Category Theory: A Gentle Introduction

Preface 9 A very short introduction 11 Why categories? 11 What do you need to bring to the party? 12 Theorems as exercises 13 Notation and terminology 13 Categories defined 14 The very idea of a category 14 The category of sets 16 More examples 18 Diagrams 23 Categories beget categories 25 Duality 25 Subcategories, product and quotient categories 26 Arrow categories and slice categories 28 Kinds of arrows 32 Monomorphisms, epimorphisms 32 Inverses 34 Isomorphisms 37 Isomorphic objects 39 Initial and terminal objects 41 Initial and terminal objects, definitions and examples 42 Uniqueness up to unique isomorphism 43 Elements and generalized elements 44 Products introduced 46 Real pairs, virtual pairs 46 Pairing schemes 47 Binary products, categorially 51 Products as terminal objects 54 Uniqueness up to unique isomorphism 55 `Universal mapping properties' 57 Coproducts 57 Products explored 61 More properties of binary products 61 And two more results 62 More on mediating arrows 64 Maps between two products 66 Finite products more generally 68 Infinite products 70 Equalizers 71 Equalizers 71 Uniqueness again 74 Co-equalizers 75 Limits and colimits defined 78 Cones over diagrams 78 Defining limit cones 80 Limit cones as terminal objects 82 Results about limits 83 Colimits defined 85 Pullbacks 85 Pushouts 89 The existence of limits 91 Pullbacks, products and equalizers related 91 Categories with all finite limits 95 Infinite limits 97 Dualizing again 98 Subobjects 99 Subsets revisited 99 Subobjects as monic arrows 100 Subobjects as isomorphism classes 101 Subobjects, equalizers, and pullbacks 102 Elements and subobjects 104 Exponentials 105 Two-place functions 105 Exponentials defined 106 Examples of exponentials 108 Exponentials are unique 111 Further results about exponentials 112 Cartesian closed categories 114 Group objects, natural number objects 118 Groups in Set 118 Groups in other categories 120 A very little more on groups 122 Natural numbers 123 The Peano postulates revisited 124 More on recursion 126 Functors introduced 130 Functors defined 130 Some elementary examples of functors 131 What do functors preserve and reflect? 133 Faithful, full, and essentially surjective functors 135 A functor from Set to Mon 137 Products, exponentials, and functors 138 An example from algebraic topology 140 Covariant vs contravariant functors 142 Categories of categories 144 Functors compose 144 Categories of categories 145 A universal category? 146 `Small' and `locally small' categories 147 Isomorphisms between categories 149 An aside: other definitions of categories 151 Functors and limits 154 Diagrams redefined as functors 154 Preserving limits 155 Reflecting limits 159 Creating limits 161 Hom-functors 162 Hom-sets 162 Hom-functors 164 Hom-functors preserve limits 165 Functors and comma categories 169 Functors and slice categories 169 Comma categories 170 Two (already familiar) types of comma category 171 Another (new) type of comma category 172 An application: free monoids again 173 A theorem on comma categories and limits 175 Natural isomorphisms 177 Natural isomorphisms between functors defined 177 Why `natural'? 178 More examples of natural isomorphormisms 181 Natural/unnatural isomorphisms between objects 186 An `Eilenberg/Mac Lane Thesis'? 187 Natural transformations and functor categories 189 Natural transformations 189 Composition of natural transformations 192 Functor categories 195 Functor categories and natural isomorphisms 196 Hom-functors from functor categories 197 Evaluation and diagonal functors 198 Limit functors 199 Equivalent categories 202 The categories Pfn and Set are `equivalent' 202 Pfn and Set are not isomorphic 204 Equivalent categories 205 Skeletons and evil 208 The Yoneda embedding 211 Natural transformations between hom-functors 211 The Restricted Yoneda Lemma 214 The Yoneda embedding 215 Yoneda meets Cayley 217 The Yoneda Lemma 221 Towards the full Yoneda Lemma 221 The generalizing move 222 Making it all natural 223 Putting everything together 225 A brief afterword on `presheaves' 226 Representables and universal elements 227 Isomorphic functors preserve the same limits 227 Representable functors 228 A first example 229 More examples of representables 231 Universal elements 232 Categories of elements 234 Limits and exponentials as universal elements 236 Galois connections 237 (Probably unnecessary) reminders about posets 237 An introductory example 238 Galois connections defined 240 Galois connections re-defined 242 Some basic results about Galois connections 244 Fixed points, isomorphisms, and closures 245 One way a Galois connection can arise 247 Syntax and semantics briefly revisited 247 Adjoints introduced 249 Adjoint functors: a first definition 249 Examples 251 Naturality 255 An alternative definition 256 Adjoints and equivalent categories 261 Adjoints further explored 264 Adjunctions reviewed 264 Two more definitions! 265 Adjunctions compose 265 The uniqueness of adjoints 267 How left adjoints can be defined in terms of right adjoints 268 Another way of getting new adjunctions from old 272 Adjoint functors and limits 274 Limit functors as adjoints 274 Right adjoints preserve limits 276 Some examples 278 The Adjoint Functor Theorems 279 Bibliography 282