nLab
topological groupoid

Context

Higher geometry

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

Contents

Definition

General

A topological groupoid is an internal groupoid in the category Top.

So this is a groupoid with a topological space of objects and one of morphisms, and all structure maps (source, target, identity, composition, inverse) are continuous maps.

A topological groupoid CC is called an open topological groupoid if the source map s:MorCObjCs : Mor C \to Obj C is an open map.

It is called an étale groupoid if in addition ss is a local homeomorphism.

Properties

Relation to toposes

Every topos (Grothendieck topos) with enough points is the classifying topos of a topological groupoid. See there for more.

Revised on June 3, 2017 05:16:05 by Urs Schreiber (178.6.236.87)