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✨ Show that Setoids is an instance of our ambient category
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2 changed files with 117 additions and 10 deletions
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@ -1,3 +1,111 @@
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<!--
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```agda
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module Category.Ambient.Setoid where
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open import Level using (lift; Lift) renaming (suc to ℓ-suc)
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open import Category.Ambient
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open import Categories.Category.Instance.Setoids
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open import Categories.Category.Instance.Properties.Setoids.Extensive
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open import Categories.Category.Instance.Properties.Setoids.CCC
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open import Categories.Category.Monoidal.Instance.Setoids
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open import Categories.Category.Instance.SingletonSet
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open import Categories.Object.NaturalNumbers
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open import Relation.Binary
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open import Data.Unit.Polymorphic using (tt; ⊤)
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open import Data.Empty.Polymorphic using (⊥)
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open import Function.Equality as SΠ renaming (id to idₛ)
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open import Categories.Object.NaturalNumbers.Properties.Parametrized
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open import Categories.Category.Cocartesian
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import Relation.Binary.Reasoning.Setoid as SetoidR
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_)
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```
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-->
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# The category of Setoids can be used as instance for ambient category
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Most of the required properties are already proven in the agda-categories library, we are only left to construct the natural numbers object.
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```agda
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module Category.Ambient.Setoid {ℓ} where
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-- equality on setoid functions
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_≋_ : ∀ {A B : Setoid ℓ ℓ} → A ⟶ B → A ⟶ B → Set ℓ
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_≋_ {A} {B} f g = Setoid._≈_ (A ⇨ B) f g
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≋-sym : ∀ {A B : Setoid ℓ ℓ} {f g : A ⟶ B} → f ≋ g → g ≋ f
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≋-sym {A} {B} {f} {g} = IsEquivalence.sym (Setoid.isEquivalence (A ⇨ B)) {f} {g}
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≋-trans : ∀ {A B : Setoid ℓ ℓ} {f g h : A ⟶ B} → f ≋ g → g ≋ h → f ≋ h
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≋-trans {A} {B} {f} {g} {h} = IsEquivalence.trans (Setoid.isEquivalence (A ⇨ B)) {f} {g} {h}
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-- we define ℕ ourselves, instead of importing it, to avoid lifting the universe lifting (builtin Nats are defined on Set₀)
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data ℕ : Set ℓ where
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zero : ℕ
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suc : ℕ → ℕ
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suc-cong : ∀ {n m} → n ≡ m → suc n ≡ suc m
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suc-cong n≡m rewrite n≡m = Eq.refl
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suc-inj : ∀ {n m} → suc n ≡ suc m → n ≡ m
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suc-inj Eq.refl = Eq.refl
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ℕ-eq : Rel ℕ ℓ
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ℕ-eq zero zero = ⊤
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ℕ-eq zero (suc m) = ⊥
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ℕ-eq (suc n) zero = ⊥
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ℕ-eq (suc n) (suc m) = ℕ-eq n m
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⊤-setoid : Setoid ℓ ℓ
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⊤-setoid = record { Carrier = ⊤ ; _≈_ = _≡_ ; isEquivalence = Eq.isEquivalence }
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ℕ-setoid : Setoid ℓ ℓ
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ℕ-setoid = record
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{ Carrier = ℕ
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; _≈_ = _≡_ -- ℕ-eq
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; isEquivalence = Eq.isEquivalence
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}
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zero⟶ : SingletonSetoid {ℓ} {ℓ} ⟶ ℕ-setoid
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zero⟶ = record { _⟨$⟩_ = λ _ → zero ; cong = λ x → Eq.refl }
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suc⟶ : ℕ-setoid ⟶ ℕ-setoid
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suc⟶ = record { _⟨$⟩_ = suc ; cong = suc-cong }
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ℕ-universal : ∀ {A : Setoid ℓ ℓ} → SingletonSetoid {ℓ} {ℓ} ⟶ A → A ⟶ A → ℕ-setoid ⟶ A
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ℕ-universal {A} z s = record { _⟨$⟩_ = app ; cong = cong' }
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where
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app : ℕ → Setoid.Carrier A
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app zero = z ⟨$⟩ tt
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app (suc n) = s ⟨$⟩ (app n)
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cong' : ∀ {n m : ℕ} → n ≡ m → Setoid._≈_ A (app n) (app m)
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cong' Eq.refl = IsEquivalence.refl (Setoid.isEquivalence A)
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ℕ-z-commute : ∀ {A : Setoid ℓ ℓ} {q : SingletonSetoid {ℓ} {ℓ} ⟶ A} {f : A ⟶ A} → q ≋ (ℕ-universal q f ∘ zero⟶)
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ℕ-z-commute {A} {q} {f} {lift tt} _ = IsEquivalence.refl (Setoid.isEquivalence A)
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ℕ-s-commute : ∀ {A : Setoid ℓ ℓ} {q : SingletonSetoid {ℓ} {ℓ} ⟶ A} {f : A ⟶ A} → (f ∘ (ℕ-universal q f)) ≋ (ℕ-universal q f ∘ suc⟶)
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ℕ-s-commute {A} {q} {f} {n} {m} n≡m rewrite n≡m = IsEquivalence.refl (Setoid.isEquivalence A)
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ℕ-unique : ∀ {A : Setoid ℓ ℓ} {q : SingletonSetoid {ℓ} {ℓ} ⟶ A} {f : A ⟶ A} {u : ℕ-setoid ⟶ A} → q ≋ (u ∘ zero⟶) → (f ∘ u) ≋ (u ∘ suc⟶) → u ≋ ℕ-universal q f
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ℕ-unique {A} {q} {f} {u} qz fs {zero} {zero} Eq.refl = (≋-sym {SingletonSetoid} {A} {q} {u ∘ zero⟶} qz) tt
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ℕ-unique {A} {q} {f} {u} qz fs {suc n} {suc m} Eq.refl = AR.begin
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u ⟨$⟩ suc n AR.≈⟨ ≋-sym {ℕ-setoid} {A} {f ∘ u} {u ∘ suc⟶} fs Eq.refl ⟩
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f ∘ u ⟨$⟩ m AR.≈⟨ cong f (ℕ-unique {A} {q} {f} {u} qz fs {n} {m} Eq.refl) ⟩
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f ∘ ℕ-universal q f ⟨$⟩ n AR.≈⟨ ℕ-s-commute {A} {q} {f} {n} Eq.refl ⟩
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ℕ-universal q f ⟨$⟩ suc m AR.∎
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where module AR = SetoidR A
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setoidNNO : NNO (Setoids ℓ ℓ) SingletonSetoid-⊤
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setoidNNO = record
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{ N = ℕ-setoid
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; isNNO = record
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{ z = zero⟶
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; s = suc⟶
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; universal = ℕ-universal
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; z-commute = λ {A} {q} {f} → ℕ-z-commute {A} {q} {f}
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; s-commute = λ {A} {q} {f} → ℕ-s-commute {A} {q} {f}
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; unique = λ {A} {q} {f} {u} qz fs → ℕ-unique {A} {q} {f} {u} qz fs
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}
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}
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-- TODO might need explicit PNNO definition if we later have to work with them
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setoidAmbient : Ambient (ℓ-suc ℓ) ℓ ℓ
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setoidAmbient = record { C = Setoids ℓ ℓ ; extensive = Setoids-Extensive ℓ ; cartesian = Setoids-Cartesian ; ℕ = NNO×CCC⇒PNNO (record { U = Setoids ℓ ℓ ; cartesianClosed = Setoids-CCC ℓ }) (Cocartesian.coproducts (Setoids-Cocartesian {ℓ} {ℓ})) setoidNNO }
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```
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@ -16,9 +16,8 @@ open import Relation.Binary
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Data.Sum.Function.Setoid
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open import Data.Sum.Relation.Binary.Pointwise
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open import Agda.Builtin.Unit using (tt)
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open import Data.Unit.Polymorphic using () renaming (⊤ to ⊤ₚ)
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open import Data.Empty.Polymorphic using () renaming (⊥ to ⊥ₚ)
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open import Data.Unit.Polymorphic using (tt; ⊤)
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open import Data.Empty.Polymorphic using (⊥)
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open import Categories.NaturalTransformation using (ntHelper)
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open import Function.Base using (id)
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```
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@ -37,9 +36,9 @@ Assuming the axiom of choice, the maybe monad is an instance of K in the categor
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just : A → Maybe A
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maybe-eq : ∀ (A : Setoid c ℓ) → Maybe (Setoid.Carrier A) → Maybe (Setoid.Carrier A) → Set ℓ
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maybe-eq _ nothing nothing = ⊤ₚ
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maybe-eq _ nothing (just y) = ⊥ₚ
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maybe-eq _ (just x) nothing = ⊥ₚ
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maybe-eq _ nothing nothing = ⊤
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maybe-eq _ nothing (just y) = ⊥
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maybe-eq _ (just x) nothing = ⊥
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maybe-eq A (just x) (just y) = Setoid._≈_ A x y
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maybeSetoid : Setoid c ℓ → Setoid c ℓ
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@ -47,7 +46,7 @@ Assuming the axiom of choice, the maybe monad is an instance of K in the categor
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where
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module A = Setoid A
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refl' : Reflexive (maybe-eq A)
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refl' {nothing} = lift tt
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refl' {nothing} = tt
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refl' {just x} = IsEquivalence.refl A.isEquivalence
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sym' : Symmetric (maybe-eq A)
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sym' {nothing} {nothing} = id
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@ -57,7 +56,7 @@ Assuming the axiom of choice, the maybe monad is an instance of K in the categor
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trans' : Transitive (maybe-eq A)
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trans' {nothing} {nothing} {nothing} = λ _ → id
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trans' {nothing} {nothing} {just z} = λ _ → id
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trans' {nothing} {just y} {nothing} = λ _ _ → lift tt
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trans' {nothing} {just y} {nothing} = λ _ _ → tt
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trans' {nothing} {just y} {just z} = λ ()
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trans' {just x} {nothing} {nothing} = λ ()
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trans' {just x} {nothing} {just z} = λ ()
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