Introduction to Category Theory

✍ Scribed by Viakalathur Sankrithi Krishnan

• Publisher: Elsevier Science Ltd
• Year: 1980
• Tongue: English
• Leaves: 186
• Edition: 1
• Category: Library

✦ Synopsis

CONTENTS
========+

Preface
CHAPTER ONE. BASICS FROM ALGEBRA AND TOPOLOGY
1.1 Set Theory
1.2 Some Typical Algebraic Structures
1.3 Algebras in General
1.4 Topological Spaces
1.5 Semimetric and Semiuniform Spaces
1.6 Completeness and the Canonical Completion
CHAPTER TWO. CATEGORIES, DEFINITIONS, AND EXAMPLES
2.1 Concrete and General Categories
2.2 Subcategories and Quotient Categories
2.3 Products and Coproducts of Categories
2.4 The Dual Category and Duality of Properties
2.5 Arrow Category and Comma Categories over a Category
CHAPTER THREE. DISTINGUISHED MORPHISMS AND OBJECTS
3.1 Distinguished Morphisms
3.2 Distinguished Objects
3.3 Equalizers and Coequalizers
3.4 Constant Morphisms and Pointed Categories
3.5 Separators and Coseparators
CHAPTER FOUR. TYPES OF FUNCTORS
4.1 Full, Faithful, Dense, Embedding Functors
4.2 Reflection and Preservation of Categorical Properties
4.3 The Feeble Functor and Reverse Quotient Functor
CHAPTER FIVE. NATURAL TRANSFORMATIONS AND EQUIVALENCES
5.1 Natural Transformations and Their Compositions
5.2 Equivalence of Categories and Skeletons
5.3 Functor Categories
5.4 Natural Transformations for Feeble Functors
CHAPTER SIX. LIMITS, COLIMITS, COMPLETENESS, COCOMPLETENESS
6.1 Predecessors and Limits of a Functor
6.2 Successors and Colimits of a Functor
6.3 Factorizations of Morphisms
6.4 Completeness
CHAPTER SEVEN. ADJOINT FUNCTORS
7.1 The Path Category
7.2 Adjointness
7.3 Near-equivalence and Adjointness
7.4 Composing and Resolving Shortest Paths or Adjoints
7.5 Adjoint Functor Theorems
7.6 Examples of Adjoints
7.7 Monads
7.8 Weak Adjoints
APPENDIX ONE. SEMIUNIFORM, BITOPOLOGICAL, AND PREORDERED ALGEBRAS
APPENDIX TWO. ALGEBRAIC FUNCTORS
APPENDIX THREE. TOPOLOGICAL FUNCTORS
Bibliography
Index

✦ Table of Contents

Front Cover......Page 1
Contents......Page 7
Preface......Page 9
1.1 Set Theory......Page 13
1.2 Some Typical Algebraic Structures......Page 15
1.3 Algebras in General......Page 17
1.4 Topological Spaces......Page 22
1.5 Semimetric and Semiuniform Spaces......Page 30
1.6 Completeness and the Canonical Completion......Page 32
2.1 Concrete and General Categories......Page 45
2.2 Subcategories and Quotient Categories......Page 49
2.3 Products and Coproducts of Categories......Page 50
2.4 The Dual Category and Duality of Properties......Page 51
2.5 Arrow Category and Comma Categories over a Category......Page 52
3.1 Distinguished Morphisms......Page 55
3.2 Distinguished Objects......Page 56
3.3 Equalizers and Coequalizers......Page 58
3.4 Constant Morphisms and Pointed Categories......Page 59
3.5 Separators and Coseparators......Page 62
4.1 Full, Faithful, Dense, Embedding Functors......Page 64
4.2 Reflection and Preservation of Categorical Properties......Page 66
4.3 The Feeble Functor and Reverse Quotient Functor......Page 68
5.1 Natural Transformations and Their Compositions......Page 74
5.2 Equivalence of Categories and Skeletons......Page 76
5.3 Functor Categories......Page 85
5.4 Natural Transformations for Feeble Functors......Page 86
6.1 Predecessors and Limits of a Functor......Page 90
6.2 Successors and Colimits of a Functor......Page 97
6.3 Factorizations of Morphisms......Page 98
6.4 Completeness......Page 103
7.1 The Path Category......Page 111
7.2 Adjointness......Page 112
7.3 Near-equivalence and Adjointness......Page 117
7.4 Composing and Resolving Shortest Paths or Adjoints......Page 119
7.5 Adjoint Functor Theorems......Page 123
7.6 Examples of Adjoints......Page 127
7.7 Monads......Page 133
7.8 Weak Adjoints......Page 139
APPENDIX ONE. SEMIUNIFORM, BITOPOLOGICAL, AND PREORDERED ALGEBRAS......Page 155
APPENDIX TWO. ALGEBRAIC FUNCTORS......Page 167
APPENDIX THREE. TOPOLOGICAL FUNCTORS......Page 174
Bibliography......Page 179
Index......Page 181
Back Cover......Page 186