Categories and Functors

CONTENTS ======== Preface 1. Preliminary Notions 1.1 Definition of a Category 1.2 Functors and Natural Transformations 1.3 Representable Functors 1.4 Duality 1.5 Monomorphisms. Epimorphisms. and Isomorphisms 1.6 Subobjects and Quotient Objects 1.7 Zero Objects and Zero Morphisms 1.8 Diagrams 1.9 Difference Kernels and Difference Cokernels 1.10 Sections and Retractions 1.11 Products and Coproducts 1.12 Intersections and Unions 1.13 Images. Coimages. and Counterimages 1.14 Multifunctors 1.15 The Yoneda Lemma 1.16 Categories as Classes Problems 2. Adjoint Functors and Limits 2.1 Adjoint Functors 2.2 Universal Problems 2.3 Monads 2.4 Reflexive Subcategories 2.5 Limits and Colimits 2.6 Special Limits and Colimits 2.7 Diagram Categories 2.8 Constructions with Limits 2.9 The Adjoint Functor Theorem 2.10 Generators and Cogenerators 2.11 Special Casesof the Adjoint Functor Theorem 2.12 Full and Faithful Functors Problems 3. Universal Algebra 3.1 Algebraic Theories 3.2 Algebraic Categories 3.3 Free Algebras 3.4 Algebraic Functors 3.5 Examples of Algebraic Theories and Functors 3.6 Algebras in Arbitrary Categories Problems 4. Abelian Categories 4.1 Additive Categories 4.2 Abelian Categories 4.3 Exact Sequences 4.4 Isomorphism Theorems 4.5 The Jordan-Holder Theorem 4.6 Additive Functors 4.7 Grothendieck Categories 4.8 The Krull.Remak.Schmidt.AzumayaTheorem 4.9 Finitely Generated Objects 4.10 Module Categories 4.11 Semisimple and Simple Rings.12 Functor Categories 4.13 Embedding Theorems Problems Injective and Projective Objects and Hulls Appendix . Fundamentals of Set Theory Bibliography Index