Conceptual mathematics: A first introduction to categories

Title ......Page 3
Copyright ......Page 4
Conents ......Page 5
Please read this ......Page 11
Note to the reader ......Page 13
Preview ......Page 15
2 Galileo and the flight of a bird ......Page 17
3 Other examples of multiplication of objects ......Page 21
Part I: The category of sets ......Page 25
Article I: Sets, maps, composition ......Page 27
1 Guide ......Page 34
Summary: Definition of category ......Page 35
1 Review of Article I ......Page 36
2 An example of different rules for a map ......Page 41
3 External diagrams ......Page 42
4 Problems on the number of maps from one set to another ......Page 43
Session 3: Composing maps and counting maps ......Page 45
Part II: The algebra of composition ......Page 51
1 Isomorphisms ......Page 53
2 General division problems: Determination and choice ......Page 59
3 Retractions, sections, and idempotents ......Page 63
4 Isomorphisms and automorphisms ......Page 68
5 Guide ......Page 72
Summary: Special properties a map may have ......Page 73
1 Division of maps versus division of numbers ......Page 74
2 Inverses versus reciprocals ......Page 75
3 Isomorphisms as 'divisors' ......Page 77
4 A small zoo of isomorphisms in other categories ......Page 78
1 Determination problems ......Page 82
2 A special case: Constant maps ......Page 84
3 Choice problems ......Page 85
4 Two special cases of division: Sections and retractions ......Page 86
5 Stacking or sorting ......Page 88
6 Stacking in a Chinese restaurant ......Page 90
1 Sorting of the domain by a property ......Page 95
2 Naming or sampling of the codomain ......Page 96
3 Philosophical explanation of the two aspects ......Page 98
1 One use of isomorphisms: Coordinate systems ......Page 100
2 Two abuses of isomorphisms ......Page 103
Session 8: Pictures of a map making its features evident ......Page 105
1 Retracts and comparisons ......Page 113
2 Idempotents as records of retracts ......Page 114
3 A puzzle ......Page 116
4 Three kinds of retract problems ......Page 117
5 Comparing infinite sets ......Page 120
Quiz ......Page 122
How to solve the quiz problems ......Page 123
Composition of opposed maps ......Page 128
Summary/quiz on pais of 'opposed' maps ......Page 130
Summary: On the equation pj=1A ......Page 131
Review of 'I-words' ......Page 132
Test 1 ......Page 133
1 Balls, spheres, fixed points, and retractions ......Page 134
3 Brouwer's proof ......Page 138
4 Relation between fixed point and retraction theorems ......Page 140
5 How to understand a proof: The objectification and 'mapification' of concents ......Page 141
6 The eye of the storm ......Page 144
7 Using maps to formulate guesses ......Page 145
Part III: Categories of structures sets ......Page 147
Article III: Examples of categories ......Page 149
1 The category So of endomaps of sets ......Page 150
2 Typical applications of So ......Page 151
4 Categories of endomaps ......Page 152
5 Irreflexive graphs ......Page 155
6 Endomaps as special graphs ......Page 157
7 The simpler category S^i: Objects are just maps of sets ......Page 158
8 Reflexive graphs ......Page 159
10 Retractions and injectivity ......Page 160
11 Types of structure ......Page 163
12 Guide ......Page 165
1 A category of richer structures: Endomaps of sets ......Page 166
2 Two subcategories: Idempotents and automorphisms ......Page 169
3 The category of graphs ......Page 170
1 Dynamical systems or automata ......Page 175
2 Family trees ......Page 176
3 Dynamical systems revisited ......Page 177
Session 13: Monoids ......Page 180
Session 14: Maps preserve positive properties ......Page 184
1 Positive properties versus negative propoerties ......Page 187
1 Structure-preserving maps from a cycle to another endomap ......Page 189
2 Naming elements that have a given period by maps ......Page 190
3 Naming arbitrary elements ......Page 191
4 The philosophical role of N ......Page 194
5 Presentations of dynamical systems ......Page 196
1 Solving exercises on idempotents and involutions ......Page 201
2 Solving exercises on maps of graphs ......Page 203
1 Paths ......Page 210
2 Graphs as diagram shapes ......Page 214
3 Commuting diagrams ......Page 215
4 Is a diagram a map? ......Page 217
Test 2 ......Page 218
Session 18: Review of Test 2 ......Page 219
Part IV: Elementary universal mapping properties ......Page 225
1 Terminal objects ......Page 227
3 Initial object ......Page 229
4 Products ......Page 230
5 Commutative, associative and identity laws for multiplication of objects ......Page 234
7 Distributive laws ......Page 236
8 Guide ......Page 237
Session 19: Terminal objects ......Page 239
Session 20: Points of an object ......Page 244
Session 21: Products in categories ......Page 250
1 A special property of the category of sets ......Page 259
2 A similar property in the category of endomaps of sets ......Page 260
3 Incidence relations ......Page 263
4 Basic figure-types, singular figures, and incidence, in the category of graphs ......Page 264
Session 23: More on universal mapping properties ......Page 268
1 A category of pairs of maps ......Page 269
2 How to calculate products ......Page 270
1 The terminal object as an identity for multiplication ......Page 275
2 The uniqueness theorem for products ......Page 277
3 Sum of two objects in a category ......Page 279
Session 25: Labelings and products of graphs ......Page 283
1 Detecting the structure of a graph by means of labelings ......Page 284
2 Calculating the graphs A x Y ......Page 287
3 The distribute law ......Page 289
1 The standard map A x B1 + A x B2 --> A x (B1 + B2) ......Page 290
3 Sum of maps in a linear category ......Page 293
4 The associative law for sums and products ......Page 195
1 Universal constructions ......Page 298
2 Can objects have negatives? ......Page 301
3 Idempotent objects ......Page 303
4 Solving equations and picturing maps ......Page 306
1 An example of a non-distributive category ......Page 309
Test 3 ......Page 313
Test 4 ......Page 314
Test 5 ......Page 315
1 Binary operations and actions ......Page 316
2 Cantor's diagonal argument ......Page 317
Part V: Higher universal mapping properties ......Page 325
1 Definition of map objects ......Page 327
2 Distributivity ......Page 329
4 Universal properties and 'observables' ......Page 330
5 Guide ......Page 333
1 Map objects, or function spaces ......Page 334
2 A fundamental example of the transformation of map objects ......Page 337
3 Laws of exponents ......Page 338
4 The distributive law in cartesian closed categories ......Page 341
Session 31: Map object versus product ......Page 342
1 Definition of map object versus definition of product ......Page 343
2 Calculating map objects ......Page 345
1 Subobjects ......Page 349
2 Truth ......Page 352
3 The truth value object ......Page 354
1 Parts and inclusions ......Page 358
2 Toposes and logic ......Page 362
Index ......Page 367