topological groupoid



Higher geometry


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

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


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory




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 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.


Relation to toposes

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


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)

and their understanding as internal categories in TopologicalSpaces may originate around:

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)

Last revised on March 24, 2021 at 03:29:59. See the history of this page for a list of all contributions to it.