Topological ring-groupoids and liftings

Abstract

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) over bar -> 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 have universal coverings, and a category UTRGdCov(pi X-1) of coverings of topological ring-groupoid pi X-1, in which X and (R) over bar (0) = (X) over bar 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.

C1 Inonu Univ, Sci & Art Fac, Dept Math, Malatya, Turkey.