Conceptual Mathematics: A First Introduction to Categories

Cover......Page 1
Title......Page 2
Contents......Page 4
Preface......Page 10
Organisation of the book......Page 12
Acknowledgements......Page 14
Preview......Page 16
SESSION 1 Galileo and multiplication of objects......Page 18
PART I The category of sets......Page 26
ARTICLE I Sets, maps, composition. A first example of a category......Page 28
SESSION 2 Sets, maps and composition......Page 37
SESSION 3 Composing maps and counting maps......Page 46
PART II The algebra of composition......Page 52
ARTICLE II Isomorphisms......Page 54
SESSION 4 Division of maps: Isomorphisms......Page 75
SESSION 5 Division of maps: Sections and retractions......Page 83
SESSION 6 Two general aspects or uses of maps......Page 96
SESSION 7 Isomorphisms and coordinates......Page 101
SESSION 8 Pictures of a map making its features evident......Page 106
SESSION 9 Retracts and idempotents......Page 114
Quiz......Page 123
How to solve the quiz problems......Page 124
Composition of opposed maps......Page 129
Summary /quiz on pairs of \'opposed\' maps......Page 131
Summary: On the equation......Page 132
SESSION 10 Brouwer\'s theorems......Page 135
PART III Categories of structured sets......Page 148
ARTICLE III Examples of categories......Page 150
SESSION 11 Ascending to categories of richer structures......Page 167
SESSION 12 Categories of diagrams......Page 176
SESSION 13 Monoids......Page 181
SESSION 14 Maps preserve positive properties......Page 185
SESSION 15 Objectification of properties in dynamical systems......Page 190
SESSION 16 Idempotents, involutions, and graphs......Page 202
SESSION 17 Some uses of graphs......Page 211
Test 2......Page 204
SESSION 18 Review of Test 2......Page 205
PART IV Elementary universal mapping properties......Page 226
ARTICLE IV Universal mapping properties......Page 228
SESSION 19 Terminal objects......Page 240
SESSION 20 Points of an object......Page 245
SESSION 21 Products in categories......Page 251
SESSION 22 Universal mapping properties. Incidence relations......Page 260
SESSION 23 More on universal mapping properties......Page 269
SESSION 24 Uniqueness of products and definition of sum......Page 276
SESSION 25 Labelings and products of graphs......Page 284
SESSION 26 Distributive categories and linear categories......Page 291
SESSION 27 Examples of universal constructions......Page 299
SESSION 28 The category of pointed sets......Page 310
Test 3......Page 314
Test 4......Page 315
Test 5......Page 316
SESSION 29 Binary operations and diagonal arguments......Page 317
PART V Higher universal mapping properties......Page 326
ARTICLE V Map objects......Page 328
SESSION 30 Exponentiation......Page 335
SESSION 31 Map object versus product......Page 343
ARTICLE VI The contravariant parts functor......Page 350
SESSION 32 Subobjects, logic, and truth......Page 354
SESSION 33 Parts of an object: Toposes......Page 363
ARTICLE VII The Connected Components Functor......Page 373
SESSION 34 Group theory and the number of types of connected objects......Page 377
SESSION 35 Constants, codiscrete objects, and many connected objects......Page 381
APPENDICES Toward Further Studies......Page 383
APPENDIX I Geometry of figures and algebra of functions......Page 384
APPENDIX II Adjoint functors with examples from graphs and dynamical systems......Page 387
APPENDIX III The emergence of category theory within mathematics......Page 393
APPENDIX IV Annotated Bibliography......Page 396
Index......Page 400
Back cover......Page 406