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Readd stuff about the yoneda embedding

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Frederik Hanghøj Iversen 2018-02-23 11:23:21 +01:00
parent 954a89f8d1
commit de1d19c442
1 changed files with 7 additions and 27 deletions
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34
src/Cat/Category/Properties.agda
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@ -72,33 +72,13 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} (unprovable : IsCategory (RawCat
[ (λ _ x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c) ] [ (λ _ x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c) ]
eqTrans = funExt λ x → funExt λ x → ℂ.ident .proj₂ eqTrans = funExt λ x → funExt λ x → ℂ.ident .proj₂
eqNat : (λ i → Natural (prshf c) (prshf c) (eqTrans i)) open import Cubical.NType.Properties
[(λ _ → funExt (λ _ → ℂ.assoc)) ≡ identityNatural (prshf c)] open import Cat.Categories.Fun
eqNat = λ i {A} {B} f → :ident: : :func→: (ℂ.𝟙 {c}) ≡ Category.𝟙 Fun {A = prshf c}
let :ident: = lemSig (naturalIsProp {F = prshf c} {prshf c}) _ _ eq
open Category 𝓢 where
lemm : (𝓢 [ eqTrans i B ∘ func→ (prshf c) f ]) ≡ eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c)
(𝓢 [ func→ (prshf c) f ∘ eqTrans i A ]) eq = funExt λ A → funExt λ B → proj₂ ℂ.ident
lemm = {!!}
lem : (λ _ → 𝓢 [ Functor.func* (prshf c) A , func* (prshf c) B ])
[ 𝓢 [ eqTrans i B ∘ func→ (prshf c) f ]
≡ 𝓢 [ func→ (prshf c) f ∘ eqTrans i A ] ]
lem
= arrowIsSet _ _ lemm _ i
-- (Sets [ eqTrans i B ∘ prshf c .func→ f ])
-- (Sets [ prshf c .func→ f ∘ eqTrans i A ])
-- lemm
-- _ i
in
lem
-- eqNat = λ {A} {B} i ℂ[B,A] i' ℂ[A,c] →
-- let
-- k : ℂ [ {!!} , {!!} ]
-- k = ℂ[A,c]
-- in {!ℂ [ ? ∘ ? ]!}
:ident: : (:func→: (ℂ.𝟙 {c})) ≡ (Category.𝟙 Fun {A = prshf c})
:ident: = Σ≡ eqTrans eqNat
yoneda : Functor ℂ (Fun {ℂ = Opposite ℂ} {𝔻 = 𝓢}) yoneda : Functor ℂ (Fun {ℂ = Opposite ℂ} {𝔻 = 𝓢})
yoneda = record yoneda = record