On the topological generalized crossed modules

In this paper, we define the topological versions of the concepts of crossed module over generalized groups, which is called generalized crossed module, and generalized group-groupoid. We construct the categories of the topological generalized crossed modules and their homomorphisms and of the topological generalized group-groupoid and homomorphisms between them. We also obtain some characterizations related to the concepts of topological generalized crossed module and topological generalized group-groupoid. Finally, at the end of the work, we prove that the category of topological generalized crossed modules is equivalent to that of topological generalized group-groupoids whose object sets are commutative topological generalized groups.