Category theory in context

Description

Preface Sample corollaries A tour of basic categorical notions Note to the reader Notational conventions Acknowledgments Chapter 1. Categories, Functors, Natural Transformations 1.1. Abstract and concrete categories 1.2. Duality 1.3. Functoriality 1.4. Naturality 1.5. Equivalence of categories 1.6. The art of the diagram chase 1.7. The 2-category of categories Chapter 2. Universal Properties, Representability, and the Yoneda Lemma 2.1. Representable functors 2.2. The Yoneda lemma 2.3. Universal properties and universal elements 2.4. The category of elements Chapter 3. Limits and Colimits 3.1. Limits and colimits as universal cones 3.2. Limits in the category of sets 3.3. Preservation, reflection, and creation of limits and colimits 3.4. The representable nature of limits and colimits 3.5. Complete and cocomplete categories 3.6. Functoriality of limits and colimits 3.7. Size matters 3.8. Interactions between limits and colimits Chapter 4. Adjunctions 4.1. Adjoint functors 4.2. The unit and counit as universal arrows 4.3. Contravariant and multivariable adjoint functors 4.4. The calculus of adjunctions 4.5. Adjunctions, limits, and colimits 4.6. Existence of adjoint functors Chapter 5. Monads and their Algebras 5.1. Monads from adjunctions 5.2. Adjunctions from monads 5.3. Monadic functors 5.4. Canonical presentations via free algebras 5.5. Recognizing categories of algebras 5.6. Limits and colimits in categories of algebras Chapter 6. All Concepts are Kan Extensions 6.1. Kan extensions 6.2. A formula for Kan extensions 6.3. Pointwise Kan extensions 6.4. Derived functors as Kan extensions 6.5. All concepts Epilogue: Theorems in Category Theory E.1. Theorems in basic category theory E.2. Coherence for symmetric monoidal categories E.3. The universal property of the unit interval E.4. A characterization of Grothendieck toposes E.5. Embeddings of abelian categories Bibliography Catalog of Categories Glossary of Notation Index