B A Y C P L E I C T S U PART I M A i 1 0 i: M : A M - - PowerPoint PPT Presentation

PART I

C U B I C A L E P Y T S

i:𝕁 ⊦ M : A

i 1 M

A

i:𝕁, j:𝕁 ⊦ M : A

i j

M

A

A

i1:𝕁, i2:𝕁, …, in:𝕁 ⊦ M : A

A

M

n-cube

M

i j

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

Γ ⊦ i : 𝕁

i:𝕁 ∈ Γ

0 : 𝕁 𝕁 is not a type! It is an alian animal r ∧ s : 𝕁 r : 𝕁 s : 𝕁 1 : 𝕁 r ∨ s : 𝕁 r : 𝕁 s : 𝕁 ~ r : 𝕁 r : 𝕁

• ptional

A IdA

i n t e r n a l i z e e x t r a p a t h s new paths

univalence funext loop

Pre-cubical Era

(induced)

A IdA

i n t e r n a l i z e n e w p a t h s

univalence loop

Cubical Era judgmental framework

• f paths

base : S1 S1 : U loopr : S1 r : 𝕁 loop0 ≡ base : S1 loop1 ≡ base : S1 loop: IdS1(base; base)

elimS1[x.C](Mbase; Mloop; N) : C[N/x] x: S1 ⊢ C : U Mbase : C[base/x] N: S1 Mloop : Mbase =loop

x.C Mbase

elimS1[x.C](Mbase; i.Mloop; N) : C[N/x] x: S1 ⊢ C : U Mbase : C[base/x] N: S1 i:𝕁 ⊢ Mloop : C[loopi/x] Mloop[0/i] ≡ Mbase : C[base/x] Mloop[1/i] ≡ Mbase : C[base/x]

elimS1[x.C](Mbase; i.Mloop; N) : C[N/x]

Mbase : C[base/x] i:𝕁 ⊢ Mloop : C[loopi/x] Mloop[0/i] ≡ Mbase : C[base/x] Mloop[1/i] ≡ Mbase : C[base/x] loopr : S1 loop1 ≡ base : S1 loop0 ≡ base : S1 base : S1

Mbase Mloop

x.C

elimS1[x.C](Mbase; j.Mloop; base) ≡ Mbase : C[base/x] (…) elimS1[x.C](Mbase; j.Mloop; loopi) ≡ Mloop[i/j] : C[loopi/x] (…)

• 1. diagonal substitution is crucial
• 2. we need to improve the framework

to enable path concatenation e.g., "loop ∙ loop"

IdA

e x t r a p a t h s

univalence funext loop

Pre-cubical

including concatenation, reversal, etc., via magical J

A

n e w p a t h s

univalence loop

Cubical

*stay tuned for the next week

Mi

Ai Path types

Pathi.A(N; O) : U i:𝕁 ⊦ A : U N : A[0/i] O : A[1/i]

"Πi:𝕁 A" with endpoints N O

λi.M : Pathi.A(N; O) i:𝕁 ⊦ M : A M[0/i] ≡ N : A[0/i] M[1/i] ≡ O : A[1/i] P@r : A[r/i] r : 𝕁 P : Pathi.A(N; O) P@0 ≡ N : A[0/i] P : Pathi.A(N; O) P@1 ≡ O : A[1/i]

Mi

Ai Path types

"Πi:𝕁 A" with endpoints N O

(λi.M)@r ≡ M[r/i] : A[r/i] i:𝕁 ⊦ M : A r : 𝕁 P ≡ λi.P@i : Pathi.A(N; O) P : Pathi.A(N; O)

Path types

internalizing i:𝕁 ⊦ M : A

Identification types

freely generated by refl

these two will become equivalent when we fix the framework (next week)

A

n e w p a t h s

univalence loop

*stay tuned for the next week

Where is function extensionality?

YES, WE CAN!

(NOT VIA UNIVALENCE)

P : Πx:APath_.B(x)(F(x); G(x)) x:A ⊦ P(x) : Path_.B(x)(F(x); G(x)) x:A, i:𝕁 ⊦ P(x)@i : B(x) i:𝕁, x:A ⊦ P(x)@i : B(x) i:𝕁 ⊦ λx.P(x)@i : B(x) λi.λx.P(x)@i : Path_.Π(x:A)B(x)(F; G)

(you need to check the boundaries F and G)

Function extensionality

i:𝕁 ⊦ λx.P(x)@i : B(x) λi.λx.P(x)@i : Path_.Π(x:A)B(x)(F; G) P : Πx:APath_.B(x)(F(x); G(x)) x:A ⊦ P(x) : Path_.B(x)(F(x); G(x)) x:A, i:𝕁 ⊦ P(x)@i : B(x) i:𝕁, x:A ⊦ P(x)@i : B(x)

• 1. Both paths and functions internalize

hypothetical judgments

• 2. You can exchange hypotheses
• 3. Paths and functions thus "commute"
• 4. Therefore, function extensionality!

Fix the framework (next week)

• 1. What are the types? (form)
• 2. What are the constructors? (intro)
• 3. How to consume an element? (elim)
• 4. What if a constructor is consumed? (β)
• 5. Uniqueness principle? (η)
• 6. How to compose stu? (Kan operators)

every type is responsible for its path concatenation

IdA

e x t r a p a t h s

univalence funext loop

Pre-cubical

including concatenation, reversal, etc., via magical J

A

n e w p a t h s

univalence loop

Cubical

*stay tuned for the next week