Deformations of Vector Bundles over Lie Groupoids

VB-groupoids are vector bundles in the category of Lie groupoids. They encompass several classical objects, including Lie group representations and 2-vector spaces. Moreover, they provide geometric pictures for 2-term representations up to homotopy of Lie groupoids. We attach to every VB-groupoid a cochain complex controlling its deformations and discuss its fundamental features, such as Morita invariance and a van Est theorem. Several examples and applications are given.


Introduction
Lie groupoids are unifying structures in differential geometry.Their theory generalizes that of Lie groups: they have an infinitesimal counterpart, Lie algebroids, and there exists a Lie differentiation functor from Lie groupoids to Lie algebroids.Moreover, Lie groupoids provide a general framework to deal with many geometric situations such as orbifolds, foliations, Lie group actions and the geometry of PDEs.In all these cases, the main difficulty lies in studying spaces that are obtained as quotients of smooth manifolds, but are not smooth: they are rather differentiable stacks [3] and it is well-known that the latter are morally just Lie groupoids up to Morita equivalence.
vector bundles can be also encoded by VB-groupoids: we study the associated deformation complexes in Subsections 2.4 and 2.5 respectively.
We assume that the reader is familiar with the basic notions on Lie groupoids and algebroids.Despite they are not as standard objects as the latter, we will also assume familiarity with VB-algebroids.The unfamiliar reader may refer e.g. to [14,4] for the details, and [16] for our notation/conventions about VB-algebroids.
Before we start, we recall from [13] the concept of homogeneity structure of a vector bundle, a basic tool that will be used throughout the paper.Let E → M be a vector bundle.The monoid R ≥0 of non-negative real numbers λ acts on E by homotheties h λ : E → E (fiber-wise scalar multiplication).The action h : R ≥0 × E → E, e → h λ (e), is called the homogeneity structure of E. Together with the smooth structure, it fully characterizes the vector bundle structure.This implies that every notion that involves the linear structure of E can be expressed in terms of h only: for example, a smooth map between the total spaces of two vector bundles is a vector bundle map if and only if it commutes with the homogeneity structures.
Remark 1.1.2.Notice that we adopt a different convention from [7], where C k def (G) is the space of smooth maps G (k) → T G satisfying properties (1) and (2), to be coherent with [16], where deformations of VB-algebroids were studied.⋄ Remark 1.1.3.We observe, for later use, that conditions (1) and ( 2) can be expressed in the following way.For every k ≥ 1, define the surjective submersions p k : G (k) → G, (g 1 , . . ., g k ) → g 1 , q k : G (k) → M, (g 1 , . . ., g k ) → s(g 1 ). (1.2) Then an element c ∈ C k def (G) is simply a section of the pull-back bundle p * k+1 T G → G (k+1) that is s-projectable, i.e. such that there exists a section s c of q * k T M → G (k) fitting into the following commutative diagram where the right arrow is the projection onto the last k elements.⋄ Recall that there is a short exact sequence of vector bundles over G: Pulling back the sequence via 1 : M → G, we get a short exact sequence of vector bundles over M 0 −→ A −→ 1 * T G −→ T M → 0.
The group of automorphisms of G acts naturally on C def (G) by pullback.Explicitly, if Ψ ∈ Aut(G), denote by ψ the induced automorphism of A: this notation will be used throughout the paper.Then the action is given by (Ψ * c)(g 0 , . . ., g k ) := T Ψ −1 (c(Ψ(g 0 ), . . ., Ψ(g k ))) for c ∈ C k def (G), k ≥ 0, and by Ψ * c := ψ * c for c ∈ C −1 def (G).It is easy to check that this action preserves the differential.Indeed, if α ∈ Γ(A), we have and so If c ∈ C k def (G), k ≥ 0, a direct computation exploiting Equation (1.1) shows that Ψ * commutes with δ.Finally, it is straightforward that the action preserves C def (G).
Recall that a representation of G on a vector bundle E → M is a morphism from G to the general linear groupoid GL(E).The isomorphism E x → E y induced by an arrow g : x → y of G is denoted v → g • v.
Moreover, C 0 (G, E) := Γ(E), and, for ε ∈ C 0 (G, E), δε is defined by The cohomology of C(G, E) is called the (differentiable) cohomology of G with coefficients in E and denoted by H(G, E).
When E is the trivial line bundle with the trivial representation, the above complex is simply called the Lie groupoid complex of G and denoted C(G).Its cohomology is called the Lie groupoid cohomology of G and denoted by H(G).The complex C(G) possesses a canonical DG-algebra structure.The product of if k > 0, l = 0, and by The same formulas define a C(G)-DG-module structure on C def (G) and on C(G, E), for every representation E.
Now we recall from [7] the deformation cohomology of a Lie groupoid in degrees −1 and 0. To do this, we need to recall the canonical representations on the isotropy and the normal bundles first.When the groupoid is regular, i.e. its orbits have all the same dimension, the latter are smooth representations.In general, they are only set-theoretic representations, but we can still define invariant sections.
Let G ⇒ M be a Lie groupoid, A ⇒ M its Lie algebroid.The isotropy bundle of G is defined by i := ker(ρ : A → T M ).
The bracket of A induces a Lie algebra structure on each fiber i x of i, x ∈ M .The Lie algebra i x is actually the Lie algebra of the isotropy group G x of G at x.For every g : x → y in G, there is an obvious conjugation map Ad g : G x → G y .Differentiating at units one obtains the isotropy action of G on i, ad g : i x → i y .
Even if i is not a honest vector bundle, we define sections of i to be and invariant sections Again, even if i is not of constant rank, one can define the differentiable cohomology of G with coefficients in i as in Definition 1.1.6,where a cochain is smooth if it is smooth as an A-valued map.One can prove that the complex obtained in this way is well defined and there is an inclusion of complexes given by where R g denotes right translation by g ∈ G. Now recall that, for any Lie groupoid G ⇒ M , one can construct the tangent prolongation Lie groupoid T G ⇒ T M : its structure maps are the tangent maps to the original ones.A vector field α is a multiplicative vector field.The flow of δα consists of inner automorphisms of G and every δα is called an inner multiplicative vector field.Proposition 1.1.8.
multiplicative vector fields on G inner multiplicative vector fields on G The next step will be the definition of the normal representation.The normal bundle of G is defined by Even if ν is not a smooth vector bundle, we can define its sections by . (1.9) There is a natural action of G on ν, defined as follows.Take an arrow g : x → y in G and v ∈ ν x , then choose a curve g(ǫ) : x(ǫ) → y(ǫ) such that g(0) = g and ẋ(0) represents v; we define ad g (v) = [ ẏ(0)].One can check that this definition does not depend on the choices involved and that the following lemma holds.
Lemma 1.1.9.Let V ∈ X(M ).Then the set-theoretic section of ν is invariant if and only if, for any s-lift X of V and any g : x → y, there exists η(g) ∈ A t(g) such that V t(g) = T t(X g ) + ρ(η(g)). (1.10) Moreover, if Equation (1.10) holds for some s-lift, it holds for all of them.Now notice that every section of ν, as defined in (1.9), induces a set-theoretic section.So it is natural to declare that a section [V ] of ν is invariant if, for some s-lift X of V , there exists a smooth section of t * A → G such that Equation (1.10) holds.But the vector field ) is also a s-lift, so we end up with the following definition.
There is a linear map π : 11) that sends the class of a multiplicative vector field to the class of its projection on M .Lemma 1.1.11(The curvature map).Let [V ] ∈ Γ(ν) inv and X an (s, t)-lift of V .Then δX ∈ C 2 (G, i) and its cohomology class does not depend on the choice of X. Therefore there is an induced linear map (1.12) Finally, we obtain: Proposition 1.1.12.There is an exact sequence This definition encodes the idea of a "smoothly varying" Lie groupoid.If B is an open interval I containing 0, we say that G is a deformation of G 0 ⇒ M 0 and we denote the latter by G ⇒ M .We will often denote by ǫ the canonical coordinate on I. Accordingly, a deformation of G is also denoted by (G ǫ ).The structure maps of G ǫ are denoted s ǫ , t ǫ , 1 ǫ , m ǫ , i ǫ .The division map is denoted mǫ .A deformation (G ǫ ) is called strict if G ǫ ∼ = G as manifolds for all ǫ.This amounts to say that G ∼ = G × I and M ∼ = M × I.In the following, for a strict deformation, we will always assume G = G × I (and M = M × I).A strict deformation is s-constant (resp.t-constant) if s ǫ (resp.t ǫ ) does not depend on ǫ.A deformation which is both s-constant and t-constant is (s, t)-constant.The constant deformation is the one with G ǫ = G as groupoids for all ǫ.
Two deformations (G ǫ ) and (G ′ ǫ ) of G are said to be equivalent if there exists a smooth family of groupoid isomorphisms Ψ ǫ : G ǫ → G ′ ǫ such that Ψ 0 = id G 0 .We say that (G ǫ ) is trivial if it is equivalent to the constant deformation.
Let (G ǫ ) be a strict deformation of the Lie groupoid G ⇒ M .Then it is natural to look at the variation of the structure maps, in particular the multiplication.However, in general, if (g, h) ∈ G (2) , there is no guaranty that g and h are composable also with respect to the groupoid structure G ǫ .We will consider this problem later: now we simply assume that (G ǫ ) is an (s, t)-constant deformation.In this case, it makes sense to consider the tangent vector for any (g, h) ∈ G (2) .It is clear that (1.13) is killed by both T s and T t.This means that it is of the form T R gh (a) with a ∈ A, and moreover a ∈ ker(ρ) = i.Hence we can define a cochain Differentiating the associativity equation , where r is the map (1.8).Differentiating at 0 the identity m ǫ ( mǫ (m 0 (g, h), h), h) = m 0 (g, h), one obtains the following expression for ξ 0 : The last computation suggests how to generalize the procedure.Let (G ǫ ) be an s-constant deformation of G ⇒ M .The deformation cocycle ξ 0 ∈ C 1 def (G) associated to (G ǫ ) is defined by Formula (1.14) (which makes sense also in the present case).
Lemma 1.1.14.ξ 0 is a cocycle and its cohomology class only depends on the equivalence class of the deformation.
Next we interpret ξ 0 in terms of the groupoid G. Proposition 1.1.15.Let G be an s-constant deformation of the Lie groupoid G.Then, if we set ξ = δ( ∂ ∂ǫ ) ∈ C 1 def ( G), we have ξ 0 = ξ| G .Notice that this statement relies heavily on the fact that (G ǫ ) is s-constant: otherwise the vector field ∂ ∂ǫ would not be s-projectable.In the general case, one has to find an analogue of ∂ ∂ǫ : this leads to the concept of transverse vector field.Definition 1.1.16.Let G be a deformation of G.A transverse vector field for G is a vector field X ∈ X( G) which is s-projectable to a vector field V ∈ X( M ) which is, in turn, π-projectable to d dǫ .Proposition 1.1.17.Let G be a deformation of G. Then: (1) there exist transverse vector fields for G; (2) if X is transverse, then δ X, when restricted to G, induces a cocycle ξ 0 ∈ C 1 def (G); (3) the cohomology class of ξ 0 does not depend on the choice of X.
The resulting cohomology class in H 1 def (G) is called the deformation class associated to the deformation G. So, in general it is not possible to find a canonical cocycle.It was possible in the case of an s-constant deformation because there was a canonical choice of a transverse vector field.Notice that from the proposition above it follows directly that the deformation class is also invariant under equivalence of deformations.
Finally, we recall a result about general families of Lie groupoids.Let G ⇒ M π −→ B be a family of Lie groupoids.Then any curve γ : I → B induces a deformation γ * G of Gγ(0) .We have the following such that there exists a bundle map σ c : ∧ k−1 E → T M , the symbol of c, satisfying the following Leibniz rule: 1-derivations are simply derivations and they are of a particular interest.The space of derivations is denoted by D(E).Recall that derivations of E are sections of a Lie algebroid DE ⇒ M , that sits in the following short exact sequence (the Spencer sequence): where σ : DE → T M is the symbol map and plays the role of the anchor.Actually, DE is the Lie algebroid of the general linear groupoid GL(E) and it is called the gauge algebroid of E.
Before going on, we recall its main properties.First of all, there is a canonical isomorphism of Lie algebroids x is uniquely determined by the condition There is an alternative description of derivations of a vector bundle π : E → M .Namely, consider the vector bundle T π : T E → T M .We denote by T E| v the fiber T π −1 (v) over a tangent vector v ∈ T M .Then a derivation δ ∈ D x E (at a point x ∈ M ) with symbol σ δ determines a linear map δ : for all e ∈ E x and ϕ ∈ Γ(E * ), where, in the lhs, ϕ is also interpreted as a fiber-wise linear function on E. The assignment δ → (σ δ , δ) establishes a one-to-one correspondence between the fiber D x E of the gauge algebroid over x, and the space of pairs (v, h) where v ∈ T x M and h : E x → T E| v is a right inverse of the projection T E| v → E x .In other words, DE is the fat algebroid [14] of the VB-algebroid (T E ⇒ E; T M ⇒ M ), and derivations are the same as linear sections of (T E ⇒ E; T M ⇒ M ), or, which is the same, linear vector fields X lin (E) on E: D(E) ∼ = X lin (E) [18].We will sometimes identify δ and the pair (σ δ , δ).
The assignment E → DE is functorial in the following sense.Let , be a regular vector bundle morphism, i.e. φ is an isomorphism on each fiber, so that E N is canonically isomorphic to the pullback bundle f * E. Then there is a pullback map φ * : Γ(E) → Γ(E N ) defined by ) for all ε ∈ Γ(E), and all y ∈ N .One can use this pull-back to define a Lie algebroid morphism Dφ : DE → DF .Specifically, for all δ ∈ D y E N we define Dφ(δ) : Γ(E) → E f (y) by It is then easy to see that Finally, the diagram / / T M is a pull-back diagram, and this induces an isomorphism DE N ∼ = T N × T M DE which is sometimes useful.From now on, we will often identify E N with the pull-back f * E. For more details about the gauge algebroid we refer to [18] (see also [10]).Now, let's go back to the main topic of this subsection and take a Lie algebroid A ⇒ M .We turn C def (A) := D • (A) [1] into a cochain complex, the deformation complex of A, using the Lie bracket for all α 0 , . . ., α k+1 ∈ Γ(A).
The group of Lie algebroid automorphisms of A naturally acts on C def (A) by pullback.
One can check that this action preserves the differential.
Remark 1.1.19.The complex C def (A) can be given a structure of differential graded Lie algebra by introducing the classical Gerstenhaber bracket.However, we will not need this additional structure.The interested reader may refer to [8] for further details.⋄ Finally, take a VB-algebroid (W ⇒ E; A ⇒ M ).Deformations of the VB-algebroid structure are controlled by a subcomplex C def,lin (W ) of C def (W ), the linear deformation complex of W , defined as follows [10,16].Let h λ be the homogeneity structure of the vector bundle W → A. We say that a deformation cochain c ∈ C def (W ) is linear if and only if for every λ > 0. The linear deformation complex consists, by definition, of linear deformation cochains.
Remark 1.1.20.A linear deformation cochain with k entries is completely determined by its action on k linear sections and k − 1 linear sections and a core section and the action of its symbol on k − 1 linear sections and k − 2 linear sections and a core section.See [16] for a proof.⋄ 1.2.VB-groupoids.In this section we recall from [18,15,4] the basic definitions and properties of VB-groupoids that will be useful later.From now on, we will need the notion of a double vector bundle (DVB for short): we refer to [18] for definitions and basic properties and to our previous paper [16] for notations.
Definition 1.2.1.A VB-groupoid is a vector bundle in the category of Lie groupoids, i.e. a diagram where V ⇒ E and G ⇒ M are Lie groupoids, V → G and E → M are vector bundles and all the vector bundle structure maps (addition, multiplication, projection and zero section) are Lie groupoid maps.We denote s, t, 1, m, ĩ the structure maps of V, s, t, 1, m, i the structure maps of G, π : V → G, π : E → M the vector bundle projections and 0 : G → V, 0 : M → E the zero sections.The VB-groupoid (1.17) will be also denoted (V ⇒ E; G ⇒ M ).The groupoid V ⇒ E is called the total groupoid, G ⇒ M is called the base groupoid.We will sometimes say that V is a VB-groupoid over G.
Remark 1.2.2.Actually, some of the conditions in the previous definition are redundant and could be omitted.For details and other equivalent definitions of VB-groupoids, see the discussion in [15].⋄ Remark 1.2.3.The definition of VB-groupoid can be greatly simplified using the concept of homogeneity structure.Indeed, consider a diagram of Lie groupoids and vector bundles like (1.17).From now on, unless otherwise stated, we denote by h the homogeneity structure of V → G.It can be shown that such a diagram is a VB-groupoid if and only if, for every λ > 0, h λ is a Lie groupoid automorphism [4].⋄ The Lie theory of VB-groupoids is studied in [4].Here the authors show that, applying the usual differentation process to the total and the base groupoids of a VB-groupoid, we end up with a VB-algebroid.
From Definition 1.2.1 it follows [17] that the map is a surjective submersion.Hence its kernel is a vector bundle Explicitly, C R is the set of elements of V that project on the units of G and s-project to the zero section of M : this is analogous to the definition of the core of a DVB.The right-core fits in a short exact sequence of vector bundles over G: where j R is defined by j R (c, g) = c • 0g .A splitting of such a sequence always exists and gives a non-canonical decomposition V ∼ = s * E ⊕ t * C R .Additionally, over the submanifold of units of G there is a natural splitting, given by 1 : E → V, and, following [15], we give the following Definition 1.2.4.A right-horizontal lift of the VB-groupoid (1.17) is a splitting h : s * E → V of (1.18) that satisfies h(e, 1 x ) = 1e for all x ∈ M , and e ∈ E x .A right-decomposition of (1.17) is a direct sum decomposition V ∼ = s * E ⊕ t * C R that comes from a right-horizontal lift.
The existence of right-horizontal lifts can be proved by a partitions of unity argument, hence every VB-groupoid admits a (non-canonical) right-decomposition [15].
By exchanging the role of the source and the target, one can similarly define a left-core C L .The analogue short exact sequence of (1.18) is The splittings of (1.19) that restrict to the natural splitting over the units are called lefthorizontal lifts.Moreover, the involution induces an isomorphism of vector bundles between the right-core and the left-core.
In the following, we will always consider the right-core: it will be referred to simply as "core" and denoted C. The core-anchor is the vector bundle map ∂ : C → E defined by ∂c = t(c).Now we review some examples of VB-groupoids that will appear later on in the paper.
Example 1.2.5.Let G ⇒ M be a Lie groupoid and T G ⇒ T M be its tangent (prolongation) groupoid.Then it is easy to check that (1.20) It is easy to prove that / / M is a VB-groupoid.Its core is trivial.Actually every VB-groupoid with trivial core arises in this way, up to a canonical isomorphism.
Let us briefly recall how duality works for VB-groupoids.Let V be a VB-groupoid like in (1.17), with core C, and let V * → G be the dual vector bundle of V → G. Then we can define the dual VB-groupoid as follows.The source and the target š, ť : , while the multiplication is defined by Here and in the following, •, • denotes the duality pairing.For details and proofs see [18].
Example 1.2.7.Let G ⇒ M be a Lie groupoid, and let A ⇒ M be its Lie algebroid.The dual of the tangent VB-groupoid is the cotangent VB-groupoid Finally, we describe the linear complex and the VB-complex of a VB-groupoid (V ⇒ E, G ⇒ M ) [15].
We know that the total groupoid comes with its Lie groupoid complex (C(V), δ).It is easy to check that there is an induced vector bundle structure on V (k) → G (k) , so there is a natural subcomplex C lin (V) of C(V), whose k-cochains are functions on V (k) that are linear over G (k) .Inside C lin (V), there is a distinguished subcomplex C proj (V) consisting of left-projectable linear cochains [15].By definition, a linear cochain The complexes C lin (V) and C proj (V) are called the linear complex and the VB-complex of V, respectively.Their cohomologies are denoted H lin (V) and H proj (V) and called the linear cohomology and the VB-cohomology of V.
Remark 1.2.8.We are adopting the terminology of [9].In [15] and [7], instead, the VB-complex and the VB-cohomology of V are defined to be C proj (V * ) and H proj (V * ), respectively.⋄ Notice that C(G) can be identified with the subcomplex of C(V) of fiberwise constant cochains.Then, one can check that the product (1.7)gives C lin (V) and C proj (V) a C(G)-DGmodule structure.
It turns out that the linear and the VB-cohomology are isomorphic: Lemma 1.2.9.[5, Lemma 3.1] The inclusion C proj (V) ֒→ C lin (V) induces an isomorphism of H(G)-modules in cohomology.
For our purposes, the VB-complex is particularly important because it gives another description of the deformation complex of a Lie groupoid G. Indeed, we have: Proposition 1.2.10.[7, Proposition 3.9] There is an isomorphism of C(G)-DG-modules given by φ(c)(θ 0 , . . ., θ k ) = θ 0 , c(g 0 , . . ., g k ) The linear deformation complex of a VB-groupoid.In this subsection we introduce the main object of this paper: the linear deformation complex of a VB-groupoid, first introduced in [10] (for different purposes from the present ones).The definition is entirely analogous to the one recalled in Subsection 1.1 for VB-algebroids.
Let (V ⇒ E; G ⇒ M ) be a VB-groupoid, and let (W ⇒ E; A ⇒ M ) be its VB-algebroid.By Remark 1.2.3, h λ is a Lie groupoid automorphism for every λ > 0, so it acts on the deformation complex C def (V) of V ⇒ E. We say that a deformation cochain c is linear if for every λ > 0. Hence, linear cochains are those which are invariant under the homogeneity structure.
We know that h * λ commutes with δ, for all λ > 0, so linear deformation cochains form a subcomplex of C def (V).We denote the latter by C def,lin (V) and we call it the linear deformation complex of V. Its cohomology is called the linear deformation cohomology of V and denoted H def,lin (V).Formula (1.7) also shows that The action of h λ on C −1 def (V) coincides, by definition, with the action induced by the homogeneity structure of W → A on Γ(W, E), so C −1 def,lin (V) is simply the space Γ lin (W, E) of linear sections of W → E. For k ≥ 0, Equation (1.24) is equivalent to saying that c : V (k+1) → T V intertwines the homogeneity structures of V (k+1) → G (k+1) and T V → T G. Again by Remark 1.2.3, this means that c is a vector bundle map over some map c : G (k+1) → T G.In this way we recover the definition in [10].
For k ≥ 0, a linear k-cochain c can also be seen as an s-projectable, linear section of the where we denote pk : V (k) → V, and qk : There is another way to describe the linear deformation complex of a VB-groupoid.Let (V ⇒ E; G ⇒ M ) be a VB-groupoid, let C be its core, and let (W ⇒ E; A ⇒ M ) be its VB-algebroid.Consider the cotangent VB-groupoid of V ⇒ E, (T * V ⇒ W * E ; V ⇒ E) (here we denote by W * E → E the dual of a vector bundle W → E).Actually, T * V ⇒ W * E is the top groupoid of another VB-groupoid.To see this, first take the dual of Finally, recall from [18] that there is a canonical isomorphism of both DVBs and Lie groupoids, covering an isomorphism of DVBs.Combining all these maps, we obtain a diagram: The maps B and β are isomorphisms of vector bundles over the identity, so the back face in the diagram (1.26) is also a VB-groupoid.
It follows that inside C(T * V) there are two distinguished subcomplexes, those of cochains that are linear over V and over V * : we denote them by C lin,• (T * V) and C •,lin (T * V) respectively.Moreover, denote C proj,• (T * V) the subcomplex of left-projectable linear cochains over V and define C lin,lin (T * V) We denote their cohomologies H lin,lin (V) and H proj,lin (V) respectively.
From Proposition 1.2.10, there is an isomorphism of C(V)-modules It is easy to check that this isomorphism takes linear deformation cochains to cochains on T * V that are linear over V * .So we get the following Proposition 1.3.1.There is an isomorphism of C(G)-modules C def,lin (V) ∼ = C proj,lin (T * V) [1]. (1.28) For later use, we notice that a "linear version" of Lemma 1.2.9 holds.Namely, we have Lemma 1.3.2.The inclusion C proj,lin (T * V) ֒→ C lin,lin (T * V) induces an isomorphism in cohomology.
Proof.The proof of [5, Lemma 3.1] works identically in our setting without significant modifications.
1.3.1.Deformations of G from linear deformations of V. We have the following and δc projects to δc.It follows that there exists a natural cochain map: (1.29) In degree k = −1, this is simply the projection Γ lin (W, E) → Γ(A) and we have the well-known short exact sequence: where Hom(E, C) is the C ∞ (M )-module of vector bundle morphisms E → C.
We now show that the map (1.29) is surjective for all k ≥ 0. Let c ∈ C k def (G).By Remark 1.1.3,we have a diagram: We can lift s c to a linear section , where the left and the right faces are DVBs and the horizontal arrows form a surjective DVB morphism.Thus we are in the situation of Lemma A.0.1 and we conclude that there exists a linear section c : V (k+1) → p * k+1 T V that projects on c, as desired.Summarizing, there is a canonical short exact sequence of cochain maps: (1.31) Now we compute ker Π.By definition, a k-cochain c : V (k+1) → T V is killed by Π if and only if it takes values in the vertical bundle T πV of π : V → G.It is easy to check that T πV ⇒ T π E is a subgroupoid of T V ⇒ T E.Moreover, T πV ∼ = V × G V and T π E ∼ = E × M E canonically as vector bundles.Under these isomorphisms, T πV is identified with the groupoid V × G V ⇒ E × M E with the component-wise structure maps.We will understand this identification.
It is clear that c(v 0 , . . ., v k ) has v 0 as first component, so one can think of c as a map V (k+1) → V that is linear over p k+1 : G (k+1) → G. Then ker Π is given by elements c ∈ Hom(V (k+1) , p * k+1 V) such that s(c(v 0 , . . ., v k )) does not depend on v 0 for any (v 0 , . . ., v k ) ∈ V (k+1) .Finally, we observe that, on ker Π, the differential is simply given by its core is the zero-vector bundle 0 M → M .We want to show that the linear deformation complex of a trivial-core VB-groupoid has a particularly simple shape.First recall that, in Example 1.2.6, we have observed that the total groupoid of a trivial-core VB-groupoid is canonically isomorphic to the action groupoid G ⋉ E ⇒ E associated to a representation of the base groupoid G ⇒ M on the side bundle E. Therefore, without loss of generality, we will assume that V = G ⋉ E. As a vector bundle over G, it is the pull-back s * E, and we denote it also by E G .
Consider a linear cochain c ∈ C k def,lin (E G ).By definition, it gives a commutative diagram: But, for every k, there is a canonical isomorphism where the fibered product is wrt the projection G (k) → M , (g 1 , . . ., g k ) → s(g k ).We also have that where the fibered product is wrt to T s : T G → T M .So, we get the following alternative description of (1.32): where the vertical arrows, except for the front right one, are projections onto the first factor.

Notice the analogy between (1.38) and (1.1).
Finally, in this case, the projection (1.29) is the map c → c1 .From the condition (TC2), a cochain in the kernel of this projection is equivalent to a map c2 : G (k) → DE such that σ • c2 = 0, so c2 takes values in End E, and it is easy to see that the sequence (1.31) takes the form: .39) where C(G, End E) is the Lie groupoid complex of G with ceofficients in the representation of G on End E induced by that on E. 1.3.3.Low-degree cohomology groups.Next we describe low-degree cohomology groups.The entire discussion of Subsection 1.1 goes through, with minor changes.We report it here for completeness.
Let (V ⇒ E; G ⇒ M ) be a VB-groupoid, and let (W ⇒ E; A ⇒ M ) be its VB-algebroid.Consider the isotropy bundle i of V and its sections.It is natural to define linear sections of i: Γ lin (i) := Γ(i) ∩ Γ lin (W, E) and linear invariant sections: Now we consider the complex C(V, i).It is clear that there is a distinguished subcomplex C lin (V, i), the one of cochains V (k) → W that are linear over some map G (k) → A. From a direct computation it follows that the map (1.8) takes C lin (V, i) to C def,lin (V), so we have a map r : C lin (V, i) ֒→ C def,lin (V).
H 0 def,lin (V) = linear multiplicative vector fields on V inner linear multiplicative vector fields on V .
Proof.Let X ∈ C 0 def,lin (V).Then X is a linear vector field, and from Proposition 1.1.8it follows that X is closed if and only if it is multiplicative, and is exact if and only if is inner multiplicative, as desired.
Recall that also the normal bundle ν of V is defined.Observe that the anchor ρ : W → T E of the Lie algebroid W is a morphism of DVBs, hence it takes linear sections to linear vector fields X lin (E).We set Γ lin (ν) := X lin (E)/ρ(Γ lin (W, E)).Following the discussion in Subsection 1.1, we declare that a section [V ] ∈ Γ lin (V ) is invariant if it possesses an (s, t)-lift X ∈ X lin (V).The space of invariant linear sections is denoted H 0 lin (V, ν) or Γ lin (ν) inv .Observing that the projection on E of a linear multiplicative vector field is linear, we obtain a linear map From Lemma 1.1.11,Proposition 1.1.12and their proofs in [7], the "linear versions" follow immediately.
Lemma 1.3.8.Let [V ] ∈ Γ lin (ν) inv and X ∈ X lin (V) an (s, t)-lift of V. Then δX ∈ C 2 lin (V, i) and its cohomology class does not depend on the choice of X, hence there is an induced linear map K : Γ lin (ν) inv → H 2 lin (V, i).Proposition 1.3.9.There is an exact sequence If B is an open interval I containing 0, the family is said to be a deformation of V 0 and the latter is denoted simply by (V ⇒ E; G ⇒ M ).A deformation of V is also denoted (V ǫ ).
The structure maps of V ǫ are denoted sǫ , tǫ , 1ǫ , mǫ , ĩǫ , the division map is denoted mǫ .Strict, s-constant, t-constant, (s, t)-constant and constant deformations are defined as in the plain groupoid case.
Two deformations (V ǫ ) and (V ′ ǫ ) of V are called equivalent if there exists a smooth family of VB-groupoid isomorphisms Ψ ǫ : V ǫ → V ′ ǫ such that Ψ 0 = id.We say that (V ǫ ) is trivial if it is equivalent to the constant deformation.Proposition 1.3.11((s, t)-constant deformations).Let (V ǫ ) be an (s, t)-constant deformation of the VB-groupoid V. Then formula Proof.We only need to show that u 0 is linear, but this follows from a direct computation exploiting that R h λ v = h λ • R v for all λ, and the linearity property of m.
We pass to s-constant deformations.
Proposition 1.3.12(s-constant deformations).Let (V ǫ ) be an s-constant deformation of the VB-groupoid V. Then defines a cocycle in C 1 def,lin (V) and its cohomology class only depends on the equivalence class of the deformation.
Proof.We know that ξ 0 is a linear cocycle.Moreover, if Ψ ǫ : The next step will be the proof of a linear version of Proposition 1.1.17.Recall from [1] that an Ehresmann connection on a Lie groupoid G ⇒ M is a splitting of the short exact sequence (1.3) that restricts to the canonical splitting (1.4) over M .Notice that an Ehresmann connection on G is exactly the same as a right-horizontal lift of the VB-groupoid T G.In particular, such a connection always exists.Now let (V ⇒ E, G ⇒ M ) be a VB-groupoid.Then there is a diagram of morphisms of DVBs 0 where the top rows are short exact sequences of vector bundles.
Definition 1.3.13.A linear Ehresmann connection on V is a morphism of DVBs such that the maps s * T E → T V and s * T M → T G are Ehresmann connections on V and G, respectively.
Proof.First of all, choose local coordinates on G adapted to the submersion s : G → M .Up to a translation, one can also assume that they are adapted to the immersion 1 : M → G.
Using a right-decomposition V ∼ = s * E ⊕ t * C, we find fiber coordinates on V with analogous properties.Now it is easy to see that linear Ehresmann connections exist locally, and one can conclude with a partition of unity argument.
Proposition 1.3.15.Let Ṽ be a deformation of V. Then: (1) there exist transverse linear vector fields for Ṽ; (2) if X is a transverse linear vector field, then δ X, when restricted to V, induces a cocycle ξ 0 ∈ C 1 def,lin (V); (3) the cohomology class of ξ 0 does not depend on the choice of X. Proof.
(1) Take a vector field Y on M that projects on d dǫ .Choosing a linear connection on Ẽ → M , one can lift it to a linear vector field Ỹ on Ẽ, that obviously projects on d dǫ .Now the choice of a linear Ehresmann connection on V gives a linear vector field X on V that projects on Ỹ , as desired.
(2) Let X be a transverse linear vector field.Then δ X ∈ C 1 def,lin ( Ṽ) and, by Proposition 1.1.17,it restricts to V, so it belongs to C 1 def,lin (V).
(3) This can be proved as in [7,Proposition 5.12].The cohomology class [ξ 0 ] ∈ H 1 def,lin (V) is called the linear deformation class associated to the deformation Ṽ.From the last proposition, it follows directly that this class is also independent of the equivalence class of the deformation.
The first statement is proved as in [7,Proposition 5.15], while the second statement trivially follows from Remark 1.3.16.1.3.5.Deformations of the dual VB-groupoid.We conclude this section noticing that the linear deformation cohomology of a VB-groupoid is canonically isomorphic to that of its dual.
Proof.Using Proposition 1.3.1 and Lemma 1.3.2,we get: For the same reason, H def,lin (V * ) ∼ = H lin,lin (T * V * ) [1].But we have already noticed that T * V ∼ = T * V * as double vector bundles and Lie groupoids, so we obtain (1.42).
1.4.The linearization map.Let (V ⇒ E; G ⇒ M ) be a VB-groupoid.We have shown that deformations of the VB-groupoid structure are controlled by a subcomplex C def,lin (V) of the deformation complex C def (V) of the top Lie groupoid V ⇒ E. In this section, we prove that there is a canonical splitting of the inclusion C def,lin (V) ֒→ C def (V) in the category of cochain complexes, the linearization map.This will imply, in particular, that the inclusion induces an injection in cohomology H def,lin (V) ֒→ H def (V).
The procedure we are going to describe is analogous to the one we used in [16] to define the linearization of sections of a DVB.Following Remark 1.2.3, denote by h the homogeneity structure of V → G.For every λ > 0, h λ is a groupoid automorphism of ), and a direct computation shows that the action of h λ on C k def (V) coincides with that induced by the homogeneity structure of the vector bundle p * k+1 T V → p * k+1 T G on sections of p * k+1 T V → V (k+1) .Therefore, by [ ) is a cochain map that splits the inclusion C def,lin (V) ֒→ C def (V).Hence C def,lin (V) is a direct summand of C def (V) and the inclusion induces an injection in cohomology: Proof.We only need to prove that the linearization map respects the differential.As h λ is an automorphism of V ⇒ E for every λ, we have that h * λ commutes with δ.It is also clear that δ preserves limits, so we compute and we are done.
Remark 1.4.2.Applying the isomorphisms (1.27) and (1.28), we obtain that H proj,lin (T * V) is a direct summand of H proj,• (T * V): it identifies with classes in H proj,• (T * V) which can be represented by cochains that are linear over V * .⋄ Finally, we discuss a first consequence of Theorem 1.4.1.We call C def,lin (V) := C def,lin (V) ∩ C def (V) the linear normalized deformation complex of V.
Applying the linearization map, we get c − c lin = δc ′ lin and c lin ∈ C k def,lin (V), as desired.
Other applications of the linearization map will be considered in the next sections.
1.5.The van Est map.The van Est theorem is a classical result relating the differentiable cohomology of a Lie group and the Chevalley-Eilenberg cohomology of its Lie algebra [11,12].It was later extended to differentiable cohomology [24,6] and deformation cohomology [7] of a Lie groupoid, and to the VB-cohomology of a VB-groupoid [5].In this subsection, we want to prove an analogous theorem for the linear deformation cohomology of a VB-groupoid.
Let G ⇒ M be a Lie groupoid, and let A ⇒ M be its Lie algebroid.The normalized deformation complex of G and the deformation complex of A are intertwined by the van Est map, defined as follows.Given a section α ∈ Γ(A), we define a map R α : if k = 0, and is given by: ).The van Est map VE is a cochain map.Moreover, if G has k-connected s-fibers, it induces an isomorphism in cohomology in all degrees p < k.
We are going to prove an analogous theorem for the linear deformation complex of a VBgroupoid.To do this, we need a simple preliminary lemma.
Let G ⇒ M be a Lie groupoid, and let A ⇒ M be its Lie algebroid.We know that the group Aut(G) of automorphisms of G acts on C def (G).Clearly, it also acts on C def (A), by As expected, we have the following Lemma 1.5.2.The van Est map (1.46) is equivariant with respect to the action of Aut(G).
Proof.We will prove that Using (1.6), we compute: So, from (1.49) we have: Finally, by applying repeatedly formula (1.48) in (1.47), we obtain for every c ∈ C def (G), as desired.
Now we are ready for the main theorem of this section.Notice that the first part of the following statement has been already proved in [10] in the special case where the rank of E is greater than zero.Here we provide an alternative proof exploiting Lemma 1.5.2 which is valid in all cases.
Theorem 1.5.3 (Linear van Est map).Let (V ⇒ E; G ⇒ M ) be a VB-groupoid, and let (W ⇒ E; A ⇒ M ) be its VB-algebroid.Then the van Est map for the Lie groupoid V ⇒ E restricts to a cochain map VE : C def,lin (V) → C def,lin (W ), which we call the linear van Est map.If G has k-connected s-fibers, this map induces an isomorphism in cohomology in all degrees p < k.
Proof.As before, we denote by h the homogeneity structure of V → G. Then the last proposition shows that h * λ (VE(c)) = VE(h * λ c) for every c ∈ C def (V), λ > 0. But the VB-algebroid automorphism corresponding to h λ is exactly the one induced by the homogeneity structure of W → A, so the last equation implies that the van Est map preserves linear cochains.Now we prove the second part of the theorem.First, we have to observe that the van Est map commutes with linearization, i.e.VE(c) lin = VE(c lin ). (1.50) To see this, take c ∈ C k def,lin (V) and w 0 , . . ., w k ∈ Γ(W, E) and compute We would like to swap the limit and the derivatives that appear in definitions (1.44) and (1.45).This is ultimately possible because of smoothness, and we get Finally, suppose that G has k-connected s-fibers.Then V has k-connected s-fibers (they are vector bundles over the s-fibers of G).Take p < k.We want to prove that the induced map VE : def (V).Applying the linearization map, we get c = δγ lin and γlin ∈ C p−1 def,lin (V), i.e.VE is injective in degree p cohomology.To conclude, take Applying again the linearization map and using (1.50), we get c − VE(c lin ) = δγ lin , i.e.VE is also surjective in degree p cohomology, as desired.
1.6.Morita invariance.The notion of Morita equivalence of VB-groupoids first appears in [9].In that reference, the authors prove that the VB-cohomologies of Morita equivalent VBgroupoids are isomorphic.As a corollary, they give a conceptual and very simple proof of the fact, first appeared in [7], that Morita equivalent Lie groupoids have isomorphic deformation cohomologies.This second result means that the deformation cohomology of a Lie groupoid is in fact an invariant of the associated differentiable stack.
In this paragraph, we want to prove an analogous result for the linear deformation cohomology of a VB-groupoid.We start recalling the necessary definitions.Let G 1 ⇒ M 1 and G 2 ⇒ M 2 be Lie groupoids.
Notice that Definition 1.6.1(1) of fully faithful morphism is slightly different from the one in [9], where an additional property is required.However, it is easy to see that for an essentially surjective morphism the two definitions are equivalent.In the literature, Morita equivalence is often expressed in terms of principal bibundles: we refer to [21] for this notion and many more details.Now, let ( [9] if the Lie groupoid morphism Ψ is a Morita map.The VB-groupoids V 1 and V 2 are Morita equivalent if there exist a VB-groupoid W and VB-Morita maps Here are some basic properties of VB-Morita maps. Proposition 1.6.2([9, Corollary 3.7]).Let Ψ : G 1 → G 2 be a Morita map and let V be a VB-groupoid over G 2 .Then the canonical map Ψ * V → V is VB-Morita.
Proposition 1.6.4([9, Corollary 3.9]).A map Ψ : V 1 → V 2 over the identity is VB-Morita if and only if its dual is so.
Morita invariance of the VB-cohomology is expressed by the following theorem.
) is an isomorphism.Now we are ready to prove Morita invariance of the linear deformation cohomology.
Proof.It is enough to show that H proj,lin (T * V 1 ) ∼ = H proj,lin (T * V 2 ).To do this we will use Propositions 1.6.2-1.6.4 and linearization.
Recall that both T * V 1 and T * V 2 have two VB-groupoid structures, as discussed in Subsection 1.3, and observe that Ψ * (T * V 2 ) possesses also two VB-groupoid structures, that fit in the following commuting diagram: We denote by C proj,• (Ψ * (T * V 2 )) the VB-complex of the VB-groupoid upstairs and by C proj,lin (Ψ * (T * V 2 )) its subcomplex of cochains that are linear with respect to the vertical projections.As usual, we denote their cohomologies by H proj,• (Ψ * (T * V 2 )) and H proj,lin (Ψ * (T * V 2 )) respectively.In analogy to Theorem 1.4.1 and Remark 1.4.2, one can prove that there is a linearization map lin : The map Ψ is VB-Morita, so, from Proposition 1.6.3,T Ψ : T V 1 → T V 2 is VB-Morita again.It follows from Propositions 1.6.2 and 1.6.4 that the dual map (T Ψ) * : Ψ * (T * V 2 ) → T * V 1 is VB-Morita as well.Remember that also the canonical map Ψ * (T * V 2 ) → T * V 2 is VB-Morita (Proposition 1.6.2again).As a result, we get isomorphisms in VB-cohomology: (1.51) Now, the maps are also DVB morphisms.This implies, on one hand, that the maps (1.51) preserve linear cohomologies, on the other hand that they commute with the respective linearization maps.
From this last property we deduce, as in Theorem 1.5.3, that the maps (1.51) induce isomorphisms on linear cohomologies, so as desired.

Examples and applications
In this section we provide several examples.Examples in Subsections 2.1, 2.4 and 2.5 parallel the analogous examples in [7], connecting our linear deformation cohomology to known cohomologies, while examples in Subsections 2.2 and 2.3 are specific to VB-groupoids.The infinitesimal counterparts of all these examples were discussed in our previous paper [16].
2.1.VB-groups and their duals.A VB-group is a vector bundle object in the category of Lie groups.In other words, it is a VB-groupoid of the form In particular, H and G are Lie groups.Let C := ker(H → G) be the core of (H ⇒ 0; G ⇒ * ).It easily follows from the definition of VB-groupoid that is the projection onto the first factor.It is then natural to study the relationship between the linear deformation complex of H and the classical complex C(G, End C) (of the Lie group G with coefficients in the representation End C) that controls deformations of the G-module C [22].To do this, we notice that the dual of H is the VB-groupoid i.e. it is the action VB-groupoid associated to the dual representation of G on C * .In particular, it is a trivial-core VB-groupoid, so there is a short exact sequence of cochain complexes: But C(G, End C * ) ∼ = C(G, End C) canonically, so the latter is recovered as the subcomplex of C def,lin (G ⋉ C * ) controlling deformations of the representation C * that fix the Lie group structure on G.Moreover, H def,lin (G ⋉ C * ) ∼ = H def,lin (H), so there is a long exact sequence in cohomology: 2-vector spaces.A 2-vector space is a (Lie) groupoid object in the category of vector spaces.In other words, it is a VB-groupoid of the form In [2] it is proved that, if s and t are the source and the target maps of Notice that C does not act by linear isomorphisms, but by translations.We will identify V 1 with C ⋉ V 0 .Now we compute the linear deformation complex of ( We will understand this isomorphism.Recall also that s : C ⋉ V 0 → V 0 is just the projection onto the second factor.It follows that, for k ≥ 0, C k def,lin (V 1 ) is the set of linear maps C k+1 ⊕ V 0 → C ⊕ V 0 such that the second component does not depend on the first arrow (c 1 , (c 2 +• • •+c k )•v), hence these maps are equivalent to couples of linear maps We will identify a deformation cochain γ with the corresponding pair of linear maps (γ 1 , γ 2 ).
A direct computation shows that the differential is given by the following formulas.For So the linear deformation cohomology of V 1 is: Finally, notice that the VB-algebroid of V 1 is an LA-vector space [16] of the form (V 1 ⇒ V 0 ; 0 ⇒ * ).In [16] we showed that the linear deformation complex of (V 1 ⇒ V 0 ; 0 ⇒ * ) is again (2.2), and it is easy to show, for example in coordinates, that the van Est map is simply the identity.

2.3.
Tangent VB-groupoid.Let G ⇒ M be a Lie groupoid.We want to relate the linear deformation cohomology of T G with the deformation cohomology of G. First recall that there is a projection pr :
A direct computation shows that ι is a cochain map.Finally, we observe that the following diagram commutes: .
This shows that ι inverts the projection (2.4).It follows that as cochain complexes, hence

Representations of foliation groupoids.
A foliation groupoid is a Lie groupoid whose anchor map is injective.This condition ensures that the connected components of the orbits of the groupoid are the leaves of a regular foliation of the base manifold, whence the name.On the other hand, foliation groupoids encompass several classical groupoids associated to a foliated manifold, such as the holonomy and the monodromy groupoids.By Example 1.2.6, representations of a foliation groupoids G are equivalent to trivial core VB-groupoids over G.Here we want to study the linear deformation cohomology of such VBgroupoids.First of all, let G ⇒ M be a foliation groupoid, let A ⇒ M be its Lie algebroid, ρ : A → T M the (injective) anchor map, ν = T M/ im ρ the normal bundle and let π : T M → ν be the projection.In this case, ν has constant rank and the normal representation is a plain representation of G on ν.Moreover, we recall from [7] that the map Consider a representation E → M of G and construct the associated trivial core VB-groupoid (G ⋉ E ⇒ E; G ⇒ M ).At the infinitesimal level, there is an induced representation of A on E, i.e. an A-flat connection ∇ : A → DE, and ∇ is injective because ρ is so.Therefore, the cokernel ν = DE/ im ∇ is a vector bundle over M .Denote by π : DE → ν, δ → δ the projection.We want to show that, in this situation, G acts on ν.To see this, recall that the group Bis(G) of bisections of G acts on Γ(E) via so it also acts on derivations of E by If β is a local bisection around x and β(x) = g : x → y, these formulas still make sense: Equation (2.6) shows that the action of β takes local sections around y to local sections around x, Equation (2.7) shows that β acts on derivations locally defined around x (to give a derivation locally defined around y). Now, let g : x → y be an arrow in G and δ ∈ D x E. Choose a local bisection β of G passing through g and a derivation ∆ ∈ D(E) such that ∆ x = δ.Our action is then defined by A routine computation shows that the definition does not depend on the choice of ∆.Let us prove that this definition is also independent of the choice of β.This is equivalent to prove that, if β ∈ Bis(G), β z = 1 z for some z ∈ M , then there exists α ∈ Γ(A) such that Consider the vector field σ ∆−β•∆ .We have (2.8) But t • β preserves the orbits of G, hence it maps a sufficiently small neighborhood of z in the leaf L z of im ρ through z to itself.It then follows from (2.8) that ξ kills all the functions that are constant along L z , hence it belongs to im ρ.Now, let α ∈ Γ(A) be any section such that ρ(α z ) = ξ and put D := ∆ − β • ∆ − ∇ α .By construction, σ Dz = 0, so it suffices to show that D z vanishes on ∇-flat sections.If ε is such a section, then (2.9)But the hypothesis ∇ε = 0 implies that ε is invariant under the action of Bis(G), at least locally around z, and the claim follows from (2.9).
Notice that the symbol map σ : DE → T M descends to a G-equivariant map ν → ν, and the fact that End E ∩ im ∇ = 0 implies that its kernel is again End E. Hence we have a short exact sequence of vector bundles with G-action: The rows are short exact sequences of DG-modules and the vertical arrows are DG-module surjections; additionally, p is a quasi-isomorphism.Hence, it immediately follows from the Snake Lemma and the Five Lemma that p is a quasi-isomorphism as well.We have thus proved the following Proposition 2.4.1.There is a canonical isomorphism between the linear deformation cohomology of the VB-groupoid (G ⋉ E ⇒ E; G ⇒ M ) and the leafwise cohomology with coefficients in ν: H def,lin (G ⋉ E) = H(G, ν).
2.5.Lie group actions on vector bundles.Let G be a Lie group with Lie algebra g.Assume that G acts on a vector bundle E → M by vector bundle automorphisms.Then G acts also on M and (G ⋉ E ⇒ E; G ⋉ M ⇒ M ) is a trivial-core VB-groupoid.We want to discuss its linear deformation cohomology.
Of course the action of G on M induces an infinitesimal action of g on M , and the Lie algebroid of G ⋉ M is the action algebroid g ⋉ M .We recall from [7] that G ⋉ M acts naturally on g ⋉ M , by extending the adjoint action of G on g, and on T M by differentiating the action of G on M .Moreover, there is a short exact sequence of complexes: (2.11) The projection p is defined as follows.Take a k-cochain c : (G ⋉ M ) (k+1) → T (G ⋉ M ) in the deformation complex of G ⋉ M and let (h 0 , . . ., h k ) ∈ (G ⋉ M ) (k+1) , h 0 = (g, x).Then Moreover, c is in the kernel of (2.13) if and only if its T G-component is 0. In this case, it is clear that c is determined by s c, that is in turn equivalent to the map c2 : G k × M → DE defined by (1.35).Therefore, the kernel of (2.

Proposition 1 . 1 . 18 ( 1 . 1 . 2 .
The variation map).Let b ∈ B. For any curve γ : I → B with γ(0) = b, the deformation class of γ * G at time 0 does only depend on γ(0).This defines a linear map Var G b : T b B → H 1 def ( Gb ), called the variation map of G at b. Deformations of Lie algebroids and VB-algebroids.Let E → M be a vector bundle.A multiderivation with k entries of E (and

Example 1 . 2 . 6 .
by definition, the Lie algebroid A of G and the core-anchor is the anchor map ρ : A → T M .Let G ⇒ M be a Lie groupoid and let π : E → M be a representation of G. Out of these data, we can define the action groupoid G ⋉ E ⇒ E. As a manifold, G ⋉ E = G × s π E; the structure maps are given by s(g, e) := e, t(g, e) := g • e, (h, ge) • (g, e) := (hg, e).
V) and we can proceed as in the proof of[7, Lemma  5.3].

Remark 1 .
3.16.Clearly, a deformation Ṽ of the VB-groupoid V induces a deformation G of the base groupoid G. Now, it is easy to check that the projection (1.29) sends the linear deformation class of Ṽ to the deformation class of G. ⋄ Finally, we discuss the variation map associated to deformations of VB-groupoids.Let ( Ṽ ⇒ Ẽ; G ⇒ M ) be a family of VB-groupoids over a smooth manifold B. Then any curve γ : I → B induces a deformation γ * Ṽ of Ṽγ(0) , and we have: Proposition 1.3.17(The linear variation map).Let b ∈ B. For any curve γ : I → B with γ(0) = b, the deformation class of γ * Ṽ at time 0 does only depend on γ(0).This defines a linear map Var Ṽ lin,b : T b B → H 1 def,lin ( Ṽb ), called the linear variation map of Ṽ at b, that makes the following diagram commutative:

Definition 1 . 6 . 1 .
A morphism of Lie groupoids Ψ : G 1 → G 2 over a smooth map F : M 1 → M 2 is a Morita map (or a weak equivalence) if it is (1) fully faithful, i.e. the diagram
c(h 0 , . . ., h k ) ∈ T (g 0 ,x 0 ) (G ⋉ M ) ∼ = T g 0 G × T x 0 M ∼ = g × T x 0 Mvia right translations, and we compose with the projection T M → M to get an element in g ⋉ M .The kernel of p is given by T M -valued cochains.Since the T M -component is the projection by the source map, it does not depend on the first component, and we conclude that the kernel is C(G ⋉ M, T M ).We want to construct a similar sequence for C def,lin (G ⋉ E) taking into account the linear nature of the action.First of all, there is an obvious induced action of G on DE and the symbol map σ : DE → T M is G-equivariant.Hence there is a short exact sequence of cochain complexes:0 −→ C(G ⋉ M, End E) −→ C(G ⋉ M, DE) −→ C(G ⋉ M, T M ) −→ 0.In this case, the sequence (1.39) reads0 −→ C(G ⋉ M, End E) −→ C def,lin (G ⋉ E) Π −→ C def (G ⋉ M ) −→ 0 (2.12)and, composing Π with p, we get a cochain mapC def,lin (G ⋉ E) −→ C(G ⋉ M, g ⋉ M )[1] −→ 0. (2.13) Applying the isomorphism (1.33), we get (G ⋉ E) (k) ∼ = G k × E, and similarly (G ⋉ M ) (k) ∼ = G k × M .Then if c ∈ C k def,lin (G ⋉ E) is a linear cochain, the diagram (1.34) takes the following form:

CProposition 2 . 5 . 1 .
13) is C(G ⋉ M, DE).Summarizing there is an exact diagram of cochain complexes0 0 0 / / C(G ⋉ M, End E) C(G ⋉ M, End E) def,lin (G ⋉ E) / / C(G ⋉ M, g ⋉ M )[1] / / 0 0 / / C(G ⋉ M, T M ) / / C def (G ⋉ M ) / / C(G ⋉ M, g ⋉ M )[1]Let G be a Lie group acting on a vector bundle E → M by vector bundle automorphisms.The linear deformation cohomology of the VB-groupoid (G ⋉ E ⇒ E, G ⋉ M ⇒ M ) fits in the exact diagram: such that the first square is a VB-groupoid and the rows are families of Lie groupoids.In particular, for every M 2 is a pull-back diagram, and (2) essentially surjective, i.e. the mapG 2 × s 1 F M 1 → M 2 , (g, x) → t 2 (g) is a surjective submersion.Two groupoids G 1 and G 2 are said to be Morita (or weakly) equivalent if and only if there exist a Lie groupoid H and Morita maps Ψ 1