78 lines
2 KiB
Agda
78 lines
2 KiB
Agda
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{-# OPTIONS --allow-unsolved-metas #-}
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module Cat.Categories.Cube where
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open import Level
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open import Data.Bool hiding (T)
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open import Data.Sum hiding ([_,_])
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open import Data.Unit
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open import Data.Empty
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open import Data.Product
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open import Cubical
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open import Function
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open import Relation.Nullary
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open import Relation.Nullary.Decidable
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open import Cat.Category hiding (Hom)
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open import Cat.Functor
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open import Cat.Equality
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open Equality.Data.Product
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-- See chapter 1 for a discussion on how presheaf categories are CwF's.
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-- See section 6.8 in Huber's thesis for details on how to implement the
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-- categorical version of CTT
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open Category hiding (_∘_)
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open Functor
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module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
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-- Ns is the "namespace"
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ℓo = (suc zero ⊔ ℓ)
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FiniteDecidableSubset : Set ℓ
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FiniteDecidableSubset = Ns → Dec ⊤
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isTrue : Bool → Set
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isTrue false = ⊥
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isTrue true = ⊤
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elmsof : FiniteDecidableSubset → Set ℓ
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elmsof P = Σ Ns (λ σ → True (P σ)) -- (σ : Ns) → isTrue (P σ)
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𝟚 : Set
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𝟚 = Bool
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module _ (I J : FiniteDecidableSubset) where
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private
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Hom' : Set ℓ
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Hom' = elmsof I → elmsof J ⊎ 𝟚
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isInl : {ℓa ℓb : Level} {A : Set ℓa} {B : Set ℓb} → A ⊎ B → Set
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isInl (inj₁ _) = ⊤
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isInl (inj₂ _) = ⊥
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Def : Set ℓ
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Def = (f : Hom') → Σ (elmsof I) (λ i → isInl (f i))
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rules : Hom' → Set ℓ
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rules f = (i j : elmsof I)
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→ case (f i) of λ
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{ (inj₁ (fi , _)) → case (f j) of λ
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{ (inj₁ (fj , _)) → fi ≡ fj → i ≡ j
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; (inj₂ _) → Lift ⊤
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}
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; (inj₂ _) → Lift ⊤
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}
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Hom = Σ Hom' rules
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-- The category of names and substitutions
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ℂ : Category ℓ ℓ -- ℓo (lsuc lzero ⊔ ℓo)
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ℂ = record
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{ Object = FiniteDecidableSubset
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-- { Object = Ns → Bool
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; Arrow = Hom
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; 𝟙 = λ { {o} → inj₁ , λ { (i , ii) (j , jj) eq → Σ≡ eq {!refl!} } }
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; _∘_ = {!!}
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; isCategory = {!!}
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}
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