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Frederik Hanghøj Iversen 2018-02-23 10:44:23 +01:00
parent 3032dc6130
commit 9a4d79fa4e
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src/Cat/Categories/Cat.agda
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@ -309,109 +309,110 @@ module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where
open module 𝔻 = Category 𝔻 open module 𝔻 = Category 𝔻
open module F = IsFunctor (F .isFunctor) open module F = IsFunctor (F .isFunctor)
-- module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
-- F = F×A .proj₁ F = F×A .proj₁
-- A = F×A .proj₂ A = F×A .proj₂
-- G = G×B .proj₁ G = G×B .proj₁
-- B = G×B .proj₂ B = G×B .proj₂
-- H = H×C .proj₁ H = H×C .proj₁
-- C = H×C .proj₂ C = H×C .proj₂
-- -- Not entirely clear what this is at this point: -- Not entirely clear what this is at this point:
-- _P⊕_ = Category._∘_ (Product.obj (:obj: ×p ℂ)) {F×A} {G×B} {H×C} _P⊕_ = Category._∘_ (Product.obj (:obj: ×p ℂ)) {F×A} {G×B} {H×C}
-- module _ module _
-- -- NaturalTransformation F G × ℂ .Arrow A B -- NaturalTransformation F G × ℂ .Arrow A B
-- {θ×f : NaturalTransformation F G × ℂ [ A , B ]} {θ×f : NaturalTransformation F G × ℂ [ A , B ]}
-- {η×g : NaturalTransformation G H × ℂ [ B , C ]} where {η×g : NaturalTransformation G H × ℂ [ B , C ]} where
-- private private
-- θ : Transformation F G θ : Transformation F G
-- θ = proj₁ (proj₁ θ×f) θ = proj₁ (proj₁ θ×f)
-- θNat : Natural F G θ θNat : Natural F G θ
-- θNat = proj₂ (proj₁ θ×f) θNat = proj₂ (proj₁ θ×f)
-- f : ℂ [ A , B ] f : ℂ [ A , B ]
-- f = proj₂ θ×f f = proj₂ θ×f
-- η : Transformation G H η : Transformation G H
-- η = proj₁ (proj₁ η×g) η = proj₁ (proj₁ η×g)
-- ηNat : Natural G H η ηNat : Natural G H η
-- ηNat = proj₂ (proj₁ η×g) ηNat = proj₂ (proj₁ η×g)
-- g : ℂ [ B , C ] g : ℂ [ B , C ]
-- g = proj₂ η×g g = proj₂ η×g
-- ηθNT : NaturalTransformation F H ηθNT : NaturalTransformation F H
-- ηθNT = Category._∘_ Fun {F} {G} {H} (η , ηNat) (θ , θNat) ηθNT = Category._∘_ Fun {F} {G} {H} (η , ηNat) (θ , θNat)
-- ηθ = proj₁ ηθNT ηθ = proj₁ ηθNT
-- ηθNat = proj₂ ηθNT ηθNat = proj₂ ηθNT
-- :distrib: : :distrib: :
-- 𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ func→ F ( ℂ [ g ∘ f ] ) ] 𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ func→ F ( ℂ [ g ∘ f ] ) ]
-- ≡ 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ] ≡ 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ]
-- :distrib: = begin :distrib: = begin
-- 𝔻 [ (ηθ C) ∘ func→ F (ℂ [ g ∘ f ]) ] 𝔻 [ (ηθ C) ∘ func→ F (ℂ [ g ∘ f ]) ]
-- ≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩ ≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩
-- 𝔻 [ func→ H (ℂ [ g ∘ f ]) ∘ (ηθ A) ] 𝔻 [ func→ H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
-- ≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.distrib) ⟩ ≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.distrib) ⟩
-- 𝔻 [ 𝔻 [ func→ H g ∘ func→ H f ] ∘ (ηθ A) ] 𝔻 [ 𝔻 [ func→ H g ∘ func→ H f ] ∘ (ηθ A) ]
-- ≡⟨ sym assoc ⟩ ≡⟨ sym assoc ⟩
-- 𝔻 [ func→ H g ∘ 𝔻 [ func→ H f ∘ ηθ A ] ] 𝔻 [ func→ H g ∘ 𝔻 [ func→ H f ∘ ηθ A ] ]
-- ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) assoc ⟩ ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) assoc ⟩
-- 𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ func→ H f ∘ η A ] ∘ θ A ] ] 𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ func→ H f ∘ η A ] ∘ θ A ] ]
-- ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩ ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
-- 𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ η B ∘ func→ G f ] ∘ θ A ] ] 𝔻 [ func→ H g ∘ 𝔻 [ 𝔻 [ η B ∘ func→ G f ] ∘ θ A ] ]
-- ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym assoc) ⟩ ≡⟨ cong (λ φ → 𝔻 [ func→ H g ∘ φ ]) (sym assoc) ⟩
-- 𝔻 [ func→ H g ∘ 𝔻 [ η B ∘ 𝔻 [ func→ G f ∘ θ A ] ] ] 𝔻 [ func→ H g ∘ 𝔻 [ η B ∘ 𝔻 [ func→ G f ∘ θ A ] ] ]
-- ≡⟨ assoc ⟩ ≡⟨ assoc ⟩
-- 𝔻 [ 𝔻 [ func→ H g ∘ η B ] ∘ 𝔻 [ func→ G f ∘ θ A ] ] 𝔻 [ 𝔻 [ func→ H g ∘ η B ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
-- ≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ func→ G f ∘ θ A ] ]) (sym (ηNat g)) ⟩ ≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ func→ G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
-- 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ func→ G f ∘ θ A ] ] 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ func→ G f ∘ θ A ] ]
-- ≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ φ ]) (sym (θNat f)) ⟩ ≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ φ ]) (sym (θNat f)) ⟩
-- 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ] ∎ 𝔻 [ 𝔻 [ η C ∘ func→ G g ] ∘ 𝔻 [ θ B ∘ func→ F f ] ] ∎
-- where where
-- open Category 𝔻 open Category 𝔻
-- module H = IsFunctor (H .isFunctor) module H = IsFunctor (H .isFunctor)
-- :eval: : Functor ((:obj: ×p ℂ) .Product.obj) 𝔻 :eval: : Functor ((:obj: ×p ℂ) .Product.obj) 𝔻
-- :eval: = record :eval: = record
-- { raw = record { raw = record
-- { func* = :func*: { func* = :func*:
-- ; func→ = λ {dom} {cod} → :func→: {dom} {cod} ; func→ = λ {dom} {cod} → :func→: {dom} {cod}
-- } }
-- ; isFunctor = record ; isFunctor = record
-- { ident = λ {o} → :ident: {o} { ident = λ {o} → :ident: {o}
-- ; distrib = λ {f u n k y} → :distrib: {f} {u} {n} {k} {y} ; distrib = λ {f u n k y} → :distrib: {f} {u} {n} {k} {y}
-- } }
-- } }
-- module _ (𝔸 : Category ℓ ℓ) (F : Functor ((𝔸 ×p ℂ) .Product.obj) 𝔻) where module _ (𝔸 : Category ℓ ℓ) (F : Functor ((𝔸 ×p ℂ) .Product.obj) 𝔻) where
-- open HasProducts (hasProducts {ℓ} {ℓ}) renaming (_|×|_ to parallelProduct) open HasProducts (hasProducts {ℓ} {ℓ} unprovable) renaming (_|×|_ to parallelProduct)
-- postulate postulate
-- transpose : Functor 𝔸 :obj: transpose : Functor 𝔸 :obj:
-- eq : Catℓ [ :eval: ∘ (parallelProduct transpose (𝟙 Catℓ {A = ℂ})) ] ≡ F eq : Catℓ [ :eval: ∘ (parallelProduct transpose (𝟙 Catℓ {A = ℂ})) ] ≡ F
-- -- eq : Catℓ [ :eval: ∘ (HasProducts._|×|_ hasProducts transpose (𝟙 Catℓ {o = ℂ})) ] ≡ F -- eq : Catℓ [ :eval: ∘ (HasProducts._|×|_ hasProducts transpose (𝟙 Catℓ {o = ℂ})) ] ≡ F
-- -- eq' : (Catℓ [ :eval: ∘ -- eq' : (Catℓ [ :eval: ∘
-- -- (record { product = product } HasProducts.|×| transpose) -- (record { product = product } HasProducts.|×| transpose)
-- -- (𝟙 Catℓ) -- (𝟙 Catℓ)
-- -- ]) -- ])
-- -- ≡ F -- ≡ F
-- -- For some reason after `e8215b2c051062c6301abc9b3f6ec67106259758` -- For some reason after `e8215b2c051062c6301abc9b3f6ec67106259758`
-- -- `catTranspose` makes Agda hang. catTranspose : ∃![ F~ ] (Catℓ [ -- `catTranspose` makes Agda hang. catTranspose : ∃![ F~ ] (Catℓ [
-- -- :eval: ∘ (parallelProduct F~ (𝟙 Catℓ {o = ℂ}))] ≡ F) catTranspose = -- :eval: ∘ (parallelProduct F~ (𝟙 Catℓ {o = ℂ}))] ≡ F) catTranspose =
-- -- transpose , eq -- transpose , eq
-- :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval: postulate :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
-- :isExponential: = {!catTranspose!} -- :isExponential: : IsExponential Catℓ ℂ 𝔻 :obj: :eval:
-- where -- :isExponential: = {!catTranspose!}
-- open HasProducts (hasProducts {ℓ} {ℓ}) using (_|×|_) -- where
-- -- :isExponential: = λ 𝔸 F → transpose 𝔸 F , eq' 𝔸 F -- open HasProducts (hasProducts {ℓ} {ℓ} unprovable) using (_|×|_)
-- :isExponential: = λ 𝔸 F → transpose 𝔸 F , eq' 𝔸 F
-- -- :exponent: : Exponential (Cat ℓ ℓ) A B -- :exponent: : Exponential (Cat ℓ ℓ) A B
-- :exponent: : Exponential Catℓ ℂ 𝔻 :exponent: : Exponential Catℓ ℂ 𝔻
-- :exponent: = record :exponent: = record
-- { obj = :obj: { obj = :obj:
-- ; eval = :eval: ; eval = :eval:
-- ; isExponential = :isExponential: ; isExponential = :isExponential:
-- } }
-- hasExponentials : HasExponentials (Cat ℓ ℓ) hasExponentials : HasExponentials Catℓ
-- hasExponentials = record { exponent = :exponent: } hasExponentials = record { exponent = :exponent: }