Morita theory in abelian, derived and stable model categories

This is a survey paper, based on lectures given at the Workshop on “Structured ring spectra and their applications” which took place January 21-25, 2002, at the University of Glasgow. The term Morita theory' is usually used for results concerning equivalences of various kinds of module categories. We focus on the covariant form of Morita theory, so the basic question is: When do two rings’ have equivalent' module categories ? We discuss this question in different contexts and illustrate it by examples: (Classical) When are the module categories of two rings equivalent as categories ? (Derived) When are the derived categories of two rings equivalent as triangulated categories ? (Homotopical) When are the module categories of two ring spectra Quillen equivalent as model categories ? There is always a related question, which is in a sense more general: What characterizes the category of modules over a ring’ ? The answer is, mutatis mutandis, always the same: modules over a ring' are characterized by the existence of a small generator’, which plays the role of the free module of rank one. The precise meaning of `small generator’ depends on the context, be it an abelian category, a derived category or a stable model category.