This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small

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. We give two general constructions for the passage from unstable to stable homotopy that apply to the known example of topological spaces, but also to new situations, such as the A1-homotopy theory of Morel-Voevodsky [16, 23]. One is based on the standard notion of spectra originated by Boardman [24]. Its input is a well-behaved model category C and an endofunctor

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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.

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

For any field k of characteristic 0 the Adams spectral sequence for the sphere spectrum based on Suslin-Voevodsky mod 2 motivic cohomology [8] converges to the graded ring associated to the filtration of the Grothendieck-Witt ring of quadratic forms over k by powers of the ideal generated by even dimensional forms. Moreover, some property of the mod 2 motivic cohomology of k, which is a consequence of Voevodsky's proof of Milnor's conjecture on mod 2 Galois cohomology of k [9], implies that the spectral sequence degenerates in the critical area. This allows us to give a new proof of the Milnor conjecture on the graded ring of of the Witt ring of k [4] which differs from [11].

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

In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.