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

• étale

• stable?

• Lie

Definition

(ep (étale proper) groupoid )

An ep groupoid is a groupoid being

• étale

• stable?

• Lie

• proper

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 $C$ be a groupoid, let $|C|=C_0/C_1$ denote its orbit space.

$C$ is called

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

• effective / faithful if for ever $c\in C_0$ and for every $f\in C(c,c)$ and for every neighborhood $f\in V\subset C_1$ of $f$ there is an $f^'\in V$ such that $s(f^')\not =t(f^')$

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

Definition

(wnb (weighted nonsingular branched) groupoid )

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

For each $p\in |C|_H$ there is an open neighborhood $p\in N:=N(p)\subset |C|_H$ of $p$ and disjoint open subsets $U_1 ,...,U_j\subset \pi^{-1}_H(N)\subset C_0$ -called local branches and positive weights $m_1 ,...,m_j$ such that

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

2. (local regularity) all projections $\pi_H:U_i\to |U_i|_H$ is a homeomorphism onto a relatively closed subset of $N$.

3. (weighting) $\Lambda(q)=\Sigma_{i:q\in |U_i|_H}m_i$ for all $q\in N$

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

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

Related concepts

• stack

• Lie groupoid

• orbifold

References

The original paper is

• Ieke Moerdijk, Dorette A. Pronk, Orbifolds, Sheaves and Groupoids. K-Theory 12:1 (1997), 3-21. doi.

An expository account is in

• Ieke Moerdijk, Orbifolds as groupoids: an introduction. Orbifolds in Mathematics and Physics, Contemporary Mathematics (2002), 205-222. doi:10.1090/conm/310/05405, arXiv:math/0203100.

Other sources:

category: Lie theory

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1. Dusa McDuff, Groupoids, branched manifolds and multisections. Journal of Symplectic Geometry 4:3 (2006), 259-315.

   doi:10.4310/jsg.2006.v4.n3.a2. ↩

2. Helmut Hofer, Polyfolds and Fredholm Theory. Oxford University Press (2017) (doi:10.1093/oso/9780198784913.003.0004) ↩