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Frederik Hanghøj Iversen 2018-02-23 10:44:23 +01:00
parent 3032dc6130
commit 9a4d79fa4e

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