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>