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