Minimize dependency on category of categories
This commit is contained in:
parent
8f8800cb67
commit
77006011d3
|
@ -65,7 +65,6 @@ module _ (ℓ ℓ' : Level) where
|
|||
module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
|
||||
module _ (ℂ 𝔻 : Category ℓ ℓ') where
|
||||
private
|
||||
Catt = Cat ℓ ℓ' unprovable
|
||||
:Object: = Object ℂ × Object 𝔻
|
||||
:Arrow: : :Object: → :Object: → Set ℓ'
|
||||
:Arrow: (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
|
||||
|
@ -105,19 +104,19 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
|
|||
:product: : Category ℓ ℓ'
|
||||
Category.raw :product: = :rawProduct:
|
||||
|
||||
proj₁ : Catt [ :product: , ℂ ]
|
||||
proj₁ : Functor :product: ℂ
|
||||
proj₁ = record
|
||||
{ raw = record { func* = fst ; func→ = fst }
|
||||
; isFunctor = record { isIdentity = refl ; isDistributive = refl }
|
||||
}
|
||||
|
||||
proj₂ : Catt [ :product: , 𝔻 ]
|
||||
proj₂ : Functor :product: 𝔻
|
||||
proj₂ = record
|
||||
{ raw = record { func* = snd ; func→ = snd }
|
||||
; isFunctor = record { isIdentity = refl ; isDistributive = refl }
|
||||
}
|
||||
|
||||
module _ {X : Object Catt} (x₁ : Catt [ X , ℂ ]) (x₂ : Catt [ X , 𝔻 ]) where
|
||||
module _ {X : Category ℓ ℓ'} (x₁ : Functor X ℂ) (x₂ : Functor X 𝔻) where
|
||||
x : Functor X :product:
|
||||
x = record
|
||||
{ raw = record
|
||||
|
@ -133,29 +132,31 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
|
|||
open module x₁ = Functor x₁
|
||||
open module x₂ = Functor x₂
|
||||
|
||||
isUniqL : Catt [ proj₁ ∘ x ] ≡ x₁
|
||||
isUniqL : F[ proj₁ ∘ x ] ≡ x₁
|
||||
isUniqL = Functor≡ eq* eq→
|
||||
where
|
||||
eq* : (Catt [ proj₁ ∘ x ]) .func* ≡ x₁ .func*
|
||||
eq* : (F[ proj₁ ∘ x ]) .func* ≡ x₁ .func*
|
||||
eq* = refl
|
||||
eq→ : (λ i → {A : Object X} {B : Object X} → X [ A , B ] → ℂ [ eq* i A , eq* i B ])
|
||||
[ (Catt [ proj₁ ∘ x ]) .func→ ≡ x₁ .func→ ]
|
||||
[ (F[ proj₁ ∘ x ]) .func→ ≡ x₁ .func→ ]
|
||||
eq→ = refl
|
||||
|
||||
isUniqR : Catt [ proj₂ ∘ x ] ≡ x₂
|
||||
isUniqR : F[ proj₂ ∘ x ] ≡ x₂
|
||||
isUniqR = Functor≡ refl refl
|
||||
|
||||
isUniq : Catt [ proj₁ ∘ x ] ≡ x₁ × Catt [ proj₂ ∘ x ] ≡ x₂
|
||||
isUniq : F[ proj₁ ∘ x ] ≡ x₁ × F[ proj₂ ∘ x ] ≡ x₂
|
||||
isUniq = isUniqL , isUniqR
|
||||
|
||||
uniq : ∃![ x ] (Catt [ proj₁ ∘ x ] ≡ x₁ × Catt [ proj₂ ∘ x ] ≡ x₂)
|
||||
uniq : ∃![ x ] (F[ proj₁ ∘ x ] ≡ x₁ × F[ proj₂ ∘ x ] ≡ x₂)
|
||||
uniq = x , isUniq
|
||||
|
||||
Catℓ = Cat ℓ ℓ' unprovable
|
||||
|
||||
instance
|
||||
isProduct : IsProduct Catt proj₁ proj₂
|
||||
isProduct : IsProduct Catℓ proj₁ proj₂
|
||||
isProduct = uniq
|
||||
|
||||
product : Product {ℂ = Catt} ℂ 𝔻
|
||||
product : Product {ℂ = Catℓ} ℂ 𝔻
|
||||
product = record
|
||||
{ obj = :product:
|
||||
; proj₁ = proj₁
|
||||
|
|
Loading…
Reference in a new issue