Category Theory A Gentle Introduction Peter Smith University of Cambridge Version of January 29, 2018 c Peter Smith, 2018 This PDF is an early incomplete version of work still very much in progress. For the latest and most complete version of this Gentle Introduction and for related materials see the Category Theory page at the Logic Matters website. Corrections, please, to ps218 at cam dot ac dot uk. Contents Preface ix 1 The categorial imperative 1 1.1 Why category theory?1 1.2 From a bird's eye view2 1.3 Ascending to the categorial heights3 2 One structured family of structures 4 2.1 Groups4 2.2 Group homomorphisms and isomorphisms5 2.3 New groups from old8 2.4 `Identity up to isomorphism' 11 2.5 Groups and sets 13 2.6 An unresolved tension 16 3 Categories defined 17 3.1 The very idea of a category 17 3.2 Monoids and pre-ordered collections 20 3.3 Some rather sparse categories 21 3.4 More categories 23 3.5 The category of sets 24 3.6 Yet more examples 26 3.7 Diagrams 27 4 Categories beget categories 30 4.1 Duality 30 4.2 Subcategories, product and quotient categories 31 4.3 Arrow categories and slice categories 33 5 Kinds of arrows 37 5.1 Monomorphisms, epimorphisms 37 5.2 Inverses 39 5.3 Isomorphisms 42 5.4 Isomorphic objects 44 iii Contents 6 Initial and terminal objects 46 6.1 Initial and terminal objects, definitions and examples 47 6.2 Uniqueness up to unique isomorphism 48 6.3 Elements and generalized elements 49 7 Products introduced 51 7.1 Real pairs, virtual pairs 51 7.2 Pairing schemes 52 7.3 Binary products, categorially 56 7.4 Products as terminal objects 59 7.5 Uniqueness up to unique isomorphism 60 7.6 `Universal mapping properties' 62 7.7 Coproducts 62 8 Products explored 66 8.1 More properties of binary products 66 8.2 And two more results 67 8.3 More on mediating arrows 69 8.4 Maps between two products 71 8.5 Finite products more generally 73 8.6 Infinite products 75 9 Equalizers 76 9.1 Equalizers 76 9.2 Uniqueness again 79 9.3 Co-equalizers 80 10 Limits and colimits defined 83 10.1 Cones over diagrams 83 10.2 Defining limit cones 85 10.3 Limit cones as terminal objects 87 10.4 Results about limits 88 10.5 Colimits defined 90 10.6 Pullbacks 91 10.7 Pushouts 94 11 The existence of limits 96 11.1 Pullbacks, products and equalizers related 96 11.2 Categories with all finite limits 100 11.3 Infinite limits 102 11.4 Dualizing again 103 12 Subobjects 104 12.1 Subsets revisited 104 12.2 Subobjects as monic arrows 105 iv Contents 12.3 Subobjects as isomorphism classes 106 12.4 Subobjects, equalizers, and pullbacks 107 12.5 Elements and subobjects 109 13 Exponentials 110 13.1 Two-place functions 110 13.2 Exponentials defined 111 13.3 Examples of exponentials 113 13.4 Exponentials are unique 116 13.5 Further results about exponentials 117 13.6 Cartesian closed categories 119 14 Group objects, natural number objects 123 14.1 Groups in Set 123 14.2 Groups in other categories 125 14.3 A very little more on groups 127 14.4 Natural numbers 128 14.5 The Peano postulates revisited 129 14.6 More on recursion 131 15 Functors introduced 135 15.1 Functors defined 135 15.2 Some elementary examples of functors 136 15.3 What do functors preserve and reflect? 138 15.4 Faithful, full, and essentially surjective functors 140 15.5 A functor from Set to Mon 142 15.6 Products, exponentials, and functors 143 15.7 An example from algebraic topology 145 15.8 Covariant vs contravariant functors 147 16 Categories of categories 149 16.1 Functors compose 149 16.2 Categories of categories 150 16.3 A universal category? 151 16.4 `Small' and `locally small' categories 152 16.5 Isomorphisms between categories 154 16.6 An aside: other definitions of categories 156 17 Functors and limits 159 17.1 Diagrams redefined as functors 159 17.2 Preserving limits 160 17.3 Reflecting limits 164 17.4 Creating limits 166 18 Hom-functors 167 v Contents 18.1 Hom-sets 167 18.2 Hom-functors 169 18.3 Hom-functors preserve limits 170 19 Functors and comma categories 174 19.1 Functors and slice categories 174 19.2 Comma categories 175 19.3 Two (already familiar) types of comma category 176 19.4 Another (new) type of comma category 177 19.5 An application: free monoids again 178 19.6 A theorem on comma categories and limits 180 20 Natural isomorphisms 182 20.1 Natural isomorphisms between functors defined 182 20.2 Why `natural'? 183 20.3 More examples of natural isomorphormisms 186 20.4 Natural/unnatural isomorphisms between objects 191 20.5 An `Eilenberg/Mac Lane Thesis'? 193 21 Natural transformations 195 21.1 Natural transformations 195 21.2 Vertical composition of natural transformations 198 21.3 Horizontal composition of natural transformations 199 22 Functor categories 202 22.1 Functor categories defined 202 22.2 Functor categories and natural isomorphisms 203 22.3 Hom-functors from functor categories 204 22.4 Evaluation and diagonal functors 205 22.5 Cones as natural transformations 206 22.6 Limit functors 207 23 Equivalent categories 210 23.1 The categories Pfn and Set‹ are `equivalent' 210 23.2 Pfn and Set‹ are not isomorphic 212 23.3 Equivalent categories 213 23.4 Skeletons and evil 216 24 The Yoneda embedding 219 24.1 Natural transformations between hom-functors 219 24.2 The Restricted Yoneda Lemma 222 24.3 The Yoneda embedding 223 24.4 Yoneda meets Cayley 225 25 The Yoneda Lemma 229 vi Contents 25.1 Towards the full Yoneda Lemma 229 25.2 The generalizing move 230 25.3 Making it all natural 231 25.4 Putting everything together 233 25.5 A brief afterword on `presheaves' 234 26 Representables and universal elements 235 26.1 Isomorphic functors preserve the same limits 235 26.2 Representable functors 236 26.3 A first example 237 26.4 More examples of representables 239 26.5 Universal elements 240 26.6 Categories of elements 242 26.7 Limits and exponentials as universal elements 244 27 Galois connections 245 27.1 (Probably unnecessary) reminders about posets 245 27.2 An introductory example 246 27.3 Galois connections defined 248 27.4 Galois connections re-defined 251 27.5 Some basic results about Galois connections 252 27.6 Fixed points, isomorphisms, and closures 253 27.7 One way a Galois connection can arise 255 27.8 Syntax and semantics briefly revisited 255 28 Adjoints introduced 257 28.1 Adjoint functors: a first definition 257 28.2 Examples 259 28.3 Naturality 263 28.4 An alternative definition 264 28.5 Adjoints and equivalent categories 269 29 Adjoints further explored 272 29.1 Adjunctions reviewed 272 29.2 Two more theorems! 273 29.3 Adjunctions compose 273 29.4 The uniqueness of adjoints 275 29.5 How left adjoints can be defined in terms of right adjoints 276 29.6 Another way of getting new adjunctions from old 280 30 Adjoint functors and limits 282 30.1 Limit functors as adjoints 282 30.2 Right adjoints preserve limits 284 30.3 Some examples 286 30.4 The Adjoint Functor Theorems 287 vii Contents Bibliography 290 viii Preface The project This Gentle Introduction is very much still work in progress, so there are chapters at different levels of development and with different degrees of integration with what's around them. So far, at least in a rough and ready way, we cover the basic notions of elementary category theory { explaining the very idea of a category, then treating limits, functors, natural transformations, representables, adjunctions. The long-term plan is (possibly) to say something about categorial logic, explore categories of sets, and even edge towards some initial themes in topos theory. But considerations of length will soon begin to weigh, because we do take things pretty slowly. Experience suggests that getting a really secure under- standing by going at a rather gentle pace when first encountering categorial ways of thinking makes later adventures exploring beyond the basics very much more manageable. I imagine one reader to be a mathematics student who wants a clear introduc- tion to categorial ideas without having to take on an industrial-strength graduate course (or else who wants a helping hand while tackling the beginnings of such a course). Another reader might be a philosopher interested in the foundations of mathematics (and knowing a smidgin of mathematics) who wants to know what the categorial fuss is about. What do you need to bring to the party? You obviously can't be well placed to appreciate how category theory gives us a story about the ways in which different parts of modern abstract mathematics hang together if you really know nothing beforehand about modern mathematics! But don't be scared off. In this Gentle Introduction we try to presuppose a bare minimum. If you know just a little e.g. about what a group is, what a Boolean algebra is, what a topological space is, and some similar bits and pieces, then you should cope fairly easily. And if a few later illustrative examples pass you by, don't panic. I usually try to give multiple illustrations of important concepts and constructs; so feel free simply to skip those examples that happen not to work so well for you.