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.
C1 [Icen, Ilhan; Gursoy, M. Habil; Ozcan, A. Fatih] Inonu Univ, Fac Sci & Art, Dept Math, TR-44280 Malatya, Turkey.