Lie covering groupoids

If M is a differentiable connected manifold then there exists an universal covering manifold M having unique differentiable structure such that the covering map p : M -> M is differentiable. This fact is also true for connected Lie groups By using this fact, it is proved that the category LGdCov(G) of coverings of connected Lie groupoids which is a generalization of connected Lie groups, and the category LGdOp(G) of actions on some M differentiable manifold are equivalent. Secondly, by introducing Lie group-groupoids the category LGGdCov(G) of coverings of some G Lie group-groupoid and the category LGGdOp(G) of actions of G on connected Lie group M are established. Further, it is shown that these categories are equivalent. Finally, it is presented by launching the notion Lie ring-groupoids, a generalization of Lie group-groupoids, that the category LRGdCov(R) of coverings of R Lie ring-groupoids and the category LRGdOp(R) of actions of R on connected Lie ring M are equivalent. KEY WORDS: Groupoid, Lie groupoid, covering groupoid, Lie group-groupoid, Lie ring-groupoid.