nLab
proper topological groupoid

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

Definition

A topological groupoid X 1tsX 0X_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} X_0 is called proper if the continuous map

(s,t):X 1X 0×X 0 (s,t):X_1 \to X_0\times X_0

is a proper map.

Properties

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

Examples

A Lie groupoid is called a proper Lie groupoid if its underlying topological groupoid is proper.

An orbifold is a proper Lie groupoid which is also an étale groupoid.

Revised on February 2, 2012 11:44:25 by Urs Schreiber (82.169.65.155)