A groupoid presenting an orbifold (as a stack) is called an orbifold groupoid if it satisfies certain properties. Since in the literature there are different notions of orbifolds, there are different sets of properties the presenting groupoids are required to satisfy.
The most common definition requires the presenting groupoid to be étale and proper. Note that these properties can be defined not only in smooth- or topological context. See for the moment open map, the section on infinitesimal cohesion in cohesive (infinity,1)-topos, and proper geometric morphism.
(sse (smooth stable étale) groupoid (1)
An sse groupoid is a groupoid being
(ep (étale proper) groupoid )
An ep groupoid is a groupoid being
Since this is the most common choice of axioms a groupoid presenting an orbifold is meant to satisfy it is the default class of orbifold groupoids.
Note that this definition is redundant since properness implies stability.
Note that what 2 calls ‘’ep-groupoid’‘ is ‘’étale and proper’‘ but not Lie in the ordinary sense: Since the underlying category is that of sc-spaces, étale becomes sc-étale and also ‘’proper’‘ is understood in some modified sense.
There is further terminology applicable to orbifold groupoids:
() Let be a groupoid, let denote its orbit space.
nonsingular if every is trivial
effective / faithful if for ever and for every and for every neighborhood of there is an such that
(path)connected if is (path)connected.
(wnb (weighted nonsingular branched) groupoid )
A wnb (weighted nonsingular branched) groupoid is a pair where is a oriented nonsingular sse Lie groupoid and is a weighting function satisfying:
For each there is an open neighborhood of and disjoint open subsets -called local branches and positive weights such that
(local regularity) all projections is a homeomorphism onto a relatively closed subset of .
(weighting) for all
where denotes the maximal Hausdorff quotient of (If is proper we have and the canonical projection. Points having more than one inverse image are called branch points.
The tuple is called a local branching structure at . is called compact if is.
Note that here the properness axiom is relaxed.
Given an ep groupoid, the properness axiom implies that the orbit space of the orbifold groupoid is Hausdorff.
Properness implies stability.