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Category of categories #642

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merged 3 commits into from
Dec 15, 2021
Merged

Category of categories #642

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59a7856
Add CAT
barrettj12 Nov 22, 2021
60e08eb
Make `Cat` a precategory
barrettj12 Nov 24, 2021
05d399e
Name change
barrettj12 Dec 15, 2021
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25 changes: 25 additions & 0 deletions Cubical/Categories/Instances/Categories.agda
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-- The (pre)category of (small) categories
{-# OPTIONS --safe #-}

module Cubical.Categories.Instances.Categories where

open import Cubical.Categories.Category.Base
open import Cubical.Categories.Category.Precategory
open import Cubical.Categories.Functor.Base
open import Cubical.Categories.Functor.Properties
open import Cubical.Foundations.Prelude


module _ (ℓ ℓ' : Level) where
open Precategory

CatPrecategory : Precategory (ℓ-suc (ℓ-max ℓ ℓ')) (ℓ-max ℓ ℓ')
CatPrecategory .ob = Category ℓ ℓ'
CatPrecategory .Hom[_,_] = Functor
CatPrecategory .id = 𝟙⟨ _ ⟩
CatPrecategory ._⋆_ G H = H ∘F G
CatPrecategory .⋆IdL _ = F-lUnit
CatPrecategory .⋆IdR _ = F-rUnit
CatPrecategory .⋆Assoc _ _ _ = F-assoc

-- TODO: what is required for Functor C D to be a set?