TOPOLOGICAL RING-GROUPOIDS AND LIFTINGS

We prove that the set of homotopy classes of the paths in a topological ring is a topological ring object (called topological ring-groupoid). Let p : X?? ? X be a covering map and let X be a topological ring. We define a category UTRCov(X) of coverings of X in which both X and X?? have universal coverings, and a category UTRGdCov( ?1X ) of coverings of topological ring-groupoid ?1X , in which X and R??0 = X?? have universal coverings, and then prove the equivalence of these categories. We also prove that the topological ring structure of a topological ring-groupoid lifts to a universal topological covering groupoid.