Coverings and crossed modules of topological groups with operations

It is a well-known result of the covering groups that a subgroup G of the fundamental group at the identity of a semilocally simply connected topological group determines a covering morphism of topological groups with characteristic group G . In this paper we generalize this result to a large class of algebraic objects called topological groups with operations, including topological groups. We also prove that the crossed modules and internal categories within topological groups with operations are equivalent. This equivalence enables us to introduce the cover of crossed modules within topological groups with operations. Finally, we draw relations between the coverings of an internal groupoid within topological groups with operations and those of the corresponding crossed module.