2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/120160In general a universal covering of a non connected topological group need not admit a topological group structure such that the covering map is a morphism of topological groups. This result is due to R.L. Taylor (1953). We generalise this result and relate it to the theory of obstructions to group extensions. The methods use: the equivalence between covering maps of X and covering groupoids of the fundamental groupoid of X; the equivalence between group groupoids and crossed modules; and descriptions of cohomology in terms of crossed complexes.13 pages, A4, Paul Taylors' diagrams. archived here for availability Some improvements to diagrams and new and updated referencesAlgebraic TopologyCategory Theory22A05 (18G55 20J05 20L10 20L17)text