Categories for the working mathematician类别为工作的数学家
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作者： Saunders Mac Lane著
出 版 社：
出版时间： 1998-9-1字数：版次： 1页数： 314印刷时间： 1998/09/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387984032包装： 精装内容简介
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 on symmetric 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.
目录
Preface to the Second Edition
Preface to the First Edition
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 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
IX. Special Limits
X. Kan Extensions
XI. Symmetry and Braiding in Monoidal Categories
XII. Structures in Categories
Appendix. Foundations
Table of Standard Categories: Objects and Arrows
Table of Terminology
Bibliography
Index