Orbifolds and groupoids

2010/08/05 by Tommasini, Matteo
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1008.0988

Abstract

We define a 2-category structure (Pre-Orb) on the category of reduced complex orbifold atlases. We construct a 2-functor F from (Pre-Orb) to the 2-category (Grp) of proper étale effective groupoid objects over the complex manifolds. Both on (Pre-Orb) and (Grp) there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases in (Pre-Orb) and Morita equivalences in (Grp). We prove that F induces a bijection between the equivalence classes of its source and target.

Related

• Orbifolds as a localization of the 2-category of groupoids
• The category of reduced orbifolds in local charts
• A bicategory of reduced orbifolds from the point of view of differential geometry - I
• Atlases for Ineffective Orbifolds
• Orbifolds as Groupoids: an Introduction