In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations

are submersions.

A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries.[1] Extending the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids.

Lie groupoids were introduced by Charles Ehresmann[2][3] under the name differentiable groupoids.

Definition and basic concepts

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A Lie groupoid consists of

  • two smooth manifolds   and  
  • two surjective submersions   (called, respectively, source and target projections)
  • a map   (called multiplication or composition map), where we use the notation  
  • a map   (called unit map or object inclusion map), where we use the notation  
  • a map   (called inversion), where we use the notation  

such that

  • the composition satisfies   and   for every   for which the composition is defined
  • the composition is associative, i.e.   for every   for which the composition is defined
  •   works as an identity, i.e.   for every   and   and   for every  
  •   works as an inverse, i.e.   and   for every  .

Using the language of category theory, a Lie groupoid can be more compactly defined as a groupoid (i.e. a small category where all the morphisms are invertible) such that the sets   of objects and   of morphisms are manifolds, the maps  ,  ,  ,   and   are smooth and   and   are submersions. A Lie groupoid is therefore not simply a groupoid object in the category of smooth manifolds: one has to ask the additional property that   and   are submersions.

Lie groupoids are often denoted by  , where the two arrows represent the source and the target. The notation   is also frequently used, especially when stressing the simplicial structure of the associated nerve.

In order to include more natural examples, the manifold   is not required in general to be Hausdorff or second countable (while   and all other spaces are).

Alternative definitions

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The original definition by Ehresmann required   and   to possess a smooth structure such that only   is smooth and the maps   and   are subimmersions (i.e. have locally constant rank). Such definition proved to be too weak and was replaced by Pradines with the one currently used.[4]

While some authors[5] introduced weaker definitions which did not require   and   to be submersions, these properties are fundamental to develop the entire Lie theory of groupoids and algebroids.

First properties

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The fact that the source and the target map of a Lie groupoid   are smooth submersions has some immediate consequences:

  • the  -fibres  , the  -fibres  , and the set of composable morphisms   are submanifolds;
  • the inversion map   is a diffeomorphism;
  • the unit map   is a smooth embedding;
  • the isotropy groups   are Lie groups;
  • the orbits   are immersed submanifolds;
  • the  -fibre   at a point   is a principal  -bundle over the orbit   at that point.

Subobjects and morphisms

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A Lie subgroupoid of a Lie groupoid   is a subgroupoid   (i.e. a subcategory of the category  ) with the extra requirement that   is an immersed submanifold. As for a subcategory, a (Lie) subgroupoid is called wide if  . Any Lie groupoid   has two canonical wide subgroupoids:

  • the unit/identity Lie subgroupoid  ;
  • the inner subgroupoid  , i.e. the bundle of isotropy groups (which however may fail to be smooth in general).

A normal Lie subgroupoid is a wide Lie subgroupoid   inside   such that, for every   with  , one has  . The isotropy groups of   are therefore normal subgroups of the isotropy groups of  .

A Lie groupoid morphism between two Lie groupoids   and   is a groupoid morphism   (i.e. a functor between the categories   and  ), where both   and   are smooth. The kernel   of a morphism between Lie groupoids over the same base manifold is automatically a normal Lie subgroupoid.

The quotient   has a natural groupoid structure such that the projection   is a groupoid morphism; however, unlike quotients of Lie groups,   may fail to be a Lie groupoid in general. Accordingly, the isomorphism theorems for groupoids cannot be specialised to the entire category of Lie groupoids, but only to special classes.[6]

A Lie groupoid is called abelian if its isotropy Lie groups are abelian. For similar reasons as above, while the definition of abelianisation of a group extends to set-theoretical groupoids, in the Lie case the analogue of the quotient   may not exist or be smooth.[7]

Bisections

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A bisection of a Lie groupoid   is a smooth map   such that   and   is a diffeomorphism of  . In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold   such that   and   are diffeomorphisms; the relation between the two definitions is given by  .[8]

The set of bisections forms a group, with the multiplication   defined as and inversion defined as Note that the definition is given in such a way that, if   and  , then   and  .

The group of bisections can be given the compact-open topology, as well as an (infinite-dimensional) structure of Fréchet manifold compatible with the group structure, making it into a Fréchet-Lie group.

A local bisection   is defined analogously, but the multiplication between local bisections is of course only partially defined.

Examples

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Trivial and extreme cases

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  • Lie groupoids   with one object are the same thing as Lie groups.
  • Given any manifold  , there is a Lie groupoid   called the pair groupoid, with precisely one morphism from any object to any other.
  • The two previous examples are particular cases of the trivial groupoid  , with structure maps  ,  ,  ,   and  .
  • Given any manifold  , there is a Lie groupoid   called the unit groupoid, with precisely one morphism from one object to itself, namely the identity, and no morphisms between different objects.
  • More generally, Lie groupoids with   are the same thing as bundle of Lie groups (not necessarily locally trivial). For instance, any vector bundle is a bundle of abelian groups, so it is in particular a(n abelian) Lie groupoid.

Constructions from other Lie groupoids

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  • Given any Lie groupoid   and a surjective submersion  , there is a Lie groupoid  , called its pullback groupoid or induced groupoid, where   contains triples   such that   and  , and the multiplication is defined using the multiplication of  . For instance, the pullback of the pair groupoid of   is the pair groupoid of  .
  • Given any two Lie groupoids   and  , there is a Lie groupoid  , called their direct product, such that the groupoid morphisms   and   are surjective submersions.
  • Given any Lie groupoid  , there is a Lie groupoid  , called its tangent groupoid, obtained by considering the tangent bundle of   and   and the differential of the structure maps.
  • Given any Lie groupoid  , there is a Lie groupoid  , called its cotangent groupoid obtained by considering the cotangent bundle of  , the dual of the Lie algebroid   (see below), and suitable structure maps involving the differentials of the left and right translations.
  • Given any Lie groupoid  , there is a Lie groupoid  , called its jet groupoid, obtained by considering the k-jets of the local bisections of   (with smooth structure inherited from the jet bundle of  ) and setting  ,  ,  ,   and  .

Examples from differential geometry

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  • Given a submersion  , there is a Lie groupoid  , called the submersion groupoid or fibred pair groupoid, whose structure maps are induced from the pair groupoid   (the condition that   is a submersion ensures the smoothness of  ). If   is a point, one recovers the pair groupoid.
  • Given a Lie group   acting on a manifold  , there is a Lie groupoid  , called the action groupoid or translation groupoid, with one morphism for each triple   with  .
  • Given any vector bundle  , there is a Lie groupoid  , called the general linear groupoid, with morphisms between   being linear isomorphisms between the fibres   and  . For instance, if   is the trivial vector bundle of rank  , then   is the action groupoid.
  • Any principal bundle   with structure group   defines a Lie groupoid  , where   acts on the pairs   componentwise, called the gauge groupoid. The multiplication is defined via compatible representatives as in the pair groupoid.
  • Any foliation   on a manifold   defines two Lie groupoids,   (or  ) and  , called respectively the monodromy/homotopy/fundamental groupoid and the holonomy groupoid of  , whose morphisms consist of the homotopy, respectively holonomy, equivalence classes of paths entirely lying in a leaf of  . For instance, when   is the trivial foliation with only one leaf, one recovers, respectively, the fundamental groupoid and the pair groupoid of  . On the other hand, when   is a simple foliation, i.e. the foliation by (connected) fibres of a submersion  , its holonomy groupoid is precisely the submersion groupoid   but its monodromy groupoid may even fail to be Hausdorff, due to a general criterion in terms of vanishing cycles.[9] In general, many elementary foliations give rise to monodromy and holonomy groupoids which are not Hausdorff.
  • Given any pseudogroup  , there is a Lie groupoid  , called its germ groupoid, endowed with the sheaf topology and with structure maps analogous to those of the jet groupoid. This is another natural example of Lie groupoid whose arrow space is not Hausdorff nor second countable.

Important classes of Lie groupoids

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Note that some of the following classes make sense already in the category of set-theoretical or topological groupoids.

Transitive groupoids

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A Lie groupoid is transitive (in older literature also called connected) if it satisfies one of the following equivalent conditions:

  • there is only one orbit;
  • there is at least a morphism between any two objects;
  • the map   (also known as the anchor of  ) is surjective.

Gauge groupoids constitute the prototypical examples of transitive Lie groupoids: indeed, any transitive Lie groupoid is isomorphic to the gauge groupoid of some principal bundle, namely the  -bundle  , for any point  . For instance:

  • the trivial Lie groupoid   is transitive and arise from the trivial principal  -bundle  . As particular cases, Lie groups   and pair groupoids   are trivially transitive, and arise, respectively, from the principal  -bundle  , and from the principal  -bundle  ;
  • an action groupoid   is transitive if and only if the group action is transitive, and in such case it arises from the principal bundle   with structure group the isotropy group (at an arbitrary point);
  • the general linear groupoid of   is transitive, and arises from the frame bundle  ;
  • pullback groupoids, jet groupoids and tangent groupoids of   are transitive if and only if   is transitive.

As a less trivial instance of the correspondence between transitive Lie groupoids and principal bundles, consider the fundamental groupoid   of a (connected) smooth manifold  . This is naturally a topological groupoid, which is moreover transitive; one can see that   is isomorphic to the gauge groupoid of the universal cover of  . Accordingly,   inherits a smooth structure which makes it into a Lie groupoid.

Submersions groupoids   are an example of non-transitive Lie groupoids, whose orbits are precisely the fibres of  .

A stronger notion of transitivity requires the anchor   to be a surjective submersion. Such condition is also called local triviality, because   becomes locally isomorphic (as Lie groupoid) to a trivial groupoid over any open   (as a consequence of the local triviality of principal bundles).[6]

When the space   is second countable, transitivity implies local triviality. Accordingly, these two conditions are equivalent for many examples but not for all of them: for instance, if   is a transitive pseudogroup, its germ groupoid   is transitive but not locally trivial.

Proper groupoids

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A Lie groupoid is called proper if   is a proper map. As a consequence

  • all isotropy groups of   are compact;
  • all orbits of   are closed submanifolds;
  • the orbit space   is Hausdorff.

For instance:

  • a Lie group is proper if and only if it is compact;
  • pair groupoids are always proper;
  • unit groupoids are always proper;
  • an action groupoid is proper if and only if the action is proper;
  • the fundamental groupoid is proper if and only if the fundamental groups are finite.

As seen above, properness for Lie groupoids is the "right" analogue of compactness for Lie groups. One could also consider more "natural" conditions, e.g. asking that the source map   is proper (then   is called s-proper), or that the entire space   is compact (then   is called compact), but these requirements turns out to be too strict for many examples and applications.[10]

Étale groupoids

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A Lie groupoid is called étale if it satisfies one of the following equivalent conditions:

  • the dimensions of   and   are equal;
  •   is a local diffeomorphism;
  • all the  -fibres are discrete

As a consequence, also the  -fibres, the isotropy groups and the orbits become discrete.

For instance:

  • a Lie group is étale if and only if it is discrete;
  • pair groupoids are never étale;
  • unit groupoids are always étale;
  • an action groupoid is étale if and only if   is discrete;
  • germ groupoids of pseudogroups are always étale.

Effective groupoids

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An étale groupoid is called effective if, for any two local bisections  , the condition   implies  . For instance:

  • Lie groups are effective if and only if are trivial;
  • unit groupoids are always effective;
  • an action groupoid is effective if the  -action is free and   is discrete.

In general, any effective étale groupoid arise as the germ groupoid of some pseudogroup.[11] However, a (more involved) definition of effectiveness, which does not assume the étale property, can also be given.

Source-connected groupoids

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A Lie groupoid is called  -connected if all its  -fibres are connected. Similarly, one talks about  -simply connected groupoids (when the  -fibres are simply connected) or source-k-connected groupoids (when the  -fibres are k-connected, i.e. the first   homotopy groups are trivial).

Note that the entire space of arrows   is not asked to satisfy any connectedness hypothesis. However, if   is a source- -connected Lie groupoid over a  -connected manifold, then   itself is automatically  -connected.

For instanceː

  • Lie groups are source  -connected if and only if they are  -connected;
  • a pair groupoid is source  -connected if and only if   is  -connected;
  • unit groupoids are always source  -connected;
  • action groupoids are source  -connected if and only if   is  -connected;
  • monodromy groupoids (hence also fundamental groupoids) are source simply connected;
  • a gauge groupoid associated to a principal bundle   is source  -connected if and only if the total space   is.
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Actions and principal bundles

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Recall that an action of a groupoid   on a set   along a function   is defined via a collection of maps   for each morphism   between  . Accordingly, an action of a Lie groupoid   on a manifold   along a smooth map   consists of a groupoid action where the maps   are smooth. Of course, for every   there is an induced smooth action of the isotropy group   on the fibre  .

Given a Lie groupoid  , a principal  -bundle consists of a  -space   and a  -invariant surjective submersion   such that is a diffeomorphism. Equivalent (but more involved) definitions can be given using  -valued cocycles or local trivialisations.

When   is a Lie groupoid over a point, one recovers, respectively, standard Lie group actions and principal bundles.

Representations

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A representation of a Lie groupoid   consists of a Lie groupoid action on a vector bundle  , such that the action is fibrewise linear, i.e. each bijection   is a linear isomorphism. Equivalently, a representation of   on   can be described as a Lie groupoid morphism from   to the general linear groupoid  .

Of course, any fibre   becomes a representation of the isotropy group  . More generally, representations of transitive Lie groupoids are uniquely determined by representations of their isotropy groups, via the construction of the associated vector bundle.

Examples of Lie groupoids representations include the following:

  • representations of Lie groups   recover standard Lie group representations
  • representations of pair groupoids   are trivial vector bundles
  • representations of unit groupoids   are vector bundles
  • representations of action groupoid   are  -equivariant vector bundles
  • representations of fundamental groupoids   are vector bundles endowed with flat connections

The set   of isomorphism classes of representations of a Lie groupoid   has a natural structure of semiring, with direct sums and tensor products of vector bundles.

Differentiable cohomology

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The notion of differentiable cohomology for Lie groups generalises naturally also to Lie groupoids: the definition relies on the simplicial structure of the nerve   of  , viewed as a category.

More precisely, recall that the space   consists of strings of   composable morphisms, i.e.

 

and consider the map  .

A differentiable  -cochain of   with coefficients in some representation   is a smooth section of the pullback vector bundle  . One denotes by   the space of such  -cochains, and considers the differential  , defined as

 

Then   becomes a cochain complex and its cohomology, denoted by  , is called the differentiable cohomology of   with coefficients in  . Note that, since the differential at degree zero is  , one has always  .

Of course, the differentiable cohomology of   as a Lie groupoid coincides with the standard differentiable cohomology of   as a Lie group (in particular, for discrete groups one recovers the usual group cohomology). On the other hand, for any proper Lie groupoid  , one can prove that   for every  .[12]

The Lie algebroid of a Lie groupoid

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Any Lie groupoid   has an associated Lie algebroid  , obtained with a construction similar to the one which associates a Lie algebra to any Lie groupː

  • the vector bundle   is the vertical bundle with respect to the source map, restricted to the elements tangent to the identities, i.e.  ;
  • the Lie bracket is obtained by identifying   with the left-invariant vector fields on  , and by transporting their Lie bracket to  ;
  • the anchor map   is the differential of the target map   restricted to  .

The Lie group–Lie algebra correspondence generalises to some extends also to Lie groupoids: the first two Lie's theorem (also known as the subgroups–subalgebras theorem and the homomorphisms theorem) can indeed be easily adapted to this setting.

In particular, as in standard Lie theory, for any s-connected Lie groupoid   there is a unique (up to isomorphism) s-simply connected Lie groupoid   with the same Lie algebroid of  , and a local diffeomorphism   which is a groupoid morphism. For instance,

  • given any connected manifold   its pair groupoid   is s-connected but not s-simply connected, while its fundamental groupoid   is. They both have the same Lie algebroid, namely the tangent bundle  , and the local diffeomorphism   is given by  .
  • given any foliation   on  , its holonomy groupoid   is s-connected but not s-simply connected, while its monodromy groupoid   is. They both have the same Lie algebroid, namely the foliation algebroid  , and the local diffeomorphism   is given by   (since the homotopy classes are smaller than the holonomy ones).

However, there is no analogue of Lie's third theoremː while several classes of Lie algebroids are integrable, there are examples of Lie algebroids, for instance related to foliation theory, which do not admit an integrating Lie groupoid.[13] The general obstructions to the existence of such integration depend on the topology of  .[14]

Morita equivalence

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As discussed above, the standard notion of (iso)morphism of groupoids (viewed as functors between categories) restricts naturally to Lie groupoids. However, there is a more coarse notion of equivalence, called Morita equivalence, which is more flexible and useful in applications.

First, a Morita map (also known as a weak equivalence or essential equivalence) between two Lie groupoids   and   consists of a Lie groupoid morphism from G to H which is moreover fully faithful and essentially surjective (adapting these categorical notions to the smooth context). We say that two Lie groupoids   and   are Morita equivalent if and only if there exists a third Lie groupoid   together with two Morita maps from G to K and from H to K.

A more explicit description of Morita equivalence (e.g. useful to check that it is an equivalence relation) requires the existence of two surjective submersions   and   together with a left  -action and a right  -action, commuting with each other and making   into a principal bi-bundle.[15]

Morita invariance

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Many properties of Lie groupoids, e.g. being proper, being Hausdorff or being transitive, are Morita invariant. On the other hand, being étale is not Morita invariant.

In addition, a Morita equivalence between   and   preserves their transverse geometry, i.e. it induces:

  • a homeomorphism between the orbit spaces   and  ;
  • an isomorphism   between the isotropy groups at corresponding points   and  ;
  • an isomorphism   between the normal representations of the isotropy groups at corresponding points   and  .

Last, the differentiable cohomologies of two Morita equivalent Lie groupoids are isomorphic.[12]

Examples

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  • Isomorphic Lie groupoids are trivially Morita equivalent.
  • Two Lie groups are Morita equivalent if and only if they are isomorphic as Lie groups.
  • Two unit groupoids are Morita equivalent if and only if the base manifolds are diffeomorphic.
  • Any transitive Lie groupoid is Morita equivalent to its isotropy groups.
  • Given a Lie groupoid   and a surjective submersion  , the pullback groupoid   is Morita equivalent to  .
  • Given a free and proper Lie group action of   on   (therefore the quotient   is a manifold), the action groupoid   is Morita equivalent to the unit groupoid  .
  • A Lie groupoid   is Morita equivalent to an étale groupoid if and only if all isotropy groups of   are discrete.[16]

A concrete instance of the last example goes as follows. Let M be a smooth manifold and   an open cover of  . Its Čech groupoid   is defined by the disjoint unions   and  , where  . The source and target map are defined as the embeddings   and  , and the multiplication is the obvious one if we read the   as subsets of M (compatible points in   and   actually are the same in   and also lie in  ). The Čech groupoid is in fact the pullback groupoid, under the obvious submersion  , of the unit groupoid  . As such, Čech groupoids associated to different open covers of   are Morita equivalent.

Smooth stacks

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Investigating the structure of the orbit space of a Lie groupoid leads to the notion of a smooth stack. For instance, the orbit space is a smooth manifold if the isotropy groups are trivial (as in the example of the Čech groupoid), but it is not smooth in general. The solution is to revert the problem and to define a smooth stack as a Morita-equivalence class of Lie groupoids. The natural geometric objects living on the stack are the geometric objects on Lie groupoids invariant under Morita-equivalence: an example is the Lie groupoid cohomology.

Since the notion of smooth stack is quite general, obviously all smooth manifolds are smooth stacks. Other classes of examples include orbifolds, which are (equivalence classes of) proper étale Lie groupoids, and orbit spaces of foliations.

References

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  1. ^ Weinstein, Alan (1996-02-03). "Groupoids: unifying internal and external symmetry" (PDF). Notices of the American Mathematical Society. 43: 744–752. arXiv:math/9602220.
  2. ^ Ehresmann, Charles (1959). "Catégories topologiques et categories différentiables" [Topological categories and differentiable categories] (PDF). Colloque de Géométrie différentielle globale (in French). CBRM, Bruxelles: 137–150.
  3. ^ Ehresmann, Charles (1963). "Catégories structurées" [Structured categories]. Annales scientifiques de l'École Normale Supérieure (in French). 80 (4): 349–426. doi:10.24033/asens.1125.
  4. ^ Pradines, Jean (1966). "Théorie de Lie pour les groupoïdes dif́férentiables. Relations entre propriétés locales et globales" [Lie theory for differentiable groupoids. Relations between local and global properties]. C. R. Acad. Sci. Paris (in French). 263: 907–910 – via Gallica.
  5. ^ Kumpera, Antonio; Spencer, Donald Clayton (2016-03-02). Lie Equations, Vol. I. Princeton University Press. doi:10.1515/9781400881734. ISBN 978-1-4008-8173-4.
  6. ^ a b Mackenzie, K. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511661839. ISBN 978-0-521-34882-9.
  7. ^ Contreras, Ivan; Fernandes, Rui Loja (2021-06-28). "Genus Integration, Abelianization, and Extended Monodromy". International Mathematics Research Notices. 2021 (14): 10798–10840. arXiv:1805.12043. doi:10.1093/imrn/rnz133. ISSN 1073-7928.
  8. ^ Albert, Claude; Dazord, Pierre; Weinstein, Alan (1987). "Groupoïdes Symplectiques" [Symplectic Groupoids]. Pub. Dept. Math. Lyon (in French) (2A): 1–62 – via NUMDAM [fr].
  9. ^ Cuesta, F. Alcalde; Hector, G. (1997-09-01). "Feuilletages en surfaces, cycles évanouissants et variétés de Poisson" [Foliations on surfaces, vanishing cycles and Poisson manifolds]. Monatshefte für Mathematik (in French). 124 (3): 191–213. doi:10.1007/BF01298244. ISSN 1436-5081. S2CID 119369484.
  10. ^ Crainic, Marius; Loja Fernandes, Rui; Martínez Torres, David (2019-11-01). "Poisson manifolds of compact types (PMCT 1)". Journal für die reine und angewandte Mathematik (Crelle's Journal). 2019 (756): 101–149. arXiv:1510.07108. doi:10.1515/crelle-2017-0006. ISSN 1435-5345. S2CID 7668127.
  11. ^ Haefliger, André (1958-12-01). "Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes" [Foliated structures and cohomology taking values in a sheaf of groupoids]. Commentarii Mathematici Helvetici (in French). 32 (1): 248–329. doi:10.1007/BF02564582. ISSN 1420-8946. S2CID 121138118.
  12. ^ a b Crainic, Marius (2003-12-31). "Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes". Commentarii Mathematici Helvetici. 78 (4): 681–721. arXiv:math/0008064. doi:10.1007/s00014-001-0766-9. ISSN 0010-2571.
  13. ^ Almeida, Rui; Molino, Pierre (1985). "Suites d'Atiyah et feuilletages transversalement complets" [Atiyah sequences and transversely complete foliations]. Comptes Rendus de l'Académie des Sciences, Série I (in French). 300: 13–15 – via Gallica.
  14. ^ Crainic, Marius; Fernandes, Rui (2003-03-01). "Integrability of Lie brackets". Annals of Mathematics. 157 (2): 575–620. arXiv:math/0105033. doi:10.4007/annals.2003.157.575. ISSN 0003-486X.
  15. ^ del Hoyo, Matias (2013). "Lie groupoids and their orbispaces". Portugaliae Mathematica. 70 (2): 161–209. arXiv:1212.6714. doi:10.4171/PM/1930. ISSN 0032-5155.
  16. ^ Crainic, Marius; Moerdijk, Ieke (2001-02-10). "Foliation Groupoids and Their Cyclic Homology". Advances in Mathematics. 157 (2): 177–197. arXiv:math/0003119. doi:10.1006/aima.2000.1944. ISSN 0001-8708.

Books

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