Crossed modules of Bigroups and BiCrossed modules of Bigroups

This paper is devoted to introduce the notions of crossed modules of bigroups and bicrossed modules of bigroups as a suitable generalization of the notions of crossed modules of groups and bicrossed modules of groups, respectively. This is done by embedding the categories of crossed modules of groups and bicrossed modules of groups in the category of crossed modules of bigroups and bicrossed modules of bigroups, respectively, (via isomorphism of categories). The notion of extended group and extended group homomorphism had also been introduced in the beginning of this paper. Finally, two adjoint pair of functors had been given between the category of bicrossed modules of bigroups and the category of bigroups.