Cubical.HITs.SmashProduct.Base
module Cubical.HITs.SmashProduct.Base where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Pointed.Homogeneous
open import Cubical.Foundations.Path
open import Cubical.Foundations.Function
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Equiv
open import Cubical.Data.Unit
open import Cubical.Data.Sigma
open import Cubical.Data.Nat
open import Cubical.Data.Fin
open import Cubical.HITs.Pushout.Base
open import Cubical.HITs.Wedge
open import Cubical.HITs.Susp renaming (toSusp to σ)
open import Cubical.HITs.Join
open import Cubical.Homotopy.Loopspace
data Smash {ℓ ℓ'} (A : Pointed ℓ) (B : Pointed ℓ') : Type (ℓ-max ℓ ℓ') where
basel : Smash A B
baser : Smash A B
proj : (x : typ A) → (y : typ B) → Smash A B
gluel : (a : typ A) → proj a (pt B) ≡ basel
gluer : (b : typ B) → proj (pt A) b ≡ baser
private
variable
ℓ ℓ' : Level
A B C D : Pointed ℓ
infixl 30 _⋀∙_
SmashPt : (A : Pointed ℓ) (B : Pointed ℓ') → Pointed (ℓ-max ℓ ℓ')
SmashPt A B = (Smash A B , basel)
SmashPtProj : (A : Pointed ℓ) (B : Pointed ℓ') → Pointed (ℓ-max ℓ ℓ')
SmashPtProj A B = Smash A B , (proj (snd A) (snd B))
Smash-map : (f : A →∙ C) (g : B →∙ D) → Smash A B → Smash C D
Smash-map f g basel = basel
Smash-map f g baser = baser
Smash-map (f , fpt) (g , gpt) (proj x y) = proj (f x) (g y)
Smash-map (f , fpt) (g , gpt) (gluel a i) = ((λ j → proj (f a) (gpt j)) ∙ gluel (f a)) i
Smash-map (f , fpt) (g , gpt) (gluer b i) = ((λ j → proj (fpt j) (g b)) ∙ gluer (g b)) i
comm : Smash A B → Smash B A
comm basel = baser
comm baser = basel
comm (proj x y) = proj y x
comm (gluel a i) = gluer a i
comm (gluer b i) = gluel b i
commK : (x : Smash A B) → comm (comm x) ≡ x
commK basel = refl
commK baser = refl
commK (proj x y) = refl
commK (gluel a x) = refl
commK (gluer b x) = refl
_⋀_ : Pointed ℓ → Pointed ℓ' → Type (ℓ-max ℓ ℓ')
A ⋀ B = Pushout {A = (A ⋁ B)} (λ _ → tt) ⋁↪
_⋀∙_ : Pointed ℓ → Pointed ℓ' → Pointed (ℓ-max ℓ ℓ')
A ⋀∙ B = (A ⋀ B) , (inl tt)
⋀^ : (n : ℕ) (A : Fin (suc n) → Pointed ℓ) → Pointed ℓ
⋀^ zero A = A fzero
⋀^ (suc n) A = ⋀^ n (predFinFamily∙ A) ⋀∙ (A flast)
⋀comm→ : A ⋀ B → B ⋀ A
⋀comm→ (inl x) = inl x
⋀comm→ (inr (x , y)) = inr (y , x)
⋀comm→ (push (inl x) i) = push (inr x) i
⋀comm→ (push (inr x) i) = push (inl x) i
⋀comm→ (push (push a i₁) i) = push (push tt (~ i₁)) i
⋀comm→² : {A : Pointed ℓ} {B : Pointed ℓ' }
(x : A ⋀ B) → ⋀comm→ (⋀comm→ {A = A} {B = B} x) ≡ x
⋀comm→² (inl x) = refl
⋀comm→² (inr x) = refl
⋀comm→² (push (inl x) i) = refl
⋀comm→² (push (inr x) i) = refl
⋀comm→² (push (push a i₁) i) = refl
⋀CommIso : Iso (A ⋀ B) (B ⋀ A)
Iso.fun ⋀CommIso = ⋀comm→
Iso.inv ⋀CommIso = ⋀comm→
Iso.rightInv ⋀CommIso = ⋀comm→²
Iso.leftInv ⋀CommIso = ⋀comm→²
⋀comm→∙ : A ⋀∙ B →∙ B ⋀∙ A
fst ⋀comm→∙ = ⋀comm→
snd ⋀comm→∙ = refl
SmashAdjIso : Iso ((A ⋀∙ B) →∙ C) (A →∙ (B →∙ C ∙))
SmashAdjIso {A = A} {B = B} {C = C} =
compIso is₃ (compIso iso₄ (invIso is₂))
where
is₁ : Iso (A →∙ (B →∙ C ∙))
(Σ[ f ∈ (fst A → fst B → fst C) ]
Σ[ l ∈ ((x : fst A) → f x (pt B) ≡ pt C) ]
Σ[ r ∈ ((b : fst B) → f (pt A) b ≡ pt C) ]
PathP (λ i → r (snd B) i ≡ snd C) (l (snd A)) refl)
Iso.fun is₁ f = (λ x y → f .fst x .fst y)
, (λ x → f .fst x .snd)
, (λ x i → f .snd i .fst x)
, λ i j → f .snd i .snd j
fst (fst (Iso.inv is₁ (f , l , r , p)) x) = f x
snd (fst (Iso.inv is₁ (f , l , r , p)) x) = l x
fst (snd (Iso.inv is₁ (f , l , r , p)) i) b = r b i
snd (snd (Iso.inv is₁ (f , l , r , p)) i) j = p i j
Iso.rightInv is₁ _ = refl
Iso.leftInv is₁ _ = refl
is₂ : Iso (A →∙ (B →∙ C ∙)) (
(Σ[ f ∈ (fst A → fst B → fst C) ]
Σ[ l ∈ ((x : fst A) → f x (pt B) ≡ pt C) ]
Σ[ r ∈ ((b : fst B) → f (pt A) b ≡ pt C) ]
l (pt A) ≡ r (pt B)))
is₂ = compIso is₁ (Σ-cong-iso-snd
λ f → Σ-cong-iso-snd
λ l → Σ-cong-iso-snd
λ r → pathToIso (PathP≡doubleCompPathʳ _ _ _ _
∙ cong (l (snd A) ≡_)
(sym (compPath≡compPath' (r (snd B)) refl)
∙ sym (rUnit (r (pt B))))))
is₃ : Iso ((A ⋀∙ B) →∙ C)
(Σ[ f ∈ (fst A → fst B → fst C) ]
Σ[ p ∈ singl (snd C) ]
Σ[ l ∈ ((x : fst A) → f x (pt B) ≡ fst p) ]
Σ[ r ∈ ((b : fst B) → f (pt A) b ≡ fst p) ]
l (pt A) ≡ r (pt B))
fst (Iso.fun is₃ f) x y = fst f (inr (x , y))
fst (fst (snd (Iso.fun is₃ f))) = fst f (inl tt)
snd (fst (snd (Iso.fun is₃ f))) = sym (snd f)
fst (snd (snd (Iso.fun is₃ f))) x = cong (fst f) (sym (push (inl x)))
fst (snd (snd (snd (Iso.fun is₃ f)))) x = cong (fst f) (sym (push (inr x)))
snd (snd (snd (snd (Iso.fun is₃ f)))) i j = fst f (push (push tt i) (~ j))
fst (Iso.inv is₃ (f , (c* , p) , l , r , q)) (inl x) = c*
fst (Iso.inv is₃ (f , (c* , p) , l , r , q)) (inr (x , y)) = f x y
fst (Iso.inv is₃ (f , (c* , p) , l , r , q)) (push (inl x) i) = l x (~ i)
fst (Iso.inv is₃ (f , (c* , p) , l , r , q)) (push (inr x) i) = r x (~ i)
fst (Iso.inv is₃ (f , (c* , p) , l , r , q)) (push (push a j) i) = q j (~ i)
snd (Iso.inv is₃ (f , (c* , p) , l , r , q)) = sym p
Iso.rightInv is₃ _ = refl
Iso.leftInv is₃ f =
ΣPathP ((funExt (λ { (inl x) → refl
; (inr x) → refl
; (push (inl x) i) → refl
; (push (inr x) i) → refl
; (push (push a i₁) i) → refl}))
, refl)
isContrIso : ∀ {ℓ ℓ'} {A : Type ℓ} (a : A) (B : singl a → Type ℓ')
→ Iso (Σ _ B) (B (a , refl))
isContrIso a B =
compIso (invIso
(Σ-cong-iso-fst (isContr→Iso isContrUnit (isContrSingl a))))
lUnit×Iso
iso₄ : Iso (isoToPath is₃ i1)
(isoToPath is₂ i1)
iso₄ = Σ-cong-iso-snd λ f → isContrIso (snd C) _
⋀→∙Homogeneous≡ : isHomogeneous C
→ {f g : (A ⋀∙ B) →∙ C}
→ ((x : fst A) (y : fst B) → fst f (inr (x , y)) ≡ fst g (inr (x , y)))
→ f ≡ g
⋀→∙Homogeneous≡ C {f = f} {g = g} p =
sym (Iso.leftInv SmashAdjIso f)
∙∙ cong (Iso.inv SmashAdjIso) main
∙∙ Iso.leftInv SmashAdjIso g
where
main : Iso.fun SmashAdjIso f ≡ Iso.fun SmashAdjIso g
main =
→∙Homogeneous≡ (isHomogeneous→∙ C)
(funExt λ x → →∙Homogeneous≡ C (funExt (p x)))
prod→⋀^ : (n : ℕ) (A : Fin (suc n) → Pointed ℓ)
→ prodFinFamily n (fst ∘ A) → ⋀^ n A .fst
prod→⋀^ zero A x = x
prod→⋀^ (suc n) A x =
inr ((prod→⋀^ n (predFinFamily∙ A) (fst x)) , (snd x))
⋀→Homogeneous≡ : {A B : Pointed ℓ} {D : Type ℓ'}
→ {f g : A ⋀ B → D}
→ (isHomogeneous (D , f (inl tt)))
→ ((x : _) (y : _) → f (inr (x , y)) ≡ g (inr (x , y)))
→ f ≡ g
⋀→Homogeneous≡ {A = A} {B} {D} {f = f} {g} hom p = cong fst f^≡g^
where
f^ g^ : A ⋀∙ B →∙ (D , f (inl tt))
f^ = f , refl
g^ = g , (cong g (push (inr (pt B)))
∙∙ sym (p (pt A) (pt B))
∙∙ cong f (sym (push (inr (pt B)))))
f^≡g^ : f^ ≡ g^
f^≡g^ = ⋀→∙Homogeneous≡ hom p
⋀^→Homogeneous≡ : (n : ℕ) (A : Fin (suc n) → Pointed ℓ) {B : Type ℓ'}
→ {f g : ⋀^ n A .fst → B}
→ isHomogeneous (B , f (⋀^ n A .snd))
→ ((x : _) → f (prod→⋀^ n A x) ≡ g (prod→⋀^ n A x))
→ f ≡ g
⋀^→Homogeneous≡ zero A hom ind = funExt ind
⋀^→Homogeneous≡ (suc n) A {B = B} {f = f} {g} hom ind =
⋀→Homogeneous≡ hom λ x y
→ funExt⁻ (⋀^→Homogeneous≡ n (predFinFamily∙ A)
{f = λ x → f (inr (x , y))} {g = λ x → g (inr (x , y))}
(subst (isHomogeneous ∘ (B ,_))
(cong f (push (inr y))) hom )
λ _ → ind _) x
⋀^→Homogeneous : (n : ℕ) (A : Fin (suc n) → Pointed ℓ) (B : Pointed ℓ')
→ (isHomogeneous B)
→ (f g : ⋀^ n A →∙ B)
→ ((x : _) → f .fst (prod→⋀^ n A x) ≡ g .fst (prod→⋀^ n A x))
→ f ≡ g
⋀^→Homogeneous zero A B ishom f g ind = →∙Homogeneous≡ ishom (funExt ind)
⋀^→Homogeneous (suc n) A B ishom f g ind =
⋀→∙Homogeneous≡ ishom λ x y i
→ ⋀^→Homogeneous n (predFinFamily∙ A) B ishom
(f' y) (g' y) (λ x → ind (x , y)) i .fst x
where
module _ (y : A flast .fst) where
f' g' : ⋀^ n (predFinFamily∙ A) →∙ B
f' = (λ x → f .fst (inr (x , y)))
, cong (fst f) (sym (push (inr y))) ∙ snd f
g' = (λ x → g .fst (inr (x , y)))
, cong (fst g) (sym (push (inr y))) ∙ snd g
_⋀→_ : (f : A →∙ C) (g : B →∙ D) → A ⋀ B → C ⋀ D
(f ⋀→ g) (inl tt) = inl tt
((f , fpt) ⋀→ (g , gpt)) (inr (x , x₁)) = inr (f x , g x₁)
_⋀→_ {B = B} {D = D} (f , fpt) (b , gpt) (push (inl x) i) = (push (inl (f x)) ∙ (λ i → inr (f x , gpt (~ i)))) i
_⋀→_ (f , fpt) (g , gpt) (push (inr x) i) = (push (inr (g x)) ∙ (λ i → inr (fpt (~ i) , g x))) i
_⋀→_ {A = A} {C = C} {B = B} {D = D} (f , fpt) (g , gpt) (push (push tt j) i) =
hcomp (λ k → λ { (i = i0) → inl tt
; (i = i1) → inr (fpt (~ k) , gpt (~ k))
; (j = i0) → compPath-filler (push (inl (fpt (~ k))))
((λ i → inr (fpt (~ k) , gpt (~ i)))) k i
; (j = i1) → compPath-filler (push (inr (gpt (~ k))))
((λ i → inr (fpt (~ i) , gpt (~ k)))) k i})
(push (push tt j) i)
_⋀→∙_ : (f : A →∙ C) (g : B →∙ D) → A ⋀∙ B →∙ C ⋀∙ D
fst (f ⋀→∙ g) = f ⋀→ g
snd (f ⋀→∙ g) = refl
_⋀→refl_ : ∀ {ℓ ℓ'} {C : Type ℓ} {D : Type ℓ'}
→ (f : typ A → C)
→ (g : typ B → D)
→ (A ⋀ B) → ((C , f (pt A)) ⋀ (D , g (pt B)))
(f ⋀→refl g) (inl x) = inl tt
(f ⋀→refl g) (inr (x , y)) = inr (f x , g y)
(f ⋀→refl g) (push (inl x) i) = push (inl (f x)) i
(f ⋀→refl g) (push (inr x) i) = push (inr (g x)) i
(f ⋀→refl g) (push (push a i₁) i) = push (push tt i₁) i
_⋀∙→refl_ : ∀ {ℓ ℓ'} {C : Type ℓ} {D : Type ℓ'}
→ (f : typ A → C)
→ (g : typ B → D)
→ (A ⋀∙ B) →∙ ((C , f (pt A)) ⋀∙ (D , g (pt B)))
fst (f ⋀∙→refl g) = f ⋀→refl g
snd (f ⋀∙→refl g) = refl
⋀≃ : ∀ {ℓ ℓ'} {A B : Pointed ℓ} {C D : Pointed ℓ'}
→ (f : A ≃∙ B) (g : C ≃∙ D)
→ (A ⋀ C) ≃ (B ⋀ D)
⋀≃ {ℓ = ℓ} {ℓ'} {B = B} {D = D} f g = _ , ⋀≃-isEq f g
where
help : (x : _) → (idfun∙ B ⋀→ idfun∙ D) x ≡ x
help (inl x) = refl
help (inr x) = refl
help (push (inl x) i) j = rUnit (push (inl x)) (~ j) i
help (push (inr x) i) j = rUnit (push (inr x)) (~ j) i
help (push (push a i) j) k =
hcomp (λ r → λ {(i = i0) → rUnit (push (inl (snd B))) (~ k ∧ r) j
; (i = i1) → rUnit (push (inr (snd D))) (~ k ∧ r) j
; (j = i0) → inl tt
; (j = i1) → inr (snd B , snd D)
; (k = i1) → push (push tt i) j})
(push (push tt i) j)
⋀≃-isEq : {A : Pointed ℓ} {C : Pointed ℓ'}
(f : A ≃∙ B) (g : C ≃∙ D) → isEquiv (≃∙map f ⋀→ ≃∙map g)
⋀≃-isEq {C = C} =
Equiv∙J (λ A f → (g : C ≃∙ D)
→ isEquiv (≃∙map f ⋀→ ≃∙map g))
(Equiv∙J (λ _ g → isEquiv (idfun∙ _ ⋀→ ≃∙map g))
(subst isEquiv (sym (funExt help)) (idIsEquiv _)))
⋀→Smash : A ⋀ B → Smash A B
⋀→Smash (inl x) = basel
⋀→Smash (inr (x , x₁)) = proj x x₁
⋀→Smash (push (inl x) i) = gluel x (~ i)
⋀→Smash {A = A} {B = B} (push (inr x) i) =
(sym (gluel (snd A)) ∙∙ gluer (snd B) ∙∙ sym (gluer x)) i
⋀→Smash {A = A} {B = B} (push (push a j) i) =
hcomp (λ k → λ { (i = i0) → gluel (snd A) (k ∨ ~ j)
; (i = i1) → gluer (snd B) (~ k ∧ j)
; (j = i0) → gluel (snd A) (~ i)})
(invSides-filler (gluel (snd A)) (gluer (snd B)) j (~ i))
Smash→⋀ : Smash A B → A ⋀ B
Smash→⋀ basel = inl tt
Smash→⋀ baser = inl tt
Smash→⋀ (proj x y) = inr (x , y)
Smash→⋀ (gluel a i) = push (inl a) (~ i)
Smash→⋀ (gluer b i) = push (inr b) (~ i)
module _ {ℓ ℓ' ℓ'' : Level} (A : Pointed ℓ) (B : Pointed ℓ') (C : Pointed ℓ'') where
data ⋀×2 : Type (ℓ-max ℓ (ℓ-max ℓ' ℓ'')) where
base : ⋀×2
proj : typ A → typ B → typ C → ⋀×2
gluel : (x : typ A) (y : typ B) → proj x y (pt C) ≡ base
gluem : (x : typ A) (z : typ C) → proj x (pt B) z ≡ base
gluer : (y : typ B) (z : typ C) → proj (pt A) y z ≡ base
gluel≡gluem : (a : typ A) → gluel a (pt B) ≡ gluem a (pt C)
gluel≡gluer : (y : typ B) → Path (Path (⋀×2) _ _) (gluel (pt A) y) (gluer y (pt C))
gluem≡gluer : (z : typ C) → gluem (pt A) z ≡ gluer (pt B) z
coh : Cube (gluel≡gluer (snd B)) (gluem≡gluer (pt C))
(gluel≡gluem (pt A)) (λ _ → gluer (snd B) (pt C))
refl refl
filler₁ : typ B → (i j k : I) → ⋀×2
filler₁ a i j r =
hfill (λ k → λ {(i = i0) → gluer a (pt C) (j ∧ k)
; (i = i1) → base
; (j = i0) → gluel (pt A) a i
; (j = i1) → gluer a (pt C) (i ∨ k)})
(inS (gluel≡gluer a j i))
r
filler₂ : typ C → (i j k : I) → ⋀×2
filler₂ c i j r =
hfill (λ k → λ {(i = i0) → gluer (pt B) c (j ∧ k)
; (i = i1) → base
; (j = i0) → gluem (pt A) c i
; (j = i1) → gluer (pt B) c (i ∨ k)})
(inS (gluem≡gluer c j i))
r
filler₃ : typ B → (i j r : I) → A ⋀ (B ⋀∙ C)
filler₃ b i j r =
hfill (λ k → λ {(i = i0) → compPath-filler'
(λ j → inr (pt A , (push (inl b) (~ j))))
(sym (push (inl (pt A)))) k j
; (i = i1) → push (inr (push (inl b) k)) (~ j)
; (j = i0) → inr (pt A , push (inl b) k)
; (j = i1) → inl tt})
(inS (push (push tt i) (~ j)))
r
filler₄ : typ C → (i j r : I) → A ⋀ (B ⋀∙ C)
filler₄ c i j r =
hfill (λ k → λ {(i = i0) → compPath-filler'
(λ j → inr (pt A , (push (inr c) (~ j))))
(sym (push (inl (pt A)))) k j
; (i = i1) → push (inr (push (inr c) k)) (~ j)
; (j = i0) → inr (pt A , push (inr c) k)
; (j = i1) → inl tt})
(inS (push (push tt i) (~ j)))
r
filler₅ : (i j k : I) → A ⋀ (B ⋀∙ C)
filler₅ i j r =
hfill (λ k → λ {(i = i0) → push (inl (pt A)) j
; (i = i1) → push (inr (inl tt)) (j ∧ ~ k)
; (j = i0) → inl tt
; (j = i1) → push (inr (inl tt)) (~ i ∨ ~ k)})
(inS (push (push tt i) j))
r
coh-filler : (i j k r : I) → ⋀×2
coh-filler i j k r =
hfill (λ r → λ {(i = i0) → filler₁ (pt B) j k r
; (i = i1) → filler₂ (pt C) j k r
; (j = i0) → gluer (snd B) (snd C) (k ∧ r)
; (j = i1) → base
; (k = i0) → gluel≡gluem (pt A) i j
; (k = i1) → gluer (snd B) (snd C) (j ∨ r)})
(inS (coh i k j))
r
coh-filler₂ : (i j k r : I) → A ⋀ (B ⋀∙ C)
coh-filler₂ i j k r =
hfill (λ r → λ {(i = i0) → filler₃ (snd B) j k r
; (i = i1) → filler₄ (pt C) j k r
; (j = i0) → compPath-filler'
(λ k₁ → inr (pt A , push (push tt i) (~ k₁)))
(sym (push (inl (pt A)))) r k
; (j = i1) → push (inr (push (push tt i) r)) (~ k)
; (k = i0) → inr (pt A , push (push tt i) r)
; (k = i1) → inl tt})
(inS (push (push tt j) (~ k)))
r
⋀→⋀×2 : A ⋀ (B ⋀∙ C) → ⋀×2
⋀→⋀×2 (inl x) = base
⋀→⋀×2 (inr (x , inl y)) = base
⋀→⋀×2 (inr (x , inr (y , z))) = proj x y z
⋀→⋀×2 (inr (x , push (inl a) i)) = gluel x a (~ i)
⋀→⋀×2 (inr (x , push (inr b) i)) = gluem x b (~ i)
⋀→⋀×2 (inr (x , push (push a i) j)) = gluel≡gluem x i (~ j)
⋀→⋀×2 (push (inl x) i) = base
⋀→⋀×2 (push (inr (inl x)) i) = base
⋀→⋀×2 (push (inr (inr (x , y))) i) = gluer x y (~ i)
⋀→⋀×2 (push (inr (push (inl x) i)) j) = filler₁ x (~ i) (~ j) i1
⋀→⋀×2 (push (inr (push (inr x) i)) j) = filler₂ x (~ i) (~ j) i1
⋀→⋀×2 (push (inr (push (push a i) j)) k) = coh-filler i (~ j) (~ k) i1
⋀→⋀×2 (push (push a i₁) i) = base
⋀×2→⋀ : ⋀×2 → A ⋀ (B ⋀∙ C)
⋀×2→⋀ base = inl tt
⋀×2→⋀ (proj x x₁ x₂) = inr (x , inr (x₁ , x₂))
⋀×2→⋀ (gluel x y i) =
((λ i → inr (x , (push (inl y) (~ i)))) ∙ sym (push (inl x))) i
⋀×2→⋀ (gluem x z i) =
((λ i → inr (x , (push (inr z) (~ i)))) ∙ sym (push (inl x))) i
⋀×2→⋀ (gluer y z i) = push (inr (inr (y , z))) (~ i)
⋀×2→⋀ (gluel≡gluem a i j) =
((λ k → inr (a , (push (push tt i) (~ k)))) ∙ sym (push (inl a))) j
⋀×2→⋀ (gluel≡gluer b i j) = filler₃ b i j i1
⋀×2→⋀ (gluem≡gluer c i j) = filler₄ c i j i1
⋀×2→⋀ (coh i j k) = coh-filler₂ i j k i1
gluel-fill : (x : typ A) (b : typ B) (i j k : I) → ⋀×2
gluel-fill x y i j k =
hfill (λ k → λ {(i = i0) → gluel x y (~ k)
; (i = i1) → base
; (j = i0) →
⋀→⋀×2 (compPath-filler'
(λ i → (inr (x , (push (inl y) (~ i)))))
(sym (push (inl x))) k i)
; (j = i1) → gluel x y (i ∨ ~ k) })
(inS base)
k
gluem-fill : (x : typ A) (z : typ C) (i j k : I) → ⋀×2
gluem-fill x z i j k =
hfill (λ k → λ {(i = i0) → gluem x z (~ k)
; (i = i1) → base
; (j = i0) → ⋀→⋀×2 (compPath-filler'
(λ i → (inr (x , (push (inr z) (~ i)))))
(sym (push (inl x))) k i)
; (j = i1) → gluem x z (i ∨ ~ k)})
(inS base)
k
gluel≡gluer₁ : (y : typ B) (i j r k : I) → ⋀×2
gluel≡gluer₁ y i j r k =
hfill (λ k → λ {(r = i0) → base
; (r = i1) → gluer y (snd C) (i ∧ k)
; (i = i0) → gluel≡gluer y j (~ r)
; (i = i1) → gluer y (snd C) (~ r ∨ k)
; (j = i0) → filler₁ y (~ r) i k
; (j = i1) → gluer y (snd C) ((i ∧ k) ∨ ~ r)})
(inS (gluel≡gluer y (j ∨ i) (~ r)))
k
gluem≡gluer₁ : (y : typ C) (i j r k : I) → ⋀×2
gluem≡gluer₁ z i j r k =
hfill (λ k → λ {(i = i0) → gluem≡gluer z j (~ r)
; (i = i1) → gluer (snd B) z (~ r ∨ k)
; (j = i0) → filler₂ z (~ r) i k
; (j = i1) → gluer (snd B) z (~ r ∨ (k ∧ i))
; (r = i0) → base
; (r = i1) → gluer (snd B) z (i ∧ k)})
(inS (gluem≡gluer z (j ∨ i) (~ r)))
k
gluel≡gluer₂ : (y : typ B) (k i j r : I) → ⋀×2
gluel≡gluer₂ y k i j r =
hfill (λ r → λ {(i = i0) → gluel≡gluer y (k ∧ j) (~ r)
; (i = i1) → base
; (j = i0) → ⋀→⋀×2 (filler₃ y k i r)
; (j = i1) → gluel≡gluer y k (i ∨ ~ r)
; (k = i0) → gluel-fill (pt A) y i j r
; (k = i1) → gluel≡gluer₁ y i j r i1})
(inS base)
r
gluem≡gluer₂ : (y : typ C) (k i j r : I) → ⋀×2
gluem≡gluer₂ y k i j r =
hfill (λ r → λ {(i = i0) → gluem≡gluer y (k ∧ j) (~ r)
; (i = i1) → base
; (j = i0) → ⋀→⋀×2 (filler₄ y k i r)
; (j = i1) → gluem≡gluer y k (i ∨ ~ r)
; (k = i0) → gluem-fill (pt A) y i j r
; (k = i1) → gluem≡gluer₁ y i j r i1})
(inS base)
r
gluel≡gluem-fill : (a : typ A) (i j k r : I) → ⋀×2
gluel≡gluem-fill a i j k r =
hfill (λ r → λ {(i = i0) → gluel≡gluem a k (~ r)
; (i = i1) → base
; (j = i0) → ⋀→⋀×2 (compPath-filler'
(λ i → inr (a , push (push tt k) (~ i))) (sym (push (inl a))) r i)
; (j = i1) → gluel≡gluem a k (i ∨ ~ r)
; (k = i0) → gluel-fill a (pt B) i j r
; (k = i1) → gluem-fill a (pt C) i j r})
(inS base)
r
⋀×2→⋀→⋀×2 : (x : ⋀×2) → ⋀→⋀×2 (⋀×2→⋀ x) ≡ x
⋀×2→⋀→⋀×2 base = refl
⋀×2→⋀→⋀×2 (proj x x₁ x₂) = refl
⋀×2→⋀→⋀×2 (gluel x y i) j = gluel-fill x y i j i1
⋀×2→⋀→⋀×2 (gluem x z i) j = gluem-fill x z i j i1
⋀×2→⋀→⋀×2 (gluer y z i) = refl
⋀×2→⋀→⋀×2 (gluel≡gluem a k i) j = gluel≡gluem-fill a i j k i1
⋀×2→⋀→⋀×2 (gluel≡gluer y k i) j = gluel≡gluer₂ y k i j i1
⋀×2→⋀→⋀×2 (gluem≡gluer z k i) j = gluem≡gluer₂ z k i j i1
⋀×2→⋀→⋀×2 (coh i j k) r =
hcomp (λ l → λ {(i = i0) → gluel≡gluer₂ (snd B) j k r l
; (i = i1) → gluem≡gluer₂ (pt C) j k r l
; (j = i0) → gluel≡gluem-fill (pt A) k r i l
; (j = i1) → coh-lem l i k r
; (k = i0) → coh i (j ∧ r) (~ l)
; (k = i1) → base
; (r = i0) → ⋀→⋀×2 (coh-filler₂ i j k l)
; (r = i1) → coh i j (k ∨ ~ l)})
base
where
coh-lem : PathP
(λ l → Cube (λ k r → gluel≡gluer₂ (snd B) i1 k r l)
(λ k r → gluem≡gluer₂ (pt C) i1 k r l)
(λ i r → coh i r (~ l))
(λ i r → base)
(λ i k → coh-filler i (~ l) k i1)
λ i k → gluer (snd B) (snd C) (k ∨ ~ l))
(λ _ _ _ → base)
λ i k r → gluer (snd B) (pt C) k
coh-lem l i k r =
hcomp (λ j → λ {(i = i0) → gluel≡gluer₁ (pt B) k r l j
; (i = i1) → gluem≡gluer₁ (pt C) k r l j
; (l = i0) → base
; (l = i1) → gluer (snd B) (pt C) (k ∧ j)
; (k = i0) → coh i r (~ l)
; (k = i1) → gluem≡gluer₁ (pt C) k r l j
; (r = i0) → coh-filler i (~ l) k j
; (r = i1) → gluer (snd B) (snd C) (~ l ∨ (j ∧ k))})
(hcomp (λ j → λ {(i = i0) → gluel≡gluer (snd B) (r ∨ k) (~ l)
; (i = i1) → gluem≡gluer (snd C) (r ∨ k) (~ l)
; (l = i0) → base
; (l = i1) → proj (pt A) (pt B) (snd C)
; (k = i0) → coh i r (~ l)
; (k = i1) → gluer (snd B) (snd C) (~ l)
; (r = i0) → coh i k (~ l)
; (r = i1) → gluer (snd B) (snd C) (~ l)})
(coh i (r ∨ k) (~ l)))
filler₆ : (x : typ A) (a : typ B) (i j k : I) → A ⋀ (B ⋀ C , inl tt)
filler₆ x a i j k =
hfill (λ k → λ {(i = i0) → inr (x , push (inl a) k)
; (i = i1) → push (inl x) j
; (j = i0) → compPath-filler'
(λ i₁ → inr (x , (push (inl a) (~ i₁))))
(sym (push (inl x))) k i
; (j = i1) → inr (x , push (inl a) (~ i ∧ k)) })
(inS (push (inl x) (j ∨ ~ i)))
k
filler₇ : (x : typ A) (a : typ C) (i j k : I) → A ⋀ (B ⋀ C , inl tt)
filler₇ x a i j k =
hfill (λ k → λ {(i = i0) → inr (x , push (inr a) k)
; (i = i1) → push (inl x) j
; (j = i0) → compPath-filler'
(λ i₁ → inr (x , (push (inr a) (~ i₁))))
(sym (push (inl x))) k i
; (j = i1) → inr (x , push (inr a) (~ i ∧ k)) })
(inS (push (inl x) (j ∨ ~ i)))
k
filler₈ : (x : typ A) (i j k r : I) → A ⋀ (B ⋀ C , inl tt)
filler₈ x i j k r =
hfill (λ r → λ {(i = i0) → inr (x , push (push tt k) r)
; (i = i1) → push (inl x) j
; (j = i0) → compPath-filler'
(λ j → inr (x , push (push tt k) (~ j)))
(sym (push (inl x))) r i
; (j = i1) → inr (x , push (push tt k) (~ i ∧ r)) })
(inS ((push (inl x) (j ∨ ~ i))))
r
btm-fill : (i j k r : I) → A ⋀ (B ⋀∙ C)
btm-fill i j k r =
hfill (λ r → λ {(i = i0) → push (inr (inl tt)) (~ j ∨ (r ∧ ~ k))
; (i = i1) → filler₅ j k i1
; (j = i1) → push (inr (inl tt)) (~ i ∧ (r ∧ ~ k))
; (j = i0) → push (inl (pt A)) (~ i ∨ k)
; (k = i0) → filler₅ j (~ i) (~ r)
; (k = i1) → push (inr (inl tt)) (~ j)})
(inS (filler₅ j (~ i ∨ k) i1))
r
lr-fill₁ : (b : typ C) (i j k r : I) → A ⋀ (B ⋀∙ C)
lr-fill₁ a i j k r =
hfill (λ r → λ {(i = i0) → push (inr (push (inr a) r)) (~ j ∨ ~ k)
; (i = i1) → filler₅ j k i1
; (j = i1) → push (inr (push (inr a) r)) (~ i ∧ ~ k)
; (j = i0) → filler₇ (pt A) a i k r
; (k = i0) → filler₄ a j i r
; (k = i1) → push (inr (push (inr a) (~ i ∧ r))) (~ j)})
(inS (btm-fill i j k i1))
r
ll-fill₁ : (a : typ B) (i j k r : I) → A ⋀ (B ⋀∙ C)
ll-fill₁ a i j k r =
hfill (λ r → λ {(i = i0) → push (inr (push (inl a) r)) (~ j ∨ ~ k)
; (i = i1) → filler₅ j k i1
; (j = i1) → push (inr (push (inl a) r)) (~ i ∧ ~ k)
; (j = i0) → filler₆ (pt A) a i k r
; (k = i0) → filler₃ a j i r
; (k = i1) → push (inr (push (inl a) (~ i ∧ r))) (~ j)})
(inS (btm-fill i j k i1))
r
ll-fill₂ : (a : typ B) (i j k r : I) → A ⋀ (B ⋀∙ C)
ll-fill₂ a i j k r =
hfill (λ r → λ {(i = i0) → push (inr (inr (a , pt C))) (~ j ∨ (~ r ∧ ~ k))
; (i = i1) → filler₅ j k i1
; (j = i1) → push (inr (inr (a , (snd C)))) ((~ r ∧ ~ i) ∧ ~ k)
; (j = i0) → filler₆ (pt A) a i k i1
; (k = i0) → ⋀×2→⋀ (filler₁ a i j r)
; (k = i1) → push (inr (push (inl a) (~ i))) (~ j) })
(inS (ll-fill₁ a i j k i1))
r
lr-fill₂ : (a : typ C) (i j k r : I) → A ⋀ (B ⋀∙ C)
lr-fill₂ a i j k r =
hfill (λ r → λ {(i = i0) → push (inr (inr (pt B , a))) (~ j ∨ (~ r ∧ ~ k))
; (i = i1) → filler₅ j k i1
; (j = i1) → push (inr (inr (pt B , a))) ((~ r ∧ ~ i) ∧ ~ k)
; (j = i0) → filler₇ (pt A) a i k i1
; (k = i0) → ⋀×2→⋀ (filler₂ a i j r)
; (k = i1) → push (inr (push (inr a) (~ i))) (~ j) })
(inS (lr-fill₁ a i j k i1))
r
⋀→⋀×2→⋀ : (x : A ⋀ (B ⋀∙ C))
→ ⋀×2→⋀ (⋀→⋀×2 x) ≡ x
⋀→⋀×2→⋀ (inl x) = refl
⋀→⋀×2→⋀ (inr (x , inl x₁)) = push (inl x)
⋀→⋀×2→⋀ (inr (x , inr x₁)) = refl
⋀→⋀×2→⋀ (inr (x , push (inl a) i)) j = filler₆ x a (~ i) j i1
⋀→⋀×2→⋀ (inr (x , push (inr b) i)) j = filler₇ x b (~ i) j i1
⋀→⋀×2→⋀ (inr (x , push (push a r) i)) j = filler₈ x (~ i) j r i1
⋀→⋀×2→⋀ (push (inl x) i) j = push (inl x) (j ∧ i)
⋀→⋀×2→⋀ (push (inr (inl x)) i) j = filler₅ (~ i) j i1
⋀→⋀×2→⋀ (push (inr (inr x)) i) j = push (inr (inr x)) i
⋀→⋀×2→⋀ (push (inr (push (inl x) i)) j) k = ll-fill₂ x (~ i) (~ j) k i1
⋀→⋀×2→⋀ (push (inr (push (inr x) i)) j) k = lr-fill₂ x (~ i) (~ j) k i1
⋀→⋀×2→⋀ (push (inr (push (push a r) i)) j) k =
hcomp (λ s → λ {(i = i0) → filler₅ (~ j) k i1
; (i = i1) → push (inr (inr (snd B , snd C))) (j ∨ ~ s ∧ ~ k)
; (j = i0) → push (inr (inr (pt B , pt C))) ((~ s ∧ i) ∧ ~ k)
; (j = i1) → filler₈ (pt A) (~ i) k r i1
; (k = i0) → ⋀×2→⋀ (coh-filler r (~ i) (~ j) s)
; (k = i1) → push (inr (push (push tt r) i)) j
; (r = i0) → ll-fill₂ (pt B) (~ i) (~ j) k s
; (r = i1) → lr-fill₂ (pt C) (~ i) (~ j) k s })
(hcomp (λ s → λ {(i = i0) → filler₅ (~ j) k i1
; (i = i1) → push (inr (push (push tt r) s)) (j ∨ ~ k)
; (j = i0) → push (inr (push (push tt r) s)) (i ∧ ~ k)
; (j = i1) → filler₈ (pt A) (~ i) k r s
; (k = i0) → coh-filler₂ r (~ j) (~ i) s
; (k = i1) → push (inr (push (push tt r) (i ∧ s))) j
; (r = i0) → ll-fill₁ (pt B) (~ i) (~ j) k s
; (r = i1) → lr-fill₁ (pt C) (~ i) (~ j) k s})
(hcomp (λ s → λ {(i = i0) → filler₅ (~ j) k i1
; (i = i1) → push (inr (inl tt)) (j ∨ (s ∧ ~ k))
; (j = i0) → push (inr (inl tt)) (i ∧ s ∧ ~ k)
; (j = i1) → push (inl (snd A)) (i ∨ k)
; (k = i0) → filler₅ (~ j) i (~ s)
; (k = i1) → push (inr (inl tt)) j
; (r = i0) → btm-fill (~ i) (~ j) k s
; (r = i1) → btm-fill (~ i) (~ j) k s})
(filler₅ (~ j) (i ∨ k) i1)))
⋀→⋀×2→⋀ (push (push a i) j) k =
hcomp (λ r → λ {(i = i0) → push (inl (pt A)) (k ∧ j ∨ ~ r)
; (i = i1) → filler₅ (~ j) k r
; (j = i0) → push (push tt i) (k ∧ ~ r)
; (j = i1) → push (inl (snd A)) k
; (k = i0) → inl tt
; (k = i1) → push (push tt i) (j ∨ ~ r)})
(push (push tt (~ j ∧ i)) k)
Iso-⋀-⋀×2 : Iso (A ⋀ (B ⋀∙ C)) ⋀×2
Iso.fun Iso-⋀-⋀×2 = ⋀→⋀×2
Iso.inv Iso-⋀-⋀×2 = ⋀×2→⋀
Iso.rightInv Iso-⋀-⋀×2 = ⋀×2→⋀→⋀×2
Iso.leftInv Iso-⋀-⋀×2 = ⋀→⋀×2→⋀
module _ {ℓ ℓ' ℓ'' : Level} (A : Pointed ℓ) (B : Pointed ℓ') (C : Pointed ℓ'') where
permute-fill→ : (i j k r : I) → ⋀×2 C A B
permute-fill→ i j k r =
hfill (λ r → λ {(i = i0) → gluem≡gluer (snd B) (~ j ∨ ~ r) k
; (i = i1) → gluel≡gluem (pt C) j k
; (j = i0) → gluel≡gluer (pt A) (~ i) k
; (j = i1) → gluem≡gluer (snd B) (~ i ∧ ~ r) k
; (k = i0) → proj (pt C) (pt A) (snd B)
; (k = i1) → base})
(inS (coh j (~ i) k))
r
permute-fill← : (i j k r : I) → ⋀×2 A B C
permute-fill← i j k r =
hfill (λ r → λ {(i = i0) → gluel≡gluem (snd A) (~ j) k
; (i = i1) → gluel≡gluer (pt B) (~ j ∨ ~ r) k
; (j = i0) → gluem≡gluer (pt C) i k
; (j = i1) → gluel≡gluer (pt B) (i ∧ ~ r) k
; (k = i0) → proj (snd A) (pt B) (pt C)
; (k = i1) → base})
(inS (coh (~ j) i k))
r
⋀×2-permuteFun : ⋀×2 A B C → ⋀×2 C A B
⋀×2-permuteFun base = base
⋀×2-permuteFun (proj x x₁ x₂) = proj x₂ x x₁
⋀×2-permuteFun (gluel x y i) = gluer x y i
⋀×2-permuteFun (gluem x z i) = gluel z x i
⋀×2-permuteFun (gluer y z i) = gluem z y i
⋀×2-permuteFun (gluel≡gluem a i j) = gluel≡gluer a (~ i) j
⋀×2-permuteFun (gluel≡gluer y i j) = gluem≡gluer y (~ i) j
⋀×2-permuteFun (gluem≡gluer z i j) = gluel≡gluem z i j
⋀×2-permuteFun (coh i j k) =
hcomp (λ r → λ {(i = i0) → gluem≡gluer (snd B) (~ j ∨ ~ r) k
; (i = i1) → gluel≡gluem (pt C) j k
; (j = i0) → gluel≡gluer (pt A) (~ i) k
; (j = i1) → gluem≡gluer (snd B) (~ i ∧ ~ r) k
; (k = i0) → proj (pt C) (pt A) (snd B)
; (k = i1) → base})
(coh j (~ i) k)
⋀×2-permuteInv : ⋀×2 C A B → ⋀×2 A B C
⋀×2-permuteInv base = base
⋀×2-permuteInv (proj x x₁ x₂) = proj x₁ x₂ x
⋀×2-permuteInv (gluel x y i) = gluem y x i
⋀×2-permuteInv (gluem x z i) = gluer z x i
⋀×2-permuteInv (gluer y z i) = gluel y z i
⋀×2-permuteInv (gluel≡gluem a i j) = gluem≡gluer a i j
⋀×2-permuteInv (gluel≡gluer y i j) = gluel≡gluem y (~ i) j
⋀×2-permuteInv (gluem≡gluer z i j) = gluel≡gluer z (~ i) j
⋀×2-permuteInv (coh i j k) = permute-fill← i j k i1
⋀×2-permuteIso : Iso (⋀×2 A B C) (⋀×2 C A B)
Iso.fun ⋀×2-permuteIso = ⋀×2-permuteFun
Iso.inv ⋀×2-permuteIso = ⋀×2-permuteInv
Iso.rightInv ⋀×2-permuteIso base = refl
Iso.rightInv ⋀×2-permuteIso (proj x x₁ x₂) = refl
Iso.rightInv ⋀×2-permuteIso (gluel x y i) = refl
Iso.rightInv ⋀×2-permuteIso (gluem x z i) = refl
Iso.rightInv ⋀×2-permuteIso (gluer y z i) = refl
Iso.rightInv ⋀×2-permuteIso (gluel≡gluem a i i₁) = refl
Iso.rightInv ⋀×2-permuteIso (gluel≡gluer y x x₁) = refl
Iso.rightInv ⋀×2-permuteIso (gluem≡gluer z i i₁) = refl
Iso.rightInv ⋀×2-permuteIso (coh i j k) r =
hcomp (λ l → λ { (i = i0) → gluel≡gluer (snd A) j k
; (i = i1) → gluem≡gluer (snd B) (j ∧ (r ∨ l)) k
; (j = i0) → gluel≡gluem (snd C) i k
; (j = i1) → gluem≡gluer (snd B) (~ i ∨ (l ∨ r)) k
; (k = i0) → proj (snd C) (snd A) (snd B)
; (k = i1) → base
; (r = i0) → ⋀×2-permuteFun (permute-fill← i j k l)
; (r = i1) → coh i j k})
(hcomp (λ l → λ { (i = i0) → gluel≡gluer (snd A) j k
; (i = i1) → gluem≡gluer (snd B) (j ∧ (~ l ∨ r)) k
; (j = i0) → gluel≡gluem (snd C) i k
; (j = i1) → gluem≡gluer (snd B) (~ i ∨ (~ l ∨ r)) k
; (k = i0) → proj (snd C) (snd A) (snd B)
; (k = i1) → base
; (r = i0) → permute-fill→ (~ j) i k l
; (r = i1) → coh i j k})
(coh i j k))
Iso.leftInv ⋀×2-permuteIso base = refl
Iso.leftInv ⋀×2-permuteIso (proj x x₁ x₂) = refl
Iso.leftInv ⋀×2-permuteIso (gluel x y i) = refl
Iso.leftInv ⋀×2-permuteIso (gluem x z i) = refl
Iso.leftInv ⋀×2-permuteIso (gluer y z i) = refl
Iso.leftInv ⋀×2-permuteIso (gluel≡gluem a i i₁) = refl
Iso.leftInv ⋀×2-permuteIso (gluel≡gluer y x x₁) = refl
Iso.leftInv ⋀×2-permuteIso (gluem≡gluer z i i₁) = refl
Iso.leftInv ⋀×2-permuteIso (coh i j k) r =
hcomp (λ l → λ { (i = i0) → gluel≡gluer (snd B) (j ∧ (l ∨ r)) k
; (i = i1) → gluem≡gluer (snd C) j k
; (j = i0) → gluel≡gluem (snd A) i k
; (j = i1) → gluel≡gluer (snd B) (i ∨ (l ∨ r)) k
; (k = i0) → proj (pt A) (pt B) (pt C)
; (k = i1) → base
; (r = i0) → ⋀×2-permuteInv (permute-fill→ i j k l)
; (r = i1) → coh i j k})
(hcomp (λ l → λ { (i = i0) → gluel≡gluer (snd B) (j ∧ (~ l ∨ r)) k
; (i = i1) → gluem≡gluer (snd C) j k
; (j = i0) → gluel≡gluem (snd A) i k
; (j = i1) → gluel≡gluer (snd B) (i ∨ (~ l ∨ r)) k
; (k = i0) → proj (pt A) (pt B) (pt C)
; (k = i1) → base
; (r = i0) → permute-fill← j (~ i) k l
; (r = i1) → coh i j k})
(coh i j k))
SmashAssocIso : Iso (A ⋀ (B ⋀∙ C)) ((A ⋀∙ B) ⋀ C)
SmashAssocIso {A = A} {B = B} {C = C} =
compIso
(Iso-⋀-⋀×2 A B C)
(compIso
(⋀×2-permuteIso A B C)
(compIso
(invIso (Iso-⋀-⋀×2 C A B))
⋀CommIso))
SmashAssocEquiv∙ : A ⋀∙ (B ⋀∙ C) ≃∙ (A ⋀∙ B) ⋀∙ C
fst SmashAssocEquiv∙ = isoToEquiv SmashAssocIso
snd SmashAssocEquiv∙ = refl
module _ {C : Type ℓ} (f g : A ⋀ B → C)
(bp : f (inl tt) ≡ g (inl tt))
(proj : (x : _) → f (inr x) ≡ g (inr x))
(pl : (x : typ A) → PathP (λ i → f (push (inl x) i) ≡ g (push (inl x) i))
bp (proj (x , pt B)))
(p-r : (x : typ B) → PathP (λ i → f (push (inr x) i) ≡ g (push (inr x) i))
bp (proj (pt A , x)))
where
private
⋆act : bp ≡ bp
⋆act i j =
hcomp (λ k → λ { (i = i0) → pl (pt A) (~ k) j
; (i = i1) → p-r (pt B) (~ k) j
; (j = i0) → f (push (push tt i) (~ k))
; (j = i1) → g (push (push tt i) (~ k))})
(proj (snd A , snd B) j)
⋀-fun≡ : (x : _) → f x ≡ g x
⋀-fun≡ (inl x) = bp
⋀-fun≡ (inr x) = proj x
⋀-fun≡ (push (inl x) i) = pl x i
⋀-fun≡ (push (inr x) i) j =
hcomp (λ r → λ {(i = i0) → bp j
; (i = i1) → p-r x r j
; (j = i0) → f (push (inr x) (r ∧ i))
; (j = i1) → g (push (inr x) (r ∧ i)) })
(⋆act i j)
⋀-fun≡ (push (push a i) j) k =
hcomp (λ r → λ { (i = i0) → pl (snd A) (j ∧ r) k
; (j = i0) → bp k
; (j = i1) → side i k r
; (k = i0) → f (push (push a i) (j ∧ r))
; (k = i1) → g (push (push a i) (j ∧ r))})
(⋆act (i ∧ j) k)
where
side : Cube (λ k r → pl (snd A) r k)
(λ k r → p-r (snd B) r k)
(λ i r → f (push (push a i) r))
(λ i r → g (push (push a i) r))
⋆act λ i → proj (snd A , snd B)
side i k r =
hcomp (λ j → λ { (i = i0) → pl (pt A) (~ j ∨ r) k
; (i = i1) → p-r (snd B) (~ j ∨ r) k
; (k = i0) → f (push (push a i) (~ j ∨ r))
; (k = i1) → g (push (push a i) (~ j ∨ r))
; (r = i1) → proj (snd A , snd B) k})
(proj (snd A , snd B) k)
module ⋀-fun≡' {C : Type ℓ} (f g : A ⋀ B → C)
(pr : (x : _) → f (inr x) ≡ g (inr x)) where
p : f (inl tt) ≡ g (inl tt)
p = cong f (push (inr (pt B)))
∙∙ pr (pt A , pt B)
∙∙ sym (cong g (push (inr (pt B))))
p' : f (inl tt) ≡ g (inl tt)
p' = cong f (push (inl (pt A)))
∙∙ pr (pt A , pt B)
∙∙ sym (cong g (push (inl (pt A))))
p≡p' : p ≡ p'
p≡p' i = (cong f (push (push tt (~ i))))
∙∙ pr (pt A , pt B)
∙∙ sym (cong g (push (push tt (~ i))))
Fₗ : B →∙ ((f (inl tt) ≡ g (inl tt)) , p)
fst Fₗ b = cong f (push (inr b)) ∙∙ pr (pt A , b) ∙∙ sym (cong g (push (inr b)))
snd Fₗ = refl
Fᵣ : B →∙ ((f (inl tt) ≡ g (inl tt)) , p)
fst Fᵣ b = p
snd Fᵣ = refl
module _
(lp : (x : fst A) → PathP (λ i → f (push (inl x) i) ≡ g (push (inl x) i))
p (pr (x , pt B)))
(q : Fₗ ≡ Fᵣ) where
private
lem : (b : fst B)
→ Square p (pr (snd A , b))
(cong f (push (inr b))) (cong g (push (inr b)))
lem b i j =
hcomp (λ k → λ {(i = i0) → p j
; (i = i1) → doubleCompPath-filler
(cong f (push (inr b)))
(pr (pt A , b))
(sym (cong g (push (inr b)))) (~ k) j
; (j = i0) → f (push (inr b) (i ∧ k))
; (j = i1) → g (push (inr b) (i ∧ k))})
(q (~ i) .fst b j)
main : (x : _) → f x ≡ g x
main = ⋀-fun≡ {A = A} {B = B} f g p pr lp lem
module _ {ℓ ℓ' : Level} {A : Pointed ℓ} {B : Pointed ℓ'} where
private
sm-fillᵣ : ∀ {ℓ} {A : Type ℓ} {x* : A} (y* : A) (p* : x* ≡ y*)
(y : A) → (p : x* ≡ y)
→ sym p* ≡ (sym p* ∙∙ p ∙∙ sym p)
sm-fillᵣ y* p* y p j i =
hcomp (λ r → λ {(i = i0) → p* r
; (i = i1) → p (~ r ∧ j)
; (j = i0) → p* (~ i ∧ r)
; (j = i1) → doubleCompPath-filler
(sym p*) p (sym p) r i})
(p (i ∧ j))
sm-fillₗ : ∀ {ℓ} {A : Type ℓ} {x* : A} (y* : A) (p* : x* ≡ y*)
(y : A) (p : x* ≡ y)
→ p* ≡ (p ∙∙ sym p ∙∙ p*)
sm-fillₗ y* p* y p j i =
hcomp (λ r → λ {(i = i0) → p (~ r ∧ j)
; (i = i1) → p* r
; (j = i0) → p* (r ∧ i)
; (j = i1) → doubleCompPath-filler
p (sym p) p* r i})
(p (~ i ∧ j))
sm-fillₗᵣ≡ : ∀ {ℓ} {A : Type ℓ} {x* : A} (y* : A) (p* : x* ≡ y*)
→ sm-fillₗ _ (sym p*) _ (sym p*) ≡ sm-fillᵣ _ p* _ p*
sm-fillₗᵣ≡ = J> refl
SuspSmash→Join : Susp (A ⋀ B) → (join (typ A) (typ B))
SuspSmash→Join north = inr (pt B)
SuspSmash→Join south = inl (pt A)
SuspSmash→Join (merid (inl x) i) =
push (pt A) (pt B) (~ i)
SuspSmash→Join (merid (inr (x , b)) i) =
(sym (push x (pt B)) ∙∙ push x b ∙∙ sym (push (pt A) b)) i
SuspSmash→Join (merid (push (inl x) j) i) =
sm-fillₗ {A = join (typ A) (typ B)} _
(sym (push (pt A) (pt B))) _ (sym (push x (pt B))) j i
SuspSmash→Join (merid (push (inr x) j) i) =
sm-fillᵣ {A = join (typ A) (typ B)} _
(push (pt A) (pt B)) _ (push (pt A) x) j i
SuspSmash→Join (merid (push (push a k) j) i) =
sm-fillₗᵣ≡ _ (push (pt A) (pt B)) k j i
Join→SuspSmash : join (typ A) (typ B) → Susp (A ⋀ B)
Join→SuspSmash (inl x) = north
Join→SuspSmash (inr x) = south
Join→SuspSmash (push a b i) = merid (inr (a , b)) i
Join→SuspSmash→Join : (x : join (typ A) (typ B))
→ SuspSmash→Join (Join→SuspSmash x) ≡ x
Join→SuspSmash→Join (inl x) = sym (push x (pt B))
Join→SuspSmash→Join (inr x) = push (pt A) x
Join→SuspSmash→Join (push a b i) j =
doubleCompPath-filler
(sym (push a (pt B))) (push a b) (sym (push (pt A) b)) (~ j) i
SuspSmash→Join→SuspSmash : (x : Susp (A ⋀ B))
→ Join→SuspSmash (SuspSmash→Join x) ≡ x
SuspSmash→Join→SuspSmash north = sym (merid (inr (pt A , pt B)))
SuspSmash→Join→SuspSmash south = merid (inr (pt A , pt B))
SuspSmash→Join→SuspSmash (merid a i) j =
hcomp (λ r
→ λ {(i = i0) → merid (inr (pt A , pt B)) (~ j ∨ ~ r)
; (i = i1) → merid (inr (pt A , pt B)) (j ∧ r)
; (j = i0) → Join→SuspSmash (SuspSmash→Join (merid a i))
; (j = i1) → doubleCompPath-filler
(sym (merid (inr (pt A , pt B))))
(merid a)
(sym (merid (inr (pt A , pt B)))) (~ r) i})
(f₁₂ j .fst a i)
where
f₁ f₂ : A ⋀∙ B →∙ (Path (Susp (A ⋀ B)) south north
, sym (merid (inr (snd A , snd B))))
(fst f₁) a i = Join→SuspSmash (SuspSmash→Join (merid a i))
snd f₁ = refl
(fst f₂) a =
sym (merid (inr (pt A , pt B)))
∙∙ merid a
∙∙ sym (merid (inr (pt A , pt B)))
snd f₂ = cong₂ (λ x y → sym x ∙∙ y ∙∙ sym x)
refl (cong merid (push (inl (pt A))))
∙ doubleCompPath≡compPath
(sym (merid (inr (pt A , pt B)))) _ _
∙ cong₂ _∙_ refl (rCancel (merid (inr (pt A , pt B))))
∙ sym (rUnit _)
f₁₂ : f₁ ≡ f₂
f₁₂ = ⋀→∙Homogeneous≡ (isHomogeneousPath _ _)
λ x y → cong-∙∙ Join→SuspSmash
(sym (push x (pt B)))
(push x y)
(sym (push (pt A) y))
∙ (λ i → sym (merid ((sym (push (inl x))
∙ push (inl (pt A))) i))
∙∙ merid (inr (x , y))
∙∙ sym (merid ((sym (push (inr y))
∙ push (inl (pt A))) i)))
SmashJoinIso : Iso (Susp (A ⋀ B)) (join (typ A) (typ B))
Iso.fun SmashJoinIso = SuspSmash→Join
Iso.inv SmashJoinIso = Join→SuspSmash
Iso.rightInv SmashJoinIso = Join→SuspSmash→Join
Iso.leftInv SmashJoinIso = SuspSmash→Join→SuspSmash
module _ {ℓ ℓ' : Level} {A : Pointed ℓ} {B : Pointed ℓ'} where
merid-fill : typ A → I → I → I → Susp (A ⋀ B)
merid-fill a i j k =
hfill (λ k → λ {(i = i0) → north
; (i = i1) → merid (push (inl a) k) j
; (j = i0) → north
; (j = i1) → merid (inl tt) i})
(inS (merid (inl tt) (i ∧ j))) k
inl-fill₁ : typ A → I → I → I → (Susp∙ (typ A)) ⋀ B
inl-fill₁ a i j k =
hfill (λ k → λ {(i = i0) → inl tt
; (i = i1) → inr (σ A a j , snd B)
; (j = i0) → push (push tt k) i
; (j = i1) → push (push tt k) i})
(inS (push (inl (σ A a j)) i)) k
inl-fill : typ A → I → I → I → (Susp∙ (typ A)) ⋀ B
inl-fill a i j k =
hfill (λ k → λ {(i = i0) → inl tt
; (i = i1) → doubleCompPath-filler
(push (inr (snd B)))
(λ i₂ → inr (σ A a i₂ , snd B))
(sym (push (inr (snd B)))) k j
; (j = i0) → push (inr (snd B)) (~ k ∧ i)
; (j = i1) → push (inr (snd B)) (~ k ∧ i)})
(inS (inl-fill₁ a i j i1)) k
inr-fill₁ : typ B → I → I → I → (Susp∙ (typ A)) ⋀ B
inr-fill₁ b i j k =
hfill (λ k → λ {(i = i0) → inl tt
; (i = i1) → inr (rCancel (merid (pt A)) (~ k) j , b)
; (j = i0) → push (inr b) i
; (j = i1) → push (inr b) i})
(inS (push (inr b) i)) k
inr-fill : typ B → I → I → I → (Susp∙ (typ A)) ⋀ B
inr-fill b i j k =
hfill (λ k → λ {(i = i0) → inl tt
; (i = i1) → doubleCompPath-filler
(push (inr b))
(λ i₂ → inr (σ A (pt A) i₂ , b))
(sym (push (inr b))) k j
; (j = i0) → push (inr b) (~ k ∧ i)
; (j = i1) → push (inr b) (~ k ∧ i)})
(inS (inr-fill₁ b i j i1)) k
SuspL→Susp⋀ : (Susp∙ (typ A)) ⋀ B → Susp (A ⋀ B)
SuspL→Susp⋀ (inl x) = north
SuspL→Susp⋀ (inr (north , y)) = north
SuspL→Susp⋀ (inr (south , y)) = south
SuspL→Susp⋀ (inr (merid a i , y)) = merid (inr (a , y)) i
SuspL→Susp⋀ (push (inl north) i) = north
SuspL→Susp⋀ (push (inl south) i) = merid (inl tt) i
SuspL→Susp⋀ (push (inl (merid a j)) i) = merid-fill a i j i1
SuspL→Susp⋀ (push (inr x) i) = north
SuspL→Susp⋀ (push (push a i₁) i) = north
Susp⋀→SuspL : Susp (A ⋀ B) → (Susp∙ (typ A)) ⋀ B
Susp⋀→SuspL north = inl tt
Susp⋀→SuspL south = inl tt
Susp⋀→SuspL (merid (inl x) i) = inl tt
Susp⋀→SuspL (merid (inr (x , y)) i) =
(push (inr y) ∙∙ (λ i → inr (σ A x i , y)) ∙∙ sym (push (inr y))) i
Susp⋀→SuspL (merid (push (inl x) i₁) i) = inl-fill x i₁ i i1
Susp⋀→SuspL (merid (push (inr x) i₁) i) = inr-fill x i₁ i i1
Susp⋀→SuspL (merid (push (push a k) j) i) =
hcomp (λ r → λ {(i = i0) → inl-fill (snd A) j i r
; (i = i1) → inr-fill (snd B) j i r
; (j = i0) → inl tt
; (j = i1) → doubleCompPath-filler
(push (inr (pt B)))
(λ i₂ → inr (σ A (pt A) i₂ , (pt B)))
(sym (push (inr (pt B)))) r i
; (k = i0) → inl-fill (snd A) j i r
; (k = i1) → inr-fill (snd B) j i r})
(hcomp (λ r → λ {(i = i0) → push (push tt (r ∨ k)) j
; (i = i1) → push (push tt (r ∨ k)) j
; (j = i0) → inl tt
; (j = i1) → inr (rCancel (merid (pt A))
(~ r ∧ k) i , pt B)
; (k = i0) → inl-fill₁ (snd A) j i r
; (k = i1) → inr-fill₁ (snd B) j i r})
(hcomp (λ r → λ {(i = i0) → push (push tt k) j
; (i = i1) → push (push tt k) j
; (j = i0) → inl tt
; (j = i1) → inr (rCancel (merid (pt A))
(~ r ∨ k) i , pt B)
; (k = i0) → push (inl (rCancel (merid (pt A))
(~ r) i)) j
; (k = i1) → push (inr (snd B)) j})
(push (push tt k) j)))
SuspSmashCommIso : Iso (Susp∙ (typ A) ⋀ B) (Susp (A ⋀ B))
Iso.fun SuspSmashCommIso = SuspL→Susp⋀
Iso.inv SuspSmashCommIso = Susp⋀→SuspL
Iso.rightInv SuspSmashCommIso north = refl
Iso.rightInv SuspSmashCommIso south = merid (inl tt)
Iso.rightInv SuspSmashCommIso (merid a i) j =
hcomp (λ r → λ {(i = i0) → north
; (i = i1) → merid (inl tt) (j ∧ r)
; (j = i0) → f₁≡f₂ (~ r) .fst a i
; (j = i1) → compPath-filler
(merid a) (sym (merid (inl tt))) (~ r) i })
(f₂ .fst a i)
where
f₁ f₂ : (A ⋀∙ B) →∙ Ω (Susp∙ (A ⋀ B))
fst f₁ x = cong SuspL→Susp⋀ (cong Susp⋀→SuspL (merid x))
snd f₁ = refl
fst f₂ = σ (A ⋀∙ B)
snd f₂ = rCancel (merid (inl tt))
inr' : (Susp (typ A)) × (typ B) → (Susp∙ (typ A)) ⋀ B
inr' = inr
f₁≡f₂ : f₁ ≡ f₂
f₁≡f₂ =
⋀→∙Homogeneous≡ (isHomogeneousPath _ _)
λ x y
→ cong (cong SuspL→Susp⋀) (cong (push (inr y) ∙∙_∙∙ sym (push (inr y)))
(cong-∙ (λ x → inr' (x , y)) (merid x) (sym (merid (pt A)))))
∙∙ cong-∙∙ SuspL→Susp⋀ (push (inr y))
(cong (λ x → inr' (x , y)) (merid x)
∙ cong (λ x → inr' (x , y)) (sym (merid (pt A)))) (sym (push (inr y)))
∙∙ (sym (rUnit _)
∙ cong-∙ SuspL→Susp⋀
(λ i → inr' (merid x i , y)) (λ i → inr' (merid (pt A) (~ i) , y))
∙ cong (merid (inr (x , y)) ∙_)
λ j i → merid (push (inr y) (~ j)) (~ i) )
Iso.leftInv SuspSmashCommIso =
⋀-fun≡ _ _ refl
(λ x → main (snd x) (fst x))
(λ { north i j → sₙ i j i1
; south i j → sₛ i j i1
; (merid a k) i j → cube a j k i})
λ x i j → push (inr x) (i ∧ j)
where
inr' : Susp (typ A) × (typ B) → (Susp∙ (typ A)) ⋀ B
inr' = inr
sₙ : I → I → I → (Susp∙ (typ A)) ⋀ B
sₙ i j k =
hfill (λ k → λ {(i = i0) → inl tt
; (i = i1) → push (inr (pt B)) (j ∨ ~ k)
; (j = i0) → push (inr (pt B)) (i ∧ ~ k)
; (j = i1) → push (inl north) i})
(inS (push (push tt (~ j)) i))
k
sₛ : I → I → I → (Susp∙ (typ A)) ⋀ B
sₛ i j k =
hfill (λ k → λ {(i = i0) → inl tt
; (i = i1) → compPath-filler
(push (inr (pt B)))
(λ i → inr (merid (pt A) i , pt B)) k j
; (j = i0) → inl tt
; (j = i1) → push (inl (merid (pt A) k)) i})
(inS (sₙ i j i1))
k
filler : fst A → fst B → I → I → I → (Susp∙ (typ A)) ⋀ B
filler a y i j k =
hfill (λ k → λ {(i = i0) → push (inr y) j
; (i = i1) → compPath-filler
(push (inr y))
(λ i₁ → inr (merid (pt A) i₁ , y)) k j
; (j = i0) → Susp⋀→SuspL (SuspL→Susp⋀ (inr (merid a i , y)))
; (j = i1) → inr (compPath-filler
(merid a)
(sym (merid (pt A))) (~ k) i , y)})
(inS (doubleCompPath-filler
(push (inr y))
(λ i₁ → inr' (σ A a i₁ , y))
(sym (push (inr y))) (~ j) i)) k
cube₁ : (a : typ A)
→ Cube (λ k i → Iso.inv SuspSmashCommIso (merid-fill a k i i1))
(λ k i → inl-fill a k i i1)
(λ _ _ → inl tt)
refl
(λ _ _ → inl tt)
λ _ _ → inl tt
cube₁ a j k i =
hcomp (λ r → λ {(i = i0) → inl tt
; (i = i1) → inl tt
; (j = i0) → Iso.inv SuspSmashCommIso (merid-fill a k i r)
; (j = i1) → inl-fill a k i i1
; (k = i0) → inl tt
; (k = i1) → inl-fill a (j ∨ r) i i1})
(inl-fill a (j ∧ k) i i1)
cube : (a : typ A)
→ Cube (λ k i → Iso.inv SuspSmashCommIso
(Iso.fun SuspSmashCommIso (push (inl (merid a k)) i)))
(λ k i → push (inl (merid a k)) i)
(λ j i → sₙ i j i1) (λ j i → sₛ i j i1)
(λ j k → inl tt) (λ j k → filler a (pt B) k j i1)
cube a = (λ j k i → cube₁ a j i k) ◁
(λ j k i →
hcomp (λ r
→ λ {(i = i0) → inl tt
; (i = i1) → filler a (pt B) k j r
; (j = i0) → inl-fill a i k i1
; (j = i1) → push (inl (compPath-filler
(merid a) (sym (merid (pt A))) (~ r) k)) i
; (k = i0) → sₙ i j i1
; (k = i1) → sₛ i j r})
(hcomp (λ r
→ λ {(i = i0) → inl tt
; (i = i1) → doubleCompPath-filler
(push (inr (pt B)))
(λ i₁ → inr' (σ A a i₁ , (pt B)))
(sym (push (inr (pt B)))) (~ j ∧ r) k
; (j = i0) → inl-fill a i k r
; (j = i1) → push (inl (σ A a k)) i
; (k = i0) → sₙ i j r
; (k = i1) → sₙ i j r})
(hcomp (λ r
→ λ {(i = i0) → inl tt
; (i = i1) → inl-fill₁ a i k r
; (j = i0) → inl-fill₁ a i k r
; (j = i1) → push (inl (σ A a k)) i
; (k = i0) → push (push tt (r ∧ (~ j))) i
; (k = i1) → push (push tt (r ∧ (~ j))) i})
(push (inl (σ A a k)) i))))
main : (y : typ B) (x : Susp (typ A))
→ Susp⋀→SuspL (SuspL→Susp⋀ (inr (x , y))) ≡ inr (x , y)
main y north = push (inr y)
main y south = push (inr y) ∙ λ i → inr (merid (pt A) i , y)
main y (merid a i) j = filler a y i j i1