Pseudo simplicial groups and crossed modules

Using the homotopy groups of pseudosimplicial groups, Inasaridze, [10], construct nonabelian derived functors with values in the category of groupsThis thesis consists of three chapters. In the first chapter we gave some basic information.In the second chapter. Using the Inasaridze, [10], define pseudosimplicial groups. We show that the category of crossed module is equivalent to the category of pseudosimplicial groups with Moore complexes of lenght 1. Also we define pseudo 2-crossed module and we prove that the category of pseudo 2-crossed module is equivalent to the category of pseudo simplicial groups with Moore complex of lenght 2.In the third chapter using the Inasaridze, [10],[11] constructed homotopy groups of pseudosimplicial groups and non abelian derived functors with values in the category of group. We define pseudosimplicial algebra considering pseudosimplicial groups. We show that the category of crossed module is equivalent to the category of pseudosimplicial algebras with Moore complexes of lenght 1. Also we define pseudo 2-crossed module and we prove that the category of pseudo 2-crossed module is equivalent to the category of pseudo simplicial algebras with Moore complex of lenght 2.