The Stable Homotopy Category Has a Unique Model at the Prime 2

Abstract

The stable homotopy category has been extensively studied by algebraic topologists for a long time. For many applications it is convenient or even necessary to work with point set level models of spectra as opposed to working up-to-homotopy, and the outcome of a calculation can depend on the choice of model. In recent years many new models for the stable homotopy category have been constructed. It is especially useful to have the structure of a closed model category in the sense of Quillen [Qui] and many examples of spectra categories t into this context [BF, Rob87, EKMM, HSS, Lyd, MMSS]. Moreover all known examples capture the ‘same homotopy theory’ { in technical terms one speaks of Quillen equivalent model categories [Hov, Def. 1.3.12]. Hence not only the homotopy categories, but also higher order information such as Toda brackets, homotopy colimits and homotopy types of function spaces coincide. In two Quillen equivalent model categories the answer to every homotopy theoretic question comes out the same. In a model category one can pass to the associated homotopy category by formally inverting the class of weak equivalences. However, passage to the homotopy category loses information and in general the ‘homotopy theory’ can not be recovered from the homotopy category, see 2.1 and 2.2 for two examples. In this paper we show that in contrast to the general case, the stable homotopy category completely determines the stable homotopy theory 2-locally. We prove a uniqueness theorem which says that there is essentially only one model category structure underlying the stable homotopy category of 2-local spectra | the stable homotopy category has no ‘exotic’ models at the prime 2.