NON-EFFECTIVE ORBIFOLDS AND DOUBLE CATEGORIES

Abstract

Satake introduced Orbifolds in terms of an underlying space with an atlas of charts consisting of an open subset of Euclidean space with an action of a finite group. The representation of orbifolds by topological groupoids introduced by Moerdijk and Pronk has greatly helped in developing their study: orbifolds are then proper effective ´etale groupoids. Motivated by examples from physics we have started considering orbifolds with non-effective group actions. This has led us to consider orbifolds as proper ´etale groupoids. We introduce a new notion of non-effective orbifold atlas which corresponds closely to the groupoid representation of orbifolds. This definition uses the language of double categories, and in particular the double category Bimod of finite groups with group homomorphisms as vertical morphisms and bimodules as horizontal morphisms. We describe how this definition of orbifold atlas generalizes Satake’s definition. Then we introduce a notion of atlas refinement which is defined in terms of profunctors over Bimod. We show that atlas equivalence corresponds to Morita equivalence of groupoids. We define two notions of map. The first one is the vertical maps which is defined in terms of group homomorphisms. The second one is the horizontal maps which is defined in terms of poset-valued profunctors over Bimod. As an application, we define a pseudo double category OAts of orbifolds in terms of atlases and OGpd in terms of orbigroupoids. We would like to show that the pseudo double categories of orbifold atlases and orbigroupoids are weakly equivalent. We encounter two issues. The first one is that certain double cells in OAts that are not equal get sent to exactly the same double cells in OGpd. We fix this issue by introducing an equivalence relation on the double cells in OAts. The second problem is that certain cells in the double category of atlases which are not vertically invertible get sent to a vertical isomorphism in the double category of orbigroupoids. We fix that by adding inverses for those cells. This leads us to define a double category of atlases with refinements, rOAts. Then we show that rOAts and OGpd are weakly biequivalent.