Introduction |
The Language of Categories / 1: |
Preliminaries: Sets and Universes / 1.1: |
Categories and Functors / 1.2: |
Morphisms of Functors / 1.3: |
The Yoneda Lemma / 1.4: |
Adjoint Functors / 1.5: |
Exercises |
Limits / 2: |
Examples / 2.1: |
Kan Extension of Functors / 2.3: |
Inductive Limits in the Category Set / 2.4: |
Cofinal Functors / 2.5: |
Ind-lim and Pro-lim / 2.6: |
Yoneda Extension of Functors / 2.7: |
Filtrant Limits / 3: |
Filtrant Inductive Limits in the Category Set / 3.1: |
Filtrant Categories / 3.2: |
Exact Functors / 3.3: |
Categories Associated with Two Functors / 3.4: |
Tensor Categories / 4: |
Projectors / 4.1: |
Rings, Modules and Monads / 4.2: |
Generators and Representability / 5: |
Strict Morphisms / 5.1: |
Strictly Generating Subcategories / 5.2: |
Indization of Categories / 6: |
Indization of Categories and Functors / 6.1: |
Representable Ind-limits / 6.2: |
Indization of Categories Admitting Inductive Limits / 6.3: |
Finite Diagrams in Ind(C) / 6.4: |
Localization / 7: |
Localization of Categories / 7.1: |
Localization of Subcategories / 7.2: |
Localization of Functors / 7.3: |
Indization and Localization / 7.4: |
Additive and Abelian Categories / 8: |
Group Objects / 8.1: |
Additive Categories / 8.2: |
Abelian Categories / 8.3: |
Injective Objects / 8.4: |
Ring Action / 8.5: |
Indization of Abelian Categories / 8.6: |
Extension of Exact Functors / 8.7: |
[pi]-accessible Objects and F-injective Objects / 9: |
Cardinals / 9.1: |
[pi]-filtrant Categories and [pi]-accessible Objects / 9.2: |
[pi]-accessible Objects and Generators / 9.3: |
Quasi-Terminal Objects / 9.4: |
F-injective Objects / 9.5: |
Applications to Abelian Categories / 9.6: |
Triangulated Categories / 10: |
Localization of Triangulated Categories / 10.1: |
Localization of Triangulated Functors / 10.3: |
Extension of Cohomological Functors / 10.4: |
The Brown Representability Theorem / 10.5: |
Complexes in Additive Categories / 11: |
Differential Objects and Mapping Cones / 11.1: |
The Homotopy Category / 11.2: |
Simplicial Constructions / 11.3: |
Double Complexes / 11.5: |
Bifunctors / 11.6: |
The Complex Hom / 11.7: |
Complexes in Abelian Categories / 12: |
The Snake Lemma / 12.1: |
Abelian Categories with Translation / 12.2: |
Example: Koszul Complexes / 12.3: |
Derived Categories / 12.5: |
Resolutions / 13.1: |
Derived Functors / 13.3: |
Unbounded Derived Categories / 13.4: |
Derived Categories of Abelian Categories with Translation / 14.1: |
Unbounded Derived Category / 14.2: |
Left Derived Functors / 14.4: |
Indization and Derivation of Abelian Categories / 15: |
Injective Objects in Ind(C) / 15.1: |
Quasi-injective Objects / 15.2: |
Derivation of Ind-categories / 15.3: |
Indization and Derivation / 15.4: |
Grothendieck Topologies / 16: |
Sieves and Local Epimorphisms / 16.1: |
Local Isomorphisms / 16.2: |
Localization by Local Isomorphisms / 16.3: |
Sheaves on Grothendieck Topologies / 17: |
Presites and Presheaves / 17.1: |
Sites / 17.2: |
Sheaves / 17.3: |
Sheaf Associated with a Presheaf / 17.4: |
Direct and Inverse Images / 17.5: |
Restriction and Extension of Sheaves / 17.6: |
Internal Hom / 17.7: |
Abelian Sheaves / 18: |
R-modules / 18.1: |
Tensor Product and Internal Hom / 18.2: |
Derived Functors for Hom and Hom / 18.3: |
Flatness / 18.5: |
Ringed Sites / 18.6: |
Cech Coverings / 18.7: |
Stacks and Twisted Sheaves / 19: |
Prestacks / 19.1: |
Simply Connected Categories / 19.2: |
Stacks / 19.3: |
Morita Equivalence / 19.5: |
Twisted Sheaves / 19.6: |
References |
List of Notations |
Index |
Introduction |
The Language of Categories / 1: |
Preliminaries: Sets and Universes / 1.1: |
Categories and Functors / 1.2: |
Morphisms of Functors / 1.3: |
The Yoneda Lemma / 1.4: |