Spectra and symmetric spectra in general model categories
Two general constructions for the passage from unstable to stable homotopy are given. The first construction is based on the standard notion of spectra [\textit{R. Vogt}, Boardman's Stable Homotopy Category, Lect. Notes Ser., Åarhus Univ. 21 (1970; Zbl 0224.55014)] and starts from a model category \({\mathcal D}\) and an endofunctor \(T\) generalizing the suspension. The second construction is based on symmetric spectra [\textit{M. Hovey}, \textit{B. Shipley} and \textit{J. Smith}, J. Am. Math. Soc. 13, No. 1, 149-208 (2000; Zbl 0931.55006)] and applies to model categories \({\mathcal C}\) witha compatible monoidal structure and an endofunctor \(T\) which is given by tensoring with a cofibrant object \({\mathcal K}\). The constructions apply not only to the homotopy theory for pointed topological spaces but also to the homotopy theory for algebraic varieties and schemes recently developed by \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45-143 (1999; Zbl 0983.14007); Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. I, 579-604 (1998; Zbl 0907.19002)] using the affine line \(\mathbb{A}^1\) instead of the unit interval \([0,1]\).
scientific article; zbMATH DE number 1698557
scientific article