## Categories for the Working MathematicianCategories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories. |

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### Contents

IV | 7 |

V | 10 |

VI | 13 |

VII | 16 |

VIII | 19 |

IX | 21 |

X | 24 |

XI | 27 |

LVII | 157 |

LVIII | 161 |

LX | 165 |

LXI | 170 |

LXII | 174 |

LXIII | 175 |

LXIV | 180 |

LXV | 184 |

XII | 31 |

XIII | 33 |

XIV | 36 |

XV | 40 |

XVI | 42 |

XVII | 45 |

XVIII | 48 |

XIX | 51 |

XX | 55 |

XXII | 59 |

XXIII | 62 |

XXIV | 68 |

XXV | 72 |

XXVI | 75 |

XXVII | 76 |

XXVIII | 79 |

XXIX | 86 |

XXX | 90 |

XXXI | 92 |

XXXII | 95 |

XXXIII | 97 |

XXXIV | 99 |

XXXV | 103 |

XXXVI | 105 |

XXXVII | 106 |

XXXVIII | 109 |

XL | 112 |

XLI | 115 |

XLII | 116 |

XLIII | 118 |

XLIV | 120 |

XLV | 126 |

XLVI | 128 |

XLVII | 132 |

XLVIII | 137 |

L | 139 |

LI | 142 |

LII | 144 |

LIII | 147 |

LIV | 149 |

LV | 151 |

LVI | 156 |

LXVI | 185 |

LXVII | 188 |

LXVIII | 191 |

LXX | 194 |

LXXI | 198 |

LXXII | 202 |

LXXIII | 211 |

LXXV | 214 |

LXXVI | 217 |

LXXVII | 218 |

LXXVIII | 222 |

LXXIX | 226 |

LXXX | 228 |

LXXXI | 230 |

LXXXII | 233 |

LXXXIV | 235 |

LXXXV | 236 |

LXXXVI | 240 |

LXXXVII | 243 |

LXXXVIII | 245 |

LXXXIX | 248 |

XC | 251 |

XCII | 255 |

XCIII | 257 |

XCIV | 260 |

XCV | 263 |

XCVI | 266 |

XCVII | 267 |

XCIX | 270 |

C | 272 |

CI | 276 |

CII | 279 |

CIII | 281 |

CIV | 283 |

CV | 285 |

CVI | 289 |

CVII | 293 |

CVIII | 295 |

CIX | 297 |

303 | |

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### Common terms and phrases

Ab-category abelian groups additive adjunction algebra applies arrow arrow f assigns associative axioms base bijection braid called canonical closed coequalizer colimits commutative complete components composite condition cone consider consists construction continuous coproduct corresponding counit defined definition described determined diagram direct domain dual elements equal equivalence exact exactly example Exercises exists extension factors familiar finite forgetful functor function given gives graph Hausdorff hence hom-sets identity implies inclusion initial inverse isomorphism left adjoint Lemma limits means monad monic monoidal category morphism natural transformation Note object operation ordered pair particular preserves projections Proof Proposition prove pullback regarded relation replaced representation result right adjoint ring sends space square strings structure subset tensor theorem theory unique unique arrow unit universal universal arrow usual written

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