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Conceptual Mathematics, Second Edition

Book 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 introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. 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. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Organization of the book
  9. Acknowledegments
  10. Preview
    1. Session 1. Galileo and multiplication of objects
      1. 1 Introduction
      2. 2 Galileo and the flight of a bird
      3. 3 Other examples of multiplication of objects
  11. Part I: The category of sets
    1. Article I: Sets, maps, composition
      1. 1 Guide
      2. Summary: Definition of category
      3. Session 2. Sets, maps, and composition
        1. 1 Review of Article I
        2. 2 An example of different rules for a map
        3. 3 External diagrams
        4. 4 Problems on the number of maps from one set to another
      4. Session 3. Composing maps and counting maps
  12. Part II: The algebra of composition
    1. Article II: Isomorphisms
      1. 1 Isomorphisms
      2. 2 General division problems: Determination and choice
      3. 3 Retractions, sections, and idempotents
      4. 4 Isomorphisms and automorphisms
      5. 5 Guide
      6. Summary: Special properties a map may have
      7. Session 4. Division of maps: Isomorphisms
        1. 1 Division of maps versus division of numbers
        2. 2 Inverses versus reciprocals
        3. 3 Isomorphisms as ‘divisors’
        4. 4 A small zoo of isomorphisms in other categories
      8. Session 5. Division of maps: Sections and retractions
        1. 1 Determination problems
        2. 2 A special case: Constant maps
        3. 3 Choice problems
        4. 4 Two special cases of division: Sections and retractions
        5. 5 Stacking or sorting
        6. 6 Stacking in a Chinese restaurant
      9. Session 6. Two general aspects or uses of maps
        1. 1 Sorting of the domain by a property
        2. 2 Naming or sampling of the codomain
        3. 3 Philosophical explanation of the two aspects
      10. Session 7. Isomorphisms and coordinates
        1. 1 One use of isomorphisms: Coordinate systems
        2. 2 Two abuses of isomorphisms
      11. Session 8. Pictures of a map making its features evident
      12. Session 9. Retracts and idempotents
        1. 1 Retracts and comparisons
        2. 2 Idempotents as records of retracts
        3. 3 A puzzle
        4. 4 Three kinds of retract problems
        5. 5 Comparing infinite sets
      13. Quiz
      14. How to solve the quiz problems
      15. Composition of opposed maps
      16. Summary/quiz on pairs of ‘opposed’ maps
      17. Summary: On the equation p ° j = 1[sub(A)]
      18. Review of ‘I-words’
      19. Test 1
      20. Session 10. Brouwer’s theorems
        1. 1 Balls, spheres, fixed points, and retractions
        2. 2 Digression on the contrapositive rule
        3. 3 Brouwer’s proof
        4. 4 Relation between fixed point and retraction theorems
        5. 5 How to understand a proof: The objectification and ‘mapification’ of concepts
        6. 6 The eye of the storm
        7. 7 Using maps to formulate guesses
  13. Part III: Categories of structured sets
    1. Article III: Examples of categories
      1. 1 The category S[arrow] of endomaps of sets
      2. 2 Typical applications of S[arrow]
      3. 3 Two subcategories of S[arrow]
      4. 4 Categories of endomaps
      5. 5 Irreflexive graphs
      6. 6 Endomaps as special graphs
      7. 7 The simpler category S[sup(↓)]: Objects are just maps of sets
      8. 8 Reflexive graphs
      9. 9 Summary of the examples and their general significance
      10. 10 Retractions and injectivity
      11. 11 Types of structure
      12. 12 Guide
      13. Session 11. Ascending to categories of richer structures
        1. 1 A category of richer structures: Endomaps of sets
        2. 2 Two subcategories: Idempotents and automorphisms
        3. 3 The category of graphs
      14. Session 12. Categories of diagrams
        1. 1 Dynamical systems or automata
        2. 2 Family trees
        3. 3 Dynamical systems revisited
      15. Session 13. Monoids
      16. Session 14. Maps preserve positive properties
        1. 1 Positive properties versus negative properties
      17. Session 15. Objectification of properties in dynamical systems
        1. 1 Structure-preserving maps from a cycle to another endomap
        2. 2 Naming elements that have a given period by maps
        3. 3 Naming arbitrary elements
        4. 4 The philosophical role of N
        5. 5 Presentations of dynamical systems
      18. Session 16. Idempotents, involutions, and graphs
        1. 1 Solving exercises on idempotents and involutions
        2. 2 Solving exercises on maps of graphs
      19. Session 17. Some uses of graphs
        1. 1 Paths
        2. 2 Graphs as diagram shapes
        3. 3 Commuting diagrams
        4. 4 Is a diagram a map?
      20. Test 2
      21. Session 18. Review of Test 2
  14. Part IV: Elementary universal mapping properties
    1. Article IV: Universal mapping properties
      1. 1 Terminal objects
      2. 2 Separating
      3. 3 Initial object
      4. 4 Products
      5. 5 Commutative, associative, and identity laws for multiplication of objects
      6. 6 Sums
      7. 7 Distributive laws
      8. 8 Guide
      9. Session 19. Terminal objects
      10. Session 20. Points of an object
      11. Session 21. Products in categories
      12. Session 22. Universal mapping properties and incidence relations
        1. 1 A special property of the category of sets
        2. 2 A similar property in the category of endomaps of sets
        3. 3 Incidence relations
        4. 4 Basic figure-types, singular figures, and incidence, in the category of graphs
      13. Session 23. More on universal mapping properties
        1. 1 A category of pairs of maps
        2. 2 How to calculate products
      14. Session 24. Uniqueness of products and definition of sum
        1. 1 The terminal object as an identity for multiplication
        2. 2 The uniqueness theorem for products
        3. 3 Sum of two objects in a category
      15. Session 25. Labelings and products of graphs
        1. 1 Detecting the structure of a graph by means of labelings
        2. 2 Calculating the graphs A × Y
        3. 3 The distributive law
      16. Session 26. Distributive categories and linear categories
        1. 1 The standard map A × B[sub(1)] + A × B[sub(2)] → A × (B[sub(1)] + B[sub(2)])
        2. 2 Matrix multiplication in linear categories
        3. 3 Sum of maps in a linear category
        4. 4 The associative law for sums and products
      17. Session 27. Examples of universal constructions
        1. 1 Universal constructions
        2. 2 Can objects have negatives?
        3. 3 Idempotent objects
        4. 4 Solving equations and picturing maps
      18. Session 28. The category of pointed sets
        1. 1 An example of a non-distributive category
      19. Test 3
      20. Test 4
      21. Test 5
      22. Session 29. Binary operations and diagonal arguments
        1. 1 Binary operations and actions
        2. 2 Cantor’s diagonal argument
  15. Part V: Higher universal mapping properties
    1. Article V: Map objects
      1. 1 Definition of map object
      2. 2 Distributivity
      3. 3 Map objects and the Diagonal Argument
      4. 4 Universal properties and ‘observables’
      5. 5 Guide
      6. Session 30. Exponentiation
        1. 1 Map objects, or function spaces
        2. 2 A fundamental example of the transformation of map objects
        3. 3 Laws of exponents
        4. 4 The distributive law in cartesian closed categories
      7. Session 31. Map object versus product
        1. 1 Definition of map object versus definition of product
        2. 2 Calculating map objects
    2. Article VI: The contravariant parts functor
      1. 1 Parts and stable conditions
      2. 2 Inverse Images and Truth
      3. Session 32. Subobject, logic, and truth
        1. 1 Subobjects
        2. 2 Truth
        3. 3 The truth value object
      4. Session 33. Parts of an object: Toposes
        1. 1 Parts and inclusions
        2. 2 Toposes and logic
    3. Article VII: The Connected Components Functor
      1. 1 Connectedness versus discreteness
      2. 2 The points functor parallel to the components functor
      3. 3 The topos of right actions of a monoid
      4. Session 34. Group theory and the number of types of connected objects
      5. Session 35. Constants, codiscrete objects, and many connected objects
        1. 1 Constants and codiscrete objects
        2. 2 Monoids with at least two constants
  16. Appendices
    1. Appendix I: Geometery of figures and algebra of functions
      1. 1 Functors
      2. 2 Geometry of figures and algebra of functions as categories themselves
    2. Appendix II: Adjoint functors with examples from graphs and dynamical systems
    3. Appendix III: The emergence of category theory within mathematics
    4. Appendix IV: Annotated Bibliography
  17. Index