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S-modules and symmetric spectra. (English) Zbl 0972.55005
The category of S-modules \(\mathcal{M}_S\) (introduced by A. D. Elmendorf, I. Kriz and J. P. May) as well as the category of symmetric spectra \(SP^\Sigma\) (introduced by J. H. Smith) are categories in which a symmetric, associative and monoidal \(\wedge\)-product is defined. On the homotopy level they are both equivalent to the Boardman category which also admits such a \(\wedge\)-product, however displaying these properties only up to homotopy. The main objective of the present paper is to compare these two categories, specifically the associated concepts of ring and module spectra. In order to accomplish this, the author introduces a functor \(\Phi: \mathcal{M}_S \rightarrow SP^\Sigma\) together with a left adjoint \(\Lambda\), satisfying:
\(\Phi\) induces an isomorphism of homotopy categories;
1) \(\text{Ho}(\mathcal{M}_S) @>\cong>> \text{Ho}(SP^\Sigma)\);
2) \(\text{Ho}(S\text{-algebras}) @>\cong>> \text{Ho(symmetric ring spectra)}\);
3) \(\text{Ho(commutative }S\)-algebras)

55P42 Stable homotopy theory, spectra
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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