Lie groupoid

Summary

A Lie groupoid is a groupoid whose objects and arrows each form a smooth manifold. All its structure maps are smooth, and the source and target maps are submersions. It generalises Lie groups, which are Lie groupoids with a single object, and also smooth group actions, foliations and principal bundles. Lie groupoids give a common language for symmetries that vary from point to point, and they have Lie algebroids as their infinitesimal versions.