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- [Chapter 4](./generated/HottBook.Chapter4.md)
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- [Chapter 4](./generated/HottBook.Chapter4.md)
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- [Chapter 5](./generated/HottBook.Chapter5.md)
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- [Chapter 5](./generated/HottBook.Chapter5.md)
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- [Chapter 6](./generated/HottBook.Chapter6.md)
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- [Chapter 6](./generated/HottBook.Chapter6.md)
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- [Chapter 7](./generated/HottBook.Chapter7.md)
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- [Chapter 8](./generated/HottBook.Chapter8.md)
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# HoTT Book (Cubical formulation)
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# HoTT Book (Cubical formulation)
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@ -96,19 +96,22 @@ pointed l = Σ (Set l) (λ A → A)
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### Definition 2.1.8 (loop space)
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### Definition 2.1.8 (loop space)
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```
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```
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Ω : {l : Level} → pointed l → pointed l
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Ω : {l : Level} → pointed l → Set l
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Ω (A , a) = (a ≡ a) , refl
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Ω (A , a) = a ≡ a
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```
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```
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### Theorem 2.1.6 (Eckmann-Hilton)
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### Theorem 2.1.6 (Eckmann-Hilton)
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```
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```
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module theorem2∙1∙6 where
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module theorem2∙1∙6 where
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Ω² : {l : Level} → pointed l → pointed l
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Ω² : {l : Level} → pointed l → Set l
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Ω² p = Ω (Ω p)
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Ω² p = Ω (Ω p , refl)
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-- compose : {l : Level} {p : pointed {l}} → (Ω² p) × (Ω² p) → Ω² p
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compose : {l : Level} {p : pointed l} → (Ω² p) × (Ω² p) → Ω² p
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-- compose a b = ?
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compose (a , b) = a ∙ b
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-- commute : {l : Level} {p : pointed l} → (α β : Ω² p) → α ∙ β ≡ β ∙ α
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-- commute {l} {p @ (A , a₀)} α β = {! !}
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```
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```
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## 2.2 Functions are functors
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## 2.2 Functions are functors
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@ -312,6 +315,7 @@ lemma2∙4∙3 {A} {B} {f} {g} H {x} {y} refl =
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-- → H (f x) ≡ ap f (H x)
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-- → H (f x) ≡ ap f (H x)
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-- corollary2∙4∙4 {A} {f} {H} x =
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-- corollary2∙4∙4 {A} {f} {H} x =
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-- let g p = H x in
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-- let g p = H x in
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-- -- let naturality = lemma2∙4∙3 H x in
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-- {! !}
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-- {! !}
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```
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```
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@ -1044,3 +1048,30 @@ theorem2∙15∙5 {X = X} {A = A} {B = B} = qinv-to-isequiv (mkQinv g forward ba
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backward f = funext λ x → refl
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backward f = funext λ x → refl
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where open axiom2∙9∙3
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where open axiom2∙9∙3
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```
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```
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### Equation 2.15.6
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```
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equation2∙15∙6 : {X : Set} {A : X → Set} {P : (x : X) → A x → Set}
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→ ((x : X) → Σ (A x) (P x))
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→ Σ ((x : X) → A x) (λ g → (x : X) → P x (g x))
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equation2∙15∙6 f = (λ x → fst (f x)) , λ x → snd (f x)
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```
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### Theorem 2.15.7
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```
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theorem2∙15∙7 : {X : Set} {A : X → Set} {P : (x : X) → A x → Set} → isequiv (equation2∙15∙6 {X = X} {A = A} {P = P})
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theorem2∙15∙7 {X} {A} {P} = qinv-to-isequiv (mkQinv g forward backward)
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where
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open axiom2∙9∙3
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g : Σ ((x : X) → A x) (λ g → (x : X) → P x (g x)) → (x : X) → Σ (A x) (P x)
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g (f1 , f2) x = f1 x , f2 x
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forward : (equation2∙15∙6 ∘ g) ∼ id
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forward (f1 , f2) = Σ-≡ (refl , refl)
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backward : (g ∘ equation2∙15∙6) ∼ id
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backward f = funext λ x → refl
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```
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@ -357,7 +357,6 @@ lemma6∙5∙1 = f , qinv-to-isequiv (mkQinv g forward backward)
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goal2 : (y : 𝟚) → {! !} ∙ refl ∙ {! !} ≡ merid true
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goal2 : (y : 𝟚) → {! !} ∙ refl ∙ {! !} ≡ merid true
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goal y = {! !}
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goal y = {! !}
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```
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```
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### Definition 6.5.2
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### Definition 6.5.2
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