Morita equivalence of Lie groupoids

Summary

Two Lie groupoids are Morita equivalent when they describe the same transverse geometry, even if their spaces of objects differ. Formally, they are joined by a third Lie groupoid that maps to both by weak equivalences, or equivalently by a principal bibundle. Morita equivalent groupoids have homeomorphic orbit spaces, isomorphic isotropy groups along corresponding orbits and the same invariants such as cohomology and K-theory. This notion makes Lie groupoids a model for singular spaces such as orbifolds, leaf spaces of foliations and quotients by group actions.