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Groupoid representations and modules over the convolution algebras

The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an etale Lie groupoid and the category of modules over its convolution algebra that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita bicategory of etale Lie groupoids and the given correspondence represents a natural equivalence between them.


Introduction
There are many phenomena in different areas of mathematics and physics that are most naturally described in the language of smooth manifolds and smooth maps between them. However, some natural constructions, coming from the theory of foliations or from Lie group actions, result in slightly more singular spaces and require a different approach. The Morita category of Lie groupoids and principal bundles [7,8,15,18,20] provides a natural framework in which to study many such singular spaces like spaces of leaves of foliations, spaces of orbits of Lie group actions or for example orbifolds [1,4,16,17]. A Lie groupoid can be considered as an atlas for the given singular space. It turns out that different Lie groupoids represent the same geometric space precisely when they are Morita equivalent, i.e. when they are isomorphic in the Morita category of Lie groupoids. For this reason we are primarily interested in those algebraic invariants of Lie groupoids that are functorially defined on the Morita category of Lie groupoids.
Tangent bundles, bundles of higher order tensors, line bundles and other vector bundles play a central role in the study of smooth manifolds. The theory of representations of Lie groupoids naturally extends all these notions to the category ofétale Lie groupoids [1,17] and shows their close connection to the theory of Lie groups representations. It encompasses many well known constructions like equivariant vector bundles [2,27], orbibundles over orbifolds [1], foliated and transversal vector bundles over spaces of leaves of foliations [17], as well as ordinary vector bundles over manifolds or representations of discrete groups. The construction of the category of representations of a Lie groupoid is invariant under the Morita equivalence and thus represents one of the basic algebraic invariants of Lie groupoids.
The Connes convolution algebra of smooth functions with compact support [4,22,23] on anétale Lie groupoid is another example of such an invariant. Smooth functions with compact support on a smooth manifold and group algebras of discrete groups are all special examples of convolution algebras. Finitely generated projective modules over the algebra of smooth functions on a compact manifold are closely connected to smooth vector bundles over that manifold by a theorem due to Serre and Swan [28,29]. Representations of a discrete group Γ on complex vector spaces on the other hand correspond bijectively to modules over the group algebra C[Γ] of the group Γ. One is thus lead to believe that both of these examples represent a similar type of objects, namely a correspondence between representations of anétale Lie groupoid and a certain class of modules over its convolution algebra.
We show in this paper (Theorem 3.2) how to extend the classical Serre-Swan's correspondence to the category ofétale Lie groupoids. The functor of smooth sections with compact support defines an equivalence between the category of representations of anétale Lie groupoid G and the category of modules over the convolution algebra of G which are of finite type and of constant rank. These modules generalize the well known finitely generated projective modules over the algebras of functions and coincide with them if the manifold of objects of G is compact and connected. For example, such a correspondence makes it possible to study the category of orbibundles over a compact orbifold with the tools of noncommutative geometry, applied to the convolution algebra of the corresponding orbifold groupoid.
The constructions of the categories of representations and of the modules of finite type and of constant rank extend to morphisms from the Morita bicategory ofétale Lie groupoids to the 2-category of additive categories. In this more general context, the given correspondence can be described (Theorem 4.3) as a natural equivalence between these two morphisms.

Basic definitions and examples
2.1. The Morita category of Lie groupoids. For the convenience of the reader, and to fix the notations, we begin by summarizing some basic definitions and results concerning Lie groupoids that will be used throughout this paper. We refer the reader to one of the books [13,17,18] for a more detailed exposition and further examples.
A Lie groupoid over a smooth, second countable, Hausdorff manifold M is given by a smooth manifold G and a structure of a category on G with objects M , in which all the arrows are invertible and all the structure maps are smooth, with the source map s a submersion. We allow the manifold G to be non-Hausdorff, but we assume that the fibers of the source map are Hausdorff. We write G(x, y) = s −1 (x) ∩ t −1 (y) for the set of arrows from x ∈ M to y ∈ M and G x = G(x, x) for the isotropy group of the groupoid G at x. The set G(x, y) is a submanifold of G and the isotropy group G x is a Lie group. If g ∈ G is any arrow from x to y, and g ′ ∈ G is an arrow from y to z, then the product g ′ g = mlt(g ′ , g) is an arrow from x to z. The map uni assigns to each x ∈ M the identity arrow 1 x = uni(x) in G, and we often identify M with its image uni(M ) in G. The map inv maps each arrow g ∈ G to its inverse g −1 .
A Lie groupoid isétale if all of its structure maps are local diffeomorphisms. A bisection of anétale Lie groupoid G is an open subset U of G such that both s| U and t| U are injective. To any such bisection U corresponds a diffeomorphism τ U : s(U ) → t(U ) defined by τ U = t| U • (s| U ) −1 . Bisections of the groupoid G form a basis for the topology on G, so in particular they can be chosen arbitrarily small.
Generalised morphisms [18,20] turn out to be the right notion of a map between Lie groupoids in the representation theory of Lie groupoids. They are closely connected to groupoid actions and principal bundles, which we briefly describe.
A smooth left action of a Lie groupoid G on a smooth manifold P along a smooth map π : P → M is a smooth map µ : G × s,π M P → P , (g, p) → g · p, which satisfies π(g · p) = t(g), 1 π(p) · p = p and g ′ · (g · p) = (g ′ g) · p, for all g ′ , g ∈ G and p ∈ P with s(g ′ ) = t(g) and s(g) = π(p). We define right actions of Lie groupoids on smooth manifolds in a similar way.
Let G be a Lie groupoid over M and let H be a Lie groupoid over N . A principal H-bundle over G is a smooth manifold P , equipped with a left action µ of G along a smooth submersion π : P → M and a right action η of H along a smooth map φ : P → N , such that (i) φ is G-invariant, π is H-invariant and both actions commute: φ(g · p) = φ(p), π(p · h) = π(p) and g · (p · h) = (g · p) · h for every g ∈ G, p ∈ P and h ∈ H with s(g) = π(p) and φ(p) = t(h), (ii) π : P → M is a principal right H-bundle: (pr 1 , η) : P × φ,t N H → P × π,π M P is a diffeomorphism. A map f : P → P ′ between principal H-bundles P and P ′ over G is equivariant if it satisfies π ′ (f (p)) = π(p), φ ′ (f (p)) = φ(p) and f (g · p · h) = g · f (p) · h, for every g ∈ G, p ∈ P and h ∈ H with s(g) = π(p) and φ(p) = t(h). Any such map is automatically a diffeomorphism. Principal H-bundles P and P ′ over G are isomorphic if there exists an equivariant diffeomorphism between them. There is a natural structure of a category (actually a groupoid) on the set of principal H-bundles over G for any two Lie groupoids G and H. It has principal H-bundles over G as objects and equivariant diffeomorphisms as morphisms between them.
To any smooth functor ψ : G → H there corresponds a principal H-bundle P (ψ) = M × ψ,t N H over G with the actions given by the maps g·(x, h) A principal bundle P is isomorphic to one induced by a functor if and only if it is trivial, i.e. if there exists a global smooth section of the bundle P .
If P is a principal H-bundle over G and if P ′ is a principal K-bundle over H, for another Lie groupoid K, one can define the composition P ⊗ H P ′ [18,19,20], which is a principal K-bundle over G. It is the quotient of P × φ,π ′ N P ′ with respect to the diagonal action of the groupoid H. So defined composition is associative only up to a natural isomorphism.
The Morita category GPD of Lie groupoids consists of Lie groupoids as objects and isomorphism classes of principal bundles as morphisms between them. The morphisms in GPD are sometimes referred to as Hilsum-Skandalis maps or generalised morphisms between Lie groupoids. Two Lie groupoids are Morita equivalent if they are isomorphic in the category GPD. The Morita category EtGPD ofétale Lie groupoids is the full subcategory of the category GPD withétale Lie groupoids as objects. If G and H areétale Lie groupoids and if P is a principal H-bundle over G, then the corresponding map π : P → M is automatically a local diffeomorphism. We will be primarily interested in the Morita bicategory of Lie groupoids which we describe later on in the paper.

2.2.
Representations of Lie groupoids. Lie groupoids admit a twofold interpretation. They can be used to describe symmetries of fibre bundles in a similar way as Lie groups are used to study symmetries of topological spaces. However, the extra transversal part of the structure, which is encoded in the manifold of objects, makes them a convenient model for singular geometric spaces such as orbifolds, spaces of leaves of foliations or spaces of orbits of Lie group actions. The theory of representations of Lie groupoids provides a unified framework for the study of vector bundles on such geometric spaces and shows their intimate connection to representations of Lie groups.
Let G be a Lie groupoid over a smooth manifold M and let E be a smooth complex vector bundle of rank k over M . A representation of the groupoid G on E is a smooth left action ρ : G × M E → E, denoted by ρ(g, v) = g · v, of G on E along the bundle projection p : E → M [17], such that for any arrow g ∈ G(x, y) the induced map g * : E x → E y , v → g · v, is a linear isomorphism. (ii) Let G be a point groupoid with only one object, i.e. G is a Lie group K. The representation theory of G then coincides with the representation theory of the Lie group K on finite dimensional complex vector spaces.
(iii) The pair groupoid G = M × M over a smooth manifold M has both projections as source and target maps and multiplication defined in a natural way. Every representation of G on a vector bundle E over M amounts to a natural identification of all the fibers of E and is thus isomorphic to a trivial representation.
(iv) Let G = K ⋉ M be the translation groupoid of a smooth left action of a Lie group K on a manifold M . In this case the representations of the groupoid G correspond to K-equivariant vector bundles over M [2,27].
(v) Anyétale Lie groupoid G over a manifold M has a natural representation on the complexified tangent bundle of the manifold M , where the action of any arrow is defined by the differential of the local diffeomorphism corresponding to some bisection through that arrow [1,17]. The cotangent bundle and tensor bundles all inherit this natural representation, so it makes sense to speak of vector fields, differential forms or riemannian metrics onétale Lie groupoids.
(vii) Let (M, F ) be a foliated manifold. Representations of the holonomy groupoid Hol(M, F ) are sometimes referred to as transversal vector bundles, while those of the monodromy groupoid Mon(M, F ) are referred to as foliated vector bundles.
A morphism between representations E and F of the groupoid G over M is a G-equivariant morphism φ : E → F of vector bundles. More precisely, φ : E → F is a fiberwise linear smooth map which commutes with bundle projections and satisfies φ(g · e) = g · φ(e) for all g ∈ G and all e ∈ E s(g) . Representations of a Lie groupoid G together with G-equivariant morphisms between them form a category Rep(G) of representations of G. Categories of (equivariant) vector bundles over a manifold, categories of orbibundles over an orbifold or categories of representations of Lie groups are some examples of categories of representations of Lie groupoids that arise naturally in various contexts. Direct sums, tensor products, duals and other operations on vector bundles generalize to representations of Lie groupoids and turn the category Rep(G) into an additive category for every groupoid G.
Generalised maps between groupoids can be used to pull back representations in the same sense as vector bundles can be pulled back along smooth maps. Let G and H be Lie groupoids over M respectively N and let P be a principal H-bundle over G. For any representation E of the groupoid H we get the pull back representation P ⊗ H E as follows (see [9] for details). The pull back bundle φ * E = P × N E has a natural structure of a vector bundle over P with projection onto the first factor as the projection map. Groupoid H acts diagonally from the right on the space φ * E along the fibers of the projection onto M and it is easy to see that the natural map P ⊗ H E = φ * E/H → M is well defined, smooth and makes P ⊗ H E a vector bundle over M . Finally, the action of the groupoid G on the space P induces a representation of the groupoid G on the bundle P ⊗ H E by acting on the first factor.
The construction of pulling back representations along a principal bundle P extends to a functor from the category of representations of H to the category of representations of G. Define a representation Rep(P )(E) = P ⊗ H E of G for any representation E of H and a morphism Rep(P )(φ) : One can use an alternative description of the above operation in the case of trivial bundles, i.e. when the principal bundle comes from a smooth functor. Suppose that ψ : G → H is a smooth functor between Lie groupoids and let E be a representation of the groupoid H. One defines a representation ψ * E of the groupoid G on the vector bundle

Convolution algebras and principal bimodules.
A smooth manifold M is closely connected with the commutative algebra C ∞ c (M ) of smooth functions with compact support on M . By replacing a manifold M with anétale Lie groupoid G over M and by defining a proper notion of the convolution product on the space of functions one obtains the convolution algebra C ∞ c (G) of the groupoid G. It is in general noncommutative but it contains the algebra C ∞ c (M ) as a commutative subalgebra. The above construction naturally extends to a covariant functor from the Morita category ofétale Lie groupoids to the Morita category of algebras, i.e. for each principal bundle between groupoids one naturally constructs a bimodule between corresponding algebras. In this subsection we briefly recall the definition of the convolution algebra [4,5,20,22] assigned to anétale, not necessarily Hausdorff, Lie groupoid and of the principal bimodule [10,20] assigned to a principal bundle.
We first recall the definition of the convolution product on the vector space C ∞ c (G) of smooth functions with compact support on a Hausdorffétale Lie groupoid G. Define a bilinear operation on the space C ∞ c (G) by the formula for any a, b ∈ C ∞ c (G). Equipped with this product the space C ∞ c (G) becomes an associative algebra called the convolution algebra [4] of theétale Lie groupoid G.
In the case of a generalétale Lie groupoid a suitable notion of a smooth function with compact support on a non-Hausdorff manifold, as given in [5], is needed. Considering that smooth functions on a Hausdorff manifold M correspond precisely to the continuous sections of the sheaf of germs of smooth complex valued functions on M it makes sense to use this alternative approach to define smooth functions with compact support on an arbitrary manifold P . One first considers the vector space of all (not-necessarily continuous) sections of the sheaf of germs of smooth functions on P . The trivial extension of any smooth function with a compact support in a Hausdorff open subset of P naturally represents a section of that sheaf. The vector space C ∞ c (P ) of smooth functions with compact support on P is then defined to be the subspace of the space of all sections, generated by such sections. This definition of the vector space C ∞ c (P ) agrees with the classical one if P is Hausdorff, but in general there exists no natural multiplicative structure on the space C ∞ c (P ). The support, i.e. the set where the values of the section are nontrivial, of any function in C ∞ c (P ) is always a compact subset of P but not necessarily closed if P is a non-Hausdorff manifold.
The stalk of the sheaf of germs of smooth functions on P at any point of P is a commutative algebra with identity, which enables us to perform pointwise operations such as addition, multiplication or pullbacks along smooth maps. In particular, for any non-Hausdorffétale Lie groupoid G a formula analogous to the formula (1) can be used to define the convolution algebra C ∞ c (G) of theétale Lie groupoid G. We refer the reader to [22] for details.
(ii) The convolution algebra C ∞ c (G) of the pair groupoid on n points coincides with the algebra of n × n complex matrices and contains the subalgebra C ∞ c (M ) of diagonal matrices. This example shows that C ∞ c (M ) need not lie in the center of the algebra C ∞ c (G). (iii) Let G be the translation groupoid of a smooth action of a discrete group Γ on a manifold M . The convolution algebra C ∞ c (G) of G is known in the literature [4] as the crossed product algebra Γ ⋉ C ∞ c (M ). One can naturally extend the convolution algebra construction to a functor from the Morita category ofétale Lie groupoids to the Morita category of algebras. Let G and H beétale Lie groupoids and let P be a principal H-bundle over G. One can define convolution actions of the algebras C ∞ c (G) and C ∞ c (H) on the vector space which is called the principal bimodule associated to the principal H-bundle P over G. The functor C ∞ c from the Morita category ofétale Lie groupoids to the Morita category of algebras assigns to everyétale Lie groupoid its convolution algebra and to an isomorphism class of a principal bundle the isomorphism class of the associated principal bimodule [20] (see also [10] for a treatment of the non-Hausdorff case).
Throughout the rest of the paper we will restrict ourselves to Hausdorff groupoids for simplicity, although essentially the same formulas apply in the non-Hausdorff case as well.

Groupoid representations and modules over convolution algebras
The vector space of sections of a smooth vector bundle E over a manifold M admits a natural action of the algebra of smooth functions on M . Additional structure of a representation of anétale Lie groupoid G on the bundle E allows a natural extension of that action to the action of the convolution algebra of the groupoid G. In this section we characterize the modules over the convolution algebra of the groupoid G that arise in this fashion from sections of representations of G.
Results of this type were first considered by Serre [28] in the the category of algebraic varieties and Swan in the category of compact Hausdorff topological spaces [29]. All our vector bundles will be assumed to be of globally constant rank, a condition which is automatically satisfied if the manifold of objects of the groupoid is connected. Similar results hold however in the case of vector bundles of globally bounded rank as well. . Now let G be anétale Lie groupoid over M . We will associate an action of the convolution algebra C ∞ c (G) on the space of sections Γ ∞ c (E) to any representation E of the groupoid G. Define a bilinear map for a ∈ C ∞ c (G) and u ∈ Γ ∞ c (E). Since the function a ∈ C ∞ c (G) has a compact support, there are only finitely many g ∈ t −1 (x) with a(g) = 0 for each x ∈ M , hence au is a well defined section of the vector bundle E. It remains to be proven that au belongs to the space Γ ∞ c (E) and that the above map really defines an action of the algebra C ∞ c (G) on the space Γ ∞ c (E). Strictly speaking, the above formula only holds for Hausdorff Lie groupoids. However, by evaluating the sections of the sheaf of germs of smooth functions, one can use virtually the same formula for non-Hausdorff groupoids as well. Proof. First we show that au represents a smooth section of the vector bundle E for any a ∈ C ∞ c (G) and any u ∈ Γ ∞ c (E). We can decompose any function a ∈ C ∞ c (G) as a sum a = a j of functions, each of which has support contained in some bisection, so we can assume right from the start that the support of a is contained in some bisection U . Let us denote W = t(U ) and V = s(U ). We then have the following commuting diagram of maps of vector bundles. The maps µ and pr E are isomorphisms of vector bundles covering the diffeomorphisms t| U : U → W respectively s| U : U → V . Let us denote by σ U = (t| U ) • (s| U ) −1 : V → W the diffeomorphism corresponding to the bisection U , and by τ U = µ • (pr E ) −1 : E| V → E| W the corresponding isomorphism of vector bundles. The smooth section u| V of the bundle E| V gets mapped by the above isomorphism to the smooth section u ′ = τ U • u • (σ U ) −1 of the bundle E| W . Furthermore, since a has compact support in U , the function a • (t| U ) −1 has compact support in W . One can now express the section au of the bundle E as (au)(x) = (a • (t| U ) −1 )(x)u ′ (x) to prove that it is a smooth section of E with compact support in W . To see that the space of sections Γ ∞ c (E) is a module over the algebra C ∞ c (G), the equality a(bu) = (ab)u must hold for all a, b ∈ C ∞ c (G) and all u ∈ Γ ∞ c (E). To this effect we compute On the other hand we have In the last line we have used the linearity of the map g * : E x → E y for each g ∈ G(x, y). Since E is a representation of G, the equality g ·(g ′ ·e) = gg ′ ·e holds for all pairs of composable arrows g, g ′ ∈ G and all e ∈ E s(g ′ ) , thus (ab)u = a(bu).
By the above procedure we obtain a left module of sections Γ ∞ c (E) over the convolution algebra C ∞ c (G) for any representation E of the groupoid G. Any morphism φ : E → F of representations of G is in particular a morphism of vector bundles and therefore produces a homomorphism Γ ∞ Considering that the map φ is fiberwise linear and G-equivariant we get the equalities a(g)(g · u(s(g))) = t(g)=x a(g)(g · φ(u(s(g)))) = (aΓ ∞ c (φ)(u))(x) , for a ∈ C ∞ c (G) and u ∈ Γ ∞ c (E). The homomorphism Γ ∞ c (φ) is therefore a homomorphism of C ∞ c (G)-modules so we have the covariant functor Γ ∞ c = (Γ ∞ c ) G : Rep(G) → G Mod from the category of representations of the groupoid G to the category of left modules over the convolution algebra C ∞ c (G) of G.

3.2.
Modules of finite type and of constant rank. According to the previous subsection we can consider a module over the convolution algebra of anétale Lie groupoid G as a module of sections of some representation of G. However, not every C ∞ c (G)-module is of this kind and it is not too hard to find counterexamples. In the following subsection we define and explain the conditions that characterize the modules of sections of representations of the groupoid G.
Let M be a smooth manifold and denote by C Suppose now that M is a smooth, Hausdorff and paracompact manifold and let E be a vector bundle over M . The module Γ ∞ c (E) of sections of the bundle E is a basic example of a module of finite type and of constant rank. One can see that as follows. Since M is finite dimensional and paracompact, there exists a vector bundle F over M such that the bundle E ⊕ F is isomorphic to some trivial vector bundle M × C k over M ; vector bundles with this property are said to be of finite type. This property basically follows from the proof of Lemma 5.9 in [14]. As a result we obtain the isomorphism Γ ∞   Proof. The crucial point in the proof of the theorem is the observation that every vector bundle over a paracompact manifold is of finite type, i.e. a subbundle of some trivial bundle. Taking this into account, basically the same proof as in the Swan's original paper [29] goes through.
Let G be anétale Lie groupoid over M . We will prove Theorem 3.2 by constructing a quasi-inverse Bundles E and E ′ are therefore isomorphic by Theorem 3.3 and in turn they define the same vector bundle structure on the space R G (M).
We will next use the extra structure of a C ∞ c (G)-module on the space M to define a representation of the groupoid G on the vector bundle R G (M). Choose any arrow g ∈ G(x, y) and any vector v ∈ M(x). We can find an element m ∈ M such that v = m(x) and a function a ∈ C ∞ c (G) with compact support in some bisection U such that a(g) = 1. Since M is a left C ∞ c (G)-module, the element am ∈ M is well defined and we define g · m(x) = am(y) ∈ M(y).
We will denote by the map defined by the above formula. (ii) Let a ∈ C ∞ c (G) be a smooth function with compact support in the bisection U of the groupoid G and denote W = t(U ) respectively V = s(U ). For any function is a smooth function with compact support in W . Moreover, af = a for every function f which is identically equal to 1 on the set s(supp(a)) ⊂ V .
Suppose now that g ∈ G(x, y) is an arrow from x to y and let v ∈ M(x) be an arbitrary vector. We will first show that the element µ M (g, v) is well defined and independent of various choices of representatives for the elements g and v. To this effect choose an arbitrary function a ∈ C ∞ c (G) with support in a bisection U such that a(g) = 1 and let m, m ′ ∈ M both satisfy m(x) = m ′ (x) = v. We then have (m − m ′ )(x) = 0, so we can find a function f ∈ I x C ∞ c (M ) and an element m ′′ ∈ M with m − m ′ = f m ′′ . Now choose a smooth function f ′ ∈ C ∞ c (M ) with compact support in V = s(U ), such that af ′ = a as in (ii). The support of the function f ′ f is then contained in V , so by (i) it follows We obtain the equality am(y) = am ′ (y) which proves the independence of µ M (g, v) of the choice of the representative for the element v. Next we prove a similar statement for the choice of the representative for the arrow g. Suppose a ′ ∈ C ∞ c (G) is another function with a ′ (g) = 1 and with support in a bisection U ′ . One can then find functions f, f ′ ∈ C ∞ c (M ) with f (y) = f ′ (y) = 1, such that f a = f ′ a ′ is a function with compact support in the bisection U ∩ U ′ . The equalities am(y) = f am(y) = f ′ a ′ m(y) = a ′ m(y) then show that the element µ M (g, v) ∈ R G (M) is well defined.
We next show that µ M defines an action of the groupoid G on the vector bundle R G (M) over M . This will be true if we show that the equalities (g ′ g) · v = g ′ · (g · v) and 1 x · v = v hold for all arrows g ∈ G(x, y), g ′ ∈ G(y, z) and all v ∈ M(x). If a ∈ C ∞ c (G) with a(g) = 1 has support in a bisection U and if a ′ ∈ C ∞ c (G) with a ′ (g ′ ) = 1 has support in a bisection U ′ , then a ′ a has support in the bisection U ′ × s,t M U and (a ′ a)(g ′ g) = 1. We then have where m ∈ M is an arbitrary element such that m(x) = v. To prove the second claim, it is enough to observe that any function The map R G (Φ) is a smooth morphism of vector bundles, since it transforms smooth sections of the bundle R G (M) to smooth sections of the bundle R G (N ). We claim that R G (Φ) defines a G-equivariant morphism of representations of the groupoid G on R G (M) respectively R G (N ) as described above. To see this, choose an arrow g ∈ G(x, y) and a vector v ∈ R G (M) x = M(x). Let a ∈ C ∞ c (G) be a function with support in a bisection U such that a(g) = 1 and suppose m ∈ M satisfies m(x) = v. The statement then follows from the equalities The functoriality of the assignment Φ → Φ(x) for each x ∈ M extends to the functoriality of the map R G , so we have the functor Let E be a representation of the groupoid G. There is a natural isomorphism Γ ∞ c (E)(x) → E x of complex vector spaces for every x ∈ M , induced by the evaluation at the point x, which induces an isomorphism η E : R G (Γ ∞ c (E)) → E of representations of the groupoid G. The natural equivalence of functors together with the equivalence ǫ shows that Γ ∞ c is an equivalence of categories.

Equivalence of morphisms between bicategories
According to the previous sections we can associate to anyétale Lie groupoid the additive categories of representations and of modules of finite type and of constant rank. We would like to extend those constructions to define functors Rep and Mod from the category ofétale Lie groupoids and principal bundles to the category of additive categories and functors such that their values at the groupoid G are Rep(G) respectively Mod(G). Furthermore, the family of functors (Γ ∞ c ) G from Theorem 3.2 should represent a natural equivalence between these two functors.
If one wants to work in the framework of categories, it is necessary to work with isomorphism classes of representations and of modules and forget the extra categorical structure on these objects. A more convenient way of describing our statement uses the language of bicategories and morphisms between them which we now briefly summarize.

Bicategories and Morphisms.
We first recall the definition and some examples of bicategories and morphisms between them as given in [3,11] (see also [12] for a very concise treatment). A bicategory C consists of (1) A collection C 0 of objects of C. Composition of 1-cells in a bicategory C resembles the usual composition of arrows in ordinary categories, although it is in general associative only up to natural associativity coherence isomorphisms, which are given as a part of the structure of C. Similarly, the identity 1-cells 1 A for A ∈ C 0 act as units only up to predefined unit coherence isomorphisms. All these coherence isomorphisms have to satisfy some natural pentagon and triangle axioms (see [12] for details) as to insure that all the diagrams constructed out of coherence isomorphisms commute. Apart from the horizontal composition of 2-cells in C there exists a vertical composition of 2-cells, defined by the composition in the categories C(A, B). Strict bicategories or 2categories are bicategories in which the composition of 1-cells is strictly associative with strict identities. For any category A ∈ Cat 0 the 1-cell 1 A is represented by the identity functor on A. All the coherence isomorphisms are identities so Cat is in fact a 2-category. We will denote by AdCat the full sub 2-category consisting of additive categories.
(ii) Next we describe the Morita bicategory GPD of Lie groupoids. For any two Lie groupoids G and H there exists a category GPD(G, H) with principal Hbundles over G as objects and equivariant diffeomorphisms as morphisms between them. Composition of 1-cells P and Q is defined by the tensor product construction P ⊗ H Q together with the natural associtativity coherence isomorphisms a P,Q,R : for P ∈ GPD(G, H), Q ∈ GPD(H, K) and R ∈ GPD(K, L). For any G ∈ GPD 0 the identity 1-cell 1 G is simply the groupoid G viewed as a principal G-bundle over G. The unit coherence isomorphisms l P : G ⊗ G P → P and r P : P ⊗ H H → P are induced by the action maps of the groupoids on the principal bundles. If f : P → P ′ and g : Q → Q ′ are equivariant diffeomorphisms, their horizontal composition is naturally defined as f ⊗ g : P ⊗ H Q → P ′ ⊗ H Q ′ . We will denote by EtGPD the full subbicategory of GPD consisting ofétale Lie groupoids. A morphism of bicategories is functorial only up to a family of natural isomorphisms for each pair of composable 1-cells of C and for each object of C. These natural isomorphisms have to satisfy some further natural coherence axioms. Morphisms of bicategories are also referred to as Lax functors in the literature. Contavariant versions of morphisms between bicategories can be defined analogously.
Our main examples of morphisms between bicategories come from the correspondence between representations of a Lie groupoid and modules of finite type and of constant rank over its convolution algebra.
We will first describe the contravariant morphism c (K)-bimodules as defined in [20], see also [10]. The identity coherence isomorphisms are provided by the actions of convolution algebras on the modules.
It is straightforward to check that so defined families of functors and natural isomorphisms satisfy the axioms for the morphisms between bicategories.

4.2.
Natural equivalence of the morphisms Rep and Mod. We will show in this section how one can interpret the Serre-Swan's correspondence as a natural transformation between the contravariant morphisms Rep and Mod from the Morita bicategory ofétale Lie groupoids to the 2-category of additive categories.
To this effect we first review the definition of a natural transformation between two morphisms of bicategories [11,12]. Let C and D be two bicategories and let F, G : C → D be morphisms between them. A natural transformation σ : F → G consists of (1) For each A ∈ C 0 a 1-cell σ A : F (A) → G(A).
(2) For each 1-cell f : such that σ f is natural in f and satisfies coherence axioms as in [12]. We begin with a proposition that is of some interest independently of our discussion.
Proposition 4.2. Let G and H beétale Lie groupoids and let P be a principal Hbundle over G. For any representation E of H there exists a natural isomorphism We first define a bilinear map σ P (E) : C ∞ c (P ) × Γ ∞ c (E) → Γ ∞ c (P ⊗ H E) by the formula (σ P (E)(f, u))(x) = π(p)=x f (p)(p ⊗ u(φ(p))) for f ∈ C ∞ c (P ) and u ∈ Γ ∞ c (E). It is not too hard to check that the map σ P (E) is well defined and that it induces a homomorphism c (G)-modules, which we claim to be an isomorphism. It suffices to show that σ P (E) is a bijective map.
We first consider the case when the bundle P is trivial, i.e. P = P (ψ) for some smooth functor ψ : G → H. The representation P ⊗ H E of G is then isomorphic to the representation ψ * E of G via the isomorphism f : P ⊗ H E → ψ * E. In particular, f is an isomorphism of vector bundles over M , so we obtain the isomorphism c (M )-modules. Now consider the representation E of H as a vector bundle over N . It is a classical result (see [6]) that the map