nLab
locally compact groupoid

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Category theory

Contents

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 1tsX 0X_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} X_0 is called locally compact if the maps ss and tt are locally proper open maps.

This is far more general than is usually assumed. One can additionally assume that either X 1X_1 is locally compact or X 0X_0 is locally compact, or that X 0X_0 is Hausdorff or X 1X_1 is locally Hausdorff, or that X 1X_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

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

Last revised on February 8, 2019 at 00:44:23. See the history of this page for a list of all contributions to it.