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The localization of orthogonal calculus with respect to homology

Niall Taggart

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.

The localization of orthogonal calculus with respect to homology

Abstract

For a set of maps of based spaces we construct a version of Weiss' orthogonal calculus which depends only on the -local homotopy type of the functor involved. We show that -local homogeneous functors of degree are equivalent to levelwise -local spectra with an action of the orthogonal group 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.

Paper Structure

This paper contains 36 sections, 52 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Local orthogonal calculus
  4. Orthogonal functors
  5. Local input functors
  6. Polynomial functors
  7. Polynomial functors
  8. Local polynomial functors
  9. Differentiation
  10. The derivatives
  11. The local $n$-stable model structure
  12. The derivatives as spectra
  13. Homogeneous functors and their classification
  14. Homogeneous functors
  15. Local homogeneous functors
  16. ...and 21 more sections

Key Result

Theorem 1

Let $S$ be a set of maps of based spaces and $n \geq 0$. There is a zigzag of Quillen equivalences where ${{\sf{Sp}}}(L_S\mathop{\mathrm{\sf{Top}_\ast}}\nolimits)[O(n)]$ is the category of levelwise $S$-local spectra with an action of $O(n)$.

Theorems & Definitions (106)

  • Theorem : Corollary \ref{['cor: zigzag']}
  • Theorem : Corollary \ref{['cor: zigzag spectra']}
  • Theorem : Theorem \ref{['thm: classification of E-local n-homog']}
  • Theorem : Example \ref{['ex: BF classes']}
  • Theorem : Corollary \ref{['cor: telescope implies agree towers']}
  • Theorem : Lemma \ref{['lem: telescope and WSS']}
  • Theorem : Proposition \ref{['prop: GR k-types in orthogonal functors']}
  • Theorem : Corollary \ref{['cor: homog holim']}
  • Proposition 2.0.1
  • Proposition 2.1.1
  • ...and 96 more