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.

These notes are based on lectures given at the Workshop on Structured ring spectra and

Roughly speaking, the stable homotopy category of algebraic topology is obtained from the

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. 1.

We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.

The two categories are not Quillen equivalent, and his proof uses systems of triangulated diagram categories rather than model categories. Our main result is that in the case n = 1 Franke’s functor maps the derived tensor product to the smash product. It can however not be an associative equivalence of monoidal categories. The first part of our paper sets up a monoidal version of Franke’s systems of triangulated diagram categories and explores its properties. The second part applies these results to the specific construction of Franke’s functor in order to prove the above result. 1.

He spent 11 years in a succession of postdoctoral positions in Canada, USA and Britain, including two years at the University of Chicago as an L. E. Dickson Instructor and 2 years as an EPSRC Advanced Fellow, before being appointed in 1991 to a Lectureship (and subsequently in 1996 to a Readership) at the University of Glasgow. His research has centred on algebraic topology, especially stable homotopy theory. In particular he has focused on applications of algebra and number theory to complex oriented and periodic cohomology theories (especially K-theory and elliptic cohomology). For a representative overview of his work see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In recent years he has been very involved in work on structured ring spectra and related topics, and organised a series of workshops in Glasgow, Bonn and Rosendal (Norway), as well as editing a book based on the first of these [13]. Sarah Whitehouse was awarded a Ph.D. from the University of Warwick in 1994. She spent several years in France, as a Marie-Curie post-doctoral researcher at the Université Paris-Nord and as a Lecturer at the Université d’Artois. She joined the University of Sheffield as a Lecturer in 2002 and was promoted to Senior Lecturer in 2005. Much of her work has involved the algebras of operations or cooperations of generalised cohomology theories [18, 19, 27, 38]. Recently, this has given new results for complex K-theory, cobordism and the Morava Ktheories

The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘non-additive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, compare Definition 3.1; this class includes the algebraic triangulated categories. The purpose of this paper is to explain some systematic differences between these two kinds of triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. These differences are all torsion phenomena, and rationally every topological triangulated category is algebraic (at least under mild size restrictions). Our main tool is a new numerical invariant, the n-order of an object in a triangulated category, for n a natural number (see Definition 1.1). The n-order is a nonnegative integer (or infinity), and an object Y has positive n-order if and only if n · Y = 0; the n-order can be thought of

Abstract. We survey various approaches to axiomatic stable homotopy theory, with examples including derived categories, categories of (possibly equivariant or localized) spectra, and stable categories of modular representations of finite groups. We focus mainly on representability theorems, localisation, Bousfield classes, and nilpotence. 1.