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 introdu...
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| Format: | eBook |
| Language: | English |
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Berlin, Heidelberg
Springer Berlin Heidelberg
1972, 1972
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| Edition: | 1st ed. 1972 |
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| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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
- 9.5Commutativity 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 theYoneda 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
- 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
- 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