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**Model categories of diagram spectra.**
*(English)*
Zbl 1017.55004

There are several different constructions of a category of spectra and a symmetric monoidal smash product which descends to the usual smash product in its homotopy category. The paper focuses on the followings ones: the category of symmetric spectra introduced by M. Horey, B. Shipley and J. Smith [J. Am. Math. Soc. 13, No. 1, 149-208 (2000; Zbl 0931.55006)], of orthogonal spectra considered by J. P. May [J. Pure Appl. Algebra 19, 299-346 (1980; Zbl 0469.18009)], of \(\Gamma\)-spaces introduced by G. Segal in [Topology 13, 293-312 (1974; Zbl 0284.55016)] and of \(W\)-spaces studied by D. W. Anderson in [Localiz. Group Theory Homotopy Theory rel. Topics, Lect. Notes Math. 418, 1-5 (1974; Zbl 0301.55001)].

Since they all arise as diagram categories there are forgetful and prolongation functors, relating the various categories to each other. It is shown that with the exception of the \(\Gamma\)-spaces all of them give Quillen equivalent model categories in a way that the induced smash products in the homotopy categories are preserved. Similar statements hold for the corresponding categories of ring spectra and module spectra.

Commutative ring spectra are more subtle. Replacing the stable model structure of symmetric spectra by a positive one, the authors establish a Quillen equivalence to the category of orthogonal commutative ring spectra. Finally, it is proved that the categories of \(\Gamma\)-(ring) spaces and \(W\)-(ring) spaces are connectively Quillen equivalent.

The paper is well written and can serve as an introduction to this brave new world of highly structured ring spectra. It relates the modern approaches among each other by using the very general, elegant and efficient perspective of diagram categories.

Since they all arise as diagram categories there are forgetful and prolongation functors, relating the various categories to each other. It is shown that with the exception of the \(\Gamma\)-spaces all of them give Quillen equivalent model categories in a way that the induced smash products in the homotopy categories are preserved. Similar statements hold for the corresponding categories of ring spectra and module spectra.

Commutative ring spectra are more subtle. Replacing the stable model structure of symmetric spectra by a positive one, the authors establish a Quillen equivalence to the category of orthogonal commutative ring spectra. Finally, it is proved that the categories of \(\Gamma\)-(ring) spaces and \(W\)-(ring) spaces are connectively Quillen equivalent.

The paper is well written and can serve as an introduction to this brave new world of highly structured ring spectra. It relates the modern approaches among each other by using the very general, elegant and efficient perspective of diagram categories.

Reviewer: Gerd Laures (Heidelberg)