Lie groupoid

In mathematics, a Lie groupoid is a groupoid where the set ${\displaystyle \operatorname {Ob} }$ of objects and the set ${\displaystyle \operatorname {Mor} }$ 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

${\displaystyle s,t:\operatorname {Mor} \to \operatorname {Ob} }$

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.