Categories for the Working Mathematician provides 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. The book 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 characterized 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 two new chapters on topics of active interest. One is onsymmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.

Review

From the reviews of the second edition:

“The book under review is an introduction to the theory of categories which, as the title suggests, is addressed to the (no-nonsense) working mathematician, thus presenting the ideas and concepts of Category Theory in a broad context of mainstream examples (primarily from algebra). … the book remains an authoritative source on the foundations of the theory and an accessible first introduction to categories. … It is very well-written, with plenty of interesting discussions and stimulating exercises.” (Ittay Weiss, MAA Reviews, July, 2014)

Second Edition

S.M. Lane

Categories for the Working Mathematician

"A very useful introduction to category theory."―INTERNATIONALE MATHEMATISCHE NACHRICHTEN

Table of Contents

I. Categories, Functors, and Natural Transformations

1. Axioms for Categories

2. Categories

3. Functors

4. Natural Transformations

5. Monies, 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

Ill. Universals and Limits

1. Universal Arrows

2. The Yoneda 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

About the Author

Saunders Mac Lane (born August 4, 1909, Taftville, Connecticut, U.S.—died April 14, 2005, San Francisco, California) was an American mathematician who was a cocreator of category theory, an architect of homological algebra, and an advocate of categorical foundations for mathematics.