Cubical Type Theory Dan Licata Wesleyan University Guillaume - PowerPoint PPT Presentation

Respect Equivalence p 0 : α a 0 = B b 0 α : A ≃ B p 0 : a 0 = A α -1 b 0 p 0 : a 0 = α b 0 p 1 : a 1 = α b 1 p 0 = α p 1 : a 0 = A a 1 ≃ b 0 = B b 1 α p α a 0 α a 1 p : a 0 = A a 1 � p 0 p 1 α p : α a 0 = B α a 1 b 0 b 1 ! p 0 ; α p ; p 1 B 21

Missing Sides 22

p : x = A y p’ : x’ = A’ y’ (p,p’) : (x , x’) = A × A’ (y , y’) 23

p : x = A y p’ : x’ = A’ y’ (p,p’) : (x , x’) = A × A’ (y , y’) (t,t’) (x , x’) (z , z’) A × A’ (l,l’) (r,r’) (y , y’) (w , w’) 23

p : x = A y p’ : x’ = A’ y’ (p,p’) : (x , x’) = A × A’ (y , y’) (t,t’) (x , x’) (z , z’) A × A’ (l,l’) (r,r’) (y , y’) (w , w’) t x z A l r y w 23

p : x = A y p’ : x’ = A’ y’ (p,p’) : (x , x’) = A × A’ (y , y’) (t,t’) (x , x’) (z , z’) A × A’ (l,l’) (r,r’) (y , y’) (w , w’) t x z A l r y w b 23

p : x = A y p’ : x’ = A’ y’ (p,p’) : (x , x’) = A × A’ (y , y’) (t,t’) (x , x’) (z , z’) A × A’ (l,l’) (r,r’) (y , y’) (w , w’) t t’ x z x’ z’ A A’ r’ l r l’ y w y’ w’ b 23

p : x = A y p’ : x’ = A’ y’ (p,p’) : (x , x’) = A × A’ (y , y’) (t,t’) (x , x’) (z , z’) A × A’ (l,l’) (r,r’) (y , y’) (w , w’) t t’ x z x’ z’ A A’ r’ l r l’ y w y’ w’ b b’ 23

p : x = A y p’ : x’ = A’ y’ (p,p’) : (x , x’) = A × A’ (y , y’) (t,t’) (x , x’) (z , z’) A × A’ (l,l’) (r,r’) (y , y’) (w , w’) t t’ (b,b’) x z x’ z’ A A’ r’ l r l’ y w y’ w’ b b’ 23

x:A ⊢ p : f x = A’ g x λ x.p : f = A $ A’ g 24

x:A ⊢ p : f x = A’ g x λ x.p : f = A $ A’ g t f h A $ A’ l r g k 24

x:A ⊢ p : f x = A’ g x λ x.p : f = A $ A’ g t t x f h f x h x A $ A’ A’ l r l x r x g k g x k x 24

x:A ⊢ p : f x = A’ g x λ x.p : f = A $ A’ g t t x f h f x h x A $ A’ A’ l r l x r x g k g x k x b[x] 24

x:A ⊢ p : f x = A’ g x λ x.p : f = A $ A’ g t t x f h f x h x A $ A’ A’ l r l x r x g k g x k x b[x] λ x.b 24

t p u l a0 = A a1 r q v p q u v : a0 = A a1 l : p = a0=a1 q t : p = a0=a1 u r : u = a0=a1 v 25

refl a0 a0 t p u p l q a1 a1 l a0 = A a1 r refl q v p q u v : a0 = A a1 l : p = a0=a1 q t : p = a0=a1 u r : u = a0=a1 v 25

refl a0 a0 p t u a1 a1 refl refl a0 a0 t p u p l q a1 a1 l a0 = A a1 r refl q v p q u v : a0 = A a1 l : p = a0=a1 q t : p = a0=a1 u r : u = a0=a1 v 25

refl a0 a0 p t u a1 a1 refl refl refl a0 a0 a0 a0 t p u u r p l q v a1 a1 a1 a1 l a0 = A a1 r refl refl q v p q u v : a0 = A a1 l : p = a0=a1 q t : p = a0=a1 u r : u = a0=a1 v 25

a0 t p u l a0 = A a1 r q v refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 t p u l a0 = A a1 r q v refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 t p u l p q l a0 = A a1 r q v refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 t p u l t p q l a0 = A a1 r q v refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 t r p u l t p u q l a0 = A a1 r v q v refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 a0 a0 t a0 a0 r p u l t p u q l a0 = A a1 r v q v refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 a0 a0 t a0 a0 r p u l t p u q l a0 = A a1 r v q v refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 a0 a0 t a0 a0 r p u l t p u q l a0 = A a1 r v a1 q v a1 a1 a1 refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 a0 a0 t a0 a0 r p u l t p u q l a0 = A a1 r v a1 q v a1 a1 a1 refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 a0 a0 t a0 a0 r p u l t b p u q l a0 = A a1 r v a1 q v a1 a1 a1 refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

a0 a0 a0 t a0 a0 r p u l t b p u q l a0 = A a1 r v a1 q v a1 b a1 a1 refl refl refl a0 a0 a0 a0 a0 a0 p t u u r v l q p a1 a1 a1 a1 a1 a1 refl refl refl 26

Kan condition: any n-dimensional open box has a lid, and an inside 27

Cubes 28

Line l l0 l1 29

Square x y s00 s10 s s01 s11 30

Square x y s-0 s00 s10 s s0- s1- s01 s11 s-1 30

Square with its boundary s x y 31

Square with its boundary s s<0/x> x y 31

Square with its boundary s s<0/x> s<1/x> x y 31

Square with its boundary s<0/y> s s<0/x> s<1/x> x y 31

Square with its boundary s<0/y> s s<0/x> s<1/x> s<1/y> x y 31

Square with its boundary s<0/y> s<0/x><0/y> s s<0/x> s<1/x> s<1/y> x y 31

Square with its boundary s<0/y> s<0/x><0/y> s s<0/x> s<1/x> s<0/x><1/y> s<1/y> x y 31

Square with its boundary s<0/y> s<0/x><0/y> s<1/x><0/y> s s<0/x> s<1/x> s<0/x><1/y> s<1/y> x y 31

Square with its boundary s<0/y> s<0/x><0/y> s<1/x><0/y> s s<0/x> s<1/x> s<0/x><1/y> s<1/x><1/y> s<1/y> x y 31

Square with its boundary s<0/y> s<0/x><0/y> s<1/x><0/y> s s<0/x> s<1/x> s<0/x><1/y> s<1/x><1/y> s<1/y> x y 31

Square with its boundary s<0/y><1/x> s<0/y> s<0/x><0/y> s<1/x><0/y> s s<0/x> s<1/x> s<0/x><1/y> s<1/x><1/y> s<1/y> x y 31

Square with its boundary s<0/y><1/x> ≡ s<0/y> s<0/x><0/y> s<1/x><0/y> s s<0/x> s<1/x> s<0/x><1/y> s<1/x><1/y> s<1/y> x y substitutions for independent variables commute 31