more progress ch4
This commit is contained in:
parent
ab09089eb9
commit
c966fa9b4f
1 changed files with 50 additions and 7 deletions
|
@ -6,6 +6,7 @@ open import HottBook.Chapter1Util
|
|||
open import HottBook.Chapter2
|
||||
open import HottBook.Chapter2Exercises
|
||||
open import HottBook.Chapter3
|
||||
open import HottBook.CoreUtil
|
||||
|
||||
private
|
||||
variable
|
||||
|
@ -119,16 +120,16 @@ There exist types A and B and a function f : A → B such that qinv( f ) is not
|
|||
```
|
||||
theorem4∙1∙3 : ∀ {l} → Σ (Set l) (λ A → Σ (Set l) (λ B → Σ (A → B) (λ f → isProp (qinv f) → ⊥)))
|
||||
-- theorem4∙1∙3 {l} = goal
|
||||
-- where
|
||||
-- goal : ∀ {l} → Σ (Set l) (λ A → Σ (Set l) (λ B → Σ (A → B) (λ f → isProp (qinv f) → ⊥)))
|
||||
-- where
|
||||
-- goal : ∀ {l} → Σ (Set l) (λ A → Σ (Set l) (λ B → Σ (A → B) (λ f → isProp (qinv f) → ⊥)))
|
||||
|
||||
-- goal2 : ∀ {l} → Σ (Set l) (λ X → isProp ((x : X) → x ≡ x) → ⊥)
|
||||
-- goal {l} = Σ.fst (goal2 {l}) , {! !} , {! !} , {! !}
|
||||
-- goal2 : ∀ {l} → Σ (Set l) (λ X → isProp ((x : X) → x ≡ x) → ⊥)
|
||||
-- goal {l} = Σ.fst (goal2 {l}) , {! !} , {! !} , {! !}
|
||||
|
||||
-- X : ∀ {l} → Set (lsuc l)
|
||||
-- X {l} = Σ (Set l) (λ A → ∥ Lift 𝟚 ≡ A ∥)
|
||||
-- X : ∀ {l} → Set (lsuc l)
|
||||
-- X {l} = Σ (Set l) (λ A → ∥ Lift 𝟚 ≡ A ∥)
|
||||
|
||||
-- goal2 {l} = X {l} , ?
|
||||
-- goal2 {l} = X {lsuc l} , ?
|
||||
```
|
||||
|
||||
## 4.2 Half adjoint equivalences
|
||||
|
@ -148,6 +149,17 @@ record ishae {A B : Set} (f : A → B) : Set where
|
|||
### Lemma 4.2.2
|
||||
|
||||
```
|
||||
lemma4∙2∙2 : ∀ {A B}
|
||||
→ (f : A → B) → (g : B → A)
|
||||
→ (η : (g ∘ f) ∼ id)
|
||||
→ (ε : (f ∘ g) ∼ id)
|
||||
→ ((x : A) → ap f (η x) ≡ ε (f x)) ≃ ((y : B) → ap g (ε y) ≡ η (g y))
|
||||
lemma4∙2∙2 f g η ε = {! !}
|
||||
```
|
||||
|
||||
```
|
||||
ishae→qinv : {A B : Set} (f : A → B) → ishae f → qinv f
|
||||
ishae→qinv f (mkIshae g η ε _) = mkQinv g ε η
|
||||
```
|
||||
|
||||
### Theorem 4.2.3
|
||||
|
@ -169,6 +181,13 @@ theorem4∙2∙3 {A} {B} f (mkQinv g ε η) = mkIshae g' η' ε' τ
|
|||
τ x = {! !}
|
||||
```
|
||||
|
||||
### Definition 4.2.4
|
||||
|
||||
```
|
||||
fib : ∀ {A B} → (f : A → B) → (y : B) → Set
|
||||
fib {A = A} f y = Σ A (λ x → f x ≡ y)
|
||||
```
|
||||
|
||||
### Definition 4.2.7
|
||||
|
||||
```
|
||||
|
@ -178,6 +197,14 @@ module definition4∙2∙7 where
|
|||
|
||||
rinv : ∀ {A B} (f : A → B) → Set
|
||||
rinv {A} {B} f = Σ (B → A) (λ g → (f ∘ g) ∼ id)
|
||||
open definition4∙2∙7
|
||||
```
|
||||
|
||||
### Lemma 4.2.9
|
||||
|
||||
```
|
||||
lemma4∙2∙9 : ∀ {A B} (f : A → B) → qinv f → isContr (linv f) × isContr (rinv f)
|
||||
lemma4∙2∙9 f q = ({! !} , {! !}) , ({! !} , {! !})
|
||||
```
|
||||
|
||||
### Definition 4.2.10
|
||||
|
@ -194,4 +221,20 @@ module definition4∙2∙10 where
|
|||
|
||||
```
|
||||
theorem4∙2∙13 : {A B : Set} (f : A → B) → isProp (ishae f)
|
||||
```
|
||||
|
||||
## 4.3 Bi-invertible maps
|
||||
|
||||
### Definition 4.3.1
|
||||
|
||||
```
|
||||
biinv : {A B : Set} (f : A → B) → Set
|
||||
biinv f = linv f × rinv f
|
||||
```
|
||||
|
||||
### Theorem 4.3.2
|
||||
|
||||
```
|
||||
theorem4∙3∙2 : ∀ {A B} → (f : A → B) → isProp (biinv f)
|
||||
theorem4∙3∙2 f = {! !}
|
||||
```
|
Loading…
Reference in a new issue