformation groupoid $k$3 (Hudson, 14 Aug 2025). The graded-bundle and filtration-based definitions are therefore parallel rather than identical.

4. Structure theory, towers, and Lie-theoretic functoriality

Weighted VB-groupoids possess a canonical layered structure. Any weighted groupoid of degree $k$4 gives rise to reduced graded bundles

$k$5

and every transition map in this tower is a Lie groupoid morphism (Grabowski et al., 2021). The fibers $k$6 are affine with graded-linear linear parts, so the resulting tower behaves as an affine analogue of an exact sequence of graded groupoids (Grabowski et al., 2021).

The weight vector field is itself multiplicative. Since $k$7 and each $k$8 is a groupoid morphism, the one-parameter family $k$9 integrates the weight vector field $1$0, and the flow of $1$1 consists of groupoid automorphisms (Grabowski et al., 2021). This is the graded generalization of the Euler vector field on a VB-groupoid.

The Lie functor is compatible with this structure. If $1$2 is a weighted groupoid of degree $1$3, then $1$4 carries a canonical weighted Lie algebroid structure, with homogeneity $1$5 and each $1$6 a Lie algebroid morphism (Grabowski et al., 2021). The 2015 weighted-groupoid paper states the differentiation theorem in the convention that a weighted groupoid of degree $1$7 differentiates to a weighted Lie algebroid of degree $1$8, with homogeneity $1$9 obtained by tangent lift along the units (Bruce et al., 2015). Both formulations encode the same Lie-theoretic principle: homogeneity differentiates and integrates together with the multiplicative structure.

Conversely, integrable weighted Lie algebroids integrate to weighted Lie groupoids under the usual source simply-connectedness hypothesis (Bruce et al., 2015). In the wide-subgroupoid setting of multiplicative weightings, Lie filtrations of a Lie algebroid classify multiplicative weightings of the integrating groupoid along its units (Hudson, 14 Aug 2025). This provides a structural bridge between graded geometry, filtered geometry, and Lie groupoid integration.

5. Cohomology, homogeneous cochains, and representations up to homotopy

The homogeneous structure on a VB-groupoid refines its differentiable cohomology by weight. If $k \ge 1$0 is a VB-groupoid, then $k \ge 1$1 is a vector bundle over $k \ge 1$2, and one may consider the subspace of $k \ge 1$3-homogeneous cochains

$k \ge 1$4

The projection onto the $k \ge 1$5-homogeneous part is

$k \ge 1$6

and it commutes with the groupoid differential, so homogeneous cochains form a subcomplex (Cabrera et al., 2016). The $k \ge 1$7-homogeneous piece recovers ordinary groupoid cochains on the base groupoid.

The Van Est map preserves this weight decomposition. For a VB-groupoid and its VB-algebroid, $k \ge 1$8, hence the Van Est map restricts to

$k \ge 1$9

with the usual connectivity bounds: if the source fibers are $(F,h)$0-connected, the induced map in cohomology is an isomorphism for $(F,h)$1 and injective for $(F,h)$2 (Cabrera et al., 2016). The weighted Lie algebroid paper states the same principle for weighted Lie groupoids in arbitrary degree, not only the regular degree-$(F,h)$3 case (Bruce et al., 2017).

On the infinitesimal side, weighted Lie algebroids produce canonical modules over their degree-$(F,h)$4 Lie algebroid. If $(F,h)$5 is a weighted Lie algebroid of degree $(F,h)$6, then each homogeneous subcomplex $(F,h)$7, $(F,h)$8, is an $(F,h)$9-module, where $k$00 is the underlying degree-$k$01 Lie algebroid (Bruce et al., 2017). After a non-canonical splitting, these modules correspond to $k$02-term representations up to homotopy. In degree $k$03, this recovers the classical correspondence between VB-algebroids and $k$04-term representations up to homotopy.

A further extension is provided by higher vector bundles. There, VB-groupoids are realized as order-$k$05 simplicial vector bundles with cleavages, and higher or weighted variants are modeled as higher vector bundles with graded cores and weakly flat cleavages; representations up to homotopy correspond to these objects via a relative Dold–Kan–Grothendieck correspondence (Hoyo et al., 2021). In that setting, “weight” is identified with simplicial or normalized degree.

6. Canonical examples, variants, and related structures

Higher tangent lifts are the fundamental examples. If $k$06 is a Lie groupoid, then $k$07 is canonically a weighted groupoid of degree $k$08, with homogeneity induced by rescaling $k$09-jets of curves (Grabowski et al., 2021). In local coordinates $k$10 on $k$11,

$k$12

The degree-$k$13 truncation of these higher tangent groupoids is the tangent VB-groupoid (Bruce et al., 2015).

Other standard examples include the tangent and cotangent VB-groupoids $k$14 and $k$15, weighted pair groupoids, weighted action groupoids, and additive groupoids of graded-linear bundles (Bruce et al., 2015). In the filtration-based theory, weighted tangent and cotangent VB-groupoids are obtained whenever the underlying groupoid carries a multiplicative weighting, and the corresponding weighted normal and deformation constructions remain compatible with the VB structure (Hudson, 14 Aug 2025).

Weighted Poisson–Lie groupoids and weighted Lie bi-algebroids enlarge the scope further. A weighted Poisson–Lie groupoid is a weighted Lie groupoid equipped with a multiplicative Poisson bivector of prescribed homogeneous weight; for degree $k$16, this recovers PVB-groupoids (Bruce et al., 2015). The same framework also yields weighted Courant algebroids and double-weighted structures on higher tangent bundles (Grabowski et al., 2021).

A recurring source of confusion is terminology. In the graded-bundle literature, “weighted Lie groupoid” means a graded bundle in the category of Lie groupoids, and “VB-groupoid” is exactly degree $k$17 (Bruce et al., 2015). This is explicitly distinguished from Mehta’s “graded groupoids” in the category of $k$18-graded supermanifolds (Bruce et al., 2015). In the more recent filtration-based literature, “weighted VB-groupoid” refers instead to a VB-groupoid equipped with multiplicative and linear filtrations compatible with subgroupoids and weighted deformation spaces (Hudson, 14 Aug 2025). The coexistence of these usages reflects two mature but different generalizations of the same underlying principle: the systematic incorporation of weights into multiplicative and vector-bundle groupoid geometry.