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Factor out category

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Frederik Hanghøj Iversen 2018-03-08 10:23:37 +01:00
parent faf4c54188
commit 181bd1af53
1 changed files with 15 additions and 14 deletions
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29
src/Cat/Category/Product.agda
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@ -6,20 +6,21 @@ open import Data.Product as P hiding (_×_ ; proj₁ ; proj₂)
open import Cat.Category hiding (module Propositionality) open import Cat.Category hiding (module Propositionality)
open Category module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
module _ {ℓa ℓb : Level} where open Category ℂ
record RawProduct (ℂ : Category ℓa ℓb) (A B : Object ℂ) : Set (ℓa ⊔ ℓb) where
record RawProduct (A B : Object) : Set (ℓa ⊔ ℓb) where
no-eta-equality no-eta-equality
field field
obj : Object ℂ obj : Object
proj₁ : ℂ [ obj , A ] proj₁ : ℂ [ obj , A ]
proj₂ : ℂ [ obj , B ] proj₂ : ℂ [ obj , B ]
record IsProduct (ℂ : Category ℓa ℓb) {A B : Object ℂ} (raw : RawProduct ℂ A B) : Set (ℓa ⊔ ℓb) where record IsProduct {A B : Object} (raw : RawProduct A B) : Set (ℓa ⊔ ℓb) where
open RawProduct raw public open RawProduct raw public
field field
isProduct : ∀ {X : Object ℂ} (x₁ : ℂ [ X , A ]) (x₂ : ℂ [ X , B ]) isProduct : ∀ {X : Object} (x₁ : ℂ [ X , A ]) (x₂ : ℂ [ X , B ])
→ ∃![ x ] (ℂ [ proj₁ ∘ x ] ≡ x₁ P.× ℂ [ proj₂ ∘ x ] ≡ x₂) → ∃![ x ] (ℂ [ proj₁ ∘ x ] ≡ x₁ P.× ℂ [ proj₂ ∘ x ] ≡ x₂)
-- | Arrow product -- | Arrow product
@ -27,27 +28,27 @@ module _ {ℓa ℓb : Level} where
→ ℂ [ X , obj ] → ℂ [ X , obj ]
_P[_×_] π₁ π₂ = P.proj₁ (isProduct π₁ π₂) _P[_×_] π₁ π₂ = P.proj₁ (isProduct π₁ π₂)
record Product (ℂ : Category ℓa ℓb) (A B : Object ℂ) : Set (ℓa ⊔ ℓb) where record Product (A B : Object) : Set (ℓa ⊔ ℓb) where
field field
raw : RawProduct ℂ A B raw : RawProduct A B
isProduct : IsProduct ℂ {A} {B} raw isProduct : IsProduct {A} {B} raw
open IsProduct isProduct public open IsProduct isProduct public
record HasProducts (ℂ : Category ℓa ℓb) : Set (ℓa ⊔ ℓb) where record HasProducts : Set (ℓa ⊔ ℓb) where
field field
product : ∀ (A B : Object ℂ) → Product ℂ A B product : ∀ (A B : Object) → Product A B
module _ (A B : Object ℂ) where module _ (A B : Object) where
open Product (product A B) open Product (product A B)
_×_ : Object ℂ _×_ : Object
_×_ = obj _×_ = obj
-- | Parallel product of arrows -- | Parallel product of arrows
-- --
-- The product mentioned in awodey in Def 6.1 is not the regular product of -- The product mentioned in awodey in Def 6.1 is not the regular product of
-- arrows. It's a "parallel" product -- arrows. It's a "parallel" product
module _ {A A' B B' : Object ℂ} where module _ {A A' B B' : Object} where
open Product open Product
open Product (product A B) hiding (_P[_×_]) renaming (proj₁ to fst ; proj₂ to snd) open Product (product A B) hiding (_P[_×_]) renaming (proj₁ to fst ; proj₂ to snd)
_|×|_ : ℂ [ A , A' ] → ℂ [ B , B' ] → ℂ [ A × B , A' × B' ] _|×|_ : ℂ [ A , A' ] → ℂ [ B , B' ] → ℂ [ A × B , A' × B' ]