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. |
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 Ab-category abelian category abelian groups adjoint functor adjunction F algebra arrow f assigns axioms bifunctor bijection binary braid called CGHaus codomain coend coequalizer Colim colimits comma category commutative diagram composite construction contravariant coproduct counit defined definition dinatural domain dual elements equivalence example Exercises exists factors forgetful functor full subcategory function f 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 Lim F limiting cone Mac Lane monad monic morphism natural isomorphism natural transformation pair of arrows parallel pair preorder preserves Proof prove pullback quotient R-Mod R-module right adjoint right Kan extension ring simplicial small hom-sets small set subobjects subset T-algebras tensor product terminal object topological space topology unique arrow universal arrow usual vector space vertex
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