On the homotopy class of morphisms of braided regular crossed modules

In this thesis, we investigate the class of homotopies between the morphisms of braided crossed modules. We define the notion of homotopy between maps of braided crossed modules of groups and commutative algebras and we give their properties. In this meaning, this thesis consists of four chapters In the first chapter, we recall the basic definitions about of homotopy between two curves and homotopy classes. In the second chapter, we introduced the corresponding cases of these structures to the notions of braided crossed modules of groups and commutative algebras.