Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.
Author(s): George M. Bergman
Series: Universitext
Edition: 2
Publisher: Springer
Year: 2015
Language: English
Pages: 574
Dedication
Contents
1 About the Course, and These Notes
1.1 Aims and prerequisites
1.2 Approach
1.3 A question a day
1.4 Homework
1.5 The name of the game
1.6 Other reading
1.7 Numeration; history; advice; web access; request for corrections
1.8 Acknowledgements
Part I. Motivation and Examples
2 Making Some Things Precise
2.1 Generalities
2.2 What is a group?
2.3 Indexed sets
2.4 Arity
2.5 Group-theoretic terms
2.6 Evaluation
2.7 Terms in other families of operations
3 Free Groups
3.1 Motivation
3.2 The logician's approach: construction from group-theoretic terms
3.3 Free groups as subgroups of big enough directproducts
3.4 The classical construction: free groups as groups of words
4 A Cook's Tour of Other Universal Constructions
4.1 The subgroup and normal subgroup of G generatedby S|G|
4.2 Imposing relations on a group. Quotient groups
4.3 Groups presented by generators and relations
4.4 Abelian groups, free abelian groups,and abelianizations
4.5 The Burnside problem
4.6 Products and coproducts of groups
4.7 Products and coproducts of abelian groups
4.8 Right and left universal properties
4.9 Tensor products
4.10 Monoids
4.11 Groups to monoids and back again
4.12 Associative and commutative rings
4.13 Coproducts and tensor products of rings
4.14 Boolean algebras and Boolean rings
4.15 Sets
4.16 Some algebraic structures we have not looked at
4.17 The Stone-Čech compactification of a topologicalspace
4.18 Universal covering spaces
Part II. Basic Tools and Concepts
5 Ordered Sets, Induction, and the Axiom of Choice
5.1 Partially ordered sets
5.2 Digression: preorders
5.3 Induction, recursion, and chain conditions
5.4 The axioms of set theory
5.5 Well-ordered sets and ordinals
5.6 Zorn's Lemma
5.7 Some thoughts on set theory
6 Lattices, Closure Operators, and Galois Connections
6.1 Semilattices and lattices
6.2 0, 1, and completeness
6.3 Closure operators
6.4 Digression: a pattern of threes
6.5 Galois connections
7 Categories and Functors
7.1 What is a category?
7.2 Examples of categories
7.3 Other notations and viewpoints
7.4 Universes
7.5 Functors
7.6 Contravariant functors, and functors of severalvariables
7.7 Category-theoretic versions of some common mathematical notions: properties of morphisms
7.8 More categorical versions of common mathematical notions: special objects
7.9 Morphisms of functors (or ``natural transformations'')
7.10 Properties of functor categories
7.11 Enriched categories (a sketch)
8 Universal Constructions in Category-Theoretic Terms
8.1 Universality in terms of initial and terminal objects
8.2 Representable functors, and Yoneda's Lemma
8.3 Adjoint functors
8.4 Number-theoretic interlude: the p-adic numbersand related constructions
8.5 Direct and inverse limits
8.6 Limits and colimits
8.7 What respects what?
8.8 Functors respecting limits and colimits
8.9 Interaction between limits and colimits
8.10 Some existence theorems
8.11 Morphisms involving adjunctions
8.12 Contravariant adjunctions
9 Varieties of Algebras
9.1 The category Ω-Alg
9.2 Generating algebras from below
9.3 Terms and left universal constructions
9.4 Identities and varieties
9.5 Derived operations
9.6 Characterizing varieties and equational theories
9.7 Lie algebras
9.8 Some instructive trivialities
9.9 Clones and clonal theories
9.10 Structure and Semantics
Part III. More on Adjunctions
10 Algebra and Coalgebra Objects in Categories, and Functors Having Adjoints
10.1 An example: SL(n)
10.2 Algebra objects in a category
10.3 Coalgebra objects in a category
10.4 Freyd's criterion for the existence of left adjoints
10.5 Some corollaries and examples
10.6 Representable endofunctors of Monoid
10.7 Functors to and from some related categories
10.8 Representable functors among categories of abelian groups and modules
10.9 More on modules: left adjoints of representablefunctors
10.10 Some general results on representable functors, mostly negative
10.11 A few ideas and techniques
10.12 Contravariant representable functors
10.13 More on commuting operations
10.14 Some further reading on representable functors, and on General Algebra
References
List of Exercises
Symbol Index
Word and Phrase Index
