On the 1-Homotopy Type of Lie Groupoids

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbifold maps. Introduction We develop a notion of 1-homotopy between generalized maps which is suitable for applications to orbifolds. There are notions of homotopy for functors in general categories that can be applied to groupoids. But they fail to be invariant of Morita equivalence when considering Lie groupoids. Since two Lie groupoids define the same orbifold if they are Morita equivalent, a notion of homotopy between generalized maps should be invariant of Morita equivalence. For general categories (no topology nor smooth structure involved) natural transformations play the role of homotopy for functors [14, 4]. Two functors are homotopic if there is a natural transformation between them. We call this notion of homotopy a natural transformation. For topological categories T and T ′ the usual notion of homotopy is just a functor which is an ordinary homotopy on objects and on arrows. We say that two continuous functors f, g : T → T ′ are homotopic if there is a continuous functor H : T × I → T ′ such that H0 = f and H1 = g and I is the unit groupoid over the interval [0, 1] . We call this notion of homotopy an ordinary homotopy. Both notions of natural transformation and ordinary homotopy can be adapted to Lie groupoids by requiring all the maps involved to be smooth. None of these two notions is invariant under Morita equivalence. For an example of Morita equivalent Lie groupoids which are not equivalent by a natural transformation nor an ordinary homotopy, consider the holonomy groupoid G = Hol(M,FS) associated to a Seifert fibration FS on a Möbius band M , and its reduced holonomy groupoid K = HolT (M,FS) to a transversal interval T . Since the double covering of the Möbius band by Date: August 23, 2009. 2000 Mathematics Subject Classification. Primary 22A22; Secondary 18D05, 55P15. 1