Locally semisimple coverings

Abstract

We introduce the notion of a locally semisimple covering with respect to a class X of objects in a given exact category, and classify these coverings in terms of internal-category actions inside X . This is similar to the classification of ordinary covering spaces of a “good” topological space in terms of its fundamental-group actions. Locally semisimple coverings are essentially the same as the maps with fibres in X ; examples are e.g. ring and group homomorphisms with semisimple kernels (with respect to a given radical) or continuous maps of compact Hausdorff spaces with totally disconnected fibres.