Transcript

• 1. The Yoneda lemma and String diagrams Ray D. Sameshima total 54 pages 1
• 2. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 2
• 3. References Handbook of Categorical Algebra (F. Borceux) The Joy of String Diagrams (P. L. Curien) Category theory (P. L. Curien) (in progress) Cat (R. D. Sameshima) 3
• 4. Categories A Category is like a network of arrows with identities and associativity. (We ignore the size problem now!) 4
• 5. Functors A functor is a structure preserving mapping between categories (homomorphisms of categories). 5
• 6. Natural transformations A homotopy of categories. 6
• 7. Natural transformations A natural transformation consists of a class (family, set, or collection) of arrows. 7 s.t.
• 8. Natural transformations A natural transformation consists of a class (family, set, or collection) of arrows. 7 s.t.
• 9. Natural transformations We call this commutativity the naturality of the natural transformations. 8
• 10. Natural transformations We call this commutativity the naturality of the natural transformations. 8
• 11. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 9
• 12. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 9 ✔
• 13. Examples 0 1 A category of sets and mappings A class change method Representable functors Natural transformations 10
• 14. An empty category The empty category: No object and no arrow. 11
• 15. A singleton category Discrete categories: objects with identities. E.g., the singleton (one-point set) can be seen as a discrete category 1. 12
• 16. Set The mappings satisfy the associativity law. ! The identities are identity mappings. 13 f : A ! B; a7! f(a) g : B ! C; b7! g(b) h : C ! D; c7! h(c) h (g f)(a) = h(g(f(a))) = (h g) f(a) 1A : A ! A; a7! a
• 17. A class change method A class change method: we can always view an arbitrary arrow as a natural transformation. 14 8f 2 C(A,B) ) 9 ¯ f 2 Nat( ¯ A, ¯B ) where ¯ A, ¯B 2 Func(1,C)
• 18. Func(1,C) This is just pointing mappings of both objects and arrows in the category that we consider. ¯ C 2 Func(1,C) ¯ C(⇤) := C 2 |C| ¯ C(1⇤) := 1C So we can identify all objects as functors from 1 to the category. 15
• 19. Nat(A,B 2 Func(1,C)) Under the identifications, the arrow in the category can be seen as the natural transformation between the objects. 16 8f 2 C(A,B) ¯ f 2 Nat(A,B) : ⇤7! ¯ f⇤ := f This is, I call, a class change method.
• 20. Representable functors The functor represented by the object C. 17 C(C,−) 2 Func(C, Set)
• 21. C(C,−) 2 Func(C, Set) Now we ignore the size problems but… 18
• 22. ↵ 2 Nat(C(C,−), F) By definition 19 8B,C 2 |C| ↵C # C(A, g) = Fg # ↵B 8f 2 C(A,B) ↵C # C(A, g)(f) = Fg # ↵B(f)
• 23. Let me see Now we get all gadgets for the Yoneda lemma. 20
• 24. Yoneda lemma A milestone of category theory. 21
• 25. Yoneda lemma A milestone of category theory. 21
• 26. An equation based proof Basically, I traces the proof in this handbook ->. See my notes. 22
• 27. So many commutative diagrams Diagram chasing are routine tasks in the category theory. 23
• 28. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 24 ✔
• 29. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 24 ✔ ✔
• 30. String diagrams Flipping the diagrams! 25
• 31. String diagrams Two categories, two functors(objects), and a n.t. (an arrow.) 26 A f! B
• 32. Point it 8f 2 C(A,B) ¯ f 2 Nat(A,B) : ⇤7! ¯ f⇤ := f From above we can see… f : ⇤ ! C(A,B) = C(A,−)B 27
• 33. Compositions These are good examples of vertical compositions. 28
• 34. Compositions These are good examples of horizontal compositions. 29
• 35. Basically, that’s all. 30
• 36. No Standard Committees … Enjoy! Category Theory Using String Diagrams 31 (Dan Marsden)
• 37. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 32 ✔ ✔
• 38. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 32 ✔ ✔ ✔
• 39. Diagrammatic proof The basic gadget is the elevator rule. 33
• 40. Yoneda lemma A milestone of category theory. 34
• 41. Yoneda lemma A milestone of category theory. 34
• 42. Choose wisely ✓F,A(↵) := ↵A(1A) 35
• 43. Flip it ⌧ (a)(f) := Ff(a) ⌧ = "xy.Fy(x); a7! "y.Fy(a); f7! Ff(a) 36
• 44. Naturality of tau The Adventure of the Dancing Men 37
• 45. 38
• 46. Step by step 39
• 47. F is a functor 40
• 48. by def. of tau 41
• 49. a composition and the def. of tau for gf 42
• 50. tricky part 43
• 51. a representable functor 44
• 52. 45
• 53. We have proved the naturality of tau: 46 ⌧ (a) 2 Nat (A(A,−), F)
• 54. The right inverse 47 ✓F,A ⌧
• 55. 48
• 56. The left inverse 49 ⌧ ✓F,A
• 57. 50
• 58. Finally, we have proved that theta and tau are the inverse pair. 51 ✓F,A ⌧ = 1FA ⌧ ✓F,A = 1Nat(A(A,),F )
• 59. String diagrams are fun! 52
• 60. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 53 ✔ ✔ ✔
• 61. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 53 ✔ ✔ ✔ ✔
• 62. Thank you! 54
• 63. 55
• 64. Godement products and elevator rules Commutativity and elevator rules 56