Orthogonal spectra and S-modules, by M.A. Mandell and J.P. May

There are two general approaches to the construction of symmetric monoidal categories of spectra, one based on an encoding of operadic structure in the definition of the smash product and the other based on the categorical observation that categories of diagrams with symmetric monoidal domain are symmetric monoidal. The first was worked out by Elmendorf, Kriz, and the authors in the theory of ``S-modules''. The second was worked out in the case of symmetric spectra by Hovey, Shipley, and Smith and, in a general topological setting, by Schwede, Shipley, and the authors. A comparison between symmetric spectra and S-modules was given by Schwede.

Orthogonal spectra are intermediate between symmetric spectra and S-modules: they are defined in the same diagrammatic fashion as symmetric spectra, but, as with S-modules, their stable weak equivalences are just the maps that induce isomorphisms on homotopy groups. We prove that the categories of orthogonal spectra and S-modules are Quillen equivalent and that this equivalence induces Quillen equivalences between the respective categories of ring spectra, of modules over a ring spectrum, and of commutative ring spectra. The equivalence is given by a functor that is closely related to an older and more intuitive functor from orthogonal spectra to S-modules, and a comparison between the two leads to a precise understanding in terms of a category of Thom spaces of the relationship between the definitions of orthogonal spectra and of S-modules.


M.A. Mandell <mandell@math.mit.edu>
J.P. May <may@math.uchicago.edu>