The nilpotent multipliers of crossed modules

The article is devoted to extending the notions of the c-nilpotent multiplier and the c-stem cover of a group to the crossed modules context. We give a five-term exact sequence in the c-nilpotent multiplier associated with an extension of crossed modules from which we deduce a generalization version of the Basic Theorem of Grandjean and Ladra (1998) [9] relating nilpotent crossed modules to their second homologies. We also study c-stem extensions and c-covers of crossed modules and prove that any c-stem extension of a perfect crossed module is a homomorphic image of its c-stem cover. Finally, we show that the c-nilpotent multipliers of a perfect crossed module are isomorphic to each other. This generalizes the work of Burns and Ellis (1997) [5] in group theory.