Spectra and symmetric spectra in general model categories
by Mark Hovey
Wesleyan University
hovey@member.ams.org
October, 2000
This is the final version, to appear in JPAA. There are several
significant notational changes, and many minor corrections in this
version.
The basic idea of the paper 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 T of C, we want to make a new model category,
the stabilization of C, where T 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 T. We
show that, under strong smallness hypotheses on T and C, the stable
equivalences coincide with the appropriate generalization of stable
homotopy isomorphisms.
If C has a tensor product, and T 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.
We offer a 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.