Preface Part ��.Motives Introduction: Part �� Chapter ��.The Motivic Category 1.The motivic DG category 2.The triangulated motivic category 3.Structure of the motivic categories Chapter ��.Motivic Cohomology and Higher Chow Groups 1.Hypercohomology in the motivic category 2.Higher Chow groups 3.The motivic cycle map Chapter ��.K-Theory and Motives 1.Chern classes 2.Push-forward 3.Riemann-Roch Chapter ��.Homology, Cohomology, and Duality 1.Duality 2.Classical constructions 3.Motives over a perfect field Chapter ��.Realization of the Motivic Category 1.Realization for geometric cohomology 2.Concrete realizations Chapter ��.Motivic Constructions and Comparisons 1.Motivic constructions 2.Comparison with the category DMom(k) Appendix A.Equi-dimensional Cycles 1.Cycles over a normal scheme 2.Cycles over a reduced scheme Appendix B.K-Theory 1.K-theory of rings and schemes 2.K-theory and homology Part ��.Categorical Algebra Introduction: Part �� Chapter ��.Symmetric Monoidal Structures 1.Foundational material 2.Constructions and computations Chapter ��.DG Categories and Triangulated Categories 1.Differential graded categories 2.Complexes and triangulated categories 3.Constructions Chapter ��.Simplicial and Cosimplicial Constructions 1.Complexes arising from simplicial and cosimplicial objects 2.Categorical cochain operations 3.Homotopy limits Chapter ��.Canonical Models for Cohomology 1.Sheaves, sites, and topoi 2.Canonical resolutions Bibliography Subject Index Index of Notation