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Coverings of Lie groupoids

In this work we constitute the category of coverings of the Lie fundamental groupoid associated with a connected smooth manifold. We show that this category is equivalent to the category of universal coverings of a connected smooth manifold. In addition, we prove the equivalence of the category of coverings of a Lie groupoid and the category of actions of this Lie groupoid on a connected smooth manifold. | Turk J Math 35 (2011) , 207 – 218. ¨ ITAK ˙ c TUB doi:10.3906/mat-0902-2 Coverings of Lie groupoids ˙ ˙cen, M. Habil G¨ ¨ Ilhan I¸ ursoy and A. Fatih Ozcan Abstract In this work we constitute the category of coverings of the Lie fundamental groupoid associated with a connected smooth manifold. We show that this category is equivalent to the category of universal coverings of a connected smooth manifold. In addition, we prove the equivalence of the category of coverings of a Lie groupoid and the category of actions of this Lie groupoid on a connected smooth manifold. Also we present two side results related to actions of Lie groupoids on the manifolds and coverings of Lie groupoids. Key Words: Lie groupoid, covering, action, lifting. 1. Introduction The theory of covering space is one of the most important theories in algebraic topology. By studying categories and groupoids, the concept of covering is meaningful by investigation of relationships between fundamental groupoids of covering spaces and those of the base spaces. These relations are studied by Brown and Higgins in [1, 2, 8]. Brown defined fundamental groupoid π1 X for given a topological space X . Thus he defined the covering → X of topological spaces. Later, he → π1 X of groupoids for a covering map p : X morphism π1 p : π1 X showed that the equivalence of the category T Cov(X) of coverings of X and the category GdCov(π1 X) of coverings of fundamental groupoid π1 X , where X has universal covering space. In this area, another algebraic study was studied by Gabriel and Zisman [5]. They showed the equivalence of the category GdCov(G) of coverings of a groupoid G and the category GdOp(G) of actions on the sets of G . The topological version of this paper is studied in [3]. → M be a topological For the smooth case, let M be a connected smooth manifold and let p : M is a topological manifold, and it has a unique smooth structure such that p is a smooth covering map. Then M covering map [9]. In .