56 lines
1.7 KiB
Agda
56 lines
1.7 KiB
Agda
module Cat.Categories.CwF where
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open import Cat.Prelude
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open import Cat.Category
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open import Cat.Category.Functor
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open import Cat.Categories.Fam
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open import Cat.Categories.Opposite
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module _ {ℓa ℓb : Level} where
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record CwF : Set (lsuc (ℓa ⊔ ℓb)) where
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-- "A category with families consists of"
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field
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-- "A base category"
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ℂ : Category ℓa ℓb
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module ℂ = Category ℂ
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-- It's objects are called contexts
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Contexts = ℂ.Object
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-- It's arrows are called substitutions
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Substitutions = ℂ.Arrow
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field
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-- A functor T
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T : Functor (opposite ℂ) (Fam ℓa ℓb)
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-- Empty context
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[] : ℂ.Terminal
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private
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module T = Functor T
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Type : (Γ : ℂ.Object) → Set ℓa
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Type Γ = fst (fst (T.omap Γ))
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module _ {Γ : ℂ.Object} {A : Type Γ} where
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-- module _ {A B : Object ℂ} {γ : ℂ [ A , B ]} where
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-- k : Σ (fst (omap T B) → fst (omap T A))
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-- (λ f →
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-- {x : fst (omap T B)} →
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-- snd (omap T B) x → snd (omap T A) (f x))
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-- k = T.fmap γ
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-- k₁ : fst (omap T B) → fst (omap T A)
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-- k₁ = fst k
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-- k₂ : ({x : fst (omap T B)} →
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-- snd (omap T B) x → snd (omap T A) (k₁ x))
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-- k₂ = snd k
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record ContextComprehension : Set (ℓa ⊔ ℓb) where
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field
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Γ&A : ℂ.Object
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proj1 : ℂ [ Γ&A , Γ ]
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-- proj2 : ????
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-- if γ : ℂ [ A , B ]
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-- then T .fmap γ (written T[γ]) interpret substitutions in types and terms respectively.
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-- field
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-- ump : {Δ : ℂ .Object} → (γ : ℂ [ Δ , Γ ])
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-- → (a : {!!}) → {!!}
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