Simplicial Crossed Modules

In this thesis, from any homomorphism of groups, the simplicial set obtained by the Bar construction method has been investigated. In the case of existence of homotopy normal map structure over this homomorphism, we see that there is a simplicial group structure over the simplicial set which is obtained from this homomorphism using the Bar construction. Infurther sections, we explore this construction over commutative algebras. From any homomorphism of commutative algebras, using the Bar construction, a simplisel module structure has been obtained. Furthermore, in the case of existence of any homotopy normal map structure over this homomorphism of commutative algebras, we see that there is a simplicial algebra structure over the simplicial module. And vice versa, if there is an ideal simplicial algebra structure over this simplicial module, then we obtain a homotopy normal map structure over the homomorphism of algebras.