By M. A. Mandell

The previous few years have visible a revolution in our figuring out of the principles of strong homotopy thought. Many symmetric monoidal version different types of spectra whose homotopy different types are such as the reliable homotopy class at the moment are recognized, while no such different types have been identified earlier than 1993. the main famous examples are the class of $S$-modules and the class of symmetric spectra. We specialise in the class of orthogonal spectra, which enjoys the very best beneficial properties of $S$-modules and symmetric spectra and that's relatively well-suited to equivariant generalization. We first entire the nonequivariant thought via evaluating orthogonal spectra to $S$-modules. We then increase the equivariant theory.For a compact Lie crew $G$, we build a symmetric monoidal version class of orthogonal $G$-spectra whose homotopy classification is similar to the classical good homotopy classification of $G$-spectra. We additionally whole the idea of $S_G$-modules and examine the kinds of orthogonal $G$-spectra and $S_G$-modules. A key function is the research of switch of universe, swap of staff, fastened element, and orbit functors in those hugely established different types for the learn of equivariant good homotopy idea.

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**Extra resources for Equivariant Orthogonal Spectra and S-Modules (Memoirs of the American Mathematical Society)**

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A wedge of level equivalences of nondegenerately based orthogonal G-spectra is a level equivalence. (iii) If i : A −→ X is an h-cofibration and f : A −→ Y is any map of orthogonal G-spectra, where A, X, and Y are nondegenerately based, then X ∪A Y is nondegenerately based and the cobase change j : Y −→ X ∪A Y is an h-cofibration. If i is a level equivalence, then j is a level equivalence. (iv) If i and i′ are h-cofibrations and the vertical arrows are level equivalences in the following commutative diagram of nondegenerately based orthogonal G-spectra, then the induced map of pushouts is a level equivalence.

Thus all maps in K are π∗ -isomorphisms. 5 implies that all retracts of relative K-cell complexes are π∗ -isomorphisms. 9]. The proof of the model axioms is completed as in [18, §9]. The properness of the model structure is implied by the following more general statements. 13. Consider the following commutative diagram: A f j i X /B g / Y. (i) If the diagram is a pushout in which i is an h-cofibration and f is a π∗ isomorphism, then g is a π∗ -isomorphism. (ii) If the diagram is a pullback in which j is a level fibration and g is a π∗ isomorphism, then f is a π∗ -isomorphism.

Model categories of ring and module G-spectra In this section and the next, we study model structures induced by the stable or positive stable model structure on GI S . We prove here that the categories of orthogonal ring spectra and of modules over an orthogonal ring spectrum are Quillen model categories. The proofs are essentially the same as those in the nonequivariant case given in [18, §§12, 14], but some of the cases covered there dictated a more complicated line of argument than is necessary here.

### Equivariant Orthogonal Spectra and S-Modules (Memoirs of the American Mathematical Society) by M. A. Mandell

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