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

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@ -72,33 +72,13 @@ module _ { : Level} { : Category } (unprovable : IsCategory (RawCat
[ (λ _ x [ .𝟙 x ]) identityTrans (prshf c) ] [ (λ _ x [ .𝟙 x ]) identityTrans (prshf c) ]
eqTrans = funExt λ x funExt λ x .ident .proj₂ eqTrans = funExt λ x funExt λ x .ident .proj₂
eqNat : (λ i Natural (prshf c) (prshf c) (eqTrans i)) open import Cubical.NType.Properties
[(λ _ funExt (λ _ .assoc)) identityNatural (prshf c)] open import Cat.Categories.Fun
eqNat = λ i {A} {B} f :ident: : :func→: (.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 [ .𝟙 x ]) identityTrans (prshf c)
(𝓢 [ func→ (prshf c) f eqTrans i A ]) eq = funExt λ A funExt λ B proj₂ .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 [B,A] i' [A,c] →
-- let
-- k : [ {!!} , {!!} ]
-- k = [A,c]
-- in {! [ ? ∘ ? ]!}
:ident: : (:func→: (.𝟙 {c})) (Category.𝟙 Fun {A = prshf c})
:ident: = Σ≡ eqTrans eqNat
yoneda : Functor (Fun { = Opposite } {𝔻 = 𝓢}) yoneda : Functor (Fun { = Opposite } {𝔻 = 𝓢})
yoneda = record yoneda = record