Category Theory for Computing Science

Category Theory for Computer Science......Page 000003.djvu
ISBN: 0-13-120486-6......Page 000004.djvu
Contents......Page 000005.djvu
Preface......Page 000009.djvu
1.1 Sets......Page 1.djvu
1.2 Functions......Page 2.djvu
1.3 Graphs......Page 7.djvu
1.4 Homomorphisms of graphs......Page 9.djvu
2.1 Basic definitions......Page 13.djvu
2.2 Functional programming languages as categories......Page 17.djvu
2.3 Mathematical structures as categories......Page 21.djvu
2.4 Categories of sets with structure......Page 24.djvu
2.5 Categories of algebraic structures......Page 28.djvu
2.6 Constructions on categories......Page 31.djvu
2.7 Properties of objects and arrows in a category......Page 36.djvu
2.8 Monomorphisms and subobjects......Page 41.djvu
2.9 Other types of arrow......Page 45.djvu
3.1 Functors......Page 51.djvu
3.2 Actions......Page 59.djvu
3.3 Types of functors......Page 64.djvu
3.4 Equivalences......Page 67.djvu
3.5 Quotient categories......Page 71.djvu
4.1 Diagrams......Page 76.djvu
4.2 Natural transformations......Page 83.djvu
4.3 Natural transformations between functors......Page 88.djvu
4.4 The Godement calculus of natural transformations......Page 94.djvu
4.5 The Yoneda Lemma and universal elements......Page 97.djvu
4.6 Linear sketches (graphs with diagrams)......Page 103.djvu
4.7 Linear sketches with constants: initial term models......Page 107.djvu
5.1 The product of two objects in a category......Page 114.djvu
5.2 Notation for and properties of products......Page 118.djvu
5.3 Finite products......Page 127.djvu
5.4 Sums......Page 133.djvu
5.5 Natural numbers objects......Page 136.djvu
5.6 Deduction systems as categories......Page 140.djvu
6.1 Cartesian closed categories......Page 142.djvu
6.2 Typed Lambda-calculus......Page 150.djvu
6.3 Lamda-calculus to category and back......Page 152.djvu
6.4 Arrows vs. terms......Page 154.djvu
6.5 Fixed points in cartesian closed categories......Page 156.djvu
7 Finite discrete sketches......Page 161.djvu
7.1 Finite product sketches......Page 162.djvu
7.2 The sketch for semigroups......Page 167.djvu
7.3 Notation for FP sketches......Page 172.djvu
7.4 Arrows between models of FP sketches......Page 175.djvu
7.5 The theory of an FP sketch......Page 177.djvu
7.6 Initial term models for FP sketches......Page 179.djvu
7.7 Signatures and FP sketches......Page 183.djvu
7.8 FD sketches......Page 187.djvu
7.9 The sketch for fields......Page 189.djvu
7.10 Term algebras for FD sketches......Page 192.djvu
8.1 Equalizers......Page 200.djvu
8.2 The general concept of limit......Page 203.djvu
8.3 Pullbacks......Page 207.djvu
8.4 Coequalizers......Page 212.djvu
8.5 Cocones......Page 214.djvu
8.6 More about sums......Page 219.djvu
8.7 Unification as coequalizer......Page 223.djvu
9.1 Finite limit sketches......Page 228.djvu
9.2 Initial term models of FL sketches......Page 233.djvu
9.3 The theory of an FL sketch......Page 236.djvu
9.4 General definition of sketch......Page 237.djvu
10.1 Homomorphisms of sketches......Page 241.djvu
10.2 Parametrized data types as pushouts......Page 242.djvu
10.3 The model category functor......Page 248.djvu
11.1 Fibrations......Page 252.djvu
11.2 The Grothendieck construction......Page 257.djvu
11.3 An equivalence of categories......Page 263.djvu
11.4 Wreath products......Page 266.djvu
12.1 Free monoids......Page 271.djvu
12.2 Adjoints......Page 274.djvu
12.3 Further topics on adjoints......Page 281.djvu
12.4 Locally cartesian closed categories......Page 285.djvu
13.1 Fixed points for a functor......Page 288.djvu
13.2 Recursive categories......Page 293.djvu
13.3 Triples......Page 297.djvu
13.4 Factorizations of a triple......Page 298.djvu
13.5 Scott domains......Page 302.djvu
14 Toposes......Page 308.djvu
14.1 Definition of topos......Page 309.djvu
14.2 Properties of toposes......Page 312.djvu
14.3 Is a two·element poset complete?......Page 316.djvu
14.4 Presheaves......Page 318.djvu
14.5 Sheaves......Page 320.djvu
14.6 Fuzzy sets......Page 325.djvu
14.7 External functors......Page 328.djvu
14.8 The realizability topos......Page 333.djvu
chapter  1......Page 337.djvu
chapter  2......Page 339.djvu
chapter  3......Page 349.djvu
chapter  4......Page 356.djvu
chapter  5......Page 367.djvu
chapter  6......Page 373.djvu
chapter  7......Page 379.djvu
chapter  8......Page 384.djvu
chapter  9......Page 394.djvu
chapter 10......Page 395.djvu
chapter 11......Page 397.djvu
chapter 12......Page 401.djvu
chapter 13......Page 404.djvu
chapter 14......Page 411.djvu
Bibliography......Page 417.djvu
Index......Page 427.djvu