# Equivariant Orthogonal Spectra and \(S\)-Modules

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*M. A. Mandell; J. P. May*

The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of \(S\)-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of \(S\)-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to \(S\)-modules. We then develop the equivariant theory. For a compact Lie group \(G\), we construct a symmetric monoidal model category of orthogonal \(G\)-spectra whose homotopy category is equivalent to the classical stable homotopy category of \(G\)-spectra. We also complete the theory of \(S_G\)-modules and compare the categories of orthogonal \(G\)-spectra and \(S_G\)-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory.

#### Table of Contents

# Table of Contents

## Equivariant Orthogonal Spectra and $S$-Modules

- Contents vii8 free
- Introduction 112 free
- Chapter I. Orthogonal spectra and S-modules 314 free
- 1. Introduction and statements of results 314
- 2. Right exact functors on categories of diagram spaces 516
- 3. The proofs of the comparison theorems 819
- 4. Further Quillen equivalences and homotopical preliminaries 1223
- 5. Model structures and homotopical proofs 1627
- 6. The construction of the functor N* 1829
- 7. The functor M and its comparison with N 2233
- 8. A revisionist view of infinite loop space theory 2637

- Chapter II. Equivariant orthogonal spectra 2940
- Chapter III. Model categories of orthogonal G-spaces 3849
- 1. The model structure on G-spaces 3849
- 2. The level model structure on orthogonal G-spectra 4152
- 3. The homotopy groups of G-prespectra 4455
- 4. The stable model structure on orthogonal G-spectra 4758
- 5. The positive stable model structure 5162
- 6. Stable equivalences of orthogonal G-spectra 5263
- 7. Model categories of ring and module G-spectra 5364
- 8. The model category of commutative ring G-spectra 5566
- 9. Level equivalences and π[sub(*)]-isomorphisms of Ω-G-spectra 5768

- Chapter IV. Orthogonal G-spectra and S[sub(G)]-modules 5970
- 1. Introduction and statements of results 5970
- 2. Model structures on the category of S[sub(G)]-modules 6172
- 3. The construction of the functors N and N[sup(#)] 6576
- 4. The proofs of the comparison theorems 6778
- 5. The functor M and its comparison with N 6879
- 6. Families, cofamilies, and Bousfield localization 6980

- Chapter V. "Change" functors for orthogonal G-spectra 7485
- Chapter VI. "Change" functors for S[sub(G)]-modules and comparisons 8899
- 1. Comparisons of change of group functors 8899
- 2. Comparisons of change of universe functors 91102
- 3. Comparisons of fixed point and orbit G-spectra functors 96107
- 4. N-free G-spectra and the Adams isomorphism 99110
- 5. The geometric fixed point functor and quotient groups 100111
- 6. Technical results on the unit map λ:JE [omitted] E 101112

- Bibliography 103114
- Index of Notation 105116 free