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commented out code with holes

This commit is contained in:
Andrea Vezzosi 2018-05-15 17:21:57 +02:00
parent 513d91ae4f
commit c75a1d5d8b
4 changed files with 74 additions and 71 deletions
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139
src/Cat/Categories/Fun.agda
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@ -1,6 +1,7 @@
{-# OPTIONS --allow-unsolved-metas --cubical --caching #-}
module Cat.Categories.Fun where
open import Cat.Prelude
open import Cat.Equivalence
@ -52,14 +53,14 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
lem : coe (pp {C}) 𝔻.identity ≡ f→g
lem = trans (𝔻.9-1-9-right {b = Functor.omap F C} 𝔻.identity p*) 𝔻.rightIdentity
idToNatTrans : NaturalTransformation F G
idToNatTrans = (λ C → coe pp 𝔻.identity) , λ f → begin
coe pp 𝔻.identity 𝔻.<<< F.fmap f ≡⟨ cong (𝔻._<<< F.fmap f) lem ⟩
-- Just need to show that f→g is a natural transformation
-- I know that it has an inverse; g→f
f→g 𝔻.<<< F.fmap f ≡⟨ {!lem!} ⟩
G.fmap f 𝔻.<<< f→g ≡⟨ cong (G.fmap f 𝔻.<<<_) (sym lem) ⟩
G.fmap f 𝔻.<<< coe pp 𝔻.identity ∎
-- idToNatTrans : NaturalTransformation F G
-- idToNatTrans = (λ C → coe pp 𝔻.identity) , λ f → begin
-- coe pp 𝔻.identity 𝔻.<<< F.fmap f ≡⟨ cong (𝔻._<<< F.fmap f) lem ⟩
-- -- Just need to show that f→g is a natural transformation
-- -- I know that it has an inverse; g→f
-- f→g 𝔻.<<< F.fmap f ≡⟨ {!lem!} ⟩
-- G.fmap f 𝔻.<<< f→g ≡⟨ cong (G.fmap f 𝔻.<<<_) (sym lem) ⟩
-- G.fmap f 𝔻.<<< coe pp 𝔻.identity ∎
module _ {A B : Functor ℂ 𝔻} where
module A = Functor A
@ -92,70 +93,70 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
U : (F : ℂ.Object → 𝔻.Object) → Set _
U F = {A B : ℂ.Object} → ℂ [ A , B ] → 𝔻 [ F A , F B ]
module _
(omap : ℂ.Object → 𝔻.Object)
(p : A.omap ≡ omap)
where
D : Set _
D = ( fmap : U omap)
→ ( let
raw-B : RawFunctor ℂ 𝔻
raw-B = record { omap = omap ; fmap = fmap }
)
→ (isF-B' : IsFunctor ℂ 𝔻 raw-B)
→ ( let
B' : Functor ℂ 𝔻
B' = record { raw = raw-B ; isFunctor = isF-B' }
)
→ (iso' : A ≊ B') → PathP (λ i → U (p i)) A.fmap fmap
-- D : Set _
-- D = PathP (λ i → U (p i)) A.fmap fmap
-- eeq : (λ f → A.fmap f) ≡ fmap
-- eeq = funExtImp (λ A → funExtImp (λ B → funExt (λ f → isofmap {A} {B} f)))
-- where
-- module _ {X : ℂ.Object} {Y : ℂ.Object} (f : ℂ [ X , Y ]) where
-- isofmap : A.fmap f ≡ fmap f
-- isofmap = {!ap!}
d : D A.omap refl
d = res
where
module _
( fmap : U A.omap )
( let
raw-B : RawFunctor ℂ 𝔻
raw-B = record { omap = A.omap ; fmap = fmap }
)
( isF-B' : IsFunctor ℂ 𝔻 raw-B )
( let
B' : Functor ℂ 𝔻
B' = record { raw = raw-B ; isFunctor = isF-B' }
)
( iso' : A ≊ B' )
where
module _ {X Y : ℂ.Object} (f : ℂ [ X , Y ]) where
step : {!!} 𝔻.≊ {!!}
step = {!!}
resres : A.fmap {X} {Y} f ≡ fmap {X} {Y} f
resres = {!!}
res : PathP (λ i → U A.omap) A.fmap fmap
res i {X} {Y} f = resres f i
-- module _
-- (omap : ℂ.Object → 𝔻.Object)
-- (p : A.omap ≡ omap)
-- where
-- D : Set _
-- D = ( fmap : U omap)
-- → ( let
-- raw-B : RawFunctor ℂ 𝔻
-- raw-B = record { omap = omap ; fmap = fmap }
-- )
-- → (isF-B' : IsFunctor ℂ 𝔻 raw-B)
-- → ( let
-- B' : Functor ℂ 𝔻
-- B' = record { raw = raw-B ; isFunctor = isF-B' }
-- )
-- → (iso' : A ≊ B') → PathP (λ i → U (p i)) A.fmap fmap
-- -- D : Set _
-- -- D = PathP (λ i → U (p i)) A.fmap fmap
-- -- eeq : (λ f → A.fmap f) ≡ fmap
-- -- eeq = funExtImp (λ A → funExtImp (λ B → funExt (λ f → isofmap {A} {B} f)))
-- -- where
-- -- module _ {X : ℂ.Object} {Y : ℂ.Object} (f : ℂ [ X , Y ]) where
-- -- isofmap : A.fmap f ≡ fmap f
-- -- isofmap = {!ap!}
-- d : D A.omap refl
-- d = res
-- where
-- module _
-- ( fmap : U A.omap )
-- ( let
-- raw-B : RawFunctor ℂ 𝔻
-- raw-B = record { omap = A.omap ; fmap = fmap }
-- )
-- ( isF-B' : IsFunctor ℂ 𝔻 raw-B )
-- ( let
-- B' : Functor ℂ 𝔻
-- B' = record { raw = raw-B ; isFunctor = isF-B' }
-- )
-- ( iso' : A ≊ B' )
-- where
-- module _ {X Y : ℂ.Object} (f : ℂ [ X , Y ]) where
-- step : {!!} 𝔻.≊ {!!}
-- step = {!!}
-- resres : A.fmap {X} {Y} f ≡ fmap {X} {Y} f
-- resres = {!!}
-- res : PathP (λ i → U A.omap) A.fmap fmap
-- res i {X} {Y} f = resres f i
fmapEq : PathP (λ i → U (omapEq i)) A.fmap B.fmap
fmapEq = pathJ D d B.omap omapEq B.fmap B.isFunctor iso
-- fmapEq : PathP (λ i → U (omapEq i)) A.fmap B.fmap
-- fmapEq = pathJ D d B.omap omapEq B.fmap B.isFunctor iso
rawEq : A.raw ≡ B.raw
rawEq i = record { omap = omapEq i ; fmap = fmapEq i }
-- rawEq : A.raw ≡ B.raw
-- rawEq i = record { omap = omapEq i ; fmap = fmapEq i }
private
f : (A ≡ B) → (A ≊ B)
f p = idToNatTrans p , idToNatTrans (sym p) , NaturalTransformation≡ A A (funExt (λ C → {!!})) , {!!}
g : (A ≊ B) → (A ≡ B)
g = Functor≡ ∘ rawEq
inv : AreInverses f g
inv = {!funExt λ p → ?!} , {!!}
iso : (A ≡ B) ≅ (A ≊ B)
iso = f , g , inv
-- private
-- f : (A ≡ B) → (A ≊ B)
-- f p = idToNatTrans p , idToNatTrans (sym p) , NaturalTransformation≡ A A (funExt (λ C → {!!})) , {!!}
-- g : (A ≊ B) → (A ≡ B)
-- g = Functor≡ ∘ rawEq
-- inv : AreInverses f g
-- inv = {!funExt λ p → ?!} , {!!}
postulate
iso : (A ≡ B) ≅ (A ≊ B)
-- iso = f , g , inv
univ : (A ≡ B) ≃ (A ≊ B)
univ = fromIsomorphism _ _ iso

2
src/Cat/Category.agda
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@ -44,7 +44,7 @@ open Cat.Equivalence
-- about these. The laws defined are the types the propositions - not the
-- witnesses to them!
record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
no-eta-equality
-- no-eta-equality
field
Object : Set ℓa
Arrow : Object → Object → Set ℓb

1
src/Cat/Category/Monad/Voevodsky.agda
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@ -4,6 +4,7 @@ This module provides construction 2.3 in [voe]
{-# OPTIONS --cubical --caching #-}
module Cat.Category.Monad.Voevodsky where
open import Cat.Prelude
open import Cat.Category

3
src/Cat/Category/Product.agda
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@ -1,6 +1,7 @@
{-# OPTIONS --cubical --caching #-}
module Cat.Category.Product where
open import Cat.Prelude as P hiding (_×_ ; fst ; snd)
open import Cat.Equivalence
@ -11,7 +12,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
module _ (A B : Object) where
record RawProduct : Set (ℓa ⊔ ℓb) where
no-eta-equality
-- no-eta-equality
field
object : Object
fst : ℂ [ object , A ]