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Model theoretic reformulation of the BaumConnes and FarrellJones conjectures
, 2003
"... Abstract. The Isomorphism Conjectures are translated into the language of homotopical algebra, where they resemble Thomason’s descent theorems. 1. Introduction and statement of the results In [8], Thomason establishes that algebraic Ktheory satisfies Zariski and Nisnevich descent. This is now consi ..."
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Abstract. The Isomorphism Conjectures are translated into the language of homotopical algebra, where they resemble Thomason’s descent theorems. 1. Introduction and statement of the results In [8], Thomason establishes that algebraic Ktheory satisfies Zariski and Nisnevich descent. This is now considered a profound algebraicogeometric property of Ktheory. In [1, 2], we have introduced the sister notion of codescent. Here, we prove that each one of the socalled Isomorphism Conjectures (see [3, 5]) among
Codescent theory II: Cofibrant approximations
, 2003
"... Abstract. We establish a general method to produce cofibrant approximations in the model category US(C, D) of Svalued Cindexed diagrams with Dweak equivalences and Dfibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial mode ..."
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Abstract. We establish a general method to produce cofibrant approximations in the model category US(C, D) of Svalued Cindexed diagrams with Dweak equivalences and Dfibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial model category and D ⊂ C are small categories. An application to the notion of homotopy colimit is presented. 1.