Functor calculus completions for retractive operadic algebras in spectra

TL;DR

This paper investigates the convergence properties of Bousfield-Kan completions and Taylor towers for operadic algebras in spectra, focusing on the retractive setting away from the null object, advancing understanding in homotopy functor calculus.

Contribution

It introduces new convergence results for functor calculus in the context of retractive operadic algebras in spectra, extending Goodwillie's calculus to this setting.

Findings

Establishes convergence of Bousfield-Kan completions in the retractive setting.

Analyzes exotic convergence of the Taylor tower for these algebras.

Provides new insights into homotopy theory in the retractive context.

Abstract

The aim of this paper is to study convergence of Bousfield-Kan completions with respect to the 1-excisive approximation of the identity functor and exotic convergence of the Taylor tower of the identity functor, for algebras over operads in spectra centered away from the null object. In Goodwillie's homotopy functor calculus, being centered away from the null object amounts to doing homotopy theory and functor calculus in the retractive setting.