nLab
proper Lie groupoid

Definition

A Lie groupoid $(C_1\stackrel{\stackrel{s}{\to }}{\underset{t}{\to }}{C}_0)$ is proper if its underlying topological groupoid is a proper topological groupoid, hence if

$$(s,t):C_1\to C_0\times C_0$$(s,t) : C_1 \to C_0 \times C_0

is a proper map.

So in particular the automorphism group of any object in a proper Lie groupoid is a compact Lie group. In this sense proper Lie groupoids generalize compact Lie groups.

Examples

• An orbifold is (presented by) a proper étale groupoid.

Related concepts

• effective Lie groupoid

References

• M.J. Pflaum, H. Posthuma, X. Tang, Geometry of orbit spaces of proper Lie groupoids (arXiv:1101.0180)