Simplicial objects and crossed complexes

In this work, in the firs chapter we Mention some basic informations such that simplicial algebras, Moore complex and higher order Peiffer elements and especially the notions of crossed moduls and 2-crossed modules of commutative algebras.In the second chapter, we explore the notions of crossed modules over groups and groupoids and we give a relation between the categories of crossed modules over groups and groupoids.In the third chapter, we give the Carresco-Cegarra funktor from simplicial groups to crossed complexes and by using Dwyer Kan path groupoid construction we define a funktor from simplicial groups to simplicial groupoids and by using this funktor we give a neat description of a passage from the category of simplicial groups to that of crossed complexes over groupoids.In the fourth chapter, we give the notion of a pullback crossed complex of groupoids.Lastly, we introduce the simplicial R- Algebroids and we give a relation between simplicial algebras and simplicial algebroids and we give a funktor from simplicial R-Algebroids to crossed complexes of R-Algebroids defined by Mosa.