rest of 2.3
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@ -3,8 +3,9 @@ module HottBook.Chapter2 where
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open import Relation.Binary.PropositionalEquality
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open import Relation.Binary.PropositionalEquality
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open import Function
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open import Function
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open import Data.Product
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open import Data.Product.Properties
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Type = Set
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Type = Set
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transport = subst
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_∙_ = trans
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_∙_ = trans
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```
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```
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@ -19,7 +20,32 @@ ap f {x} p = J (λ y p → f x ≡ f y) p refl
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## 2.3 Type families are fibrations
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## 2.3 Type families are fibrations
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### Lemma 2.3.4 (Dependent Map)
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### Lemma 2.3.1 (Transport)
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```
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transport : {A : Type} (P : A → Type) {x y : A} (p : x ≡ y) → P x → P y
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transport P p = J (λ y r → P y) p
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```
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### Lemma 2.3.2 (Path lifting property)
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```
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lift : {A : Type} (P : A → Type) {x y : A} (u : P x) → (p : x ≡ y)
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→ (x , u) ≡ (y , transport P p u)
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lift {A} P {x} {y} u p =
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J (λ the what → (x , u) ≡ (the , transport P what u)) p refl
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```
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Verifying its property:
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```
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lift-prop : {A : Type} (P : A → Type) {x y : A} (u : P x) → (p : x ≡ y)
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→ proj₁ (Σ-≡,≡←≡ (lift P u p)) ≡ p
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lift-prop P u p =
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J (λ the what → proj₁ (Σ-≡,≡←≡ (lift P u what)) ≡ what) p refl
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```
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### Lemma 2.3.4 (Dependent map)
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```
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```
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apd : {A : Type} {P : A → Type} (f : (x : A) → P x) {x y : A} (p : x ≡ y)
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apd : {A : Type} {P : A → Type} (f : (x : A) → P x) {x y : A} (p : x ≡ y)
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