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-{-# OPTIONS --without-K --safe #-}
-
-open import Level using (Level; _⊔_)
-
-module Functor.Instance.FreeMonoid {c ℓ : Level} where
-
-import Categories.Object.Monoid as MonoidObject
-
-open import Categories.Category.Construction.Monoids using (Monoids)
-open import Categories.Category.Instance.Setoids using (Setoids)
-open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory)
-open import Categories.Functor using (Functor)
-open import Categories.NaturalTransformation using (NaturalTransformation)
-open import Category.Instance.Setoids.SymmetricMonoidal {c} {c ⊔ ℓ} using (Setoids-×)
-open import Data.List.Properties using (++-assoc; ++-identityˡ; ++-identityʳ)
-open import Data.Product using (_,_)
-open import Function using (_⟶ₛ_)
-open import Functor.Instance.List {c} {ℓ} using (List)
-open import NaturalTransformation.Instance.EmptyList {c} {ℓ} using (⊤⇒[])
-open import NaturalTransformation.Instance.ListAppend {c} {ℓ} using (++)
-open import Relation.Binary using (Setoid)
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
-
-module List = Functor List
-module Setoids-× = SymmetricMonoidalCategory Setoids-×
-module ++ = NaturalTransformation ++
-module ⊤⇒[] = NaturalTransformation ⊤⇒[]
-
-open Functor
-open MonoidObject Setoids-×.monoidal using (Monoid; IsMonoid; Monoid⇒)
-open IsMonoid
-
-module _ (X : Setoid c ℓ) where
-
- private
- module X = Setoid X
- module ListX = Setoid (List.₀ X)
-
- ListMonoid : IsMonoid (List.₀ X)
- ListMonoid .μ = ++.η X
- ListMonoid .η = ⊤⇒[].η X
- ListMonoid .assoc {(x , y) , z} = ListX.reflexive (++-assoc x y z)
- ListMonoid .identityˡ {_ , x} = ListX.reflexive (++-identityˡ x)
- ListMonoid .identityʳ {x , _} = ListX.reflexive (≡.sym (++-identityʳ x))
-
-FreeMonoid₀ : (X : Setoid c ℓ) → Monoid
-FreeMonoid₀ X = record { isMonoid = ListMonoid X }
-
-FreeMonoid₁
- : {A B : Setoid c ℓ}
- (f : A ⟶ₛ B)
- → Monoid⇒ (FreeMonoid₀ A) (FreeMonoid₀ B)
-FreeMonoid₁ f = record
- { arr = List.₁ f
- ; preserves-μ = λ {x,y} → ++.sym-commute f {x,y}
- ; preserves-η = ⊤⇒[].commute f
- }
-
-FreeMonoid : Functor (Setoids c ℓ) (Monoids Setoids-×.monoidal)
-FreeMonoid .F₀ = FreeMonoid₀
-FreeMonoid .F₁ = FreeMonoid₁
-FreeMonoid .identity {X} = List.identity {X}
-FreeMonoid .homomorphism {X} {Y} {Z} {f} {g} = List.homomorphism {X} {Y} {Z} {f} {g}
-FreeMonoid .F-resp-≈ {A} {B} {f} {g} = List.F-resp-≈ {A} {B} {f} {g}