49 lines
1.6 KiB
Agda
49 lines
1.6 KiB
Agda
{-# OPTIONS --allow-unsolved-metas #-}
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module Cat.Categories.Fam 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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import Function
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open import Cat.Category
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open import Cat.Equality
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open Equality.Data.Product
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module _ (ℓa ℓb : Level) where
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private
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Obj = Σ[ A ∈ Set ℓa ] (A → Set ℓb)
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Arr : Obj → Obj → Set (ℓa ⊔ ℓb)
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Arr (A , B) (A' , B') = Σ[ f ∈ (A → A') ] ({x : A} → B x → B' (f x))
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one : {o : Obj} → Arr o o
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proj₁ one = λ x → x
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proj₂ one = λ b → b
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_∘_ : {a b c : Obj} → Arr b c → Arr a b → Arr a c
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(g , g') ∘ (f , f') = g Function.∘ f , g' Function.∘ f'
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_⟨_∘_⟩ : {a b : Obj} → (c : Obj) → Arr b c → Arr a b → Arr a c
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c ⟨ g ∘ f ⟩ = _∘_ {c = c} g f
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module _ {A B C D : Obj} {f : Arr A B} {g : Arr B C} {h : Arr C D} where
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assoc : (D ⟨ h ∘ C ⟨ g ∘ f ⟩ ⟩) ≡ D ⟨ D ⟨ h ∘ g ⟩ ∘ f ⟩
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assoc = Σ≡ refl refl
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module _ {A B : Obj} {f : Arr A B} where
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ident : B ⟨ f ∘ one ⟩ ≡ f × B ⟨ one {B} ∘ f ⟩ ≡ f
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ident = (Σ≡ refl refl) , Σ≡ refl refl
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instance
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isCategory : IsCategory Obj Arr one (λ {a b c} → _∘_ {a} {b} {c})
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isCategory = record
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{ assoc = λ {A} {B} {C} {D} {f} {g} {h} → assoc {D = D} {f} {g} {h}
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; ident = λ {A} {B} {f} → ident {A} {B} {f = f}
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}
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Fam : Category (lsuc (ℓa ⊔ ℓb)) (ℓa ⊔ ℓb)
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Fam = record
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{ Object = Obj
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; Arrow = Arr
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; 𝟙 = one
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; _∘_ = λ {a b c} → _∘_ {a} {b} {c}
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}
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