Category Theory for Computing Science

[ssirai]

Barr-Wells-ctcs.pdf

• [ ] 1 Preliminaries 1
  • [ ] 1.1 Sets 1
  • [ ] 1.2 Functions 3
  • [ ] 1.3 Graphs 8
  • [ ] 1.4 Homomorphisms of graphs 11
• [ ] 2 Categories 15
  • [ ] 2.1 Basic deØnitions 15
  • [ ] 2.2 Functional programming languages as categories 20
  • [ ] 2.3 Mathematical structures as categories 23
  • [ ] 2.4 Categories of sets with structure 27
  • [ ] 2.5 Categories of algebraic structures 32
  • [ ] 2.6 Constructions on categories 35
  • [ ] 2.7 Properties of objects and arrows in a category 40
  • [ ] 2.8 Monomorphisms and subobjects 47
  • [ ] 2.9 Other types of arrow 53
  • [ ] 2.10 Factorization systems 58
• [ ] 3 Functors 65
  • [ ] 3.1 Functors 65
  • [ ] 3.2 Actions 74
  • [ ] 3.3 Types of functors 80
  • [ ] 3.4 Equivalences 84
  • [ ] 3.5 Quotient categories 88
• [ ] 4 Diagrams, naturality and sketches 93
  • [ ] 4.1 Diagrams 93
  • [ ] 4.2 Natural transformations 101
  • [ ] 4.3 Natural transformations between functors 109
  • [ ] 4.4 The Godement calculus of natural transformations 117
  • [ ] 4.5 The Yoneda Lemma and universal elements 121
  • [ ] 4.6 Linear sketches (graphs with diagrams) 127
  • [ ] 4.7 Linear sketches with constants: initial term models 133
  • [ ] 4.8 2-categories 140
• [ ] 5 Products and sums 153
  • [ ] 5.1 The product of two objects in a category 153
  • [ ] 5.2 Notation for and properties of products 157
  • [ ] 5.3 Finite products 168
  • [ ] 5.4 Sums 178
  • [ ] 5.5 Natural numbers objects 182
  • [ ] 5.6 Deduction systems as categories 186
  • [ ] 5.7 Distributive categories 188
• [ ] 6 Cartesian closed categories 195
  • [ ] 6.1 Cartesian closed categories 195
  • [ ] 6.2 Properties of cartesian closed categories 202
  • [ ] 6.3 Typed
  • [ ] 6.4
  • [ ] 6.5 Arrows vs. terms 212
  • [ ] 6.6 Fixed points in cartesian closed categories 215
• [ ] 7 Finite product sketches 219
  • [ ] 7.1 Finite product sketches 220
  • [ ] 7.2 The sketch for semigroups 225
  • [ ] 7.3 Notation for FP sketches 231
  • [ ] 7.4 Arrows between models of FP sketches 234
  • [ ] 7.5 The theory of an FP sketch 237
  • [ ] 7.6 Initial term models for FP sketches 239
  • [ ] 7.7 Signatures and FP sketches 245
• [ ] 8 Finite discrete sketches 251
  • [ ] 8.1 Sketches with sums 251
  • [ ] 8.2 The sketch for Øelds 254
  • [ ] 8.3 Term algebras for FD sketches 257
• [ ] 9 Limits and colimits 265
  • [ ] 9.1 Equalizers 265
  • [ ] 9.2 The general concept of limit 268
  • [ ] 9.3 Pullbacks 273
  • [ ] 9.4 Coequalizers 277
  • [ ] 9.5 Cocones 280
  • [ ] 9.6 More about sums 285
  • [ ] 9.7 UniØcation as coequalizer 289
  • [ ] 9.8 Properties of factorization systems 294
• [ ] 10 More about sketches 299
  • [ ] 10.1 Finite limit sketches 299
  • [ ] 10.2 Initial term models of FL sketches 304
  • [ ] 10.3 The theory of an FL sketch 307
  • [ ] 10.4 General deØnition of sketch 309
• [ ] 11 The category of sketches 313
  • [ ] 11.1 Homomorphisms of sketches 313
  • [ ] 11.2 Parametrized data types as pushouts 315
  • [ ] 11.3 The model category functor 320
• [ ] 12 Fibrations 327
  • [ ] 12.1 Fibrations 327
  • [ ] 12.2 The Grothendieck construction 332
  • [ ] 12.3 An equivalence of categories 338
  • [ ] 12.4 Wreath products 341
• [ ] 13 Adjoints 347
  • [ ] 13.1 Free monoids 347
  • [ ] 13.2 Adjoints 350
  • [ ] 13.3 Further topics on adjoints 356
  • [ ] 13.4 Locally cartesian closed categories 360
• [ ] 14 Algebras for endofunctors 363
  • [ ] 14.1 Fixed points for a functor 363
  • [ ] 14.2 Recursive categories 368
  • [ ] 14.3 Triples 372
  • [ ] 14.4 Factorizations of a triple 374
  • [ ] 14.5 Scott domains 376
• [ ] 15 Toposes 383
  • [ ] 15.1 Definition of topos 384
  • [ ] 15.2 Properties of toposes 387
  • [ ] 15.3 Is a two-element poset complete? 391
  • [ ] 15.4 Presheaves 393
  • [ ] 15.5 Sheaves 395
  • [ ] 15.6 Fuzzy sets 400
  • [ ] 15.7 External functors 403
  • [ ] 15.8 The realizability topos 408
• [ ] 16 Categories with monoidal structure 413
  • [ ] 16.1 Closed monoidal categories 413
  • [ ] 16.2 Properties of A °± C
  • [ ] 16.3 *-autonomous categories 422
  • [ ] 16.4 The Chu construction 424

[ssirai]

moved.