Concept
In mathematics, a Lie groupoid is a groupoid where the set of s and the set of morphisms are both manifolds, all the 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. 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 , a Lie groupoid can be more compactly defined as a groupoid (i.e. a 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 : 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 .