Conceptual Mathematics: A First Introduction to Categories

Description

In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics, Second Edition, introduces the concept of category for the learning, development, and use of mathematics, to both beginning students and general readers, and to practicing mathematical scientists. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories." Cover Title Page Copyright Page Dedication Page Contents Preface Organisation of the book Acknowledgements Preview SESSION 1 Galileo and multiplication of objects PART I The category of sets ARTICLE I Sets, maps, composition. A first example of a category SESSION 2 Sets, maps and composition SESSION 3 Composing maps and counting maps PART II The algebra of composition ARTICLE II Isomorphisms SESSION 4 Division of maps: Isomorphisms SESSION 5 Division of maps: Sections and retractions SESSION 6 Two general aspects or uses of maps SESSION 7 Isomorphisms and coordinates SESSION 8 Pictures of a map making its features evident SESSION 9 Retracts and idempotents Quiz How to solve the quiz problems Composition of opposed maps Summary /quiz on pairs of opposed maps Summary: On the equation SESSION 10 Brouwers theorems PART III Categories of structured sets ARTICLE III Examples of categories SESSION 11 Ascending to categories of richer structures SESSION 12 Categories of diagrams SESSION 13 Monoids SESSION 14 Maps preserve positive properties SESSION 15 Objectification of properties in dynamical systems SESSION 16 Idempotents, involutions, and graphs SESSION 17 Some uses of graphs Test 2 SESSION 18 Review of Test 2 PART IV Elementary universal mapping properties ARTICLE IV Universal mapping properties SESSION 19 Terminal objects SESSION 20 Points of an object SESSION 21 Products in categories SESSION 22 Universal mapping properties. Incidence relations SESSION 23 More on universal mapping properties SESSION 24 Uniqueness of products and definition of sum SESSION 25 Labelings and products of graphs SESSION 26 Distributive categories and linear categories SESSION 27 Examples of universal constructions SESSION 28 The category of pointed sets Test 3 Test 4 Test 5 SESSION 29 Binary operations and diagonal arguments PART V Higher universal mapping properties ARTICLE V Map objects SESSION 30 Exponentiation SESSION 31 Map object versus product ARTICLE VI The contravariant parts functor SESSION 32 Subobjects, logic, and truth SESSION 33 Parts of an object: Toposes ARTICLE VII The Connected Components Functor SESSION 34 Group theory and the number of types of connected objects SESSION 35 Constants, codiscrete objects, and many connected objects APPENDICES Toward Further Studies APPENDIX I Geometry of figures and algebra of functions APPENDIX II Adjoint functors with examples from graphs and dynamical systems APPENDIX III The emergence of category theory within mathematics APPENDIX IV Annotated Bibliography Index Back cover