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-{-# OPTIONS --without-K --safe #-}
-
-module Functor.Free.Instance.SymmetricMonoidalPreorder where
-
-import Functor.Free.Instance.MonoidalPreorder as MP
-
-open import Categories.Category using (Category)
-open import Category.Instance.SymMonCat using (SymMonCat)
-open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory)
-open import Categories.Functor using (Functor)
-open import Categories.Functor.Monoidal.Symmetric using () renaming (module Lax to Lax₁)
-open import Categories.Functor.Monoidal.Symmetric.Properties using (∘-SymmetricMonoidal)
-open import Categories.NaturalTransformation.NaturalIsomorphism.Monoidal.Symmetric using () renaming (module Lax to Lax₂)
-open import Category.Instance.SymMonPre.Primitive using (SymMonPre; _≃_; module ≃)
-open import Data.Product using (_,_)
-open import Level using (Level)
-open import Preorder.Primitive.Monoidal using (MonoidalPreorder; SymmetricMonoidalPreorder; SymmetricMonoidalMonotone)
-
-open Lax₁ using (SymmetricMonoidalFunctor)
-open Lax₂ using (SymmetricMonoidalNaturalIsomorphism)
--- The free symmetric monoidal preorder of a symmetric monoidal category
-
-module _ {o ℓ e : Level} where
-
- symmetricMonoidalPreorder : SymmetricMonoidalCategory o ℓ e → SymmetricMonoidalPreorder o ℓ
- symmetricMonoidalPreorder C = record
- { M
- ; symmetric = record
- { symmetric = λ x y → braiding.⇒.η (x , y)
- }
- }
- where
- open SymmetricMonoidalCategory C
- module M = MonoidalPreorder (MP.Free.₀ monoidalCategory)
-
- module _ {A B : SymmetricMonoidalCategory o ℓ e} where
-
- symmetricMonoidalMonotone
- : SymmetricMonoidalFunctor A B
- → SymmetricMonoidalMonotone (symmetricMonoidalPreorder A) (symmetricMonoidalPreorder B)
- symmetricMonoidalMonotone F = record
- { monoidalMonotone = MP.Free.₁ F.monoidalFunctor
- }
- where
- module F = SymmetricMonoidalFunctor F
-
- open SymmetricMonoidalNaturalIsomorphism using (⌊_⌋)
-
- pointwiseIsomorphism
- : {F G : SymmetricMonoidalFunctor A B}
- → SymmetricMonoidalNaturalIsomorphism F G
- → symmetricMonoidalMonotone F ≃ symmetricMonoidalMonotone G
- pointwiseIsomorphism F≃G = MP.Free.F-resp-≈ ⌊ F≃G ⌋
-
-Free : {o ℓ e : Level} → Functor (SymMonCat {o} {ℓ} {e}) (SymMonPre o ℓ)
-Free = record
- { F₀ = symmetricMonoidalPreorder
- ; F₁ = symmetricMonoidalMonotone
- ; identity = λ {A} → ≃.refl {A = symmetricMonoidalPreorder A} {x = id}
- ; homomorphism = λ {f = f} {h} → ≃.refl {x = symmetricMonoidalMonotone (∘-SymmetricMonoidal h f)}
- ; F-resp-≈ = pointwiseIsomorphism
- }
- where
- open Category (SymMonPre _ _) using (id)
-
-module Free {o ℓ e} = Functor (Free {o} {ℓ} {e})