# Classification of stable model categories

with Stefan Schwede

** ABSTRACT.**
A stable model category is a setting for homotopy theory
where the suspension functor is invertible.
The prototypical examples are the category of spectra in the sense of stable
homotopy theory and the category of unbounded chain complexes of modules over
a ring.
In this paper we develop methods for deciding when two stable model
categories represent `the same homotopy theory'.
We show that stable model categories with a single compact generator are
equivalent to modules over a ring spectrum.
More generally stable model categories with a set of generators are
characterized as modules over a `ring spectrum with several objects',
i.e., as spectrum valued diagram categories.
We also prove a Morita theorem which shows how equivalences between module
categories over ring spectra can be realized by smashing with a pair of
bimodules. Finally, we characterize stable model categories which
represent the derived category of a ring. This is a slight
generalization of Rickard's work on derived equivalent rings.
We also include a proof of the model category equivalence
of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain
complexes of R-modules for a ring R.