43 lines
1.4 KiB
Agda
43 lines
1.4 KiB
Agda
module Cat.Category.Exponential where
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open import Cat.Prelude hiding (_×_)
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open import Cat.Category
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open import Cat.Category.Product
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module _ {ℓ ℓ'} (ℂ : Category ℓ ℓ') {{hasProducts : HasProducts ℂ}} where
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open Category ℂ
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open HasProducts hasProducts public
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module _ (B C : Object) where
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record IsExponential'
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(Cᴮ : Object)
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(eval : ℂ [ Cᴮ × B , C ]) : Set (ℓ ⊔ ℓ') where
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field
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uniq
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: ∀ (A : Object) (f : ℂ [ A × B , C ])
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→ ∃![ f~ ] (ℂ [ eval ∘ f~ |×| Category.identity ℂ ] ≡ f)
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IsExponential : (Cᴮ : Object) → ℂ [ Cᴮ × B , C ] → Set (ℓ ⊔ ℓ')
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IsExponential Cᴮ eval = ∀ (A : Object) (f : ℂ [ A × B , C ])
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→ ∃![ f~ ] (ℂ [ eval ∘ f~ |×| Category.identity 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
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eval : ℂ [ obj × B , C ]
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{{isExponential}} : IsExponential obj eval
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transpose : (A : Object) → ℂ [ A × B , C ] → ℂ [ A , obj ]
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transpose A f = fst (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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open Category ℂ
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open Exponential public
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field
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exponent : (A B : Object) → Exponential ℂ A B
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_⇑_ : (A B : Object) → Object
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A ⇑ B = (exponent A B) .obj
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