Retractive spaces and Bousfield-Kan completions

TL;DR

This paper extends convergence results of Bousfield-Kan completions and Taylor towers to retractive spaces, providing stronger convergence insights and a new proof of a key result without relying on Snaith splittings.

Contribution

It generalizes the Arone-Kankaanrinta convergence result to retractive spaces and introduces stronger convergence comparisons in this setting.

Findings

Stronger convergence results for retractive spaces.

A new proof of Arone-Kankaanrinta's theorem without Snaith splittings.

Extension of homotopical estimates to retractive spaces.

Abstract

In this short paper we apply some recent techniques developed by Schonsheck, and subsequently Carr-Harper, in the context of operadic algebras in spectra -- on convergence of Bousfield-Kan completions and comparisons with convergence of the Taylor tower of the identity functor in Goodwillie's functor calculus -- to the setting of retractive spaces: this arises when working with spaces centered away from the one-point space. Interestingly, in the retractive spaces context, the comparison results are stronger in terms of convergence outside of functor calculus' notion of "radius of (strong) convergence" for analytic functors. In particular, we give a new proof (and generalization to retractive spaces) of the Arone-Kankaanrinta result for convergence of the Taylor tower of the identity functor to various Bousfield-Kan completions; it's notable that no use is made of Snaith splittings -- rather, we make extensive use of the kinds of homotopical estimates that appear in earlier work of Dundas and Dundas-Goodwillie-McCarthy.