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

TL;DR

This paper generalizes the nonabelian Dold-Kan decomposition for simplicial groups, introduces a family of total orders for subgroup decompositions, and extends the results to symmetric-simplicial groups with simpler algebraic structures.

Contribution

It extends the nonabelian Dold-Kan decomposition to a broader family of total orders and applies these ideas to symmetric-simplicial groups, simplifying their algebraic structure.

Findings

Family of total orders yields Dold-Kan decompositions

New decompositions for symmetric-simplicial groups with simpler algebraic relations

Existence of a partial order characterizing all total orders for subgroup decompositions

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.