{-# OPTIONS --without-K --safe #-}
module Categories.Functor.Construction.LiftSetoids where
open import Level using (Level; _⊔_; Lift; lift)
open import Relation.Binary.Bundles using (Setoid)
open import Function.Bundles using (Func; _⟨$⟩_)
open import Function using (_$_) renaming (id to idf)
open import Categories.Category.Instance.Setoids using (Setoids)
open import Categories.Functor.Core using (Functor)
open Func
private
variable
c ℓ : Level
LiftedSetoid : ∀ c′ ℓ′ → Setoid c ℓ → Setoid (c ⊔ c′) (ℓ ⊔ ℓ′)
LiftedSetoid c′ ℓ′ S = record
{ Carrier = Lift c′ Carrier
; _≈_ = λ where (lift x) (lift y) → Lift ℓ′ $ x ≈ y
; isEquivalence = record
{ refl = lift refl
; sym = λ where (lift eq) → lift $ sym eq
; trans = λ where (lift eq) (lift eq′) → lift $ trans eq eq′
}
}
where open Setoid S
LiftSetoids : ∀ c′ ℓ′ → Functor (Setoids c ℓ) (Setoids (c ⊔ c′) (ℓ ⊔ ℓ′))
LiftSetoids c′ ℓ′ = record
{ F₀ = LiftedSetoid c′ ℓ′
; F₁ = λ f → record
{ to = λ where (lift x) → lift $ f ⟨$⟩ x
; cong = λ where (lift eq) → lift $ cong f eq
}
; identity = λ {A} → lift $ refl A
; homomorphism = λ {_} {_} {Z} → lift $ refl Z
; F-resp-≈ = λ f≈g → lift f≈g
}
where
open Setoid