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 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.

16 pages, uses xy-pic, improved exposition, submitted

References in corpus (6)

• Calculus III: Taylor Series
• Goodwillie towers and chromatic homotopy: an overview
• Bar constructions for topological operads and the Goodwillie derivatives of the identity
• Calculus of functors and model categories II
• Calculus of functors and model categories
• A chain rule for Goodwillie derivatives of functors from spectra to spectra

Cited by in corpus (5)

• Central reflections and nilpotency in exact Mal'tsev categories
• Goodwillie calculus and Whitehead products
• A Torus Theorem for homotopy nilpotent groups
• Goodwillie Calculus via Adjunction and LS Cocategory
• Derived Character Maps of Groups Representations