Homotopy nilpotent groups and their associated functors

TL;DR

This paper explores homotopy n-nilpotent groups, associating them with specific functors, and proves their properties related to looped and n-excisive functors, advancing the understanding of their algebraic and topological structure.

Contribution

It introduces a new association between homotopy n-nilpotent groups and endofunctors, demonstrating their looped and n-excisive nature, and proves key colimit commutation properties.

Findings

The associated functor is looped and n-excisive.

The functor commutes with sifted colimits of connected spaces.

Provides new tools for studying homotopy nilpotent groups.

Abstract

To every homotopy n-nilpotent group, defined in earlier work by Dwyer and the author, we associate an endofunctor of pointed spaces and prove that it is looped and n-excisive. As a tool we prove that $\Omega P_n({\rm id})$ commutes with sifted colimits of connected spaces.