1.1. Simplicial sets 5 1.2. Symmetric spectra 6

1. Preliminaries about topological model categories 5 2. Preliminaries about equivalences of model categories 9 3. The level model structure on D-spaces 10 4. Preliminaries about π∗-isomorphisms of prespectra 14

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.

The main theorem of [4] say that there is a proper closed simplicial model category

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

Abstract not found

The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic K-theory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilinson [Be] as a natural analogue of the Atiyah-Hirzebruch spectral sequence from the singular cohomology to the topological K-theory of a topological space. The expectation of such a spectral sequence has provided much of the impetus for the development of motivic cohomology (e.g., [B1], [V2]) and should facilitate many computations in algebraic K-theory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [B-L]. Our construction depends crucially upon the main result of [B-L], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative K-theory of coherent sheaves on standard simplices over F (recalled as Theorem 5.5 below). A major step in generalizing the work of Bloch and Lichtenbaum is our reinterpretation of their spectral sequence in terms of the “topological filtration ” on the K-theory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ω-prespectra K( ∆ • ) = K 0 ( ∆ • ) ← − K 1 ( ∆ • ) ← − K 2 ( ∆ • ) ← − · · · Thus, even in the special case in which X equals SpecF, we obtain a much clearer understanding of the Bloch-Lichtenbaum spectral sequence which is essential for purposes of generalization. Following this reinterpretation, we proceed using techniques introduced by V. Voevodsky in his study of motivic cohomology. In order to do this, we provide an equivalent formulation of K-theory spectra associated to coherent sheaves on X with conditions on their supports K q ( ∆ • × X) which is functorial in X. We then Partially supported by the N.S.F. and the N.S.A.

Abstract. Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley,..., Toën and Toën-Vaquié. 1.

Abstract. Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the Dwyer-Kan theory of framings, to sheaf theory, and to the homotopy theory of schemes. Contents

1.2. The axioms