S-modules and symmetric spectra

Abstract

We study a symmetric monoidal adjoint functor pair between the category of S-modules and the category of symmetric spectra. The functors induce equivalences between the respective homotopy categories of spectra, module spectra and ring spectra. Then at approximately the same time, two categories of spectra with nice smash products were discov- ered. Elmendorf, Kriz, Mandell and May constructed the category of S-modules (EKMM), and Je Smith introduced symmetric spectra (HSS). Both categories are Quillen model categories and have associated notions of ring and module spectra. However these two categories arise in completely dierent ways. And even though the homotopy categories are equivalent, it is not a priori clear if both frameworks give rise to the same homotopy theory of rings and modules. Both categories have their merits, described in detail in the introductions of (EKMM) and (HSS), and it is desirable to be able to translate results obtained in one category into conclusions valid in the other. The present paper describes an easy mechanism which facilitates such comparisons. Below we dene a lax symmetric monoidal functor :MS ! Sp from the category of S-modules to the category of symmetric spectra. The functor preserves homotopy groups and has a strong symmetric monoidal left adjoint. We show that the two functors induce inverse equivalences of the homotopy categories of spectra, ring spectra, commutative ring spectra and module spectra: Main Theorem. The functor from the category MS of S-modules to the category Sp of symmetric spectra passes to a symmetric monoidal equivalence of homotopy categories