Nonabelian Dold-Kan Decompositions for Simplicial and Symmetric-Simplicial Groups

Abstract

We extend the nonabelian Dold-Kan decomposition for simplicial groups of Carrasco and Cegarra in two ways. First, we show that the total order of the subgroups in their decomposition belongs to a family of total orders all giving rise to Dold-Kan decompositions. We exhibit a particular partial order such that the family is characterized as consisting of all total orders extending the partial order. Second, we consider symmetric-simplicial groups and show that, by using a specially chosen presentation of the category of symmetric-simplicial operators, new Dold-Kan decompositions exist which are algebraically much simpler than those of Carrasco and Cegarra in the sense that the commutator of two component subgroups lies in a single component subgroup.