---
title: Weighted VB-Groupoids
url: https://www.emergentmind.com/topics/weighted-vb-groupoids
type: topic
---

# Weighted VB-Groupoids

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\" 2108.06387 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 \(k=1\) [2108.06387]. 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 [2508.10276]. 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 [1502.06092]. 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 \(k\) equipped with a Lie groupoid structure \(F \rightrightarrows B\) such that, for all \(t \in \mathbb{R}\), the maps \(h_t: F \to F\) are Lie groupoid morphisms [2108.06387]. 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-\(1\) 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 [1502.06092]. 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 \(1\); the weighted viewpoint extends this by admitting arbitrary non-negative degrees and polynomial graded transition laws instead of strict linearity [1502.06092].

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 [2508.10276]. 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 \(F\) is a smooth action \(h: \mathbb{R} \times F \to F\) of the multiplicative monoid \((\mathbb{R},\cdot)\) such that \(h_{ts}=h_t \circ h_s\), \(h_1=\mathrm{id}_F\), and \(h_0(F)\) is a smooth submanifold \(M \subset F\) serving as the base of a graded bundle structure [2108.06387]. Locally there exist homogeneous coordinates \((x^A,y^i)\) adapted to the projection \(h_0:F \to M\) with \(\deg(x^A)=0\), \(\deg(y^i)=w_i>0\), and
\[
h_t(x^A,y^i)=(x^A,t^{w_i}y^i).
\]
The associated complete weight vector field is
\[
V_F=\sum_i w_i y^i \frac{\partial}{\partial y^i}.
\]
A smooth function is homogeneous of weight \(w\) precisely when \(L_{V_F}(f)=wf\), equivalently \(f \circ h_t=t^w f\) for \(t>0\) [2108.06387].

This local model governs the geometry of weighted VB-groupoids. In homogeneous coordinates on the arrow manifold \(F\), the weight-\(0\) coordinates are unchanged by \(h_t\), 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 [2108.06387]. This is the key difference from the classical VB case, where every positive-weight coordinate has degree \(1\) and all admissible coordinate changes are linear.

The same weight mechanism induces a grading on functions, tensors, cochains, and differential forms. For vector fields, \(L_{V_F}(Y)=wY\) is equivalent to \((h_t)_*Y=t^{-w}Y\); for differential forms, \((h_t)^*\omega=t^w\omega\) [2108.06387]. In cohomological applications, this produces \(k\)-homogeneous subcomplexes on both the groupoid and algebroid side, a feature exploited by the Van Est theory for homogeneous cochains [1602.06887].

## 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
\[
s,t:F \to B,\qquad m:F^{(2)} \to F,\qquad i:F \to F,\qquad \epsilon:B \to F,
\]
the requirement is
\[
s(h_t(g))=h_t(s(g)),\qquad t(h_t(g))=h_t(t(g)),
\]
\[
m(h_t(g_1),h_t(g_2))=h_t(m(g_1,g_2)),
\]
\[
i(h_t(g))=h_t(i(g)),\qquad \epsilon(h_t(x))=h_t(\epsilon(x))
\]
for all \(t \in \mathbb{R}\) [2108.06387]. These identities express that each \(h_t\) is a Lie groupoid morphism, and they are the direct graded analogue of linearity in a classical VB-groupoid.

In the degree-\(1\) case, this condition says exactly that source, target, inversion, and units are fiberwise linear and multiplication is bilinear on composable arrows. In local coordinates \((x^i,y^\alpha)\) of total degree \(1\), the structure maps have fiberwise linear or bilinear form, with coefficients smooth in the base variables [1502.06092]. For higher degree, the structure maps remain graded morphisms but may be polynomial in higher-weight fiber coordinates.

Several immediate consequences follow. The base \(B=s(F)=t(F)\) is itself a graded subbundle, and \(M=h_0(F)\) is a Lie subgroupoid of \(F\). Moreover, the projection \(h_0:F \to M\) is a Lie groupoid morphism [2108.06387]. The degree-\(0\) truncation thus separates the underlying ungraded groupoid from the higher-weight directions. This underlies the “tower of levels” viewpoint: projecting away coordinates of weight \(>j\) yields lower-degree weighted groupoids, and at \(j=1\) one recovers a VB-groupoid [1502.06092].

The filtration-based theory of multiplicative weightings recasts compatibility differently. A weighting of a Lie groupoid \(G \rightrightarrows M\) along a subgroupoid \(H \rightrightarrows N\) is multiplicative if \(M \subset G\) is a weighted submanifold, \(s,t\) are weighted submersions, \(m\) is a weighted morphism, and \(inv\) is a weighted morphism [2508.10276]. This criterion is equivalent to a graph characterization involving the graph of multiplication and to the existence of a weighted deformation groupoid \(\mathbb{W}(G,H)\rightrightarrows \mathbb{W}(M,N)\) [2508.10276]. 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 \(k\) gives rise to reduced graded bundles
\[
F_k \to F_{k-1} \to \cdots \to F_1 \to F_0=M,
\]
and every transition map in this tower is a Lie groupoid morphism [2108.06387]. The fibers \(F_i \to F_{i-1}\) are affine with graded-linear linear parts, so the resulting tower behaves as an affine analogue of an exact sequence of graded groupoids [2108.06387].

The weight vector field is itself multiplicative. Since \(h_{ts}=h_t \circ h_s\) and each \(h_t\) is a groupoid morphism, the one-parameter family \(h_{e^\tau}\) integrates the weight vector field \(V_F\), and the flow of \(V_F\) consists of groupoid automorphisms [2108.06387]. This is the graded generalization of the Euler vector field on a VB-groupoid.

The Lie functor is compatible with this structure. If \((G \rightrightarrows B,h)\) is a weighted groupoid of degree \(k\), then \(\mathrm{Lie}(G)\) carries a canonical weighted Lie algebroid structure, with homogeneity \(\mathrm{Lie}(h_t)\) and each \(\mathrm{Lie}(h_t)\) a Lie algebroid morphism [2108.06387]. The 2015 weighted-groupoid paper states the differentiation theorem in the convention that a weighted groupoid of degree \(k\) differentiates to a weighted Lie algebroid of degree \(k+1\), with homogeneity \(\widehat h_t=(h_t)'\) obtained by tangent lift along the units [1502.06092]. 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 [1502.06092]. In the wide-subgroupoid setting of multiplicative weightings, Lie filtrations of a Lie algebroid classify multiplicative weightings of the integrating groupoid along its units [2508.10276]. 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 \(V \rightrightarrows E\) is a VB-groupoid, then \(B_pV\) is a vector bundle over \(B_pG\), and one may consider the subspace of \(k\)-homogeneous cochains
\[
C^p_{k\text{-hom}}(V)=\{f \in C^\infty(B_pV): h_{B_p,\lambda}^*f=\lambda^k f\}.
\]
The projection onto the \(k\)-homogeneous part is
\[
P_{k\text{-hom}}(f)=\frac{1}{k!}\left.\frac{d^k}{d\lambda^k}(h_\lambda^*f)\right|_{\lambda=0},
\]
and it commutes with the groupoid differential, so homogeneous cochains form a subcomplex [1602.06887]. The \(0\)-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, \(\mathrm{VE}\circ P_{k\text{-hom}}=P_{k\text{-hom}}\circ \mathrm{VE}\), hence the Van Est map restricts to
\[
\mathrm{VE}_{k\text{-hom}}: C^\bullet_{k\text{-hom}}(V)\to CE^\bullet_{k\text{-hom}}(b),
\]
with the usual connectivity bounds: if the source fibers are \(p_0\)-connected, the induced map in cohomology is an isomorphism for \(p \le p_0\) and injective for \(p=p_0+1\) [1602.06887]. The weighted Lie algebroid paper states the same principle for weighted Lie groupoids in arbitrary degree, not only the regular degree-\(1\) case [1705.02114].

On the infinitesimal side, weighted Lie algebroids produce canonical modules over their degree-\(0\) Lie algebroid. If \(E \to B\) is a weighted Lie algebroid of degree \(k\), then each homogeneous subcomplex \((\Omega^{(i,\bullet)}(E),d_E)\), \(1 \le i \le k\), is an \(A\)-module, where \(A=h_0(E)\to M\) is the underlying degree-\(0\) Lie algebroid [1705.02114]. After a non-canonical splitting, these modules correspond to \((i+1)\)-term representations up to homotopy. In degree \(1\), this recovers the classical correspondence between VB-algebroids and \(2\)-term representations up to homotopy.

A further extension is provided by higher vector bundles. There, VB-groupoids are realized as order-\(1\) 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 [2109.01062]. 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 \(G \rightrightarrows M\) is a Lie groupoid, then \(T^kG \rightrightarrows T^kM\) is canonically a weighted groupoid of degree \(k\), with homogeneity induced by rescaling \(k\)-jets of curves [2108.06387]. In local coordinates \((x^A;x^A_{(1)},\ldots,x^A_{(k)})\) on \(T^kF\),
\[
h_t(x^A)=x^A,\qquad h_t(x^A_{(w)})=t^w x^A_{(w)}.
\]
The degree-\(1\) truncation of these higher tangent groupoids is the tangent VB-groupoid [1502.06092].

Other standard examples include the tangent and cotangent VB-groupoids \(TG \rightrightarrows TM\) and \(T^*G \rightrightarrows A^*(G)\), weighted pair groupoids, weighted action groupoids, and additive groupoids of graded-linear bundles [1502.06092]. 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 [2508.10276].

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 \(1\), this recovers PVB-groupoids [1502.06092]. The same framework also yields weighted Courant algebroids and double-weighted structures on higher tangent bundles [2108.06387].

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 \(1\) [1502.06092]. This is explicitly distinguished from Mehta’s “graded groupoids” in the category of \(\mathbb{Z}\)-graded supermanifolds [1502.06092]. 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 [2508.10276]. 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.

Source: https://www.emergentmind.com/topics/weighted-vb-groupoids