40 lines
1.4 KiB
Agda
40 lines
1.4 KiB
Agda
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module Cat.Exponential 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 import Cat.Product
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open Category
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module _ {ℓ ℓ'} (ℂ : Category ℓ ℓ') {{hasProducts : HasProducts ℂ}} where
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open HasProducts hasProducts
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open Product hiding (obj)
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private
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_×p_ : (A B : Object ℂ) → Object ℂ
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_×p_ A B = Product.obj (product A B)
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module _ (B C : Object ℂ) where
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IsExponential : (Cᴮ : Object ℂ) → ℂ [ Cᴮ ×p B , C ] → Set (ℓ ⊔ ℓ')
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IsExponential Cᴮ eval = ∀ (A : Object ℂ) (f : ℂ [ A ×p B , C ])
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→ ∃![ f~ ] (ℂ [ eval ∘ parallelProduct f~ (Category.𝟙 ℂ)] ≡ f)
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record Exponential : Set (ℓ ⊔ ℓ') where
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field
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-- obj ≡ Cᴮ
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obj : Object ℂ
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eval : ℂ [ obj ×p B , C ]
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{{isExponential}} : IsExponential obj eval
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-- If I make this an instance-argument then the instance resolution
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-- algorithm goes into an infinite loop. Why?
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exponentialsHaveProducts : HasProducts ℂ
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exponentialsHaveProducts = hasProducts
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transpose : (A : Object ℂ) → ℂ [ A ×p B , C ] → ℂ [ A , obj ]
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transpose A f = proj₁ (isExponential A f)
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record HasExponentials {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') {{_ : HasProducts ℂ}} : Set (ℓ ⊔ ℓ') where
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field
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exponent : (A B : Object ℂ) → Exponential ℂ A B
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