Cubical Type Theory favonia 1 2 t f 2 t f 2 t - PowerPoint PPT Presentation

Cubical Type Theory favonia 1

2

t 𝔺 f 2

⟙ t 𝔺 f 2

⟙ ⟘ t 𝔺 f 2

⟙ ⟘ t 𝔺 f 4 0 7 3 8 1 5 2 … 6 ℕ 2

⟙ ⟘ t 𝔺 f 4 0 7 3 8 1 5 2 … 6 … ℕ → ℕ ℕ 2

3

3

3

3

3

3

1 6 7 4 5 9 8 2 3 0 quotient 4

1 6 7 4 5 9 8 2 3 0 quotient 4

1 6 7 4 5 9 8 2 3 0 quotient 4

universe 5

id 𝔺 not universe 5

id id 𝔺 ⟙ + ⟙ not swap universe 5

⟘ id id 𝔺 ⟙ + ⟙ not swap ℕ → ℕ … ℕ universe 5

p o o l base circle (homotopy theory) 6

I. Cubes 7

M A 8

M A 1 0 8

M A i 1 0 i: 𝕁 ⊦ M : A 8

M A i j i: 𝕁 , j: 𝕁 ⊦ M : A 9

M A n-cube i 1 : 𝕁 , i 2 : 𝕁 , ..., i n : 𝕁 ⊦ M : A 10

M i j 11

M[1/j] M[1/i] M[0/i] M[0/j] M i j 11

M[1/j] M[1/i] M[i/j] M[0/i] M[0/j] M i j 11

M N N 0 1 A path types functions from 𝕁 12

M N N 0 1 A i: 𝕁 ⊦ M : A M[0/i] ≡ N 0 : A[0/i] M[1/i] ≡ N 1 : A[1/i] λ i.M : Path i.A (N 0 ,N 1 ) 13

M N N 0 1 A i: 𝕁 ⊦ M : A M[0/i] ≡ N 0 : A[0/i] M[1/i] ≡ N 1 : A[1/i] λ i.M : Path i.A (N 0 ,N 1 ) r: 𝕁 P : Path i.A (N 0 ,N 1 ) P@r : A[r/i] P@0 ≡ N 0 : A[0/i] P@1 ≡ N 1 : A[1/i] 13

M N N 0 1 A i: 𝕁 ⊦ M : A M[0/i] ≡ N 0 : A[0/i] M[1/i] ≡ N 1 : A[1/i] λ i.M : Path i.A (N 0 ,N 1 ) r: 𝕁 P : Path i.A (N 0 ,N 1 ) ( λ i.M)@r ≡ M[r/i] : A[r/i] P@r : A[r/i] P ≡ λ i.P@i : Path i.A (N 0 ,N 1 ) P@0 ≡ N 0 : A[0/i] P@1 ≡ N 1 : A[1/i] 13

function extensionality almost trivial h : Π (x:A). Path(F(x), G(x)) λ i. λ x.h(x)@i : Path(F, G) 14

II. the Book 15

Id A (M,N) N M A re fl M : Id A (M,M) the only generator 16

zero base case suc induction step mathematical induction re fl re fl case “J”: induction principle for Id 17

univalence as axiom Id U (A, B) B A A ≃ B (A ≃ B) ≃ Id U (A, B) 18

univalence as axiom Id U (A, B) B A A ≃ B (A ≃ B) ≃ Id U (A, B) (A ≃ B) → Id U (A, B) 18

p o o l base circle base : circle loop : Id circle (base, base) 19

y l loop n o h e t r w i J … fl univalence re fl case normalization in danger* J(loop) ≡ ??? 20 *most experts believe the normalization fails

Leave Id alone! Id/Path should re fl ect existing paths, not inducing new ones 21

p o o l i loop i base circle base : circle i: 𝕁 ⊦ loop i : circle loop 0 ≡ base : circle loop 1 ≡ base : circle 22

univalence as a type V i (A, B, E) B A E : A ≃ B (when i = 0)* *see [AFH] (not recommended as the fi rst paper to read) 23

Judgmental framework paths of (then internalized by Path/Id) 24

III. Compositions 25

concatenation? 26

concatenation? 26

concatenation? 26

Kan fi lling for cubes 27

Kan fi lling for cubes 27

constant concatenation Kan fi lling for cubes 27

fi llers can be done by higher-dimensional composition* Kan composition 28 *technical limitations apply; see papers for real (!) math

fi llers can be done by higher-dimensional composition* k j i 29

fi llers can be done by higher-dimensional composition* j=1 ⊦ M j=1 : A i=1 ⊦ M i=1 : A i=0 ⊦ M i=0 : A k j=0 ⊦ M j=0 : A j i 29

fi llers can be done by higher-dimensional composition* j=1 ⊦ M j=1 : A i=1 ⊦ M i=1 : A i=0 ⊦ M i=0 : A k j=0 ⊦ M j=0 : A j i k=0 ⊦ M k=0 : A 29

i=0 ↪ M i=0 i=1 ↪ M i=1 comp k.A M k=0 : A[1/k] j=0 ↪ M j=0 j=1 ↪ M j=1 M j=1 M i=1 M i=0 k M j=0 j i M k=0 30

U a composite in a universe is a type itself which has its own composition operator 31

A : M 𝕁 : i ⊦ x a t n y s models in other higher toposes? a model equivalent to “Standard” the homotopy theory? 32

Agda --cubical https://github.com/agda/cubical redtt https://github.com/RedPRL/redtt 33

One Path to Enlightenment (in this order) Homotopy Type Theory: Univalent Foundations of Mathematics Syntax and Models of Cartesian Cubical Type Theory [ABCFHL] https://github.com/dlicata335/cart-cube/blob/master/cart-cube.pdf Axioms for Modelling Cubical Type Theory in a Topos [OP] (expanded version) https://arxiv.org/abs/1712.04864 Computational Semantics of Cartesian Cubical Type Theory [A] (chapter 3, still changing everyday) https://www.cs.cmu.edu/~cangiuli/thesis/ 34