Move properties of categories to Cat.Category.Properties
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@ -53,68 +53,26 @@ record Category (ℓ ℓ' : Level) : Set (lsuc (ℓ' ⊔ ℓ)) where
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open Category
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module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : ℂ .Object } where
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private
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open module ℂ = Category ℂ
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_+_ = ℂ._⊕_
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module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} where
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module _ { A B : ℂ .Object } where
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Isomorphism : (f : ℂ .Arrow A B) → Set ℓ'
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Isomorphism f = Σ[ g ∈ ℂ .Arrow B A ] ℂ ._⊕_ g f ≡ ℂ .𝟙 × ℂ ._⊕_ f g ≡ ℂ .𝟙
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Isomorphism : (f : ℂ.Arrow A B) → Set ℓ'
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Isomorphism f = Σ[ g ∈ ℂ.Arrow B A ] g ℂ.⊕ f ≡ ℂ.𝟙 × f + g ≡ ℂ.𝟙
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Epimorphism : {X : ℂ .Object } → (f : ℂ .Arrow A B) → Set ℓ'
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Epimorphism {X} f = ( g₀ g₁ : ℂ .Arrow B X ) → ℂ ._⊕_ g₀ f ≡ ℂ ._⊕_ g₁ f → g₀ ≡ g₁
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Epimorphism : {X : ℂ.Object } → (f : ℂ.Arrow A B) → Set ℓ'
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Epimorphism {X} f = ( g₀ g₁ : ℂ.Arrow B X ) → g₀ + f ≡ g₁ + f → g₀ ≡ g₁
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Monomorphism : {X : ℂ .Object} → (f : ℂ .Arrow A B) → Set ℓ'
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Monomorphism {X} f = ( g₀ g₁ : ℂ .Arrow X A ) → ℂ ._⊕_ f g₀ ≡ ℂ ._⊕_ f g₁ → g₀ ≡ g₁
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Monomorphism : {X : ℂ.Object} → (f : ℂ.Arrow A B) → Set ℓ'
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Monomorphism {X} f = ( g₀ g₁ : ℂ.Arrow X A ) → f + g₀ ≡ f + g₁ → g₀ ≡ g₁
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-- Isomorphism of objects
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_≅_ : (A B : Object ℂ) → Set ℓ'
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_≅_ A B = Σ[ f ∈ ℂ .Arrow A B ] (Isomorphism f)
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iso-is-epi : ∀ {X} (f : ℂ.Arrow A B) → Isomorphism f → Epimorphism {X = X} f
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iso-is-epi f (f- , left-inv , right-inv) g₀ g₁ eq =
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begin
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g₀ ≡⟨ sym (fst ident) ⟩
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g₀ + ℂ.𝟙 ≡⟨ cong (_+_ g₀) (sym right-inv) ⟩
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g₀ + (f + f-) ≡⟨ assoc ⟩
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(g₀ + f) + f- ≡⟨ cong (λ x → x + f-) eq ⟩
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(g₁ + f) + f- ≡⟨ sym assoc ⟩
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g₁ + (f + f-) ≡⟨ cong (_+_ g₁) right-inv ⟩
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g₁ + ℂ.𝟙 ≡⟨ fst ident ⟩
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g₁ ∎
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where
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open IsCategory ℂ.isCategory
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iso-is-mono : ∀ {X} (f : ℂ.Arrow A B ) → Isomorphism f → Monomorphism {X = X} f
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iso-is-mono f (f- , (left-inv , right-inv)) g₀ g₁ eq =
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begin
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g₀ ≡⟨ sym (snd ident) ⟩
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ℂ.𝟙 + g₀ ≡⟨ cong (λ x → x + g₀) (sym left-inv) ⟩
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(f- + f) + g₀ ≡⟨ sym assoc ⟩
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f- + (f + g₀) ≡⟨ cong (_+_ f-) eq ⟩
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f- + (f + g₁) ≡⟨ assoc ⟩
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(f- + f) + g₁ ≡⟨ cong (λ x → x + g₁) left-inv ⟩
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ℂ.𝟙 + g₁ ≡⟨ snd ident ⟩
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g₁ ∎
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where
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open IsCategory ℂ.isCategory
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iso-is-epi-mono : ∀ {X} (f : ℂ.Arrow A B ) → Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
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iso-is-epi-mono f iso = iso-is-epi f iso , iso-is-mono f iso
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{-
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epi-mono-is-not-iso : ∀ {ℓ ℓ'} → ¬ ((ℂ : Category {ℓ} {ℓ'}) {A B X : Object ℂ} (f : Arrow ℂ A B ) → Epimorphism {ℂ = ℂ} {X = X} f → Monomorphism {ℂ = ℂ} {X = X} f → Isomorphism {ℂ = ℂ} f)
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epi-mono-is-not-iso f =
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let k = f {!!} {!!} {!!} {!!}
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in {!!}
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-}
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-- Isomorphism of objects
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_≅_ : ∀ {ℓ ℓ'} {ℂ : Category ℓ ℓ'} (A B : Object ℂ) → Set ℓ'
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_≅_ {ℂ = ℂ} A B = Σ[ f ∈ ℂ .Arrow A B ] (Isomorphism {ℂ = ℂ} f)
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IsProduct : ∀ {ℓ ℓ'} (ℂ : Category ℓ ℓ') {A B obj : Object ℂ} (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) → Set (ℓ ⊔ ℓ')
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IsProduct ℂ {A = A} {B = B} π₁ π₂
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= ∀ {X : ℂ.Object} (x₁ : ℂ.Arrow X A) (x₂ : ℂ.Arrow X B)
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→ ∃![ x ] (π₁ ℂ.⊕ x ≡ x₁ × π₂ ℂ.⊕ x ≡ x₂)
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where
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open module ℂ = Category ℂ
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module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') {A B obj : Object ℂ} where
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IsProduct : (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) → Set (ℓ ⊔ ℓ')
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IsProduct π₁ π₂
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= ∀ {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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-- 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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@ -131,19 +89,7 @@ record Product {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} (A B : ℂ .Object)
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proj₂ : ℂ .Arrow obj B
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{{isProduct}} : IsProduct ℂ proj₁ proj₂
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-- Two pairs are equal if their components are equal.
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eqpair : ∀ {ℓa ℓb} {A : Set ℓa} {B : Set ℓb} {a a' : A} {b b' : B}
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→ a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
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eqpair eqa eqb i = eqa i , eqb i
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module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
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private
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instance
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_ : IsCategory (ℂ .Object) (flip (ℂ .Arrow)) (ℂ .𝟙) (flip (ℂ ._⊕_))
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_ = record { assoc = sym assoc ; ident = swap ident }
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where
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open IsCategory (ℂ .isCategory)
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Opposite : Category ℓ ℓ'
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Opposite =
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record
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@ -151,7 +97,10 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
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; Arrow = flip (ℂ .Arrow)
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; 𝟙 = ℂ .𝟙
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; _⊕_ = flip (ℂ ._⊕_)
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; isCategory = record { assoc = sym assoc ; ident = swap ident }
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}
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where
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open IsCategory (ℂ .isCategory)
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-- A consequence of no-eta-equality; `Opposite-is-involution` is no longer
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-- definitional - i.e.; you must match on the fields:
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@ -2,10 +2,53 @@
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module Cat.Category.Properties where
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open import Agda.Primitive
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open import Data.Product
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open import Cubical.PathPrelude
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open import Cat.Category
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open import Cat.Functor
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open import Cat.Categories.Sets
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module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : ℂ .Category.Object } {X : ℂ .Category.Object} (f : ℂ .Category.Arrow A B) where
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open Category ℂ
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open IsCategory (isCategory)
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iso-is-epi : Isomorphism {ℂ = ℂ} f → Epimorphism {ℂ = ℂ} {X = X} f
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iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq =
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begin
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g₀ ≡⟨ sym (proj₁ ident) ⟩
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g₀ ⊕ 𝟙 ≡⟨ cong (_⊕_ g₀) (sym right-inv) ⟩
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g₀ ⊕ (f ⊕ f-) ≡⟨ assoc ⟩
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(g₀ ⊕ f) ⊕ f- ≡⟨ cong (λ φ → φ ⊕ f-) eq ⟩
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(g₁ ⊕ f) ⊕ f- ≡⟨ sym assoc ⟩
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g₁ ⊕ (f ⊕ f-) ≡⟨ cong (_⊕_ g₁) right-inv ⟩
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g₁ ⊕ 𝟙 ≡⟨ proj₁ ident ⟩
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g₁ ∎
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iso-is-mono : Isomorphism {ℂ = ℂ} f → Monomorphism {ℂ = ℂ} {X = X} f
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iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
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begin
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g₀ ≡⟨ sym (proj₂ ident) ⟩
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𝟙 ⊕ g₀ ≡⟨ cong (λ φ → φ ⊕ g₀) (sym left-inv) ⟩
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(f- ⊕ f) ⊕ g₀ ≡⟨ sym assoc ⟩
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f- ⊕ (f ⊕ g₀) ≡⟨ cong (_⊕_ f-) eq ⟩
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f- ⊕ (f ⊕ g₁) ≡⟨ assoc ⟩
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(f- ⊕ f) ⊕ g₁ ≡⟨ cong (λ φ → φ ⊕ g₁) left-inv ⟩
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𝟙 ⊕ g₁ ≡⟨ proj₂ ident ⟩
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g₁ ∎
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iso-is-epi-mono : Isomorphism {ℂ = ℂ} f → Epimorphism {ℂ = ℂ} {X = X} f × Monomorphism {ℂ = ℂ} {X = X} f
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iso-is-epi-mono iso = iso-is-epi iso , iso-is-mono iso
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{-
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epi-mono-is-not-iso : ∀ {ℓ ℓ'} → ¬ ((ℂ : Category {ℓ} {ℓ'}) {A B X : Object ℂ} (f : Arrow ℂ A B ) → Epimorphism {ℂ = ℂ} {X = X} f → Monomorphism {ℂ = ℂ} {X = X} f → Isomorphism {ℂ = ℂ} f)
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epi-mono-is-not-iso f =
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let k = f {!!} {!!} {!!} {!!}
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in {!!}
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-}
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module _ {ℓa ℓa' ℓb ℓb'} where
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Exponential : Category ℓa ℓa' → Category ℓb ℓb' → Category {!!} {!!}
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Exponential A B = record
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@ -13,7 +56,7 @@ module _ {ℓa ℓa' ℓb ℓb'} where
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; Arrow = {!!}
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; 𝟙 = {!!}
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; _⊕_ = {!!}
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; isCategory = ?
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; isCategory = {!!}
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}
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_⇑_ = Exponential
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