The localization of orthogonal calculus with respect to homology

TL;DR

This paper develops a version of Weiss' orthogonal calculus focused on S-local homotopy types, establishing equivalences with S-local spectra and applying the theory to localizations, nullifications, and the Telescope Conjecture.

Contribution

It introduces an S-local orthogonal calculus framework that depends solely on S-local homotopy types, connecting to spectra and homological localizations.

Findings

S-local homogeneous functors are equivalent to S-local spectra with O(n) action

The theory specializes to homological localizations and nullifications

Applications include reformulating the Telescope Conjecture and Postnikov sections

Abstract

For a set of maps of based spaces $S$ we construct a version of Weiss' orthogonal calculus which depends only on the $S$-local homotopy type of the functor involved. We show that $S$-local homogeneous functors of degree $n$ are equivalent to levelwise $S$-local spectra with an action of the orthogonal group $O(n)$ via a zigzag of Quillen equivalences between appropriate model categories. Our theory specialises to homological localizations and nullifications at a based space. We give a variety of applications including a reformulation of the Telescope Conjecture in terms of our local orthogonal calculus and a calculus version of Postnikov sections. Our results also apply when considering the orthogonal calculus for functors which take values in spectra.