56 lines
1.8 KiB
Agda
56 lines
1.8 KiB
Agda
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{-# OPTIONS --cubical #-}
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module Category.Categories.Cat where
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open import Agda.Primitive
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open import Cubical
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open import Function
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open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
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open import Category
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-- The category of categories
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module _ {ℓ ℓ' : Level} where
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private
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_⊛_ = functor-comp
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module _ {A B C D : Category {ℓ} {ℓ'}} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
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assc : h ⊛ (g ⊛ f) ≡ (h ⊛ g) ⊛ f
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assc = {!!}
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module _ {A B : Category {ℓ} {ℓ'}} where
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lift-eq : (f g : Functor A B)
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→ (eq* : Functor.func* f ≡ Functor.func* g)
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-- TODO: Must transport here using the equality from above.
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-- Reason:
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-- func→ : Arrow A dom cod → Arrow B (func* dom) (func* cod)
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-- func→₁ : Arrow A dom cod → Arrow B (func*₁ dom) (func*₁ cod)
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-- In other words, func→ and func→₁ does not have the same type.
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-- → Functor.func→ f ≡ Functor.func→ g
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-- → Functor.ident f ≡ Functor.ident g
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-- → Functor.distrib f ≡ Functor.distrib g
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→ f ≡ g
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lift-eq
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(functor func* func→ idnt distrib)
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(functor func*₁ func→₁ idnt₁ distrib₁)
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eq-func* = {!!}
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module _ {A B : Category {ℓ} {ℓ'}} {f : Functor A B} where
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idHere = identity {ℓ} {ℓ'} {A}
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lem : (Functor.func* f) ∘ (Functor.func* idHere) ≡ Functor.func* f
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lem = refl
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ident-r : f ⊛ identity ≡ f
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ident-r = lift-eq (f ⊛ identity) f refl
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ident-l : identity ⊛ f ≡ f
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ident-l = {!!}
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CatCat : Category {lsuc (ℓ ⊔ ℓ')} {ℓ ⊔ ℓ'}
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CatCat =
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record
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{ Object = Category {ℓ} {ℓ'}
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; Arrow = Functor
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; 𝟙 = identity
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; _⊕_ = functor-comp
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; assoc = {!!}
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; ident = ident-r , ident-l
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}
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