nLab topological groupoid

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.

Related entries

• Lie groupoid

• topological group

• topological stack

• amenable topological groupoid

• localic groupoid

References

The notion of topological categories, hence of topological groupoids, goes back to

• Charles Ehresmann, Catégories topologiques et categories différentiables, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 (pdf, zbMath:0205.28202)

Their understanding as internal groupoids internal to TopologicalSpaces is often attributed to

• Charles Ehresmann, Catégories structurées, Annales scientifiques de l’École Normale Supérieure, Série 3, Tome 80 (1963) no. 4, pp. 349-426 (numdam:ASENS_1963_3_80_4_349_0)

but the simple notion of internalization and internal groupoids (Grothendieck 1960, 61) is hardly recognizable in this account.

Exposition:

• Alan Weinstein, p. 6 of: Groupoids: Unifying Internal and External Symmetry – A Tour through some Examples, Notices of the AMS volume 43, Number 7 (1996) (pdf, pdf)

Textbook account:

• Kirill Mackenzie, Chapter II in: Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (doi:10.1017/CBO9780511661839, MR:896907)

Many references on topological groupoids deal with them as models for topological stacks, see there for more.

On topological groupoids 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)

On topological groupoids as presenting toposes with enough points:

• Carsten Butz, Ieke Moerdijk, Representing topoi by topological groupoids, Journal of Pure and Applied Algebra Volume 130, Issue 3, 17 September 1998, Pages 223-235 (doi:10.1016/S0022-4049(97)00107-2)

On exponential objects among topological groupoids:

• Susan B. Niefield, Dorette A. Pronk, Internal groupoids and exponentiability, Cahiers de topologie et géométrie différentielle catégoriques, LX 4 (2019) (pdf)