Table Of Content

Category Theory for Programmers Bartosz Milewski CategoryTheoryforProgrammers BartoszMilewski Version0.1,September2017 ThisworkislicensedunderaCreativeCommons Attribution-ShareAlike4.0InternationalLicense(ccby-sa4.0). ConvertedtoLaTeXfromaseriesofblogpostsbyBartoszMilewski. PDFcompiledbyIgalTabachnik. Contents Preface xiii I PartOne 1 1 Category:TheEssenceofComposition 2 1.1 ArrowsasFunctions . . . . . . . . . . . . . . . . . . . 2 1.2 PropertiesofComposition . . . . . . . . . . . . . . . . 5 1.3 CompositionistheEssenceofProgramming . . . . . . 8 1.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 TypesandFunctions 11 2.1 WhoNeedsTypes? . . . . . . . . . . . . . . . . . . . . 11 2.2 TypesAreAboutComposability . . . . . . . . . . . . . 13 2.3 WhatAreTypes? . . . . . . . . . . . . . . . . . . . . . 14 2.4 WhyDoWeNeedaMathematicalModel? . . . . . . . 17 2.5 PureandDirtyFunctions . . . . . . . . . . . . . . . . . 20 2.6 ExamplesofTypes. . . . . . . . . . . . . . . . . . . . . 21 2.7 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 25 iii 3 CategoriesGreatandSmall 27 3.1 NoObjects . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 SimpleGraphs . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 MonoidasSet . . . . . . . . . . . . . . . . . . . . . . . 29 3.5 MonoidasCategory . . . . . . . . . . . . . . . . . . . . 34 3.6 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 KleisliCategories 39 4.1 TheWriterCategory . . . . . . . . . . . . . . . . . . . 44 4.2 WriterinHaskell . . . . . . . . . . . . . . . . . . . . . 48 4.3 KleisliCategories . . . . . . . . . . . . . . . . . . . . . 50 4.4 Challenge . . . . . . . . . . . . . . . . . . . . . . . . . 51 5 ProductsandCoproducts 53 5.1 InitialObject . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 TerminalObject . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . 58 5.5 Products . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.6 Coproduct . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.7 Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . 70 5.8 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 74 6 SimpleAlgebraicDataTypes 75 6.1 ProductTypes . . . . . . . . . . . . . . . . . . . . . . . 76 6.2 Records . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 SumTypes . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.4 AlgebraofTypes . . . . . . . . . . . . . . . . . . . . . 86 iv 6.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 90 7 Functors 92 7.1 FunctorsinProgramming. . . . . . . . . . . . . . . . . 96 7.1.1 TheMaybeFunctor . . . . . . . . . . . . . . . . 96 7.1.2 EquationalReasoning . . . . . . . . . . . . . . 98 7.1.3 Optional . . . . . . . . . . . . . . . . . . . . . . 101 7.1.4 Typeclasses . . . . . . . . . . . . . . . . . . . . 103 7.1.5 FunctorinC++ . . . . . . . . . . . . . . . . . . 105 7.1.6 TheListFunctor . . . . . . . . . . . . . . . . . 106 7.1.7 TheReaderFunctor . . . . . . . . . . . . . . . . 108 7.2 FunctorsasContainers . . . . . . . . . . . . . . . . . . 110 7.3 FunctorComposition . . . . . . . . . . . . . . . . . . . 113 7.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 116 8 Functoriality 117 8.1 Bifunctors . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.2 ProductandCoproductBifunctors . . . . . . . . . . . . 121 8.3 FunctorialAlgebraicDataTypes . . . . . . . . . . . . . 122 8.4 FunctorsinC++ . . . . . . . . . . . . . . . . . . . . . . 126 8.5 TheWriterFunctor . . . . . . . . . . . . . . . . . . . . 129 8.6 CovariantandContravariantFunctors . . . . . . . . . 131 8.7 Profunctors. . . . . . . . . . . . . . . . . . . . . . . . . 134 8.8 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 136 9 FunctionTypes 138 9.1 UniversalConstruction . . . . . . . . . . . . . . . . . . 139 9.2 Currying . . . . . . . . . . . . . . . . . . . . . . . . . . 144 9.3 Exponentials . . . . . . . . . . . . . . . . . . . . . . . . 148 9.4 CartesianClosedCategories . . . . . . . . . . . . . . . 149 v 9.5 ExponentialsandAlgebraicDataTypes . . . . . . . . . 150 9.5.1 ZerothPower . . . . . . . . . . . . . . . . . . . 151 9.5.2 PowersofOne . . . . . . . . . . . . . . . . . . 152 9.5.3 FirstPower . . . . . . . . . . . . . . . . . . . . 152 9.5.4 ExponentialsofSums . . . . . . . . . . . . . . 152 9.5.5 ExponentialsofExponentials . . . . . . . . . . 153 9.5.6 ExponentialsoverProducts . . . . . . . . . . . 153 9.6 Curry-HowardIsomorphism . . . . . . . . . . . . . . . 154 9.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 157 10 NaturalTransformations 158 10.1 PolymorphicFunctions . . . . . . . . . . . . . . . . . . 162 10.2 BeyondNaturality . . . . . . . . . . . . . . . . . . . . . 168 10.3 FunctorCategory . . . . . . . . . . . . . . . . . . . . . 170 10.4 2-Categories . . . . . . . . . . . . . . . . . . . . . . . . 175 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 180 10.6 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 181 II PartTwo 183 11 DeclarativeProgramming 184 12 LimitsandColimits 192 12.1 LimitasaNaturalIsomorphism . . . . . . . . . . . . . 198 12.2 ExamplesofLimits . . . . . . . . . . . . . . . . . . . . 203 12.3 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . 211 12.4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 215 vi 13 FreeMonoids 216 13.1 FreeMonoidinHaskell . . . . . . . . . . . . . . . . . . 218 13.2 FreeMonoidUniversalConstruction . . . . . . . . . . 219 13.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 224 14 RepresentableFunctors 226 14.1 TheHomFunctor . . . . . . . . . . . . . . . . . . . . . 228 14.2 RepresentableFunctors . . . . . . . . . . . . . . . . . . 231 14.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 236 14.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 237 15 TheYonedaLemma 238 15.1 YonedainHaskell . . . . . . . . . . . . . . . . . . . . . 246 15.2 Co-Yoneda . . . . . . . . . . . . . . . . . . . . . . . . . 248 15.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 249 15.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . 249 16 YonedaEmbedding 250 16.1 TheEmbedding . . . . . . . . . . . . . . . . . . . . . . 253 16.2 ApplicationtoHaskell . . . . . . . . . . . . . . . . . . 254 16.3 PreorderExample . . . . . . . . . . . . . . . . . . . . . 255 16.4 Naturality . . . . . . . . . . . . . . . . . . . . . . . . . 257 16.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 259 III PartThree 260 17 It’sAllAboutMorphisms 261 17.1 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . 261 17.2 CommutingDiagrams . . . . . . . . . . . . . . . . . . . 262 vii 17.3 NaturalTransformations . . . . . . . . . . . . . . . . . 263 17.4 NaturalIsomorphisms. . . . . . . . . . . . . . . . . . . 265 17.5 Hom-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 265 17.6 Hom-SetIsomorphisms . . . . . . . . . . . . . . . . . . 266 17.7 AsymmetryofHom-Sets . . . . . . . . . . . . . . . . . 267 17.8 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 268 18 Adjunctions 269 18.1 AdjunctionandUnit/CounitPair . . . . . . . . . . . . 270 18.2 AdjunctionsandHom-Sets . . . . . . . . . . . . . . . . 276 18.3 ProductfromAdjunction . . . . . . . . . . . . . . . . . 281 18.4 ExponentialfromAdjunction . . . . . . . . . . . . . . 286 18.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 287 19 Free/ForgetfulAdjunctions 289 19.1 SomeIntuitions . . . . . . . . . . . . . . . . . . . . . . 293 19.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 296 20 Monads:Programmer’sDefinition 298 20.1 TheKleisliCategory . . . . . . . . . . . . . . . . . . . . 300 20.2 FishAnatomy . . . . . . . . . . . . . . . . . . . . . . . 303 20.3 ThedoNotation . . . . . . . . . . . . . . . . . . . . . . 305 21 MonadsandEffects 310 21.1 TheProblem . . . . . . . . . . . . . . . . . . . . . . . . 310 21.2 TheSolution . . . . . . . . . . . . . . . . . . . . . . . . 311 21.2.1 Partiality . . . . . . . . . . . . . . . . . . . . . 312 21.2.2 Nondeterminism . . . . . . . . . . . . . . . . . 313 21.2.3 Read-OnlyState . . . . . . . . . . . . . . . . . 316 21.2.4 Write-OnlyState . . . . . . . . . . . . . . . . . 317 viii 21.2.5 State . . . . . . . . . . . . . . . . . . . . . . . . 318 21.2.6 Exceptions. . . . . . . . . . . . . . . . . . . . . 319 21.2.7 Continuations . . . . . . . . . . . . . . . . . . . 320 21.2.8 InteractiveInput . . . . . . . . . . . . . . . . . 322 21.2.9 InteractiveOutput . . . . . . . . . . . . . . . . 325 21.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 326 22 MonadsCategorically 327 22.1 MonoidalCategories . . . . . . . . . . . . . . . . . . . 333 22.2 MonoidinaMonoidalCategory . . . . . . . . . . . . . 338 22.3 MonadsasMonoids . . . . . . . . . . . . . . . . . . . . 340 22.4 MonadsfromAdjunctions . . . . . . . . . . . . . . . . 343 23 Comonads 347 23.1 ProgrammingwithComonads . . . . . . . . . . . . . . 348 23.2 TheProductComonad . . . . . . . . . . . . . . . . . . 349 23.3 DissectingtheComposition . . . . . . . . . . . . . . . 350 23.4 TheStreamComonad . . . . . . . . . . . . . . . . . . . 353 23.5 ComonadCategorically . . . . . . . . . . . . . . . . . . 355 23.6 TheStoreComonad . . . . . . . . . . . . . . . . . . . . 358 23.7 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 361 24 F-Algebras 362 24.1 Recursion. . . . . . . . . . . . . . . . . . . . . . . . . . 366 24.2 CategoryofF-Algebras . . . . . . . . . . . . . . . . . . 368 24.3 NaturalNumbers . . . . . . . . . . . . . . . . . . . . . 372 24.4 Catamorphisms . . . . . . . . . . . . . . . . . . . . . . 373 24.5 Folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 24.6 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 377 24.7 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 380 ix 25 AlgebrasforMonads 381 25.1 T-algebras . . . . . . . . . . . . . . . . . . . . . . . . . 384 25.2 TheKleisliCategory . . . . . . . . . . . . . . . . . . . . 388 25.3 CoalgebrasforComonads . . . . . . . . . . . . . . . . 391 25.4 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 25.5 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 394 26 EndsandCoends 395 26.1 DinaturalTransformations . . . . . . . . . . . . . . . . 397 26.2 Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 26.3 EndsasEqualizers . . . . . . . . . . . . . . . . . . . . . 403 26.4 NaturalTransformationsasEnds . . . . . . . . . . . . 404 26.5 Coends . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 26.6 NinjaYonedaLemma . . . . . . . . . . . . . . . . . . . 409 26.7 ProfunctorComposition . . . . . . . . . . . . . . . . . 411 27 KanExtensions 413 27.1 RightKanExtension . . . . . . . . . . . . . . . . . . . 416 27.2 KanExtensionasAdjunction . . . . . . . . . . . . . . . 418 27.3 LeftKanExtension . . . . . . . . . . . . . . . . . . . . 420 27.4 KanExtensionsasEnds . . . . . . . . . . . . . . . . . . 424 27.5 KanExtensionsinHaskell . . . . . . . . . . . . . . . . 427 27.6 FreeFunctor . . . . . . . . . . . . . . . . . . . . . . . . 430 28 EnrichedCategories 432 28.1 WhyMonoidalCategory? . . . . . . . . . . . . . . . . 433 28.2 MonoidalCategory . . . . . . . . . . . . . . . . . . . . 434 28.3 EnrichedCategory . . . . . . . . . . . . . . . . . . . . 437 28.4 Preorders . . . . . . . . . . . . . . . . . . . . . . . . . . 439 28.5 MetricSpaces . . . . . . . . . . . . . . . . . . . . . . . 440 x

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Category Theory for Programmers PDF

510 Pages·2017·12.5 MB·english
by  Bartosz Milewski
#Mathematics - Computational Mathematics
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