Simplicial polygroups and the generalized Moore complexes

Abstract

A simplicial group is a simplicial object in the category of groups. A very nice application of simplicial group which is simplicial polygroup is given in this paper. Using polygroups instead of groups, we already had very good results from the well known properties due to Loday. Loday proved that a crossed module, a cat1-group, a group object in the category of categories and a simplicial group whose Moore complex is of length one are equivalent. Using Loday’s idea we present a functor from the category of groups to the category of polygroups and the simplicial groups to the simplicial polygroups. We show that there exist a functor from the category of cat1-polygroups to the category of groups and the category of groups to the category of polygroups. We also prove that the category of simplicial groups is equivalent to the category of simplicial polygroups and the category of simplicial polygroups with generalized Moore complex with of length one is equivalent to the category of polygroups.