Orbifolds as diffeologies
September 15, 2005, update May 21, 2010
 
Orbifolds have become ordinary objects in mathematics, but their definition remain problematic, different authors give different definitions.
Originally, orbifolds have been introduced by Ishiro Satake, as V-manifolds, (see references below). Later, Thurston changed the name from V-manifolds to orbifolds. Apart from the variety of definitions, the strangest part is that V-manifolds come to us alone, not included in a specific category, as it is usual for mathematical objects. In one of his papers, Satake's even quoted:
“The notion of C∞-map thus defined is inconvenient in the point that a composite of two C∞-maps defined in a different choice of defining families is not always a C∞-map.”
These remarks, led us to add, to the flow of existing definitions of orbifolds, our own one, introducing orbifolds into the category of diffeological spaces. Orbifolds, as diffeologies, become now members of the society of orbifolds whose relationships are defined by differentiable maps.
Definition We call diffeological orbifold any diffeological space which is locally diffeomorphic, at each point, to some quotient Rn/K, where K is a finite group of linear transformations.
We show how each Satake's V-manifold is naturally associated to a diffeological orbifold, in a one to one correspondence, and how to retrieve the V-manifold structure from the diffeological orbifold. This shows, in particular, that V-manifold structure is local, which is not completely obvious in the Satake definition. Moreover, diffeological orbifolds inherit differentiable maps from the diffeology category. In particular, the local diffeomorphisms from Rn/K (the local model) to the diffeological orbifold, which define the diffeological structure, are themselves (bi)differentiable maps between orbifolds. The notions of structure groups, singular and regular points are specified.
The paper has essentially 4 parts:
1.	Diffeologies and diffeological orbifolds
2.	Technical lemma on lifting of diffeomorphisms of Rn/K.
3.	Satake's definition of V-manifolds
4.	Equivalence between V-Manifolds and diffeological orbifolds
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This paper is a joint work with Yael Karshon and Moshe Zadka,The Hebrew University of Jerusalem (Final version September 2005), accepted for publication after few minor revisions in Transaction of the American Mathematical Society.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 362, Number 6, June 2010, Pages 2811–2831 
S 0002-9947(10)05006-3 
Article electronically published on January 7, 2010../More%20about%20orbifolds.htmlshapeimage_3_link_0shapeimage_3_link_1