nLab
orbifold groupoid

Idea

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.

Definitions

Definition

(sse (smooth stable étale) groupoid (1)

An sse groupoid is a groupoid being

Definition

(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:

Definition

() Let CC be a groupoid, let C=C 0/C 1|C|=C_0/C_1 denote its orbit space.

CC is called

  • nonsingular if every C(c,c)C(c,c) is trivial

  • effective / faithful if for ever cC 0c\in C_0 and for every fC(c,c)f\in C(c,c) and for every neighborhood fVC 1f\in V\subset C_1 of ff there is an f Vf^'\in V such that s(f )¬=t(f )s(f^')\not =t(f^')

  • (path)connected if C|C| is (path)connected.

Definition

(wnb (weighted nonsingular branched) groupoid )

A wnb (weighted nonsingular branched) groupoid is a pair (C,Λ)(C,\Lambda) where CC is a oriented nonsingular sse Lie groupoid and Λ:C H(0,)\Lambda:|C|_H\to (0,\infty) is a weighting function satisfying:

For each pC Hp\in |C|_H there is an open neighborhood pN:=N(p)C Hp\in N:=N(p)\subset |C|_H of pp and disjoint open subsets U 1,...,U jπ H 1(N)C 0U_1 ,...,U_j\subset \pi^{-1}_H(N)\subset C_0 -called local branches and positive weights m 1,...,m jm_1 ,...,m_j such that

  1. (covering) π H 1(N)=U 1...U jC\pi^{-1}_H(N)=|U_1|\cup ...\cup |U_j|\subset |C|

  2. (local regularity) all projections π H:U iU i H\pi_H:U_i\to |U_i|_H is a homeomorphism onto a relatively closed subset of NN.

  3. (weighting) Λ(q)=Σ i:qU i Hm i\Lambda(q)=\Sigma_{i:q\in |U_i|_H}m_i for all qNq\in N

where C H|C|_H denotes the maximal Hausdorff quotient of C|C| (If CC is proper we have C=C H|C|=|C|_H and π H:C 0C H\pi_H:C_0\to |C|_H the canonical projection. Points pC Hp\in |C|_H having more than one inverse image are called branch points.

The tuple (N,U i,m i)=(N p,U i p,m i p)(N,U_i,m_i)=(N^p,U^p_i,m^p_i) is called a local branching structure at pp. CC is called compact if C H|C|_H is.

Note that here the properness axiom is relaxed.

Relation of the axioms

Given an ep groupoid, the properness axiom implies that the orbit space of the orbifold groupoid is Hausdorff.

Properness implies stability.

References

category: Lie theory


  1. D. McDuff, groupoids, branched manifolds and multisections

  2. H. Hofer, polyfolds and a general Fredholm theory

Revised on September 16, 2012 12:57:16 by Tim Porter (95.147.237.41)