Categories

Categorical methods of speaking and thinking are becoming more and more widespread in mathematics because they achieve a unifi­ cation of parts of different mathematical fields; frequently they bring simplifications and provide the impetus for new developments. The purpose of this book is to introduce the reader to the central part of category theory and to make the literature accessible to the reader who wishes to go farther. In preparing the English version, I have used the opportunity to revise and enlarge the text of the original German edition. Only the most elementary concepts from set theory and algebra are assumed as prerequisites. However, the reader is expected to be mathe­ to follow an abstract axiomatic approach. matically sophisticated enough The vastness of the material requires that the presentation be concise, and careful cooperation and some patience is necessary on the part of the reader. Definitions alway precede the examples that illuminate them, and it is assumed that the reader is familiar with some of the algebraic and topological examples (he should not let the other ones confuse him). It is also hoped that he will be able to explain the con­ cepts to himself and that he will recognize the motivation.

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.