Homotopy nilpotent groups

TL;DR

This paper explores the relationship between the Goodwillie tower of the identity and the lower central series of loop groups, introducing homotopy n-nilpotent groups that bridge infinite loop spaces and loop spaces.

Contribution

It defines the theory of homotopy n-nilpotent groups and connects it to classical n-nilpotent groups, providing new insights into the structure of the Goodwillie tower.

Findings

The set-valued algebraic theory matches classical n-nilpotent groups.

The Goodwillie tower is characterized by a homotopy left Kan extension.

n-excisive functors of the form ΩF take values in homotopy n-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 the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\Omega F$ have values in homotopy n-nilpotent groups.