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."
Author(s): F. William Lawvere; Stephen H. Schanuel
Edition: Hardcover
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 408
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
