Spectra and symmetric spectra in general model categories

Abstract

We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A 1 -homotopy theory of Morel and Voevodsky (preprint, 1998) and Voevodsky (Proceedings of the International Congress of Mathematicians, Vol. I, Berlin, Doc. Math. Extra Vol. I, 1998, pp. 579–604 (electronic)). One is based on the standard notion of spectra originated by Vogt (Boardman's Stable Homotopy Category, Lecture Notes Series, Vol. 21, Matematisk Institut Aarhus Universitet, Aarhus, 1970). Its input is a well-behaved model category D and an endofunctor T, generalizing the suspension. Its output is a model category Sp N ( D,T) on which T is a Quillen equivalence. The second construction is based on symmetric spectra (Hovey et al., J. Amer. Math. Soc. 13(1) (2000) 149–208) and applies to model categories C with a compatible monoidal structure. In this case, the functor T must be given by tensoring with a cofibrant object K. The output is again a model category Sp Σ( C,K) where tensoring with K is a Quillen equivalence, but now Sp Σ( C,K) is again a monoidal model category. We study general properties of these stabilizations; most importantly, we give a sufficient condition for these two stabilizations to be equivalent that applies both in the known case of topological spaces and in the case of A 1 -homotopy theory.