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6.4.1

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Michael Zhang 2024-07-16 16:41:37 -04:00
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26
src/HottBook/Chapter2.lagda.md
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@ -488,26 +488,30 @@ module lemma2∙4∙12 where
```
definition2∙6∙1 : {A B : Set l} {x y : A × B}
→ (p : x ≡ y)
→ (Σ.fst x ≡ Σ.fst y) × (Σ.snd x ≡ Σ.snd y)
definition2∙6∙1 p = ap Σ.fst p , ap Σ.snd p
→ (fst x ≡ fst y) × (snd x ≡ snd y)
definition2∙6∙1 p = ap fst p , ap snd p
```
### Theorem 2.6.2
```
module theorem2∙6∙2 where
pair-≡ : {A B : Set l} {x y : A × B} → (Σ.fst x ≡ Σ.fst y) × (Σ.snd x ≡ Σ.snd y) → x ≡ y
private
variable
A B : Set l
x y : A × B
pair-≡ : (fst x ≡ fst y) × (snd x ≡ snd y) → x ≡ y
pair-≡ (refl , refl) = refl
theorem2∙6∙2 : {A B : Set l} {x y : A × B}
→ isequiv (definition2∙6∙1 {A = A} {B = B} {x = x} {y = y})
theorem2∙6∙2 {A} {B} {x} {y} = qinv-to-isequiv (mkQinv pair-≡ backward forward)
where
backward : (definition2∙6∙1 ∘ pair-≡) ∼ id
backward (refl , refl) = refl
backward : ((definition2∙6∙1 {A = A} {B = B} {x = x} {y = y}) ∘ pair-≡) ∼ id
backward (refl , refl) = refl
forward : (pair-≡ ∘ definition2∙6∙1) ∼ id
forward refl = refl
forward : (pair-≡ ∘ (definition2∙6∙1 {A = A} {B = B} {x = x} {y = y})) ∼ id
forward refl = refl
theorem2∙6∙2 : isequiv (definition2∙6∙1 {A = A} {B = B} {x = x} {y = y})
theorem2∙6∙2 = qinv-to-isequiv (mkQinv pair-≡ backward forward)
pair-≃ : {A B : Set l} {x y : A × B} → (x ≡ y) ≃ ((fst x ≡ fst y) × (snd x ≡ snd y))
pair-≃ = definition2∙6∙1 , theorem2∙6∙2

46
src/HottBook/Chapter3.lagda.md
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@ -84,24 +84,17 @@ isSet A = (x y : A) → (p q : x ≡ y) → p ≡ q
where
open theorem2∙6∙2
open axiom2∙10∙3
open isequiv
p' : (xa ≡ ya) × (xb ≡ yb)
p' = definition2∙6∙1 p
p' = theorem2∙6∙2 .h-id p
q' = theorem2∙6∙2 .h-id q
q' : (xa ≡ ya) × (xb ≡ yb)
q' = definition2∙6∙1 q
pr1 : ap fst p ≡ ap fst q
pr1 = A-set xa ya (ap fst p) (ap fst q)
pr2 : ap snd p ≡ ap snd q
pr2 = B-set xb yb (ap snd p) (ap snd q)
lol : {A B : Set l} {x y : A × B} → (x ≡ y) ≡ ((fst x ≡ fst y) × (snd x ≡ snd y))
lol = ua pair-≃
convert : (p : (xa , xb) ≡ (ya , yb)) → (xa ≡ ya) × (xb ≡ yb)
convert p = definition2∙6∙1 p
goal : p ≡ q -- (p : (xa, xb)≡(ya, yb)) (q : (xa, xb)≡(ya, yb))
-- (p': xa≡ya × xb≡yb)
goal2 = p' ≡ q'
goal = {! ap (λ z → ?) goal2 !}
goal = sym p' ∙ ap pair-≡ (pair-≡ (pr1 , pr2)) ∙ q'
```
### Example 3.1.6
@ -432,20 +425,13 @@ module section3∙7 where
∥_∥ : Set l → Set l
∣_∣ : {A : Set l} → A → ∥ A ∥
trunc-witness : {A : Set l} → isProp (∥ A ∥)
rec-∥_∥ : (A : Set l) → {B : Set l}
→ isProp B
→ (f : A → B)
→ Σ (∥ A ∥ → B) (λ g → (a : A) → g (∣ a ∣) ≡ f a)
trunc-rec : (A : Set) → {B : Set} → isProp B → (f : A → B)
→ ∥ A ∥ → B
rec-trunc : (A : Set) → {B : Set} → isProp B → (f : A → B) → ∥ A ∥ → B
trunc-rec-1 : {A : Set} → {B : Set} → (Bprop : isProp B) → (f : A → B)
→ (a : A) → trunc-rec A Bprop f (∣ a ∣) ≡ f a
rec-trunc-1 : {A : Set} → {B : Set} → (Bprop : isProp B) → (f : A → B) → (a : A) → rec-trunc A Bprop f (∣ a ∣) ≡ f a
{-# REWRITE rec-trunc-1 #-}
open section3∙7 public
{-# REWRITE trunc-rec-1 #-}
```
### Definition 3.7.1
@ -503,7 +489,7 @@ lemma3∙9∙1 : {P : Set} → isProp P → P ≃ ∥ P ∥
lemma3∙9∙1 {P} Pprop = lemma3∙3∙3 Pprop witness ∣_∣ rec-func
where
rec-func : ∥ P ∥ → P
rec-func = trunc-rec P Pprop id
rec-func = rec-trunc P Pprop id
witness : isProp ∥ P ∥
witness = trunc-witness
@ -673,11 +659,11 @@ module lemma3∙11∙9 where
g p = a , p
forward : (p : P a) → transport P (sym (aContr a)) p ≡ p
forward p = y (sym (aContr a))
forward p = admit
where
y : (q : a ≡ a) → transport P q p ≡ p
y q = {! !}
postulate
-- TODO
admit : transport P (sym (aContr a)) p ≡ p
backward : (g ∘ f) ∼ id
backward (x , p) =

23
src/HottBook/Chapter4.lagda.md
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@ -95,16 +95,19 @@ lemma4∙1∙1 {A} {B} f e @ (mkQinv g _ _) = goal
### Lemma 4.1.2
```
open section3∙7
lemma4∙1∙2 : {A : Set} {a : A} (q : a ≡ a)
→ isSet (a ≡ a)
→ ((x : A) → ∥ a ≡ x ∥)
→ (g : (x : A) → ∥ a ≡ x ∥)
→ ((p : a ≡ a) → p ∙ q ≡ q ∙ p)
→ Σ ((x : A) → x ≡ x) (λ f → f a ≡ q)
lemma4∙1∙2 {A} {a} q prop1 g prop3 = (λ x → {! !}) , {! !}
where
allsets : (x y : A) → isSet (x ≡ y)
allsets x .x refl refl refl refl = refl
allsets x y p q r s = {! trunc-witness ? ? !}
ctx = let
f = rec-trunc A {! !} {! g !}
in {! !}
B : (x : A) → Set
B x = Σ (x ≡ x) (λ r → (s : a ≡ x) → r ≡ (sym s) ∙ q ∙ s)
@ -188,6 +191,20 @@ fib : ∀ {A B} → (f : A → B) → (y : B) → Set
fib {A = A} f y = Σ A (λ x → f x ≡ y)
```
### Lemma 4.2.5
```
lemma4∙2∙5 : {A B : Set}
→ (f : A → B)
→ (y : B)
→ ((x , p) (x' , p') : fib f y)
→ ((x , p) ≡ (x' , p')) ≃ Σ (x ≡ x') (λ γ → ap f γ ∙ p' ≡ p)
lemma4∙2∙5 f y (x , p) (x' , p') = {! !} , {! !}
where
ff : (x , p) ≡ (x' , p') → Σ (x ≡ x') (λ γ → ap f γ ∙ p' ≡ p)
ff q = ap fst q , {! !} -- (_ : f x ≡ f x') ∙ p' ≡ p
```
### Definition 4.2.7
```

124
src/HottBook/Chapter6.lagda.md
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@ -33,35 +33,28 @@ syntax dep-path P p u v = u ≡[ P , p ] v
```
module Interval where
data #I : Set where
#0 : #I
#1 : #I
I : Set
I = #I
0I 1I : I
0I = #0
1I = #1
postulate
I : Set
0I 1I : I
seg : 0I ≡ 1I
rec-Interval : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → I → B
rec-Interval b₀ b₁ s #0 = b₀
rec-Interval b₀ b₁ s #1 = b₁
rec-Interval : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → I → B
rec-Interval-0I : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → rec-Interval b₀ b₁ s 0I ≡ b₀
rec-Interval-1I : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → rec-Interval b₀ b₁ s 1I ≡ b₁
{-# REWRITE rec-Interval-0I #-}
{-# REWRITE rec-Interval-1I #-}
postulate
rec-Interval-3 : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → apd (rec-Interval b₀ b₁ s) seg ≡ s
{-# REWRITE rec-Interval-3 #-}
{-# REWRITE rec-Interval-3 #-}
rec-Interval-d : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → (x : I) → B x
rec-Interval-d b₀ b₁ s #0 = b₀
rec-Interval-d b₀ b₁ s #1 = b₁
rec-Interval-d : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → (x : I) → B x
rec-Interval-d-0I : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → rec-Interval-d {B = B} b₀ b₁ s 0I ≡ b₀
rec-Interval-d-1I : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → rec-Interval-d {B = B} b₀ b₁ s 1I ≡ b₁
{-# REWRITE rec-Interval-d-0I #-}
{-# REWRITE rec-Interval-d-1I #-}
postulate
rec-Interval-d-3 : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → apd (rec-Interval-d {B} b₀ b₁ s) seg ≡ s
{-# REWRITE rec-Interval-d-3 #-}
{-# REWRITE rec-Interval-d-3 #-}
open Interval public
```
@ -101,25 +94,18 @@ lemma6∙3∙2 {A = A} {B = B} {f = f} {g = g} p = apd q seg
## 6.4 Circles and sphere
```
private
data #S¹ : Set where
#base : #S¹
S¹ : Set
S¹ = #S¹
base : S¹
base = #base
postulate
S¹ : Set
base : S¹
loop : base ≡ base
rec-S¹ : {l : Level} {P : S¹ → Set l} → (b : P base) → (l : b ≡[ P , loop ] b) → ((x : S¹) → P x)
rec-S¹ b l #base = b
postulate
rec-S¹ : {l : Level} {P : S¹ → Set l} → (b : P base) → (l : b ≡[ P , loop ] b) → ((x : S¹) → P x)
rec-S¹-base : {l : Level} {P : S¹ → Set l} → (b : P base) → (l : b ≡[ P , loop ] b) → rec-S¹ {P = P} b l base ≡ b
{-# REWRITE rec-S¹-base #-}
rec-S¹-loop : {l : Level} {P : S¹ → Set l} → (b : P base) → (l : b ≡[ P , loop ] b) → apd {P = P} (rec-S¹ b l) loop ≡ l
{-# REWRITE rec-S¹-loop #-}
-- TODO: Uncommenting this leads to a bug in the definition of z2 in lemma 6.4.1
-- {-# REWRITE rec-S¹-loop #-}
```
### Lemma 6.4.1
@ -128,54 +114,38 @@ postulate
lemma6∙4∙1 : loop ≢ refl
lemma6∙4∙1 loop≡refl = goal3
where
-- goal : (s : S¹) (p : s ≡ s) → p ≡ refl
-- goal s p = z1 ∙ z2 ∙ z3
-- where
-- f : {A : Set} {x : A} {p : x ≡ x} → S¹ → A
-- f {x = x} {p = p} = rec-S¹ x p
-- z1 : p ≡ apd f loop
-- z1 = refl
-- z2 : apd f loop ≡ apd f refl
-- z2 = ap (apd f) loop≡refl
-- z3 : apd f refl ≡ refl
-- z3 = refl
f : ∀ {l} {A : Set l} {x : A} {p : x ≡ x} → S¹ → A
f {A = A} {x = x} {p = p} = rec-S¹ {P = λ _ → A} x p
goal2 : ∀ {l} {A : Set l} → isSet A
goal2 x .x refl refl = refl
goal2 {l = l} {A = A} x .x p refl = z1 ∙ z2 ∙ z3
where
f' : S¹ → A
f' = f {l = l} {A = A} {x = x} {p = p}
z1 : p ≡ apd f' loop
z1 = sym (rec-S¹-loop x p)
z2 : apd f' loop ≡ apd {x = base} f' refl
z2 = ap {A = base ≡ base} (apd f') loop≡refl
z3 : apd {x = base} f' refl ≡ refl
z3 = refl
goal3 : ⊥
goal3 = example3∙1∙9 (goal2 {A = Set})
```
-- where
-- f : {l : Level} {A : Set l} {x : A} {p : x ≡ x} → (S¹ → A)
-- f {A = A} {x = x} {p = p} s =
-- let p' = transportconst A loop x
-- in (rec-S¹ x (p' ∙ p)) s
-- -- f-loop : {A : Set} {x : A} {p : x ≡ x} → apd f loop ≡ p
-- -- f-loop {x = x} = S¹-rec-loop x ?
### Lemma 6.4.2
-- goal : ⊥
```
lemma6∙4∙2 : Σ ((x : S¹) → x ≡ x) (λ y → y ≢ (λ x → refl))
lemma6∙4∙2 = f , g
where
open axiom2∙9∙3
-- goal2 : ∀ {l} → (A : Set l) (a : A) (p : a ≡ a) → isSet A
-- goal = example3∙1∙9 (goal2 Set 𝟙 refl)
f = rec-S¹ loop {! !}
-- goal3 : ∀ (s : S¹) (p : s ≡ s) → p ≡ refl
-- goal2 {l} A a p x y r s = {! !}
-- where
-- f' = f {A = A} {x = a} {p = p}
-- test = apd f' loop ∙ sym (apd f' loop)
-- z1 : p ≡ apd f' loop
-- z1 = {! !}
-- z2 : apd f' loop ≡ apd f' refl
-- z2 = {! !}
-- z3 : apd f' refl ≡ refl
-- z3 = refl
-- wtf = let wtf = z1 ∙ z2 ∙ z3 in {! !}
g : f ≡ (λ x → refl) → ⊥
g p = {! !}
```