Contents

# 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. Composition here refers to the map defined on the space of all composable morphisms.

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

It is called an étale groupoid if in addition $s$ 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.

## References

As a model for orbispaces:

• André Haefliger, Groupoides d’holonomie et classifiants, Astérisque no. 116 (1984), p. 70-97 (numdam:AST_1984__116__70_0)

• André Haefliger, Complexes of Groups and Orbihedra, in: E. Ghys, A. Haefliger, A Verjovsky (eds.), Proceedings of the Group Theory from a Geometrical Viewpoint, ICTP, Trieste, Italy , 26 March – 6 April 1990_, World Scientific 1991 (doi:10.1142/1235)

Last revised on December 18, 2020 at 10:59:30. See the history of this page for a list of all contributions to it.