## 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

2-category 2-cells abelian category abelian groups adjoint functor adjoint functor theorem algebra arrow h assigns axioms bifunctor bijection binary biproduct braid called CGHaus codomain coend coequalizer colimits comma category commutative diagram construction continuous maps coproduct counit defined definition described dinatural domain dual elements equal equivalence exact sequences example Exercises exists factors finite products forgetful functor full subcategory functor category functor F given graph Hausdorff spaces hence hom-sets homomorphism homotopy identity arrow implies initial object inverse Kan extension kernel left adjoint Lemma limiting cone Mac Lane modules monad monic morphism natural isomorphism natural transformation operation pair of arrows parallel pair preorder preserves projections Proof Proposition prove pullback quotient R-module representation right adjoint right Kan extension ring simplicial small hom-sets small set small-complete strings subobjects subset T-algebras tensor product terminal object topological space unique arrow unit universal arrow usual vector space vertex

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