Categories for the Working Mathematician

An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence. CONTENTS ========= Introduction I. Categories, Functors, and Natural Transformations 1. Axioms for Categories 2. Categories 3. Functors 4. Natural Transformations 5. Monics, Epis, and Zeros 6. Foundations 7. Large Categories 8. Hom-Sets II. Constructions on Categories 1. Duality 2. Contravariance and Opposites 3. Products of Categories 4. Functor Categories 5. The Category of All Categories 6. Comma Categories 7. Graphs and Free Categories 8. Quotient Categories III. Universals and Limits 1. Universal Arrows 2. The Y oneda Lemma 3. Coproducts and Colimits 4. Products and Limits 5. Categories with Finite Products 6. Groups in Categories 7. Colimits of Representable Functors IV. Adjoints 1. Adjunctions 2. Examples of Adjoints 3. Reflective Subcategories 4. Equivalence of Categories 5. Adjoints for Preorders 6. Cartesian Closed Categories 7. Transformations of Adjoints 8. Composition of Adjoints 9. Subsets and Characteristic Functions 10. Categories Like Sets V. Limits 1. Creation of Limits 2. Limits by Products and Equalizers 3. Limits with Parameters 4. Preservation of Limits 5. Adjoints on Limits 6. Freyd's Adjoint Functor Theorem 7. Subobjects and Generators 8. The Special Adjoint Functor Theorem 9. Adjoints in Topology VI. Monads and Algebras 1. Monads in a Category 2. Algebras for a Monad 3. The Comparison with Algebras 4. Words and Free Semigroups 5. Free Algebras for a Monad 6. Split Coequalizers 7. Beck's Theorem 8. Algebras Are T-Algebras 9. Compact Hausdorff Spaces VII. Monoids 1. Monoidal Categories 2. Coherence 3. Monoids 4. Actions 5. The Simplicial Category 6. Monads and Homology 7. Closed Categories 8. Compactly Generated Spaces 9. Loops and Suspensions VIII. Abelian Categories 1. Kernels and Cokernels 2. Additive Categories 3. Abelian Categories 4. Diagram Lemmas IX. Special Limits 1. Filtered Limits 2. Interchange of Limits 3. Final Functors 4. Diagonal Naturality 5. Ends 6. Coends 7. Ends with Parameters 8. Iterated Ends and Limits X. Kan Extensions 1. Adjoints and Limits 2. Weak Universality 3. The Kan Extension 4. Kan Extensions as Coends 5. Pointwise Kan Extensions 6. Density 7. All Concepts Are Kan Extensions XI. Symmetry and Braiding in Monoidal Categories 1. Symmetric Monoidal Categories 2. Monoidal Functors 3. Strict Monoidal Categories 4. The Braid Groups Bn and the Braid Category 5. Braided Coherence 6. Perspectives XII. Structures in Categories 1. Internal Categories 2. The Nerve of a Category 3. 2-Categories 4. Operations in 2-Categories 5. Single-Set Categories 6. Bicategories 7. Examples of Bicategories 8. Crossed Modules and Categories in Grp Appendix. Foundations Table of Standard Categories: Objects and Arrows Table of Terminology Bibliography Index