Astudy in derived algebraic geometry
            Volume I: Correspondences and duality
                  Dennis Gaitsgory
                  Nick Rozenblyum
                    2010 Mathematics Subject Classification. Primary
                                                                       Contents
                                 Preface                                                                                    xi
                                 Acknowledgements                                                                       xxvii
                                 Introduction                                                                            xxix
                                 Part I. Preliminaries                                                                       1
                                 Introduction                                                                                3
                                    Whydoweneed these preliminaries?                                                         3
                                    1.   ∞-categories and higher algebra                                                     3
                                    2.   Basics of derived algebraic geometry                                                4
                                    3.   Quasi-coherent sheaves                                                              5
                                 Chapter 1.     Some higher algebra                                                          7
                                    Introduction                                                                             7
                                    1.   ˆ∞,1-categories                                                                  16
                                    2.   Basic operations with ˆ∞,1-categories                                            26
                                    3.   Monoidal structures                                                               34
                                    4.   Duality                                                                           45
                                    5.   Stable ˆ∞,1-categories                           St,cocmpl                       50
                                    6.   The symmetric monoidal structure on 1-Cat                                         56
                                                                                           cont
                                    7.   Compactly generated stable categories                                             64
                                    8.   Algebra in stable categories                                                      70
                                    9.   Rigid monoidal categories                                                         79
                                    10.   DGcategories                                                                     86
                                 Chapter 2.     Basics of derived algebraic geometry                                       95
                                    Introduction                                                                           95
                                    1.   Prestacks                                                                         99
                                    2.   Descent and stacks                                                               110
                                    3.   (Derived) schemes                                                                121
                                    4.   (Derived) Artin stacks                                                           129
                                 Chapter 3.     Quasi-coherent sheaves on prestacks                                       141
                                    Introduction                                                                          141
                                    1.   The category of quasi-coherent sheaves                                           143
                                    2.   Direct image for QCoh                                                            151
                                    3.   The symmetric monoidal structure                                                 157
                                 Part II. Ind-coherent sheaves                                                            167
                                                                              vii