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 image-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 (Boardmanʹs Stable Homotopy Category, Lecture Notes Series, Vol. 21, Matematisk Institut Aarhus Universitet, Aarhus, 1970). Its input is a well-behaved model category image and an endofunctor T, generalizing the suspension. Its output is a model category image 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 image 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 image where tensoring with K is a Quillen equivalence, but now image 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 image-homotopy theory.
