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{-# OPTIONS --without-K --safe #-}
module Category.Instance.Preorder.Primitive.Monoidals.Lax where
import Preorder.Primitive.MonotoneMap as MonotoneMap using (_≃_; module ≃)
open import Categories.Category using (Category)
open import Categories.Category.Helper using (categoryHelper)
open import Category.Instance.Preorder.Primitive.Preorders using (Preorders)
open import Level using (Level; suc; _⊔_)
open import Preorder.Primitive.Monoidal using (MonoidalPreorder)
open import Preorder.Primitive.MonotoneMap.Monoidal.Lax using (MonoidalMonotone)
open import Relation.Binary using (IsEquivalence)
module _ {c₁ c₂ ℓ₁ ℓ₂ : Level} {A : MonoidalPreorder c₁ ℓ₁} {B : MonoidalPreorder c₂ ℓ₂} where
-- Pointwise equality of monoidal monotone maps
open MonoidalMonotone using (F)
_≃_ : (f g : MonoidalMonotone A B) → Set (c₁ ⊔ ℓ₂)
f ≃ g = F f MonotoneMap.≃ F g
infix 4 _≃_
≃-isEquivalence : IsEquivalence _≃_
≃-isEquivalence = let open MonotoneMap.≃ in record
{ refl = λ {x} → refl {x = F x}
; sym = λ {f g} → sym {x = F f} {y = F g}
; trans = λ {f g h} → trans {i = F f} {j = F g} {k = F h}
}
module ≃ = IsEquivalence ≃-isEquivalence
private
identity : {c ℓ : Level} (A : MonoidalPreorder c ℓ) → MonoidalMonotone A A
identity A = let open MonoidalPreorder A in record
{ F = Category.id (Preorders _ _)
; ε = refl
; ⊗-homo = λ p₁ p₂ → refl {p₁ ⊗ p₂}
}
compose
: {c ℓ : Level}
{P Q R : MonoidalPreorder c ℓ}
→ MonoidalMonotone Q R
→ MonoidalMonotone P Q
→ MonoidalMonotone P R
compose {R = R} G F = record
{ F = let open Category (Preorders _ _) in G.F ∘ F.F
; ε = trans G.ε (G.mono F.ε)
; ⊗-homo = λ p₁ p₂ → trans (G.⊗-homo (F.map p₁) (F.map p₂)) (G.mono (F.⊗-homo p₁ p₂))
}
where
module F = MonoidalMonotone F
module G = MonoidalMonotone G
open MonoidalPreorder R
compose-resp-≃
: {c ℓ : Level}
{A B C : MonoidalPreorder c ℓ}
{f h : MonoidalMonotone B C}
{g i : MonoidalMonotone A B}
→ f ≃ h
→ g ≃ i
→ compose f g ≃ compose h i
compose-resp-≃ {C = C} {f = f} {g} {h} {i} = ∘-resp-≈ {f = F f} {F g} {F h} {F i}
where
open Category (Preorders _ _)
open MonoidalMonotone using (F)
Monoidals : (c ℓ : Level) → Category (suc (c ⊔ ℓ)) (c ⊔ ℓ) (c ⊔ ℓ)
Monoidals c ℓ = categoryHelper record
{ Obj = MonoidalPreorder c ℓ
; _⇒_ = MonoidalMonotone
; _≈_ = _≃_
; id = λ {A} → identity A
; _∘_ = compose
; assoc = λ {f = f} {g h} → ≃.refl {x = compose (compose h g) f}
; identityˡ = λ {f = f} → ≃.refl {x = f}
; identityʳ = λ {f = f} → ≃.refl {x = f}
; equiv = ≃-isEquivalence
; ∘-resp-≈ = λ {f = f} {g h i} → compose-resp-≃ {f = f} {g} {h} {i}
}
where
open MonoidalMonotone using (F)
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