2.11.2 - 2.11.4
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7 changed files with 151 additions and 40 deletions
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@ -6,7 +6,7 @@ module HottBook.Chapter2 where
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open import Agda.Primitive
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open import HottBook.Chapter1
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open import HottBook.Chapter2Lemma221
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open import HottBook.Chapter2Lemma221 public
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open import HottBook.Util
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```
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@ -738,45 +738,67 @@ open axiom2∙10∙3
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### Theorem 2.11.1
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```
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theorem2∙11∙1 : {A B : Set}
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→ (eqv @ (f , f-eqv) : A ≃ B)
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→ (a a' : A)
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→ (a ≡ a') ≃ (f a ≡ f a')
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theorem2∙11∙1 (f , f-eqv) a a' =
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let
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open ≡-Reasoning
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mkQinv g α β = isequiv-to-qinv f-eqv
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inv : (f a ≡ f a') → a ≡ a'
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inv p = (sym (β a)) ∙ (ap g p) ∙ (β a')
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backward : (p : f a ≡ f a') → (ap f ∘ inv) p ≡ id p
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backward q = begin
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ap f ((sym (β a)) ∙ (ap g q) ∙ (β a')) ≡⟨ lemma2∙2∙2.i (sym (β a)) (ap g q ∙ β a') ⟩
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ap f (sym (β a)) ∙ ap f ((ap g q) ∙ (β a')) ≡⟨ {! !} ⟩
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ap f (sym (β a)) ∙ ap f ((ap g q) ∙ (β a')) ≡⟨ {! !} ⟩
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id q ∎
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forward : (p : a ≡ a') → (inv ∘ ap f) p ≡ id p
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forward p = begin
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(sym (β a)) ∙ (ap g (ap f p)) ∙ (β a') ≡⟨ ap (λ p → (sym (β a)) ∙ p ∙ (β a')) (lemma2∙2∙2.iii f g p) ⟩
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(sym (β a)) ∙ (ap (g ∘ f) p) ∙ (β a') ≡⟨ {! !} ⟩
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-- (sym (β a)) ∙ (ap id p) ∙ (β a') ≡⟨ {! !} ⟩
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id p ∎
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eqv = mkQinv inv backward forward
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in
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ap f , qinv-to-isequiv eqv
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-- theorem2∙11∙1 : {A B : Set}
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-- → (eqv @ (f , f-eqv) : A ≃ B)
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-- → (a a' : A)
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-- → (a ≡ a') ≃ (f a ≡ f a')
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-- theorem2∙11∙1 (f , f-eqv) a a' =
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-- let
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-- open ≡-Reasoning
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-- mkQinv g α β = isequiv-to-qinv f-eqv
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-- inv : (f a ≡ f a') → a ≡ a'
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-- inv p = (sym (β a)) ∙ (ap g p) ∙ (β a')
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-- backward : (p : f a ≡ f a') → (ap f ∘ inv) p ≡ id p
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-- backward q = begin
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-- ap f ((sym (β a)) ∙ (ap g q) ∙ (β a')) ≡⟨ lemma2∙2∙2.i (sym (β a)) (ap g q ∙ β a') ⟩
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-- ap f (sym (β a)) ∙ ap f ((ap g q) ∙ (β a')) ≡⟨ {! !} ⟩
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-- ap f (sym (β a)) ∙ ap f ((ap g q) ∙ (β a')) ≡⟨ {! !} ⟩
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-- id q ∎
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-- forward : (p : a ≡ a') → (inv ∘ ap f) p ≡ id p
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-- forward p = begin
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-- (sym (β a)) ∙ (ap g (ap f p)) ∙ (β a') ≡⟨ ap (λ p → (sym (β a)) ∙ p ∙ (β a')) (lemma2∙2∙2.iii f g p) ⟩
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-- (sym (β a)) ∙ (ap (g ∘ f) p) ∙ (β a') ≡⟨ {! !} ⟩
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-- -- (sym (β a)) ∙ (ap id p) ∙ (β a') ≡⟨ {! !} ⟩
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-- id p ∎
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-- eqv = mkQinv inv backward forward
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-- in
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-- ap f , qinv-to-isequiv eqv
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```
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### Theorem 2.11.2
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```
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-- module theorem2∙11∙2 where
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-- i : {A : Set} {a x1 x2 : A}
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-- → (p : x1 ≡ x2)
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-- → (q : a ≡ x1)
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-- → transport (λ y → a ≡ y) p q ≡ q ∙ p
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-- i {A} {a} {x1} {x2} p q =
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-- J (λ x3 x4 p1 → (q1 : a ≡ x3) → transport (λ y → a ≡ y) p1 q1 ≡ q1 ∙ p1)
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-- (λ x3 q1 → J (λ x5 x6 q2 → transport (λ y → a ≡ y) refl q1 ≡ q1 ∙ refl) (λ x4 → {! refl !}) a x3 q1)
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-- x1 x2 p q
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module theorem2∙11∙2 where
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open ≡-Reasoning
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i : {A : Set} {a x1 x2 : A}
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→ (p : x1 ≡ x2)
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→ (q : a ≡ x1)
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→ transport (λ y → a ≡ y) p q ≡ q ∙ p
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i {A} {a} {x1} {x2} p q =
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J (λ x3 x4 p1 → (q1 : a ≡ x3) → transport (λ y → a ≡ y) p1 q1 ≡ q1 ∙ p1)
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(λ x3 q1 → J (λ x5 x6 q2 → transport (λ y → a ≡ y) refl q1 ≡ q1 ∙ refl) (λ x4 → lemma2∙1∙4.i1 q1) a x3 q1)
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x1 x2 p q
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ii : {A : Set} {a x1 x2 : A}
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→ (p : x1 ≡ x2)
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→ (q : x1 ≡ a)
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→ transport (λ y → y ≡ a) p q ≡ sym p ∙ q
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ii {A} {a} {x1} {x2} p q =
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J (λ x y p → (q1 : x ≡ a) → transport (λ y → y ≡ a) p q1 ≡ sym p ∙ q1)
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(λ x q1 → J (λ x y p → transport (λ y → y ≡ a) refl q1 ≡ sym refl ∙ q1) (λ x → lemma2∙1∙4.i2 q1) x a q1)
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x1 x2 p q
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iii : {A : Set} {a x1 x2 : A}
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→ (p : x1 ≡ x2)
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→ (q : x1 ≡ x1)
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→ transport (λ y → y ≡ y) p q ≡ sym p ∙ q ∙ p
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iii {A} {a} {x1} {x2} p q =
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J (λ x y p → (q1 : x ≡ x) → transport (λ y → y ≡ y) p q1 ≡ sym p ∙ q1 ∙ p)
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(λ x q1 → J (λ x y p → transport (λ y → y ≡ y) refl q1 ≡ sym refl ∙ q1 ∙ refl)
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(λ x → let helper = ap (λ p → sym refl ∙ p) (lemma2∙1∙4.i1 q1) in lemma2∙1∙4.i2 q1 ∙ helper)
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x x q1)
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x1 x2 p q
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```
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### Theorem 2.11.3
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@ -786,7 +808,41 @@ theorem2∙11∙3 : {A B : Set} → {f g : A → B} → {a a' : A}
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→ (p : a ≡ a')
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→ (q : f a ≡ g a)
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→ transport (λ x → f x ≡ g x) p q ≡ sym (ap f p) ∙ q ∙ (ap g p)
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theorem2∙11∙3 p q = {! !}
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theorem2∙11∙3 {A} {B} {f} {g} {a} {a'} p q =
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J (λ x y p → (q1 : f x ≡ g x) → transport (λ x → f x ≡ g x) p q1 ≡ sym (ap f p) ∙ q1 ∙ (ap g p))
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(λ x q1 →
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J (λ x y p → transport (λ z → f z ≡ g z) refl q1 ≡ sym (ap f refl) ∙ q1 ∙ ap g refl)
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(λ x → let helper = ap (λ p → sym refl ∙ p) (lemma2∙1∙4.i1 q1) in lemma2∙1∙4.i2 q1 ∙ helper)
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(f a) (g a) q)
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a a' p q
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```
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### Theorem 2.11.4
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```
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theorem2∙11∙4 : {A : Set} {B : A → Set} {f g : (x : A) → B x} {a a' : A}
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→ (p : a ≡ a')
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→ (q : f a ≡ g a)
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→ transport (λ x → f x ≡ g x) p q ≡ sym (apd f p) ∙ ap (transport B p) q ∙ apd g p
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theorem2∙11∙4 {A} {B} {f} {g} {a} {a'} p q =
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let open ≡-Reasoning in
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J (λ x y p → (q1 : f x ≡ g x) → transport (λ x → f x ≡ g x) p q1 ≡ sym (apd f p) ∙ ap (transport B p) q1 ∙ apd g p)
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(λ x q1 →
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J (λ x y p → transport (λ z → f z ≡ g z) refl q1 ≡ sym (apd f refl) ∙ ap (transport B refl) q1 ∙ apd g refl)
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(λ y →
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let
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q2 = ap id q1
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q1≡q2 : q1 ≡ q2
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q1≡q2 = J (λ a b p → p ≡ ap id p) (λ x → refl) (f x) (g x) q1
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in
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begin
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transport (λ z → f z ≡ g z) refl q1 ≡⟨⟩
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q1 ≡⟨ q1≡q2 ⟩
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q2 ≡⟨ (let helper = ap (λ p → refl ∙ p) (lemma2∙1∙4.i1 q2) in lemma2∙1∙4.i2 q2 ∙ helper) ⟩
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refl ∙ ap (transport B refl) q1 ∙ refl ∎
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)
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(f a) (g a) q)
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a a' p q
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```
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## 2.12 Coproducts
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BIN
src/HottBook/Chapter2.pdf
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BIN
src/HottBook/Chapter2.pdf
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Binary file not shown.
59
src/HottBook/Chapter2.typ
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59
src/HottBook/Chapter2.typ
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@ -0,0 +1,59 @@
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#import "@preview/cetz:0.2.2"
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#set page(width: auto, height: auto, margin: .5cm)
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#let data = (
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([Belgium], 24),
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([Germany], 31),
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([Greece], 18),
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([Spain], 21),
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([France], 23),
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([Hungary], 18),
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([Netherlands], 27),
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([Romania], 17),
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([Finland], 26),
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([Turkey], 13),
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)
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#cetz.canvas({
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import cetz.chart
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import cetz.draw: *
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// let colors = gradient.linear(red, blue, green, yellow)
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// chart.piechart(
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// data,
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// value-key: 1,
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// label-key: 0,
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// radius: 4,
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// slice-style: colors,
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// inner-radius: 1,
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// outset: 3,
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// inner-label: (content: (value, label) => [#text(white, str(value))], radius: 110%),
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// outer-label: (content: "%", radius: 110%))
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let point(p, name) = {
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content((p.at(0)-0.1, p.at(1)+0.1), name)
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circle(p, radius: 0.05, fill: black)
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}
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circle((2, 1), radius: (1, 1), fill: gray, stroke: none)
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circle((2, 3), radius: (1, 0.75), fill: gray, stroke: none)
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let a = (1.6, 3)
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let b = (2.4, 3)
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let fa = (1.6, 1.5)
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let fb = (2.4, 1.5)
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let ga = (1.3, 0.5)
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let gb = (2.7, 0.5)
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point(a, "a")
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point(b, "b")
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point(fa, "f(a)")
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point(fb, "f(b)")
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point(ga, "g(a)")
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point(gb, "g(b)")
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line(a, fa)
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})
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@ -4,7 +4,6 @@ module HottBook.Chapter2Exercises where
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open import Agda.Primitive
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open import HottBook.Chapter1
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open import HottBook.Chapter2
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open import HottBook.Chapter2Lemma221
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open import HottBook.Util
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```
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@ -5,7 +5,6 @@ open import Agda.Primitive
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open import HottBook.Chapter1
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open import HottBook.Chapter1Util
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open import HottBook.Chapter2
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open import HottBook.Chapter2Lemma221
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open import HottBook.Chapter3Definition331
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open import HottBook.Chapter3Lemma333
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open import HottBook.Util
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@ -3,7 +3,6 @@ module HottBook.Chapter6 where
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open import HottBook.Chapter1
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open import HottBook.Chapter2
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open import HottBook.Chapter2Lemma221
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```
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# 6 Higher inductive types
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@ -29,5 +29,4 @@ https://en.wikipedia.org/wiki/Homomorphism
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## 4 Homotopy invariance
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```
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_⋆
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```
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