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Proove identity laws for natural transformations

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Frederik Hanghøj Iversen 2018-02-16 12:46:25 +01:00
parent a64e2484e3
commit 73ab4d1836
1 changed files with 13 additions and 2 deletions
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15
src/Cat/Categories/Fun.agda
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@ -142,10 +142,21 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
open IsCategory (isCategory 𝔻) open IsCategory (isCategory 𝔻)
module _ {A B : Functor ℂ 𝔻} {f : NaturalTransformation A B} where module _ {A B : Functor ℂ 𝔻} {f : NaturalTransformation A B} where
private
allNatural = naturalIsProp {F = A} {B}
f' = proj₁ f
module 𝔻Data = Category 𝔻
eq-r : ∀ C → (𝔻 [ f' C ∘ identityTrans A C ]) ≡ f' C
eq-r C = begin
𝔻 [ f' C ∘ identityTrans A C ] ≡⟨⟩
𝔻 [ f' C ∘ 𝔻Data.𝟙 ] ≡⟨ proj₁ (𝔻.ident {A} {B}) ⟩
f' C ∎
eq-l : ∀ C → (𝔻 [ identityTrans B C ∘ f' C ]) ≡ f' C
eq-l C = proj₂ (𝔻.ident {A} {B})
ident-r : (_:⊕:_ {A} {A} {B} f (identityNat A)) ≡ f ident-r : (_:⊕:_ {A} {A} {B} f (identityNat A)) ≡ f
ident-r = {!!} ident-r = lemSig allNatural _ _ (funExt eq-r)
ident-l : (_:⊕:_ {A} {B} {B} (identityNat B) f) ≡ f ident-l : (_:⊕:_ {A} {B} {B} (identityNat B) f) ≡ f
ident-l = {!!} ident-l = lemSig allNatural _ _ (funExt eq-l)
:ident: :ident:
: (_:⊕:_ {A} {A} {B} f (identityNat A)) ≡ f : (_:⊕:_ {A} {A} {B} f (identityNat A)) ≡ f
× (_:⊕:_ {A} {B} {B} (identityNat B) f) ≡ f × (_:⊕:_ {A} {B} {B} (identityNat B) f) ≡ f