TOPOLOGICAL RING-GROUPOIDS AND LIFTINGS

Abstract

We prove that the set of homotopy classes of the paths in a topological ring is a topological ringobject (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 acategory UTRGdCov( π1X ) of coverings of topological ring-groupoid π1X , in which X and R􀀄0 = X􀀄 haveuniversal coverings, and then prove the equivalence of these categories. We also prove that the topologicalring structure of a topological ring-groupoid lifts to a universal topological covering groupoid.