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. 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 under the name differentiable groupoids.

Definition and basic concepts

A Lie groupoid consists of such that 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 M of objects and G of morphisms are manifolds, the maps s, t, m, i and u are smooth and s and t 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 s and t 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 G is not required in general to be Hausdorff or second countable (while M and all other spaces are).

Alternative definitions

The original definition by Ehresmann required G and M to possess a smooth structure such that only m 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. While some authors introduced weaker definitions which did not require s and t to be submersions, these properties are fundamental to develop the entire Lie theory of groupoids and algebroids.

First properties

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

Subobjects and morphisms

A Lie subgroupoid of a Lie groupoid is a subgroupoid (i.e. a subcategory of the category G) with the extra requirement that is an immersed submanifold. As for a subcategory, a (Lie) subgroupoid is called wide if N = M. Any Lie groupoid has two canonical wide subgroupoids: A normal Lie subgroupoid is a wide Lie subgroupoid inside IG such that, for every with, one has. The isotropy groups of H are therefore normal subgroups of the isotropy groups of G. A Lie groupoid morphism between two Lie groupoids and is a groupoid morphism (i.e. a functor between the categories G and H), where both F and f 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, G/\ker(F) 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. 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.

Bisections

A bisection of a Lie groupoid is a smooth map b: M \to G such that and t \circ b is a diffeomorphism of M. 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 B = b(M). The set of bisections forms a group, with the multiplication defined asand inversion defined asNote 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

Trivial and extreme cases

Constructions from other Lie groupoids

Examples from differential geometry

Important classes of Lie groupoids

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

Transitive groupoids

A Lie groupoid is transitive (in older literature also called connected) if it satisfies one of the following equivalent conditions: 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 G_x-bundle, for any point x \in M. For instance: As a less trivial instance of the correspondence between transitive Lie groupoids and principal bundles, consider the fundamental groupoid \Pi_1(M) of a (connected) smooth manifold M. This is naturally a topological groupoid, which is moreover transitive; one can see that \Pi_1(M) is isomorphic to the gauge groupoid of the universal cover of M. Accordingly, \Pi_1(M) 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 \mu. A stronger notion of transitivity requires the anchor to be a surjective submersion. Such condition is also called local triviality, because G becomes locally isomorphic (as Lie groupoid) to a trivial groupoid over any open (as a consequence of the local triviality of principal bundles). When the space G 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 \Gamma is a transitive pseudogroup, its germ groupoid is transitive but not locally trivial.

Proper groupoids

A Lie groupoid is called proper if is a proper map. As a consequence For instance: 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 s: G \to M is proper (then is called s-proper), or that the entire space G is compact (then is called compact), but these requirements turns out to be too strict for many examples and applications.

Étale groupoids

A Lie groupoid is called étale if it satisfies one of the following equivalent conditions: As a consequence, also the t-fibres, the isotropy groups and the orbits become discrete. For instance:

Effective groupoids

An étale groupoid is called effective if, for any two local bisections b_1, b_2, the condition implies b_1 = b_2. For instance: In general, any effective étale groupoid arise as the germ groupoid of some pseudogroup. However, a (more involved) definition of effectiveness, which does not assume the étale property, can also be given.

Source-connected groupoids

A Lie groupoid is called s-connected if all its s-fibres are connected. Similarly, one talks about s-simply connected groupoids (when the s-fibres are simply connected) or source-k-connected groupoids (when the s-fibres are k-connected, i.e. the first k homotopy groups are trivial). Note that the entire space of arrows G is not asked to satisfy any connectedness hypothesis. However, if G is a source-k-connected Lie groupoid over a k-connected manifold, then G itself is automatically k-connected. For instanceː

Further related concepts

Actions and principal bundles

Recall that an action of a groupoid on a set P along a function is defined via a collection of maps for each morphism g \in G between x,y \in M. Accordingly, an action of a Lie groupoid on a manifold P along a smooth map consists of a groupoid action where the maps are smooth. Of course, for every x \in M there is an induced smooth action of the isotropy group G_x on the fibre \mu^{-1}(x). Given a Lie groupoid, a principal G-bundle consists of a G-space P and a G-invariant surjective submersion such thatis a diffeomorphism. Equivalent (but more involved) definitions can be given using G-valued cocycles or local trivialisations. When G is a Lie groupoid over a point, one recovers, respectively, standard Lie group actions and principal bundles.

Representations

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 G on E can be described as a Lie groupoid morphism from G to the general linear groupoid GL(E). Of course, any fibre E_x becomes a representation of the isotropy group G_x. 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: 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

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 G^{(n)} consists of strings of n composable morphisms, i.e. and consider the map. A differentiable n-cochain of with coefficients in some representation E \to M is a smooth section of the pullback vector bundle. One denotes by C^n(G,E) the space of such n-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 E \to M. 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 G 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 n > 0.

The Lie algebroid of a Lie groupoid

Any Lie groupoid has an associated Lie algebroid A \to M, obtained with a construction similar to the one which associates a Lie algebra to any Lie groupː 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 G there is a unique (up to isomorphism) s-simply connected Lie groupoid \tilde{G} with the same Lie algebroid of G, and a local diffeomorphism which is a groupoid morphism. For instance, 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. The general obstructions to the existence of such integration depend on the topology of G.

Morita equivalence

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 P \to G_0 and P \to H_0 together with a left G-action and a right H-action, commuting with each other and making P into a principal bi-bundle.

Morita invariance

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: Last, the differentiable cohomologies of two Morita equivalent Lie groupoids are isomorphic.

Examples

A concrete instance of the last example goes as follows. Let M be a smooth manifold and an open cover of M. 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 M and also lie in ). The Čech groupoid is in fact the pullback groupoid, under the obvious submersion p:G_0\to M, of the unit groupoid. As such, Čech groupoids associated to different open covers of M are Morita equivalent.

Smooth stacks

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.

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