Covering Spaces in Homotopy Type Theory Favonia Robert Harper - PowerPoint PPT Presentation

Covering Spaces in Homotopy Type Theory Favonia Robert Harper Carnegie Mellon University {favonia,rwh}@cs.cmu.edu This material is based upon work supported by the This material is based upon work supported by the 1 National Science Foundation under Grant No. 1116703. National Science Foundation under Grant No. 1116703.

Homotopy Type Theory (HoTT) (HoTT) A Type Space a : A Term Point f : A → B Function Continuous Mapping C : A → Type Dependent Type Fibration C(a) Fiber a = A b Identity Path 2

Every type is an ∞ -groupoid 3

Every type is an ∞ -groupoid b a 3

Every type is an ∞ -groupoid b p:a=b a 3

Every type is an ∞ -groupoid b p:a=b q:a=b a 3

⋮ Every type is an ∞ -groupoid b p:a=b q:a=b a h:p=q 3

⋮ f : A → B ⟼ a : A b : B ⟼ p : a 1 =a 2 q : b 1 =b 2 f a 2 b 2 p q a 1 b 1 A B 4

⋮ type A 5

⋮ ⋮ [ ] type groupoid A 5

⋮ ⋮ [ ] type groupoid A ‖ A ‖ 1 5

⋮ ⋮ ⋮ [ ] [ ] [ ] set type groupoid (UIP) A ‖ A ‖ 1 ‖ A ‖ 0 5

⋮ ⋮ ⋮ ⋮ [ ] [ ] [ ] [ ] [ ] [ ] set prop. type groupoid (UIP) (squash) A ‖ A ‖ 1 ‖ A ‖ 0 ‖ A ‖ -1 5

Covering Spaces Continuously changing families of sets Classical definition: A covering space of A is a space C together with a continuous surjective map p : C → A, such that for every a ∈ A, there exists an open neighborhood U of a, such that p -1 (U) is a union of disjoint open sets in A, each of which is mapped homeomorphically onto U by p. HoTT definition: F : A → Set  estion: Is it correct (up to homotopy)? 6

⋮ Covering Spaces F : A → Set ⟼ a : A F(a) : Set ⟼ p : a 1 =a 2 iso : F(a 1 )=F(a 2 ) ⟼ q : p 1 =p 2 (trivial) 7

Classification Theorem Suppose A is pointed (a 0 ) and connected. F : A → Set ≃ ⟼ a 0 : A F(a 0 ) : Set ⟼ loop : a 0 =a 0 auto : F(a 0 )=F(a 0 ) This is an action of ‖ a 0 =a 0 ‖ 0 on F(a 0 ). ‖ a 0 =a 0 ‖ 0 is the fundamental group π 1 (A, a 0 ). 8

Classification Theorem Suppose A is pointed (a 0 ) and connected. (A → Set) ≃ π 1 (A, a 0 )-Set Pointed (a 0 ) and connected: (a 0 : A) × ((x : A) → (y : A) → ‖ x = y ‖ -1 ) Fundamental group π 1 (A, a 0 ): ‖ a 0 = a 0 ‖ 0 G-Set: (X : Set) × ( α : G → (X → X)) × ( α unit = id) × ( α (g 1 ∙ g 2 ) = α g 1 ∘ α g 2 ) 9

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . (A → Set) ≃ π 1 (A, a 0 )-Set 10 10

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . (A → Set) ≃ π 1 (A, a 0 )-Set ⟼ F (F(a 0 ), ★ 0 , …) 10 10

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . (A → Set) ≃ π 1 (A, a 0 )-Set ⟼ F (F(a 0 ), ★ 0 , …) p ★ x x transport x along p (p ★ x) ★ : a 1 = a 2 → F(a 1 ) → F(a 2 ) F(a 2 ) F(a 1 ) ★ 0 : ‖ a 1 = a 2 ‖ 0 → F(a 1 ) → F(a 2 ) a 2 p ( ★ for set-truncated paths) a 1 A 10 10

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . (A → Set) ≃ π 1 (A, a 0 )-Set ⟼ F (F(a 0 ), ★ 0 , …) ⟼ (X, α , —, —) ? (x,p) x Idea: formal transports X a p a 0 A 11 11

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . (A → Set) ≃ π 1 (A, a 0 )-Set ⟼ F (F(a 0 ), ★ 0 , …) ⟼ (X, α , —, —) R X, α R X, α (a) : ≡ X × ‖ a 0 = a ‖ 0 quotiented by some relation ~. 12 12

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . (A → Set) ≃ π 1 (A, a 0 )-Set ⟼ F (F(a 0 ), ★ 0 , …) ⟼ (X, α , —, —) R X, α R X, α (a) : ≡ X × ‖ a 0 = a ‖ 0 quotiented by some relation ~. Goal: F = R F(a0), ★ 0 F(a) ≃ F(a 0 ) × ‖ a 0 = a ‖ 0 quotiented by some relation ~. 12 12

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . Goal: F = R F(a0), ★ 0 F(a) ≃ F(a 0 ) × ‖ a 0 = a ‖ 0 quotiented by some relation ~. ⟼ (x, p) p ★ 0 x 13 13

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . Goal: F = R F(a0), ★ 0 F(a) ≃ F(a 0 ) × ‖ a 0 = a ‖ 0 quotiented by some relation ~. ⟼ (x, p) p ★ 0 x ⟼ (q -1 ★ 0 x, q)? x We only have ‖ a 0 = a ‖ -1 but need q : ‖ a 0 = a ‖ 0 . 13 13

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . Goal: F = R F(a0), ★ 0 F(a) ≃ F(a 0 ) × ‖ a 0 = a ‖ 0 quotiented by some relation ~. ⟼ (x, p) p ★ 0 x ⟼ (q -1 ★ 0 x, q)? x We only have ‖ a 0 = a ‖ -1 but need q : ‖ a 0 = a ‖ 0 . -1 -1 Lemma: If (q 1 ★ 0 x, q 1 ) = (q 2 ★ 0 x, q 2 ) then ‖ a 0 = a ‖ -1 is fine. 13 13

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . Goal: F = R F(a0), ★ 0 F(a) ≃ F(a 0 ) × ‖ a 0 = a ‖ 0 quotiented by some relation ~. -1 -1 Wants (q 1 ★ 0 x, q 1 ) = (q 2 ★ 0 x, q 2 ). 14 14

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . Goal: F = R F(a0), ★ 0 F(a) ≃ F(a 0 ) × ‖ a 0 = a ‖ 0 quotiented by some relation ~. -1 -1 Wants (q 1 ★ 0 x, q 1 ) = (q 2 ★ 0 x, q 2 ). ( α loop x , p) ~ (x , loop ▪ p) Intuition: p ★ 0 (loop ★ 0 x) = (loop ▪ p) ★ 0 x ★ 0 x, (q 1 ▪ q 2 -1 ) ▪ q 2 ) -1 -1 (q 1 ★ 0 x, q 1 ) = (q 1 = ((q 1 ▪ q 2 -1 ) ★ 0 (q 1 ★ 0 x), q 2 ) = (q 2 -1 ★ 0 x, q 2 ) 14 14

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . (A → Set) ≃ π 1 (A, a 0 )-Set ⟼ F (F(a 0 ), ★ 0 , …) ⟼ (X, α , —, —) R X, α R X, α (a) : ≡ X × ‖ a 0 = a ‖ 0 quotiented by ( α loop x , path) ~ (x , loop ▪ path) 15 15

Suppose a 0 : A and (x : A) → (y : A) → ‖ x = y ‖ -1 . (A → Set) ≃ π 1 (A, a 0 )-Set ⟼ F (F(a 0 ), ★ 0 , …) ⟼ (X, α , —, —) R X, α R X, α (a) : ≡ X × ‖ a 0 = a ‖ 0 quotiented by ( α loop x , path) ~ (x , loop ▪ path) The other round trip is easy. The other round trip is easy. (G-sets → covering spaces → G-sets) (G-sets → covering spaces → G-sets) 15 15

Summary - A simple formulation: A → Set. - Type equivalence of A → Set and π 1 (A)-Set. 16 16

Summary - A simple formulation: A → Set. - Type equivalence of A → Set and π 1 (A)-Set. Notes - Other theorems (universal coverings, categories). - Fibers need not to be decidable types. ☞ “path-constant” spaces, not just discrete ones? - A → Groupoid? 16 16

Summary - A simple formulation: A → Set. - Type equivalence of A → Set and π 1 (A)-Set. Notes - Other theorems (universal coverings, categories). - Fibers need not to be decidable types. ☞ “path-constant” spaces, not just discrete ones? - A → Groupoid? Thank you Acknowledgements: Carlo Angiuli, Steve Awodey, Andrej Bauer, Spencer Breiner, Guillaume Brunerie, Daniel Grayson, Chris Kapulkin, Nicolai Kraus, Peter LeFanu Lumsdaine and Ed Morehouse 16 16