51 lines
2 KiB
Agda
51 lines
2 KiB
Agda
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module Cat.Product where
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open import Agda.Primitive
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open import Data.Product
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open import Cubical
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open import Cat.Category
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open Category
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module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') {A B obj : Object ℂ} where
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IsProduct : (π₁ : ℂ [ obj , A ]) (π₂ : ℂ [ obj , B ]) → Set (ℓ ⊔ ℓ')
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IsProduct π₁ π₂
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= ∀ {X : Object ℂ} (x₁ : ℂ [ X , A ]) (x₂ : ℂ [ X , B ])
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→ ∃![ x ] (ℂ [ π₁ ∘ x ] ≡ x₁ × ℂ [ π₂ ∘ x ] ≡ x₂)
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-- Tip from Andrea; Consider this style for efficiency:
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-- record IsProduct {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'})
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-- {A B obj : Object ℂ} (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) : Set (ℓ ⊔ ℓ') where
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-- field
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-- isProduct : ∀ {X : ℂ .Object} (x₁ : ℂ .Arrow X A) (x₂ : ℂ .Arrow X B)
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-- → ∃![ x ] (ℂ ._⊕_ π₁ x ≡ x₁ × ℂ. _⊕_ π₂ x ≡ x₂)
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record Product {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} (A B : Object ℂ) : Set (ℓ ⊔ ℓ') where
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no-eta-equality
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field
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obj : Object ℂ
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proj₁ : ℂ [ obj , A ]
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proj₂ : ℂ [ obj , B ]
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{{isProduct}} : IsProduct ℂ proj₁ proj₂
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arrowProduct : ∀ {X} → (π₁ : ℂ [ X , A ]) (π₂ : ℂ [ X , B ])
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→ ℂ [ X , obj ]
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arrowProduct π₁ π₂ = proj₁ (isProduct π₁ π₂)
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record HasProducts {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') : Set (ℓ ⊔ ℓ') where
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field
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product : ∀ (A B : Object ℂ) → Product {ℂ = ℂ} A B
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open Product
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objectProduct : (A B : Object ℂ) → Object ℂ
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objectProduct A B = Product.obj (product A B)
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-- The product mentioned in awodey in Def 6.1 is not the regular product of arrows.
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-- It's a "parallel" product
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parallelProduct : {A A' B B' : Object ℂ} → ℂ [ A , A' ] → ℂ [ B , B' ]
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→ ℂ [ objectProduct A B , objectProduct A' B' ]
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parallelProduct {A = A} {A' = A'} {B = B} {B' = B'} a b = arrowProduct (product A' B')
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(ℂ [ a ∘ (product A B) .proj₁ ])
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(ℂ [ b ∘ (product A B) .proj₂ ])
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