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@ -309,109 +309,110 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
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open module 𝔻 = Category 𝔻
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open module F = IsFunctor (F .isFunctor)
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-- module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
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-- F = F×A .proj₁
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-- A = F×A .proj₂
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-- G = G×B .proj₁
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-- B = G×B .proj₂
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-- H = H×C .proj₁
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-- C = H×C .proj₂
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-- -- Not entirely clear what this is at this point:
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-- _P⊕_ = Category._∘_ (Product.obj (:obj: ×p ℂ)) {F×A} {G×B} {H×C}
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-- module _
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-- -- NaturalTransformation F G × ℂ .Arrow A B
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-- {θ×f : NaturalTransformation F G × ℂ [ A , B ]}
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-- {η×g : NaturalTransformation G H × ℂ [ B , C ]} where
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-- private
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-- θ : Transformation F G
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-- θ = proj₁ (proj₁ θ×f)
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-- θNat : Natural F G θ
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-- θNat = proj₂ (proj₁ θ×f)
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-- f : ℂ [ A , B ]
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-- f = proj₂ θ×f
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-- η : Transformation G H
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-- η = proj₁ (proj₁ η×g)
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-- ηNat : Natural G H η
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-- ηNat = proj₂ (proj₁ η×g)
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-- g : ℂ [ B , C ]
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-- g = proj₂ η×g
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module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
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F = F×A .proj₁
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A = F×A .proj₂
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G = G×B .proj₁
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B = G×B .proj₂
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H = H×C .proj₁
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C = H×C .proj₂
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-- Not entirely clear what this is at this point:
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_P⊕_ = Category._∘_ (Product.obj (:obj: ×p ℂ)) {F×A} {G×B} {H×C}
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module _
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-- NaturalTransformation F G × ℂ .Arrow A B
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{θ×f : NaturalTransformation F G × ℂ [ A , B ]}
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{η×g : NaturalTransformation G H × ℂ [ B , C ]} where
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private
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θ : Transformation F G
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θ = proj₁ (proj₁ θ×f)
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θNat : Natural F G θ
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θNat = proj₂ (proj₁ θ×f)
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f : ℂ [ A , B ]
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f = proj₂ θ×f
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η : Transformation G H
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η = proj₁ (proj₁ η×g)
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ηNat : Natural G H η
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ηNat = proj₂ (proj₁ η×g)
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g : ℂ [ B , C ]
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g = proj₂ η×g
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-- ηθNT : NaturalTransformation F H
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-- ηθNT = Category._∘_ Fun {F} {G} {H} (η , ηNat) (θ , θNat)
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ηθNT : NaturalTransformation F H
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ηθNT = Category._∘_ Fun {F} {G} {H} (η , ηNat) (θ , θNat)
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-- ηθ = proj₁ ηθNT
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-- ηθNat = proj₂ ηθNT
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ηθ = proj₁ ηθNT
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ηθNat = proj₂ ηθNT
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-- :distrib: :
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-- 𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ func→ F ( ℂ [ g ∘ f ] ) ]
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-- ≡ 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ]
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-- :distrib: = begin
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-- 𝔻 [ (ηθ C) ∘ func→ F (ℂ [ g ∘ f ]) ]
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-- ≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩
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-- 𝔻 [ func→ H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
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-- ≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.distrib) ⟩
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-- 𝔻 [ 𝔻 [ func→ H g ∘ func→ H f ] ∘ (ηθ A) ]
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-- ≡⟨ sym assoc ⟩
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-- 𝔻 [ func→ H g ∘ 𝔻 [ func→ H f ∘ ηθ A ] ]
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-- ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) assoc ⟩
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-- 𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ func→ H f ∘ η A ] ∘ θ A ] ]
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-- ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
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-- 𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ η B ∘ func→ G f ] ∘ θ A ] ]
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-- ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym assoc) ⟩
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-- 𝔻 [ func→ H g ∘ 𝔻 [ η B ∘ 𝔻 [ func→ G f ∘ θ A ] ] ]
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-- ≡⟨ assoc ⟩
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-- 𝔻 [ 𝔻 [ func→ H g ∘ η B ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
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-- ≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ func→ G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
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-- 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
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-- ≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ φ ]) (sym (θNat f)) ⟩
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-- 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ] ∎
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-- where
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-- open Category 𝔻
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-- module H = IsFunctor (H .isFunctor)
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:distrib: :
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𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ func→ F ( ℂ [ g ∘ f ] ) ]
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≡ 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ]
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:distrib: = begin
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𝔻 [ (ηθ C) ∘ func→ F (ℂ [ g ∘ f ]) ]
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≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩
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𝔻 [ func→ H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.distrib) ⟩
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𝔻 [ 𝔻 [ func→ H g ∘ func→ H f ] ∘ (ηθ A) ]
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≡⟨ sym assoc ⟩
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𝔻 [ func→ H g ∘ 𝔻 [ func→ H f ∘ ηθ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) assoc ⟩
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𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ func→ H f ∘ η A ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
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𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ η B ∘ func→ G f ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym assoc) ⟩
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𝔻 [ func→ H g ∘ 𝔻 [ η B ∘ 𝔻 [ func→ G f ∘ θ A ] ] ]
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≡⟨ assoc ⟩
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𝔻 [ 𝔻 [ func→ H g ∘ η B ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ func→ G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
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𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ φ ]) (sym (θNat f)) ⟩
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𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ] ∎
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where
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open Category 𝔻
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module H = IsFunctor (H .isFunctor)
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-- :eval: : Functor ((:obj: ×p ℂ) .Product.obj) 𝔻
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-- :eval: = record
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-- { raw = record
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-- { func* = :func*:
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-- ; func→ = λ {dom} {cod} → :func→: {dom} {cod}
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-- }
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-- ; isFunctor = record
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-- { ident = λ {o} → :ident: {o}
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-- ; distrib = λ {f u n k y} → :distrib: {f} {u} {n} {k} {y}
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-- }
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-- }
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:eval: : Functor ((:obj: ×p ℂ) .Product.obj) 𝔻
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:eval: = record
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{ raw = record
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{ func* = :func*:
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; func→ = λ {dom} {cod} → :func→: {dom} {cod}
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}
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; isFunctor = record
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{ ident = λ {o} → :ident: {o}
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; distrib = λ {f u n k y} → :distrib: {f} {u} {n} {k} {y}
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}
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}
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-- module _ (𝔸 : Category ℓ ℓ) (F : Functor ((𝔸 ×p ℂ) .Product.obj) 𝔻) where
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-- open HasProducts (hasProducts {ℓ} {ℓ}) renaming (_|×|_ to parallelProduct)
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module _ (𝔸 : Category ℓ ℓ) (F : Functor ((𝔸 ×p ℂ) .Product.obj) 𝔻) where
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open HasProducts (hasProducts {ℓ} {ℓ} unprovable) renaming (_|×|_ to parallelProduct)
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-- postulate
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-- transpose : Functor 𝔸 :obj:
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-- eq : Catℓ [ :eval: ∘ (parallelProduct transpose (𝟙 Catℓ {A = ℂ})) ] ≡ F
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-- -- eq : Catℓ [ :eval: ∘ (HasProducts._|×|_ hasProducts transpose (𝟙 Catℓ {o = ℂ})) ] ≡ F
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-- -- eq' : (Catℓ [ :eval: ∘
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-- -- (record { product = product } HasProducts.|×| transpose)
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-- -- (𝟙 Catℓ)
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-- -- ])
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-- -- ≡ F
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postulate
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transpose : Functor 𝔸 :obj:
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eq : Catℓ [ :eval: ∘ (parallelProduct transpose (𝟙 Catℓ {A = ℂ})) ] ≡ F
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-- eq : Catℓ [ :eval: ∘ (HasProducts._|×|_ hasProducts transpose (𝟙 Catℓ {o = ℂ})) ] ≡ F
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-- eq' : (Catℓ [ :eval: ∘
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-- (record { product = product } HasProducts.|×| transpose)
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-- (𝟙 Catℓ)
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-- ])
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-- ≡ F
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-- -- For some reason after `e8215b2c051062c6301abc9b3f6ec67106259758`
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-- -- `catTranspose` makes Agda hang. catTranspose : ∃![ F~ ] (Catℓ [
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-- -- :eval: ∘ (parallelProduct F~ (𝟙 Catℓ {o = ℂ}))] ≡ F) catTranspose =
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-- -- transpose , eq
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-- For some reason after `e8215b2c051062c6301abc9b3f6ec67106259758`
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-- `catTranspose` makes Agda hang. catTranspose : ∃![ F~ ] (Catℓ [
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-- :eval: ∘ (parallelProduct F~ (𝟙 Catℓ {o = ℂ}))] ≡ F) catTranspose =
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-- transpose , eq
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-- :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
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-- :isExponential: = {!catTranspose!}
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-- where
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-- open HasProducts (hasProducts {ℓ} {ℓ}) using (_|×|_)
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-- -- :isExponential: = λ 𝔸 F → transpose 𝔸 F , eq' 𝔸 F
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postulate :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
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-- :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
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-- :isExponential: = {!catTranspose!}
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-- where
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-- open HasProducts (hasProducts {ℓ} {ℓ} unprovable) using (_|×|_)
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-- :isExponential: = λ 𝔸 F → transpose 𝔸 F , eq' 𝔸 F
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-- -- :exponent: : Exponential (Cat ℓ ℓ) A B
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-- :exponent: : Exponential Catℓ ℂ 𝔻
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-- :exponent: = record
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-- { obj = :obj:
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-- ; eval = :eval:
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-- ; isExponential = :isExponential:
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-- }
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-- :exponent: : Exponential (Cat ℓ ℓ) A B
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:exponent: : Exponential Catℓ ℂ 𝔻
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:exponent: = record
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{ obj = :obj:
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; eval = :eval:
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; isExponential = :isExponential:
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}
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-- hasExponentials : HasExponentials (Cat ℓ ℓ)
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-- hasExponentials = record { exponent = :exponent: }
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hasExponentials : HasExponentials Catℓ
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hasExponentials = record { exponent = :exponent: }
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