1. Categories -- 1.1 Definition of Categories -- 1.2 Examples -- 1.3 Isomorphisms -- 1.4 Further Examples -- 1.5 Additive Categories -- 1.6 Subcategories -- 1.7 Problems -- 2. Functors -- 2.1 Covariant Functors -- 2.2 Standard Examples -- 2.3 Contravariant Functors -- 2.4 Dual Categories -- 2.5 Bifunctors -- 2.6 Natural Transformations -- 2.7 Problems -- 3. Categories of Categories and Categories of Functors -- 3.1 Preliminary Remarks -- 3.2 Universes -- 3.3 Conventions -- 3.4 Functor Categories -- 3.5 The Category of Small Categories -- 3.6 Large Categories -- 3.7 The Evaluation Functor -- 3.8 The Additive Case -- 3.9 Problems -- 4. Representable Functors -- 4.1 Embeddings -- 4.2 Yoneda Lemma -- 4.3 The Additive Case -- 4.4 Representable Functors -- 4.5 Partially Representable Bifunctors -- 4.6 Problems -- 5. Some Special Objects and Morphisms -- 5.1 Monomorphisms -- 5.1° Epimorphisms -- 5.2 Retractions and Coretractions -- 5.3 Bimprphisms -- 5.4 Terminal and Initial Objects -- 5.5 Zero objects -- 5.6 Problems -- 6. Diagrams -- 6.1 Diagram Schemes and Diagrams -- 6.2 Diagrams with Commutativity Conditions -- 6.3 Diagrams as Presentations of Functors -- 6.4 Quotients of Categories -- 6.5 Classes of Mono-, resp., Epimorphisms -- 6.6 Problems -- 7 Limits -- 7.1 Definition of Limits -- 7.2 Equalizers -- 7.3 Products -- 7.4 Complete Categories -- 7.5 Limits in Functor Categories -- 7.6 Double Limits -- 7.7 Criteria for Limits -- 7.8 Pullbacks -- 7.9 Problems -- 8. Colimits -- 8.1 Definition of Colimits -- 8.2 Coequalizers -- 8.3 Coproducts -- 8.4 Cocomplete Categories -- 8.5 Colimits in Functor Categories -- 8.6 Double Colimits -- 8.7 Criteria for Colimits -- 8.8 Pushouts -- 8.9 Problems -- 9. Filtered Colimits -- 9.1 Connected Categories -- 9.2 On the Calculation of Limits and Colimits -- 9.3 Filtered Categories -- 9.4 Filtered Colimits -- 9.5 Commutativity Theorems -- 9.6 Problems -- 10. Setvalued Functors -- 10.2 Properties Inherited from the Codomain Category -- 10.2 The Yoneda Embedding H*: C ? [C0, Ens] -- 10.3 The General Representation Theorem -- 10.4 Projective and Injective Objects -- 10.5 Generators and Cogenerators -- 10.6 Well-powered Categories -- 10.7 Problems -- 11. Objects with an Algebraic Structure -- 11.1 Algebraic Structures -- 11.2 Operations of an Object on Another -- 11.3 Homomorphisms -- 11.4 Reduction to Ens -- 11.5 Limits and Filtered Colimits -- 11.6 Homomorphically Compatible Structures -- 11.7 Problems -- 12. Abelian Categories -- 12.1 Survey -- 12.2 Semi-additive Structure -- 12.3 Kernels and Cokernels -- 12.4 Factorization of Morphisms -- 12.5 The Additive Structure -- 12.6 Idempotents -- 12.7 Problems -- 13. Exact Sequences -- 13.1 Exact Sequences in Exact Categories -- 13.2 Short Exact Sequences -- 13.3 Exact and Faithful Functors -- 13.4 Exact Squares -- 13.5 Some Diagram Lemmas -- 13.6 Problems -- 14. Colimits of Monomorphisms -- 14.1 Preordered Classes -- 14.2 Unions of Monomorphisms -- 14.3 Inverse Images of Monomorphisms -- 14.4 Images of Monomorphisms -- 14.5 Constructions for Colimits -- 14.6 Grothendieck Categories -- 14.7 Problems -- 15. Injective Envelopes -- 15.1 Modules over Additive Categories -- 15.2 Essential Extensions -- 15.3 Existence of Injectives -- 15.4 An Embedding Theorem -- 15.5 Problems -- 16. Adjoint Functors -- 16.1 Composition of Functors and Natural Transformations -- 16.2 Equivalences of Categories -- 16.3 Skeletons -- 16.4 Adjoint Functors -- 16.5 Quasi-inverse Adjunction Transformations -- 16.6 Fully Faithful Adjoints -- 16.7 Tensor Products -- 16.8 Problems -- 17. Pairs of Adjoint Functors between Functor Categories -- 17.1 The Kan Construction -- 17.2 Dense Functors -- 17.3 Characterization of the Yoneda Embedding -- 17.4 Small Projective Objects -- 17.5 Finitely Generated Objects -- 17.6 Natural Transformations with Parameters -- 17.7 Tensor Products over Small Categories -- 17.8 Relatives of the Tensor Product -- 17.9 Problems -- 18. Principles of Universal Algebra -- 18.1 Algebraic Theories -- 18.2 Yoneda Embedding and Free Algebras -- 18.3 Subalgebras and Cocompleteness -- 18.4 Coequalizers and Kernel Pairs -- 18.5 Algebraic Functors and Left Adjoints -- 18.6 Semantics and Structure -- 18.7 The Kronecker Product -- 18.8 Characterization of Algebraic Categories -- 18.9 Problems -- 19. Calculus of Fractions -- 19.1 Categories of Fractions -- 19.2 Calculus of Left Fractions -- 19.3 Factorization of Functors and Saturation -- 19.4 Interrelation with Subcategories -- 19.5 Additivity and Exactness -- 19.6 Localization in Abelian Categories -- 19.7 Characterization of Grothendieck Categories with a Generator -- 19.8 Problems -- 20. Grothendieck Topologies -- 20.1 Sieves and Topologies -- 20.2 Covering Morphisms and Sheaves -- 20.3 Sheaves Associated with a Presheaf -- 20.4 Generation of Topologies -- 20.5 Pretopologies -- 20.6 Characterization of Topos -- 20.7 Problems -- 21. Triples -- 21.1 The Construction of Eilenberg and Moore -- 21.2 Full Image and Kleisli Categories -- 21.3 Limits and Colimits in Eilenberg-Moore Categories -- 21.4 Split Forks -- 21.5 Characterization of Eilenberg-Moore Situations -- 21.6 Consequences of Factorizations of Morphisms -- 21.7 Eilenberg-Moore Categories as Functor Categories -- 21.8 Problems.