1. Products and Sums http://cis.k.hosei.ac.jp/~yukita/

2. 2 Products To express the notion of function with several variables We need to talk about products of objects.

3. 3 Ex. 1. Add and Multiply

4. 4 The one point set and the empty set

5. 5 Prop. Property [2] characterizes the empty set.

6. 6 Prop. Property [1] characterizes the one point set.

7. 7 What is the use of this kind of argument? We respect specification by arrows. Properties [1] and [2] are specifications. Corresponding implementations are the one point set and the empty set. There are many cases where specification determines implementation up to isomorphism.

8. 8 Def. Initial and terminal objects

9. 9 Ex. 2. Elements of a set

10. 10 Ex. 3. The power set 2 X

11. 11

12. 12 Products

13. 13 Cartesian product of X and Y

14. 14

15. 15 Universality (existence)

16. 16 Universality (uniqueness)

17. 17 Prop. The product of two objects in a category is unique up to isomorphism.

18. 18

19. 19

20. 20

21. 21 Note

22. 22 Ex. 6. Category 2 X

23. 23 Preordered Category The product of two objects, if it exists, is their intersection. In other words, the greatest lower bound of the two objects.

24. 24 Ex. 7. The monoid with one object A and two arrows 1 A and a, satisfying a 2 =a, does not have products.

25. 25 Ex. 8. The Diagonal Function

26. 26 Def. Diagonal in an Arbitrary Category with Products

27. 27

28. 28 Def. Parallel Functions in an Arbitrary Category with Products

29. 29 Ex. 10. The Twist Function

30. 30 Def. The Twist Function in an Aribitrary Category with Products

31. 31 Object oriented view public class ObjCatA{ } public class ProdCatA extends ObjCatA{ ObjCatA x, y; public ProdCatA(ObjCatA x, ObjCatyA y){ this.x = x; this.y = y; } public ArrowCatA factArrows (ObjA z, ArrowCatA f, ArrowCatA g){ return /* the unique arrow that satisfies the property in the last slide */ }

32. 32 Ex.11. Category Circ

33. 33 Negation, And, Or

34. 34 Claim. Category Circ has products.

35. 35

36. 36 not

37. 37 &

38. 38 or

39. 39

40. 40 f g

41. 41 fg

42. 42

43. 43 Boolean Gates f(x,y,z) & or not & && x y z

44. 44

45. 45

46. 46 Summary Using wires, we can implement products. Every function B m  B n can be implemented using not, &, or, true, false, using products and composition.

47. 47

48. 48 Note. The product of the empty family The product is a terminal object. Since the family is empty, the only requirement is that, given Z, there is a unique arrow from Z to the product.

49. 49 Prop. The product of a family of objects in a category is unique up to isomorphism.

50. 50

51. 51

52. 52

53. 53 Prop. 4.2. If products of all pairs of objects exist in A and a terminal object exists then products of finite families exit.

54. 54

55. 55