E-Book Overview
The fundamental concepts of category theory are explained in this text which allows the reader to develop their understanding gradually. With over 300 exercises, students are encouraged to monitor their progression. A wide coverage of topics in category theory and computer science is developed including introductory treatments of cartesian closed categories, sketches and elementary categorical model theory, and triples. The presentation is informal with proofs included only when they are instructive, providing a broad coverage of the competing texts on category theory in computer science.
E-Book Content
Contents Preface iii 1 Finite discrete sketches 1.1 Sketches with sums 1.2 The sketch for ¯elds 1.3 Term algebras for FD sketches 1 1 3 5 2 More about sketches 2.1 Finite limit sketches 2.2 Initial term models of FL sketches 2.3 The theory of an FL sketch 2.4 General de¯nition of sketch 12 12 16 19 21 3 The 3.1 3.2 3.3 26 26 28 33 category of sketches Homomorphisms of sketches Parametrized data types as pushouts The model category functor 4 Fibrations 4.1 Fibrations 4.2 The Grothendieck construction 4.3 An equivalence of categories 4.4 Wreath products 38 38 43 48 51 5 Toposes 5.1 De¯nition of topos 5.2 Properties of toposes 5.3 Is a two-element poset complete? 5.4 Presheaves 5.5 Sheaves 5.6 Fuzzy sets 5.7 External functors 5.8 The realizability topos 56 57 60 64 66 67 72 75 79 i ii Contents Answers to Exercises Solutions for Chapter Solutions for Chapter Solutions for Chapter Solutions for Chapter Solutions for Chapter Bibliography Index 1 2 3 4 5 83 83 85 87 88 92 97 102 Preface This is the electronic supplement to Category Theory for Computing Science, second edition, Prentice-Hall International, 1995, IS