nLab locally compact groupoid

Idea

There are a number of slightly different definitions in the literature, this page will try to give a general overarching definition.

Definition

A topological groupoid $X_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} X_0$ is called locally compact if the maps $s$ and $t$ are locally proper open maps.

This is far more general than is usually assumed. One can additionally assume that either $X_1$ is locally compact or $X_0$ is locally compact, or that $X_0$ is Hausdorff or $X_1$ is locally Hausdorff, or that $X_1$ is ∞-compact

One usually wants to place a Haar system? on a locally compact groupoid. The obvious topological requirement of local compactness is further refined in order to make such a system of measures well-behaved.

Examples

• Any finite-dimensional Lie groupoid is locally compact
• The action groupoid of a locally compact group on a locally compact topological space is (?should be) a locally compact groupoid.

References

An early reference is

• Jean Renault, A Groupoid Approach to C-Algebras_, Lecture Notes in Mathematics 793 (1980) doi:10.1007/BFb0091072

A version where Hausdorffness is weakened so that only the space of objects is Hausdorff is

• Jean-Louis Tu, Non-Hausdorff groupoids, proper actions and K-theory, Documenta Mathematica, Vol. 9 (2004) pp565-597, journal page, arXiv:math/0403071