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.

Abstract. In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model structures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. We prove homotopy invariance properties for the categories of algebras and modules. In a second part we develop the theory of S-modules and algebras of [EKMM] and [KM], which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.

Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). 1.

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

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Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads.

Let M n be a closed, connected n-manifold. Let M −τ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that M −τ is homotopy equivalent to the Spanier-Whitehead dual of M with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F(M, S), where S is the sphere spectrum. F(M, S) has the structure of a commutative, symmetric ring spectrum in the sense of [7], [12] [9]. In this paper we prove that M −τ also has a natural, geometrically defined, structure of a commutative, symmetric ring spectrum, in such a way that the classical duality maps of Alexander, Spanier-Whitehead, and Atiyah define an equivalence of symmetric ring spectra, α: M −τ → F(M,S). We discuss applications of this to Hochschild cohomology representations of the Chas-Sullivan loop product in the homology of the free loop space of M.

Under a certain normalization assumption we prove that the P 1-spectrum BGL of Voevodsky which represents algebraic K-theory is unique over Spec(Z). Following an idea of Voevodsky, we equip the P 1-spectrum BGL with the structure of a commutative ring P 1-spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over Spec(Z). For an arbitrary Noetherian base scheme S we pull this structure back to get a distinguished monoidal structure on BGL. 1

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