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

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

Oct 08, 2022 418 likes β€’1.05k views

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 2 t f 4 0 7 3 8 1 5 2 6 2 3 3 3 3

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favonia

Cubical

Type Theory

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𝔺 t f

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𝔺 t f βŸ™

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𝔺 t f βŸ™ ⟘

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𝔺 t f βŸ™ ⟘ β„• 1 2 3 4 7 5 6 8 …

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𝔺 t f βŸ™ ⟘ β„• 1 2 3 4 7 5 6 8 … β„• β†’ β„• …

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2 4 6 8 1 3 5 7 9 quotient

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2 4 6 8 1 3 5 7 9 quotient

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2 4 6 8 1 3 5 7 9 quotient

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universe

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𝔺 id not universe

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𝔺 id not βŸ™+βŸ™ id swap universe

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𝔺 id not βŸ™+βŸ™ id swap β„•β†’β„• … β„• ⟘ universe

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base l

  • p

circle (homotopy theory)

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  • I. Cubes

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M A

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1 M A

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i:𝕁 ⊦ M : A

i 1 M A

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i:𝕁, j:𝕁 ⊦ M : A

i j M A

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i1:𝕁, i2:𝕁, ..., in:𝕁 ⊦ M : A

A M

n-cube

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M

i j

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M

i j

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

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M

i j

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

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N0 N

1

A M

path types

functions from 𝕁

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i:𝕁 ⊦ M : A M[0/i] ≑ N0 : A[0/i] M[1/i] ≑ N1 : A[1/i] Ξ»i.M : Pathi.A(N0,N1)

N0 N

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A M

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i:𝕁 ⊦ M : A M[0/i] ≑ N0 : A[0/i] M[1/i] ≑ N1 : A[1/i] Ξ»i.M : Pathi.A(N0,N1) P : Pathi.A(N0,N1) r:𝕁 P@r : A[r/i] P@0 ≑ N0 : A[0/i]

N0 N

1

A M

P@1 ≑ N1 : A[1/i]

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i:𝕁 ⊦ M : A M[0/i] ≑ N0 : A[0/i] M[1/i] ≑ N1 : A[1/i] Ξ»i.M : Pathi.A(N0,N1) P : Pathi.A(N0,N1) r:𝕁 P@r : A[r/i] P ≑ Ξ»i.P@i : Pathi.A(N0,N1) P@0 ≑ N0 : A[0/i]

N0 N

1

A M

P@1 ≑ N1 : A[1/i] (Ξ»i.M)@r ≑ M[r/i] : A[r/i]

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h : Ξ (x:A). Path(F(x), G(x)) Ξ»i.Ξ»x.h(x)@i : Path(F, G)

function extensionality

almost trivial

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  • II. the Book

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M A N IdA(M,N) reflM : IdA(M,M)

the only generator

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suc induction step zero base case refl refl case mathematical induction β€œJ”: induction principle for Id

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(A ≃ B) ≃ IdU(A, B)

univalence

A B IdU(A, B) A ≃ B as axiom

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(A ≃ B) ≃ IdU(A, B) (A ≃ B) β†’ IdU(A, B)

univalence

A B IdU(A, B) A ≃ B as axiom

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base circle base : circle loop : Idcircle(base, base) l

  • p

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r e fl univalence loop … J w i t h

  • n

l y

normalization in danger* J(loop) ≑ ???

*most experts believe the normalization fails

refl case

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Leave Id alone!

Id/Path should reflect existing paths, not inducing new ones

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base circle base : circle i:𝕁 ⊦ loopi : circle

loopi

loop0 ≑ base : circle loop1 ≑ base : circle

i

l

  • p

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univalence

A B Vi(A, B, E) E : A ≃ B (when i = 0)* as a type

*see [AFH] (not recommended as the first paper to read)

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Judgmental

framework paths

  • f

(then internalized by Path/Id)

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  • III. Compositions

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concatenation?

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concatenation?

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concatenation?

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Kan filling for cubes

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Kan filling for cubes

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Kan filling for cubes

constant

concatenation

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Kan composition fillers can be done by higher-dimensional composition*

*technical limitations apply; see papers for real (!) math

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i j k fillers can be done by higher-dimensional composition*

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i j k

i=0 ⊦ Mi=0 : A j=0 ⊦ Mj=0 : A j=1 ⊦ Mj=1 : A i=1 ⊦ Mi=1 : A

fillers can be done by higher-dimensional composition*

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i j k

i=0 ⊦ Mi=0 : A j=0 ⊦ Mj=0 : A j=1 ⊦ Mj=1 : A i=1 ⊦ Mi=1 : A k=0 ⊦ Mk=0 : A

fillers can be done by higher-dimensional composition*

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i j k

compk.A Mk=0

Mi=0 Mj=0 Mj=1 Mi=1 Mk=0

i=0 β†ͺ Mi=0 i=1 β†ͺ Mi=1 j=0 β†ͺ Mj=0 j=1 β†ͺ Mj=1 : A[1/k]

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

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i : 𝕁 ⊦ M : A

s y n t a x

a model equivalent to

β€œStandard”

homotopy theory? models in other higher toposes? the

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Agda --cubical redtt

https://github.com/RedPRL/redtt https://github.com/agda/cubical

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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/

One Path to Enlightenment

(in this order)

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