Homotopy nilpotent groups

Abstract

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define homotopy $n$–nilpotent groups as homotopy algebras over certain simplicial algebraic theories. This notion interpolates between infinite loop spaces and loop spaces, but backwards. We study the relation to ordinary nilpotent groups. We prove that $n$–excisive functors of the form $\Omega F$ factor over the category of homotopy $n$–nilpotent groups.