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Models of G-spectra as presheaves of spectra

Bertrand Guillou, J. P. May

Abstract

Let G be a finite group. We give Quillen equivalent models for the category of G-spectra as categories of spectrally enriched functors from explicitly described domain categories to nonequivariant spectra. Our preferred model is based on equivariant infinite loop space theory applied to elementary categorical data. It recasts equivariant stable homotopy theory in terms of point-set level categories of G-spans and nonequivariant spectra. We also give a more topologically grounded model based on equivariant Atiyah duality.

Models of G-spectra as presheaves of spectra

Abstract

Let G be a finite group. We give Quillen equivalent models for the category of G-spectra as categories of spectrally enriched functors from explicitly described domain categories to nonequivariant spectra. Our preferred model is based on equivariant infinite loop space theory applied to elementary categorical data. It recasts equivariant stable homotopy theory in terms of point-set level categories of G-spans and nonequivariant spectra. We also give a more topologically grounded model based on equivariant Atiyah duality.

Paper Structure

This paper contains 27 sections, 29 theorems, 123 equations.

Table of Contents

  1. The bicategory $G\mathscr{E}$ and $\mathscr{S}$-category $G\mathscr{A}$
  2. The bicategory $G\mathscr{E}$ of $G$-spans
  3. The precise statement of the main theorem
  4. The $G$-bicategory $\mathscr{E}_G$ of spans: intuitive definition
  5. The $G$-bicategory $\mathscr{E}_G$ of spans: working definition
  6. The categorical duality maps
  7. The proof of the main theorem
  8. The equivariant approach to \ref{['MAINNew1']}
  9. Results from equivariant infinite loop space theory
  10. The self-duality of $\Sigma^{\infty}_G(A_+)$
  11. The proof that $\mathscr{A}_G$ is equivalent to $\mathscr{D}_{\text{All}}$
  12. The identification of suspension $G$-spectra
  13. Some comparisons of functors
  14. Change of groups and fixed point functors
  15. Fixed point orbit functors
  16. ...and 12 more sections

Key Result

Theorem 1

There is a zig-zag of Quillen equivalences relating the category of $G$-spectra to the category of spectrally enriched contravariant functors $G\mathscr{A}\longrightarrow \mathscr{S}$.

Theorems & Definitions (76)

  • Theorem 1: Main theorem
  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Definition 1.3
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.1: LMS
  • Definition 1.5
  • ...and 66 more