Monoidal uniqueness of stable homotopy
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.