INTEGRATING THE NONINTEGRABLE. Alan WEINSTEIN Department of Mathematics, University of California, Berkeley, USA

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1 INTEGRATING THE NONINTEGRABLE Alan WEINSTEIN Department of Mathematics, University of California, Berkeley, USA Dominique Flament (dir) Série Documents de travail (Équipe F 2 DS) Feuilletages -quantification géométrique : textes des journées d étude des 16 et 17 octobre 2003, Paris, Fondation Maison des Sciences de l Homme, 2004

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3 Integrating the Nonintegrable Alan Weinstein Department of Mathematics University of California Berkeley, CA USA February 10, Introduction Foliations occur throughout geometric quantization and related areas of symplectic geometry. Among the symplectic manifolds which represent phase spaces in mechanics, the simplest are the cotangent bundles of configuration spaces, and these are quantized in a standard way via wave functions on the configuration space. For a general symplectic manifold, one gets something which plays the role of configuration space by introducing a local structure which looks like the decomposition of a cotangent bundle into its fibres (manifolds of constant momentum ). This structure is a lagrangian foliation, also known as a (real) polarization; its leaf space is then the natural domain of quantum wave functions. When the foliation has nicely behaved leaves (e.g. when they are the fibres of a fibration), so that the leaf space is a manifold, life is relatively easy, but new methods must be introduced when the foliation is more complicated, especially when it has leaves which are not closed. Foliations also appear as the characteristic foliations of coisotropic submanifolds. These submanifolds may be defined locally by the vanishing of Poisson commuting constraint functions ; the leaves of the characteristic foliation are swept out by the trajectories of the hamiltonian flows of these Research partially supported by NSF Grant DMS MSC2000 Subject Classification Numbers: xxxxx (Primary), xxxxx (Secondary). Keywords: 1

4 functions. Points on the same characteristic leaf are considered to be physically equivalent, so that the leaf space serves as the space of true physical degrees of freedom. When the leaf space is a manifold, it has a symplectic structure, but many constraint manifolds carry more complicated characteristic foliations, and one is required once again to introduce new methods to deal with the leaf space. Leaf spaces also appear in the solution to a fundamental problem in Poisson geometry going back to Lie [11], that of realizing a given Poisson manifold P as a quotient of a symplectic manifold. If there are any nice (i.e. complete, in a certain sense) realizations, then there is one which is a symplectic groupoid [5]. Cattaneo and Felder [2] construct a universal groupoid G(P ) by a classical analogue of the Poisson sigma model technique by which they recovered Kontsevich s [10] deformation quantization of Poisson manifolds. They start with the symplectic Banach manifold of C 1 paths in T P, restrict attention to a certain coisotropic submanifold therein, and then pass to the leaf space of its (finite codimensional) characteristic foliation. When this leaf space is not a manifold, P does not have any nice symplectic realizations. Nevertheless, one would like to work with the groupoid G(P ). For instance, Xu s [18] notion of Morita equivalence of Poisson manifolds fails to be an equivalence relation because P is Morita equivalent to itself only when G(P ) is a manifold. In addition, there is a more general notion of Morita morphism, but the composition of these morphisms is a geometric tensor product which is again the leaf space of a characteristic foliation. For all the reasons listed above, and many others, it is useful to be able to treat the leaf space of a foliation as if it were a manifold, and in particular to be able to extend the notions of Lie groupoid, symplectic manifold, and symplectic groupoid to some category including these leaf spaces. One approach to such problems is via noncommutative geometry [3]. In this paper, I will describe a different approach, based on the theory of differentiable stacks, especially as it is applied to the problem of integrating Lie algebroids to Lie groupoids, raised and partially solved by Pradines [14]. The body of the paper begins with an explanation of the path space construction (inspired by that of [2]) in the context of Poisson sigma models) by which Crainic and Fernandes [4] have precisely identified the obstructions to integration of Lie algebroids. I will then give a brief exposition of the theory of stacks and its application to leaf spaces. Finally, I will describe work on stacks as groupoids being done by Chenchang Zhu in collaboration with Hsian-Hua Tseng, both currently (winter, 2004) PhD students at Berkeley. I would like to thank them for allowing me to present their work at the 2

5 meeting and in this report. I would also like to thank Henrique Bursztyn, Marius Crainic and Rui Fernandes for their comments on this manuscript. 2 Integrating Lie algebroids by path spaces 2.1 Lie algebroids We recall (see [1] for an exposition) that a Lie algebroid over a manifold M is a vector bundle π : A M equipped with a Lie algebra structure [, ] (over R) on its space of sections Γ(A) and a bundle map ρ : A T M, called the anchor, satisfying the Leibniz rule [X, fy ](x) = f(x)[x, Y ](x) + ((ρx)f)(x)y (x) for all X, Y Γ(A), f C (M), and x M. It follows from the Leibniz rule and the Jacobi identity that the map from Γ(A) to vector fields on M induced by ρ is a Lie algebra homomorphism. Examples of Lie algebroids include tangent bundles and their integrable subbundles (with the induced bracket, and inclusion as anchor), gauge Lie algebroids T P/H where P is a principal H bundle over M, action Lie algebroids h M determined by homomorphisms from the Lie algebra h to vector fields on M, Lie algebras (when M is a point), and cotangent bundles of Poisson manifolds with a bracket on 1-forms originally introduced by Fuchssteiner [8]. It is useful to think of elements of a Lie algebroid A as generalized tangent vectors to M, so that the pair (M, A) becomes a new kind of space which we denote by M A ; i.e. we may write T (M A ) = A (and M T M is just M). The image of a Lie algebroid anchor is a family of subspaces in the fibres of T M which is integrable in the sense of Stefan and Sussmann; the integral manifolds form a singular foliation of M whose leaves may be thought of as the components of M A. In fact, these leaves may be taken as the actual components of a new differentiable structure on M, but with a topology different from the original one, and with (usually) an uncountable number of connected components. If we look only at the decomposition of M A into its components, though, we lose an important part of the information contained in A, namely the kernel of ρ. Elements of this kernel may be thought of as internal tangent vectors to M A ; the set of such internal vectors based at a single point inherits a Lie algebra structure from the bracket on sections. 3

6 2.2 Lie groupoids Lie algebras are the linearization of Lie groups. Since linearization involves differentiation, the reverse notion, passing from a Lie algebra to a Lie group, may be thought of as a kind of integration. In these terms, the geometric objects which integrate Lie algebroids are Lie groupoids (sometimes called differentiable groupoids). We recall that a groupoid over a set M is a set G whose elements are the morphisms in a category whose objects are the points of M, where all morphisms are invertible. M is sometimes called the base of the groupoid. More concretely, there is a target-source map (t, s) from G to M M (sometimes called the anchor of the groupoid, by analogy with the terminology for Lie algebroids), a units map ɛ : M G, an inversion map g g 1 from G to itself, and a multiplication map (g, h) gh with values in G, defined on the set of G (2) of pairs (g, h) for which s(g) = t(h). (Think of the elements g G as arrows going from right to left with head at s(g) and tail at t(g).) The algebraic axioms state that multiplication is associative whenever defined, t(gh) = t(g), s(gh) = s(h), ɛ(t(g))g = g = gɛ(s(g)), gg 1 = ɛ(t(g)), and g 1 g = ɛ(s(g)). G is a Lie groupoid if M and G are manifolds, if (t, s) is smooth and transverse to the diagonal so that G (2) is a manifold, and if the units, inversion, and multiplication maps are smooth. Examples of Lie groupoids (corresponding to the examples of Lie algebroids above) include M M with product (x, y)(y, z) = (x, z), those equivalence relations on manifolds which are submanifolds transverse to the diagonal, gauge groupoids P P/H of principal bundles, and action Lie groupoids H M associated with actions of the Lie group H on M. The counterparts of cotangent bundles of Poisson manifolds are symplectic groupoids, about which we will have say more at the end of this paper. We may think of the elements of G as generalized ordered pairs in M, so that (M, G) becomes a new kind of space which we could denote by M G. The image of (t, s) in M M is an equivalence relation whose equivalence classes are (not necessarily connected) submanifolds of M which may be thought of as the components of M G. The extra information in G is found in the isotropy subgroupoid, defined as the inverse under (t, s) of the diagonal in M M. This kernel of (t, s) is a family of groups G x lying over the points of M. G x is called the isotropy group of x; its elements may be thought of as internal symmetries of x. 4

7 2.3 From Lie groupoids to Lie algebroids The Lie algebroid of a given Lie groupoid is defined in same way as for Lie groups, where the Lie algebra consists of the left invariant vector fields, identified with their values at the unit element. On a groupoid, the left translations by elements of G act between fibres of t, so the left invariant vector fields are necessarily sections of the subbundle ker T t T G tangent to these fibres. As on a group, these vector fields are closed under the usual bracket operation and can be identified with their values along the unit section ɛ(m). Thus, the pulled back bundle ɛ (ker T t) becomes a Lie algebroid A(G) over M, with anchor the restriction of T s Development of paths Integration of left-invariant vector fields yields an exponential map from compactly supported (and certain other nicely behaved) sections of A(G) to the group B of submanifolds in G which map under (t, s) to the graphs of diffeomorphisms. (These submanifolds are known as bisections, or admissible sections.) There is, though, no exponential map from A(G) itself to G which might allow one to recover G from A(G) by a Campbell-Hausdorff construction as in the case of groups. However, there is a development bijection between certain paths in A(G) and the paths in G which start at unit elements and remain in a single fibre of t; this map makes it possible to transport algebraic structure from G to A(G). Defining the analogue of this transported structure for an arbitrary Lie algebroid A is not hard, and is a key step in integrating A. The inverse of the development map is easier to describe. (We follow [4], with slight changes in notation.) Given the groupoid G over M, we define the space P (G) of G-paths to consist of the C 2 (twice continuously differentiable) paths γ : [0, 1] G for which γ(0) is a unit element and 1 This construction illustrates a persistent headache in the theory of groupoids and Lie algebroids. In addition to the usual question of sign conventions, there is also a question of handedness conventions. For Lie groups, one defines the Lie algebra by left rather than right invariant vector fields in order to have the Lie bracket for GL(n) coincide with the usual commutator bracket of matrices. But when we adopt this convention for a Lie groupoid like the pair groupoid M M, the typical Lie algebroid element looks like an infinitesimal arrow pointing from its target to its source, rather than the other way around. This problem can be corrected if we think of the groupoid arrows as pointing from left to right, defining the anchor as (s, t) rather than (t, s), so that the condition for the product gh to exist is that t(g) = s(h). But this does not agree with the usual conventions for compositions of functions. If only function evaluation were commonly denoted (x)g instead of g(x)! (We could pronounce it x g d, as in x squared.) 5

8 t is constant along γ. The lift (or logarithm ) L(γ) is the path in A(G) which assigns to each τ [0, 1] the left-translate of γ (τ) to the unit element s(γ(τ)). The result is an A(G)-path, i.e. a C 1 path σ : [0, 1] A(G) satisfying the compatibility condition ρ σ = (π σ), where π : A(G) M is the vector bundle projection. The inverse map (development, a sort of exponentiation) from the space P (A(G)) of A(G)-paths to P (G) will be denoted by E. To develop an A(G)-path σ into G, we first extend σ to a compactly supported time-dependent section of A(G), which we may consider as a time-dependent left-invariant vector field on G. E(σ) is then defined as the integral curve of this vector field whose initial value is the unit at π(σ(0)). (Existence of the integral curve for all time is guaranteed by compact support of the section and left-invariance of the vector field.) Checking that E inverts L also assures us that the result is independent of the choice of extension vector field. When G is a group and σ is a constant path in its Lie algebra A(G), E(σ) is in fact the initial segment of the one-parameter group generated by the image of σ; i.e. it really is an exponential. But when G is a general Lie groupoid, a constant path in A(G) is an A(G)-path only when its value belongs to the kernel of the anchor, so that the exponentiation of individual elements of A(G) only reaches the isotropy subgroupoid of G. 2.5 Product structure on the path space The space G can be recovered from P (G): we identify two paths if they have the same value at τ = 1. To recover the multiplication on G, we need a product structure on paths. When G is a group, there are two such structures. The first is pointwise multiplication, and the second is concatenation, modified to be consistent with the condition that all paths start at the unit element, and so that the concatenated path is smooth. (The usual homotopy theory of Lie groups teaches us that these two products are equivalent up to homotopy.) Pointwise multiplication was used by Duistermaat and Kolk [6] to recover groups from their Lie algebras, but it is not available for paths in general groupoids, since the pointwise products generally do not exist. Therefore, we must use concatenation. When G is a groupoid over M, the concatenation product will make P (G) into something which resembles a groupoid over the same M. The compatibility condition for forming the product γ 1 γ 2 of two G-paths is that s(γ 1 (1)) = t(γ 2 (0)). When this condition is satisfied, we can replace γ 2 by its left translate 2 γ 2 = γ 1 (1)γ 2, which has the property that, since γ 2 (0) 2 This construction suggests that it might also be useful to think of P (G) as the space 6

9 is a unit element, γ 2 (0) = γ 1 (1). We can now define γ 1 γ 2 to be the usual concatenation of γ 1 and γ 2. Notice that (γ 1 γ 2 )(1) = γ 2 (1) = γ 1 (1)γ 2 (1), so evaluation at τ = 1 recovers the product on G from that on P (G). If we define the units of P (G) to be the constant curves with values in the units of G, and inversion of a path to be the same path traversed in the opposite direction and left-translated to begin at a unit, these operations, too, project to the corresponding operations in G. Although they project to the groupoid operations in G, the operations in P (G) do not satisfy the groupoid identities. The product is not associative, the units are not units, and the inversion of a path is not its inverse. We will deal with this issue later. An additional technical problem is that, although we have arranged via the left translations to make the concatenation of two paths continuous, there is no reason for it to be differentiable. To remedy the latter problem, we must either restrict to paths which have second-order contact with constant paths at their endpoints, or enlarge the path space to include piecewise smooth paths. The former approach seems to be the more effective one, and we will from now on use the notation P (G) to denote this restricted class of paths. The operations on P (G) which project to the groupoid structure on G can now be transfered by the development map to P (A(G)). The transferred product on P (A(G)) is simply concatenation, without any remnant of the left-translations. Inversion becomes time reversal combined with multiplication by 1, and the units are the constant paths with zero value. The boundary condition which allows for smooth concatenation is that the paths in P (A(G)) should vanish at their endpoints, along with their first derivatives. There is one more structure on P (G) which we must transfer to P (A(G)): the equivalence relation determined by the projection from P (G) to G. We first note that, although there may be many groupoids with the same A(G) (with different connectivity of their s and t fibres), they all have isomorphic P (G) s. In order to have G determined uniquely by P (G), we will assume that G has simply connected (i.e. connected with trivial fundamental group) s and t fibres. In this case, the fibres of the projection from P (G) to G may be thought of as homotopy classes of paths in the restructured space M G. They are the leaves of a foliation which, when transferred to P (A), may be described strictly in terms of a notion of homotopy of A-paths which makes of all paths in G on which t is constant, modulo left translation, instead of requiring the condition that the paths have unit initial values. The projection from P (G) to G would then be given by γ γ(0) 1 γ(1). 7

10 no reference to G. 2.6 From Lie algebroids to Lie groupoids We are now ready to go in the other direction. Starting with an arbitrary Lie algebroid A over M, we will attempt to realize it as (isomorphic to) A(G) for a groupoid G. Start with the space P (A) of A-paths, defined as above by the condition ρ γ = (π γ) and the boundary condition of vanishing at least to second order at the endpoints. Multiplication is defined by concatenation, inversion by reversal of direction and multiplication by 1; the units are the constant zero paths. We think of P (A) as diffeomorphic via a development map to P (G(A)) for a groupoid G(A) which is still to be defined. If G(A) exists, it must be the quotient of P (G(A)), and hence of P (A), by the foliation defined in the previous section. In other words, we define on P (A) the homotopy foliation H mentioned above. We now define G(A) as the leaf space P (A)/H. One can show that the algebraic structure on P (A) descends to G(A) and that all the anomalies vanish, so that G(A) becomes a true groupoid. What is harder is to put a geometric structure on this leaf space. When H is a simple foliation, then G(A) has a (unique) structure of smooth manifold for which the projection from P (A) is a smooth submersion. In this case (which is precisely the case when A is integrable), G(A) is a Lie groupoid whose Lie algebroid is naturally isomorphic to A. In the general case, G(A) has been described in [4] simply as a topological groupoid, but this is not enough geometric structure from which to recover the Lie algebroid A. It turns out that a suitable structure to put on G(A) is that of a differentiable stack. This kind of structure will be described in the next section. 3 Differentiable stacks Stacks originated in algebraic geometry, and their definition in that field is the basis for the differentiable version. We refer to the articles of Pronk [16] and Metzler [12] for careful treatments of the theory of differentiable stacks. What follows is a very rough, cartoon version of that theory. The most abstract definition of a differentiable stack is as a category X with a covariant functor F from X to the category C of smooth manifolds and smooth mappings. We will not state the axioms which must be satisfied by these objects, but we merely mention that the most important of them is a gluing property. From here, we proceed immediately to some examples. 8

11 First of all, for each ordinary manifold M, there is a stack X M for which the objects are smooth maps N M, where N ranges over the objects of C. The morphisms from N 1 M to N 2 M are smooth maps N 1 N 2 forming a commutative triangle with the other two, and the functor F is the one which forgets about M and looks only at the domains N and the maps between them. Another class of examples are the classifying stacks. For each Lie group H, BH is the stack whose objects are smooth principal bundles with structure group H and whose morphisms are H-equivariant maps. The functor F assigns to each bundle its base. In fact, there is a notion of principal bundle with structure groupoid, and BG can be defined in the same way when G is any Lie groupoid. In particular, when G is a groupoid over M consisting entirely of unit morphisms, BG is isomorphic to X M. Care must taken with respect to morphisms of stacks. The most obvious definition is as a functor between the categories, forming a commutative diagram with the functors to C. But two functors between stacks should be considered as giving the same morphism when they are related by a natural equivalence. Strictly speaking, the stacks should be regarded as objects in a 2-category in which these natural equivalences are the 2-morphisms. We refer to the references cited above for a careful discussion of this point. 3.1 Atlases as groupoids A somewhat less abstract description of stacks is via their presentations by groupoids. These presentations are known as atlases to see why, we will explain how to view ordinary smooth atlases in terms of groupoids. If M is a manifold with an atlas of coordinate charts U α V α, where the U α s are open subsets of M and the V α s are open subsets of R n, we can define a groupoid G M over the disjoint union Q of the V α s whose morphism space is the disjoint union of the pairwise intersections U α U β, with source and target maps given by the charts. As a set, M is isomorphic to the orbit space of this groupoid; its differentiable structure is the unique one for which the quotient map from Q to M is a local diffeomorphism. (The classifying stack BG M is isomorphic to X M.) If we have a second atlas for the same manifold, the corresponding groupoid G M over a manifold Q is not isomorphic to G M, but the two groupoids are equivalent by a (G M, G M )- bibundle, a notion which we now explain. In general, an equivalence bibundle for Lie groupoids G and G over manifolds Q and Q is a manifold B on which G and G act from the left and right respectively in such a way that, if we think of the elements of B as 9

12 morphisms from objects in Q to objects in Q and introduce another copy B of B to represent the inverses of these morphisms, then the disjoint union of G, G, B, and B becomes a groupoid over the disjoint union of Q and Q in such a way that its restrictions to Q and Q are G and G. Returning to our two atlases, we define the bibundle B to be the disjoint union of the intersections U α U β, with projections to Q and Q again given by the charts. Note that this is essentially the same as a common definition of compatibility between smooth atlases: their union should also be a smooth atlas. We can also define mappings between different manifolds in terms of a more general kind of bibundle. If G M and G M are groupoids associated with atlases for M and M, and if f : M M is a smooth map, then we define the bibundle B f to be the disjoint union of the open subsets f 1 (U α ) U β, with projection to Q given by the charts for M composed with f, while projection to Q is given by the charts for M. In other words, this bibundle encodes the coordinate representation of the map f. In this case, we do not introduce inverses for the morphisms in B, and the resulting object over the union of Q and Q is not a groupoid but a category having certain divisibility properties. Of course, even having chosen atlases for M and M, we should identify the map f not with a particular bibundle, but with an isomorphism class of bibundles. 3.2 Groupoids as atlases Following the example above, we may consider any differentiable groupoid as a presentation of (or atlas for ) some stack, with the stack itself defined by all presentations which are related by equivalence bibundles. Morphisms between stacks are then defined, with respect to given presentations, as isomorphism classes of bibundles. (Again we have a 2-category, with morphisms between bibundles as the 2-morphisms.) 3.3 Etale stacks and foliations A stack is called étale if it has a presentation by an étale groupoid (i.e. one whose source and target maps are étale). Such stacks arise naturally from foliations. If F is a foliation of a manifold P, there is an associated holonomy groupoid. (One may also use the monodromy groupoid.) Restricting to a global (not necessarily connected, of course) cross section Σ to the foliation gives an equivalent étale groupoid. Thus, the étale groupoids attached to different cross sections are all equivalent and are therefore presentations of 10

13 a single differential stack known as the leaf stack of the foliation. If the foliation is simple, the leaf space P/F is a manifold, and the leaf stack is just X P/F. (If we use the monodromy groupoid instead of the holonomy groupoid, the leaf stack incorporates the fundamental groups of the leaves.) The ideas in the paragraph above were already clearly expressed in work of Haefliger [9], van Est [7], and Pradines [15] from the mid 1980 s, and further developed by Moerdijk [13]. 4 Stacks as groupoids We now resume the discussion which was temporarily abandoned at the end of Section 2. Starting with a Lie algebroid A over M, we have on the Banach manifold P (A) of A-paths a collection of operations resembling those of a groupoid, and a foliation H such that these operations project to a groupoid structure on G(A) = P (A)/H. We now consider G(A) as a differentiable stack, the leaf stack of H. (Notice that, since H has finite codimension, the etale groupoids representing G(A) are purely finite-dimensional objects.) Tseng and Zhu [17] show that the groupoid operations on G(A) are morphisms of differentiable stacks. This is not the end of the story though. When we check the groupoid identities, they are not satisfied exactly, but only up to equivalence. A further analysis of the situation shows that all the equivalences involved here are consistent with one another. Finally, one defines the Lie algebroid for a general stack groupoid and shows that the Lie algebroid of G(A) is isomorphic to A, thus completing the integration of A. The main result of [4] is the establishment of computable criteria which determine precisely when G(A) is an ordinary manifold. When these constructions are applied in Poisson geometry to the Lie algebroid T P over a Poisson manifold P, one recovers the universal groupoid G(P ) of [2], which now becomes a differentiable stack. The notion of symplectic structure on a differentiable stack is easy to define, and G(P ) now becomes a symplectic stack groupoid. In particular, it provides a Morita equivalence from P to itself. To see that we have not gone too far by introducing these exotic objects, we note that, if P and Q are integrable Poisson manifolds which are Morita equivalent via a symplectic bibundle which is a differentiable stack, then this stack is actually isomorphic to a manifold, so the Poisson manifolds were already Morita equivalent in the usual sense. Thus, we have accomplished the goal of extending Xu s Morita equivalence to an equivalence relation among all Poisson manifolds. 11

14 References [1] Cannas da Silva, A., and Weinstein, A., Geometric Models for Noncommutative Algebras, Berkeley Math. Lecture Notes, Amer. Math. Soc., Providence, [2] Cattaneo, A.S., and Felder, G., Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, (ed. N. P. Landsman, M. Pflaum, M. Schlichenmaier), Progress in Mathematics 198 (2001), [3] Connes, A., Noncommutative Geometry, Academic Press, San Diego, [4] Crainic, M., and Fernandes, R., Integrability of Lie brackets, Ann. Math. 157 (2003), [5] Crainic, M., and Fernandes, R., Integrability of Poisson brackets, preprint math.dg/ [6] Duistermaat, J.J., and Kolk, J.A.C., Lie Groups, Springer-Verlag, Berlin-Heidelberg-New York, [7] Est, E.T. van, Rapport sur les S-atlas, in Structure Transverse des Feuilletages, Toulouse fév 1982, Astérisque 116 (1982), [8] Fuchssteiner, B., The Lie algebra structure of degenerate Hamiltonian and bi-hamiltonian systems, Progr. Theoret. Phys. 68 (1982), [9] Haefliger, A., Groupoïdes d holonomie et classifiants, in Structure Transverse des Feuilletages, Toulouse fév 1982, Astérisque 116 (1984), [10] Kontsevich, M., Deformation quantization of Poisson manifolds, I, preprint q-alg/ [11] Lie, S., Theorie der Transformationsgruppen, (Zweiter Abschnitt, unter Mitwirkung von Prof. Dr. Friedrich Engel), Teubner, Leipzig, [12] Metzler, D., Topological and smooth stacks, preprint math.dg/ [13] Moerdijk, I., Foliations, groupoids and Grothendieck étenndues, Rev. Acad. Cienc. Zaragoza (2) 48 (1993),

15 [14] Pradines, J., Troisième théorème de Lie sur les groupoïdes différentiables, C. R. Acad. Sc. Paris, 267 (1968), [15] Pradines, J., How to define the differentiable graph of a singular foliation, Cahiers de Top. et Géom. Diff. 26 (1985), [16] Pronk, D.A., Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996), [17] Tseng, H.-H., and Zhu, C., Integrating Lie algebroids via stacks, in preparation. [18] Xu, P. Morita equivalence of Poisson manifolds, Comm. Math. Phys. 142 (1991),