Spectra and symmetric spectra in general model categories

by Mark Hovey
Wesleyan University
April, 2000

This is a revised version.  The basic idea is to automate the passage
from unstable to stable homotopy theory, so that it applies in
particular to the A^1 category of Voevodsky.  So if we start with a model
category C and a left Quillen endofunctor G of C, we want to make a new
model category, the stabilization of C, where G becomes a Quillen
equivalence.  The simplest way to do this is with ordinary spectra.
Thanks to Hirschhorn's localization technology, we can construct the
stable model structure on ordinary spectra with almost no hypotheses on
C and G.  A new feature of this revision is that we show that, under
strong smallness hypotheses on G and C, the stable equivalences coincide
with the appropriate generalization of stable homotopy isomorphisms.  In
particular, this holds for the A^1 category.  

If C has a tensor product, and G is given by tensoring with a cofibrant
object K, then we also can construct symmetric spectra.  The
localization techniques apply here as well, so we get a stable model
structure of symmetric spectra without having to assume anything like the
Freudenthal suspension theorem.  In particular, this is a new
construction of the stable model structure on simplicial symmetric
spectra.  Symmetric spectra form a monoidal model category, unlike
ordinary spectra, but we are unable to prove that the monoid axiom holds
in general. 

Also new to this revision is a much more careful comparison between
symmetric spectra and ordinary spectra when both are defined.  Symmetric
spectra and ordinary spectra are not always Quillen equivalent; we need
the cyclic permutation map on K tensor K tensor K to be homotopic to the
identity.  Under some additional technical hypotheses (which again are
satisfied in the A^1 category), we construct a zigzag of Quillen
equivalences between symmetric spectra and ordinary spectra.