Preface Part Ⅰ.Motives Introduction: Part Ⅰ Chapter Ⅰ.The Motivic Category 1.The motivic DG category 2.The triangulated motivic category 3.Structure of the motivic categories Chapter Ⅱ.Motivic Cohomology and Higher Chow Groups 1.Hypercohomology in the motivic category 2.Higher Chow groups 3.The motivic cycle map Chapter Ⅲ.K-Theory and Motives 1.Chern classes 2.Push-forward 3.Riemann-Roch Chapter Ⅳ.Homology, Cohomology, and Duality 1.Duality 2.Classical constructions 3.Motives over a perfect field Chapter Ⅴ.Realization of the Motivic Category 1.Realization for geometric cohomology 2.Concrete realizations Chapter Ⅵ.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 Ⅱ.Categorical Algebra Introduction: Part Ⅱ Chapter Ⅰ.Symmetric Monoidal Structures 1.Foundational material 2.Constructions and computations Chapter Ⅱ.DG Categories and Triangulated Categories 1.Differential graded categories 2.Complexes and triangulated categories 3.Constructions Chapter Ⅲ.Simplicial and Cosimplicial Constructions 1.Complexes arising from simplicial and cosimplicial objects 2.Categorical cochain operations 3.Homotopy limits Chapter Ⅳ.Canonical Models for Cohomology 1.Sheaves, sites, and topoi 2.Canonical resolutions Bibliography Subject Index Index of Notation