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.
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.
is also frequently used, especially when stressing the simplicial structure of the associated nerve.
to be submersions, these properties are fundamental to develop the entire Lie theory of groupoids and algebroids.
The fact that the source and the target map of a Lie groupoid
is a groupoid morphism; however, unlike quotients of Lie groups,
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
In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold
Note that some of the following classes make sense already in the category of set-theoretical or topological groupoids.
inherits a smooth structure which makes it into a Lie groupoid.
are an example of non-transitive Lie groupoids, whose orbits are precisely the fibres of
One could also consider more "natural" conditions, e.g. asking that the source map
is called compact), but these requirements turns out to be too strict for many examples and applications.
[10] A Lie groupoid is called étale if it satisfies one of the following equivalent conditions: As a consequence, also the
For instance: An étale groupoid is called effective if, for any two local bisections
[11] However, a (more involved) definition of effectiveness, which does not assume the étale property, can also be given.
consists of a Lie groupoid action on a vector bundle
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
has a natural structure of semiring, with direct sums and tensor products of vector bundles.
The notion of differentiable cohomology for Lie groups generalises naturally also to Lie groupoids: the definition relies on the simplicial structure of the nerve
as a Lie groupoid coincides with the standard differentiable cohomology of
there is a unique (up to isomorphism) s-simply connected Lie groupoid
[13] The general obstructions to the existence of such integration depend on the topology of
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).
[15] Many properties of Lie groupoids, e.g. being proper, being Hausdorff or being transitive, are Morita invariant.
preserves their transverse geometry, i.e. it induces: Last, the differentiable cohomologies of two Morita equivalent Lie groupoids are isomorphic.
Investigating the structure of the orbit space of a Lie groupoid leads to the notion of a smooth stack.
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.