Crossed Modules with Action

Following the idea of a group with action in a higher dimension, we consider crossed modules with a given crossed-module action upon themselves. A category of these objects XMod* is given by replacing the set of morphisms in the category Gr* of groups with action by the set of φ-crossed homomorphisms of a modified form. Moreover, the connections of XMod* with some other known categories, such as Gr* and XMod, are given by describing some related functors.