On the internal categories within the category of crossed modules

Crossed modules are algebraic models of connected topological spaces of homotopy type 2 while groups are algebraic models of connected topological spaces of homotopy type 1. Crossed modules were originally defined by Whitehead. Brown and Spencer showed that the category of the crossed modules on the groups is equal to the category of the internal groupoids (also called group-groupoids) in the category of groups. In this thesis study by a similar thought, first of all internal categories in the category of crossed modules are characterized and it has been shown that that these are actually internal groupoids. Finally, it has been shown that the category of crossed modules of the crossed modules called crossed squares equivalent to the category of internal groupoids in the category of crossed modules.