From simplicial groups to crossed squares

Abstract

Simplicial groups are defined to be contravariant functors from the simplex category to the category of groups. The truncation functor maps a simplicial group to a [2,0]-simplicial group, satisfying the Conduché condition. A functor from the category of [2,0]-simplicial groups to the category of crossed squares is constructed, following Porter. It is shown that the latter functor is not an equivalence of categories. In addition, Loday's variant of the resulting crossed square is constructed and shown to be isomorphic to Porter's variant.