Lie Groupoid

In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, the source and target operations

are submersions, and all the category operations (source and target, composition, and identity-assigning map) are smooth.

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. Just as every Lie group has a Lie algebra, every Lie groupoid has a Lie algebroid.