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H I T H I T I H H T T I T T T I T T H I T H T - PowerPoint PPT Presentation

Feb 15, 2023 •134 likes •415 views

H H H I T H I T I H H T T I T T T I T T H I T H T I T H I H H I I H I I pushout suspension join sphere circle Higher Inductive Types PUSHOUT g(c) f(c) c C A B g C f B A data Pushout {A B C : U}

  1. H H H I T H I T I H H T T I T T T I T T H I T H T I T H I H H I I H I I

  2. pushout suspension join sphere circle Higher Inductive Types

  3. PUSHOUT g(c) f(c) c C A B

  4. g C f B A data Pushout {A B C : U} (f : C → A) (g : C → B) : U where inl : A → Pushout f g inr : B → Pushout f g push : (c : C) → inl (f c) ≡ inr (g c)

  5. g push f inr inl g f

  6. pushout suspension sphere circle join

  7. COEQUALIZER g(b) f(b) b B A

  8. g(b) f(b) b B A data Coequalizer {A B : U} (f g : B → A) : U where inc : A → Coequalizer f g eq : (b : B) → inc (f b) ≡ inc (g b)

  9. f inc g

  10. C C A+B A B B B+B B A A

  11. SUSPENSION A

  12. north data Susp (A : U) : U where north : Susp A merid south : Susp A A merid : (a : A) → north ≡ south south

  13. north = merid A S n := Susp n (2) See the lecture on truncation levels south

  14. WEDGE a b B A sum types for pointed types

  15. A data Wedge (A B : U) (a : A) (b : B) : U where a inl : A → Wedge A B a b inr : A → Wedge A B a b b glue : inl a ≡ inr b B

  16. SMASH A ∧ B := A × B / A ∨ B smash wedge sum

  17. b B a A A×B/ wedge SMASH sum

  18. B A SMASH

  19. baser B basel A data Smash (A B : U) (a : A) (b : B) : U where pair : A → B → Smash A B a b basel : Smash A B a b baser : Smash A B a b gluel : (a' : A) → inc a' b ≡ basel gluer : (b' : B) → inc a b' ≡ baser

  20. JOIN Paths between all pairs

  21. A B data Join (A B : U) : U where inl : A → Join A B inr : B → Join A B join : (a : A) (b : B) → inl a ≡ inr b

  22. X, Y and Z are pointed types X ★ Y ≃ Susp (X ∧ Y) smash join Susp (X ∧ Y) ≃ (Susp X) ∧ Y ≃ X ∧ (Susp Y) S n ∧ S m ≃ S n+m A × B → C ≃ A → (B → C) S n ★ S m ≃ S n+m+1 X ∧ Y ∙→ Z ≃ X ∙→ (Y ∙→ Z) point- CURRYING preserving functons

  23. n-TRUNCATION Best n-type approximation

  24. n-TRUNCATION Fill every image of S n+1 with a cone See the lecture on truncation levels also [HoTT, 7.3] A

  25. hub l spoke l x l data Trunc n (A : U) : U where inc : A → Trunc n A hub : (S n+1 → A) → Trunc n A spoke : (l : S n+1 → A) (x : S n+1 ) → hub f ≡ f x e ff ectively has-level n (Trunc n A) [HoTT, 7.3]

  26. f any n-type inc Any function to an n-type n-truncation factors through n-truncation

  27. More: set quotients coequalizer + 0-truncation More 2 : sequential colimits e.g. define S ∞ as lim S n More 3 : textbooks or ask Favonia All definable using pushouts

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