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Weighted limits in simplicial homotopy theory
 Journal of Pure and Applied Algebra vol
, 2010
"... Abstract. We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functo ..."
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Abstract. We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor. 1.
The additivity of traces in monoidal derivators
 the Journal of Ktheory. arXiv:1212.3277
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SHEAVES AND HOMOTOPY THEORY
"... The purpose of this note is to describe the homotopytheoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel ..."
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The purpose of this note is to describe the homotopytheoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the LichtenbaumQuillen conjecture for Bottperiodic algebraic Ktheory. He termed his constructions ‘hypercohomology spectra’, and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic Ktheory or étale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason’s and Jardine’s machinery can be built, stepbystep, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopytheoretic situation as a generalization of the classical case. In some sense the approach here will be the reverse of this: we will instead assume a general familiarity with homotopy theory, and show how the theory of sheaves fits in with perspectives already offered by the field.
On compactly generated torsion pairs and the classification of cotstructures for commutative Noetherian rings
, 2014
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