nLab
topological groupoid
Contents
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

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

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 object s and one of morphism s, 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.
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