Monoidal uniqueness of stable homotopy

ABSTRACT. We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences produced here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, W-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. The strong monoidal left adjoints constructed here give rise to an action of symmetric spectra and hence also an enrichment over symmetric spectra on any simplicial, strong monoidal model category with a certain condition on the unit.