### Model categories of diagram spectra, by M. A. Mandell, J. P. May, S. Schwede, and B. Shipley

In this sequel to our paper ``Diagram spaces, diagram spectra,
and FSP's", we construct and compare model structures on the
categories of prespectra, symmetric spectra, orthogonal spectra,
Gamma-spaces, and W-spaces defined there. With the caveat that
Gamma-spaces are always connective, the homotopy categories
associated to all of these model categories are equivalent to
the classical stable homotopy category.

In all cases, there is a levelwise model structure, in which the
weak equivalences and fibrations are defined levelwise. Actually,
it is often convenient or necessary to modify this by considering
some but not all levels. There is then a stable model structure
in which the cofibrations are the cofibrations in the level model
structure and the weak equivalences are the stable weak equivalences.
In the cases of prespectra, orthogonal spectra, Gamma-spaces, and
W-spaces, stable weak equivalences are just maps whose associated
maps of prespectra induce isomorphisms of homotopy groups. In the
case of symmetric spectra, a stable weak equivalence f: X >--> Y is
a map such that f^*:[Y,E] >--> [X,E] is an isomorphism for all
symmetric Omega-spectra E, where the brackets refer to the levelwise
homotopy category. Modulo the caveat about Gamma-spaces, the model
categories of prespectra, symmetric spectra, orthogonal spectra,
Gamma-spaces, and W-spaces are Quillen equivalent and thus have
equivalent homotopy categories.

In favorable cases, the subcategories of ring spectra, module
spectra over a ring spectrum, and commutative ring spectra are
also model categories. Prespectra do not form a symmetric monoidal
category, this being the main reason for interest in the other
categories. In all other cases, the respective categories of ring
spectra are model categories and, with the caveat about Gamma-spaces,
they are all Quillen equivalent and thus have equivalent homotopy
categories. A similar statement holds for module spectra over ring
spectra. The categories of commutative symmetric ring spectra and
commutative orthogonal ring spectra are model categories and are
Quillen equivalent.

M. A. Mandell <mandell@math.mit.edu>

J. P. May <may@math.uchicago.edu>

S. Schwede <schwede@mathematik.uni-bielefeld.de>

B. Shipley <bshipley@math.purdue.edu>