### 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>