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Acknowledgements
Preview Session 1 Galileo and multiplication of objects
1. Part I The category of sets Article I Sets, maps, composition
1.1. Guide
2. Session 2 Sets, maps, and composition
2.1. Review of Article I
2.2. An example of different rules for a map
2.3. External diagrams
2.4. Problems on the number of maps from one set to another
3. Session 3 Composing maps and counting maps
4. Part II The algebra of composition Article II Isomorphisms
4.1. Isomorphisms
4.2. General division problems: Determination and choice
4.3. Retractions, sections, and idempotents
4.4. Isomorphisms and automorphisms
4.5. Guide
5. Session 4 Division of maps: Isomorphisms
5.1. Division of maps versus dilision of numbers
5.2. Inverses versus reciprocals
5.3. Isomorphisms as 'divisors'
5.4. A small zoo of isomorphisms in other categories
6. Session 5 Division of maps: Sections and retractions
6.1. Determination problems
6.2. A special case: Constant maps
6.3. Choice problems
6.4. Two special cases of division: Sections and retractions
6.5. Stacking or sorting
6.6. Stacking in a Chinese restaurant
7. Session 6 Two general aspects or uses of maps
7.1. Sorting of the domain by a property
7.2. Naming or sampling of the codomain
7.3. Philosophical explanation of the two aspects
8. Session 7 Isomorphisms and coordinates
8.1. One use of isomorphisms: Coordinate systems
8.2. Two abuses of isomorphisms
9. Session 8 Pictures of a map making its features evident
10. Session 9 Retracts and idempotents
10.1. Retracts and comparisons
10.2. Idempotents as records of retracts
10.3. A puzzle
10.4. Three kinds of retract problems
10.5. Comparing infinite sets
11. Quiz
12. How to solve the quiz problems
13. Composition of opposed maps
14. Summary/quiz on pairs of 'opposed' maps
15. Summary: On the equation poj =1A
16. Review of 'I-words'
17. Test 1
18. Session 10 Brouwer's theorems
18.1. Balls, spheres, fixed points, and retractions
18.2. Digression on the contrapositive rule
18.3. Brouwer's proof
18.4. Relation between fi)*1 point and retraction theorems
18.5. How to understand a proof: The objectification and `mapification' of concepts
18.6. The eye of the storm
18.7. Using maps to formulate guesses
19. Part III Categories of structured sets Article III Examples of categories
19.1. The category .50 of endomaps of sets
19.2. Typical applications of .50
19.3. Two subcategories of S°
19.4. Categories of endomaps
19.5. Irreflexive graphs
19.6. Endomaps as special graphs
19.7. The simpler category S1-: Objects are just maps of sets
19.8. Reflexive graphs
19.9. Summary of the examples and their general significance
19.10. Retractions and injectivity
19.11. Types of structure
19.12. Guide
20. Session 11 Ascending to categories of richer structures
20.1. A category of richer structures: Endomaps of sets
20.2. Two subcategories: Idempotents and automorphisms
20.3. The category of graphs
21. Session 12 Categories of diagrams
21.1. Dynamical systems or automata
21.2. Family trees
21.3. Dynamical systems revisited
22. Session 13 Monoids
23. Session 14 Maps preserve positive properties
23.1. Positive properties versus negative properties
24. Session 15 Objectification of properties in dynamical systems
24.1. Structure-preserving maps from a cycle to another endomap
24.2. Naming elements that have a given period by maps
24.3. Naming arbitrary elements
24.4. The philosophical role of N
24.5. Presentations of dynamical systems
25. Session 16 Idempotents, involutions, and graphs
25.1. Solving exercises on idempotents and involutions
25.2. Solving exercises on maps of graphs
26. Session 17 Some uses of graphs
26.1. Paths
26.2. Graphs as diagram shapes
26.3. Commuting diagrams
26.4. Is a diagram a map?
27. Test 2
28. Session 18 Review of Test 2
29. Part IV Elementary universal mapping properties Article IV Universal mapping properties
29.1. Terminal objects
29.2. Separating
29.3. Initial object
29.4. Products
29.5. Commutative, associative, and identity laws for multiplication of objects
29.6. Sums
29.7. Distributive laws
29.8. Guide
30. Session 19 Terminal objects
31. Session 20 Points of an object
32. Session 21 Products in categories
33. Session 22 Universal mapping properties and incidence relations
33.1. A special property of the category of sets
33.2. A similar property in the category of endomaps of sets
33.3. Incidence relations
33.4. Basic figure-types, singular figures, and incidence, in the category of graphs
34. Session 23 More on universal mapping properties
34.1. A category of pairs of maps
34.2. How to calculate products
35. Session 24 Uniqueness of products and definition of sum
35.1. The terminal object as an identity for multiplication
35.2. The uniqueness theorem for products
35.3. Sum of two objects in a category
36. Session 25 Labelings and products of graphs
36.1. Detecting the structure of a graph by means of labelings
36.2. Calculating the graphs A x Y
36.3. The distributive law
37. Session 26 Distributive categories and linear categories
37.1. The standard map Ax B 1 + A X B2 -> A x (B 1 + B2 )
37.2. Matrix multiplication in linear categories
37.3. Sum of maps in a linear category
37.4. The associative law for sums and products
38. Session 27 Examples of universal constructions
38.1. Universal constructions
38.2. Can objects have negatives?
38.3. Idempotent objects
38.4. Solving equations and picturing maps
39. Session 28 The category of pointed sets
39.1. An example of a non-distributive category
40. Test 3
41. Test 4
42. Test 5
43. Session 29 Binary operations and diagonal arguments
43.1. Binary operations and actions
43.2. Cantor's diagonal argument
44. Part V Higher universal mapping properties Article V Map objects
44.1. Definition of map object
44.2. Distributivity
44.3. Map objects and the Diagonal Argument
44.4. Universal properties and `observables'
44.5. Guide
45. Session 30 Exponentiation
45.1. Map objects, or function spaces
45.2. A fundamental example of the transformation of map objects
45.3. Laws of exponents
45.4. The distributive law in cartesian closed categories
46. Session 31 Map object versus product
46.1. Definition of map object versus definition of product
46.2. Calculating map objects
47. Session 32 Subobjects, logic, and truth
47.1. Subobjects
47.2. Truth
47.3. The truth value object
48. Session 33 Parts of an object: Toposes
48.1. Parts and inclusions
48.2. Toposes and logic
Index
