Conceptual Mathematics · A First Introduction to Categories by Lawvere, F. William -- Read -- Imperial Library of Trantor

Index

Cover Half Title Title Page Copyright Dedication Contents Preface Organization of the book Acknowledegments Preview

Session 1. Galileo and multiplication of objects

1 Introduction 2 Galileo and the flight of a bird 3 Other examples of multiplication of objects

Part I: The category of sets

Article I: Sets, maps, composition

1 Guide Summary: Definition of category Session 2. Sets, maps, and composition

1 Review of Article I 2 An example of different rules for a map 3 External diagrams 4 Problems on the number of maps from one set to another

Session 3. Composing maps and counting maps

Part II: The algebra of composition

Article II: Isomorphisms

1 Isomorphisms 2 General division problems: Determination and choice 3 Retractions, sections, and idempotents 4 Isomorphisms and automorphisms 5 Guide Summary: Special properties a map may have Session 4. Division of maps: Isomorphisms

1 Division of maps versus division of numbers 2 Inverses versus reciprocals 3 Isomorphisms as ‘divisors’ 4 A small zoo of isomorphisms in other categories

Session 5. Division of maps: Sections and retractions

1 Determination problems 2 A special case: Constant maps 3 Choice problems 4 Two special cases of division: Sections and retractions 5 Stacking or sorting 6 Stacking in a Chinese restaurant

Session 6. Two general aspects or uses of maps

1 Sorting of the domain by a property 2 Naming or sampling of the codomain 3 Philosophical explanation of the two aspects

Session 7. Isomorphisms and coordinates

1 One use of isomorphisms: Coordinate systems 2 Two abuses of isomorphisms

Session 8. Pictures of a map making its features evident Session 9. Retracts and idempotents

1 Retracts and comparisons 2 Idempotents as records of retracts 3 A puzzle 4 Three kinds of retract problems 5 Comparing infinite sets

Quiz How to solve the quiz problems Composition of opposed maps Summary/quiz on pairs of ‘opposed’ maps Summary: On the equation p ° j = 1[sub(A)] Review of ‘I-words’ Test 1 Session 10. Brouwer’s theorems

1 Balls, spheres, fixed points, and retractions 2 Digression on the contrapositive rule 3 Brouwer’s proof 4 Relation between fixed point and retraction theorems 5 How to understand a proof: The objectification and ‘mapification’ of concepts 6 The eye of the storm 7 Using maps to formulate guesses

Part III: Categories of structured sets

Article III: Examples of categories

1 The category S[arrow] of endomaps of sets 2 Typical applications of S[arrow] 3 Two subcategories of S[arrow] 4 Categories of endomaps 5 Irreflexive graphs 6 Endomaps as special graphs 7 The simpler category S[sup(↓)]: Objects are just maps of sets 8 Reflexive graphs 9 Summary of the examples and their general significance 10 Retractions and injectivity 11 Types of structure 12 Guide Session 11. Ascending to categories of richer structures

1 A category of richer structures: Endomaps of sets 2 Two subcategories: Idempotents and automorphisms 3 The category of graphs

Session 12. Categories of diagrams

1 Dynamical systems or automata 2 Family trees 3 Dynamical systems revisited

Session 13. Monoids Session 14. Maps preserve positive properties

1 Positive properties versus negative properties

Session 15. Objectification of properties in dynamical systems

1 Structure-preserving maps from a cycle to another endomap 2 Naming elements that have a given period by maps 3 Naming arbitrary elements 4 The philosophical role of N 5 Presentations of dynamical systems

Session 16. Idempotents, involutions, and graphs

1 Solving exercises on idempotents and involutions 2 Solving exercises on maps of graphs

Session 17. Some uses of graphs

1 Paths 2 Graphs as diagram shapes 3 Commuting diagrams 4 Is a diagram a map?

Test 2 Session 18. Review of Test 2

Part IV: Elementary universal mapping properties

Article IV: Universal mapping properties

1 Terminal objects 2 Separating 3 Initial object 4 Products 5 Commutative, associative, and identity laws for multiplication of objects 6 Sums 7 Distributive laws 8 Guide Session 19. Terminal objects Session 20. Points of an object Session 21. Products in categories Session 22. Universal mapping properties and incidence relations

1 A special property of the category of sets 2 A similar property in the category of endomaps of sets 3 Incidence relations 4 Basic figure-types, singular figures, and incidence, in the category of graphs

Session 23. More on universal mapping properties

1 A category of pairs of maps 2 How to calculate products

Session 24. Uniqueness of products and definition of sum

1 The terminal object as an identity for multiplication 2 The uniqueness theorem for products 3 Sum of two objects in a category

Session 25. Labelings and products of graphs

1 Detecting the structure of a graph by means of labelings 2 Calculating the graphs A × Y 3 The distributive law

Session 26. Distributive categories and linear categories

1 The standard map A × B[sub(1)] + A × B[sub(2)] → A × (B[sub(1)] + B[sub(2)]) 2 Matrix multiplication in linear categories 3 Sum of maps in a linear category 4 The associative law for sums and products

Session 27. Examples of universal constructions

1 Universal constructions 2 Can objects have negatives? 3 Idempotent objects 4 Solving equations and picturing maps

Session 28. The category of pointed sets

1 An example of a non-distributive category

Test 3 Test 4 Test 5 Session 29. Binary operations and diagonal arguments

1 Binary operations and actions 2 Cantor’s diagonal argument

Part V: Higher universal mapping properties

Article V: Map objects

1 Definition of map object 2 Distributivity 3 Map objects and the Diagonal Argument 4 Universal properties and ‘observables’ 5 Guide Session 30. Exponentiation

1 Map objects, or function spaces 2 A fundamental example of the transformation of map objects 3 Laws of exponents 4 The distributive law in cartesian closed categories

Session 31. Map object versus product

1 Definition of map object versus definition of product 2 Calculating map objects

Article VI: The contravariant parts functor

1 Parts and stable conditions 2 Inverse Images and Truth Session 32. Subobject, logic, and truth

1 Subobjects 2 Truth 3 The truth value object

Session 33. Parts of an object: Toposes

1 Parts and inclusions 2 Toposes and logic

Article VII: The Connected Components Functor

1 Connectedness versus discreteness 2 The points functor parallel to the components functor 3 The topos of right actions of a monoid Session 34. Group theory and the number of types of connected objects Session 35. Constants, codiscrete objects, and many connected objects

1 Constants and codiscrete objects 2 Monoids with at least two constants

Appendices

Appendix I: Geometery of figures and algebra of functions

1 Functors 2 Geometry of figures and algebra of functions as categories themselves

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

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