The localization of orthogonal calculus with respect to homology

Abstract

For a set of maps of based spaces S we construct a version of Weiss' orthogonal calculus which only depends 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.