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The localization of orthogonal calculus with respect to homology

Niall Taggart

Algebraic & Geometric Topology 24 (2024) 1505–1549
Abstract

For a set of maps of based spaces S we construct a version of Weiss’s 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.

Keywords
orthogonal calculus, Bousfield localization, homological localization, nullification, calculus of functors
Mathematical Subject Classification
Primary: 55P60, 55P65
Secondary: 55N20, 55P42
References
Publication
Received: 14 February 2022
Revised: 11 October 2022
Accepted: 20 November 2022
Published: 28 June 2024
Authors
Niall Taggart
Mathematical Institute
Utrecht University
Utrecht
Netherlands

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