diff --git a/src/Category/Rel.agda b/src/Category/Rel.agda index 7624da9..42ed17e 100644 --- a/src/Category/Rel.agda +++ b/src/Category/Rel.agda @@ -7,21 +7,6 @@ open import Cubical.GradLemma open import Agda.Primitive open import Category --- Sets are built-in to Agda. The set of all small sets is called Set. - -Fun : {ℓ : Level} → ( T U : Set ℓ ) → Set ℓ -Fun T U = T → U - -𝕊et-as-Cat : {ℓ : Level} → Category {lsuc ℓ} {ℓ} -𝕊et-as-Cat {ℓ} = record - { Object = Set ℓ - ; Arrow = λ T U → Fun {ℓ} T U - ; 𝟙 = λ x → x - ; _⊕_ = λ g f x → g ( f x ) - ; assoc = refl - ; ident = funExt (λ x → refl) , funExt (λ x → refl) - } - -- Subsets are predicates over some type. Subset : {ℓ : Level} → ( A : Set ℓ ) → Set (ℓ ⊔ lsuc lzero) Subset A = A → Set @@ -166,8 +151,8 @@ module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset ≡ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q)) assocc = equivToPath equi -Rel-as-Cat : Category -Rel-as-Cat = record +Rel : Category +Rel = record { Object = Set ; Arrow = λ S R → Subset (S × R) ; 𝟙 = λ {S} → Diag S @@ -175,17 +160,3 @@ Rel-as-Cat = record ; assoc = funExt assocc ; ident = funExt ident-l , funExt ident-r } - -module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ}} where - private - C-Obj = Object ℂ - _+_ = Arrow ℂ - - RepFunctor : Functor ℂ 𝕊et-as-Cat - RepFunctor = - record - { F = λ A → (B : C-Obj) → Hom {ℂ = ℂ} A B - ; f = λ { {c' = c'} f g → {!HomFromArrow {ℂ = } c' g!}} - ; ident = {!!} - ; distrib = {!!} - }