Add adjunction between free monoid and forget

1 files changed, 85 insertions, 0 deletions

diff --git a/Functor/Free/Instance/Monoid.agda b/Functor/Free/Instance/Monoid.agda
new file mode 100644
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+++ b/Functor/Free/Instance/Monoid.agda
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+{-# OPTIONS --without-K --safe #-}
+
+open import Level using (Level; _⊔_)
+
+module Functor.Free.Instance.Monoid {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.Opaque.List using ([]ₛ; Listₛ; ++ₛ; mapₛ)
+open import Data.Product using (_,_)
+open import Data.Setoid using (∣_∣)
+open import Function using (_⟶ₛ_; _⟨$⟩_)
+open import Functor.Instance.List {c} {ℓ} using (List)
+open import NaturalTransformation.Instance.EmptyList {c} {ℓ} using (⊤⇒[])
+open import NaturalTransformation.Instance.ListAppend {c} {ℓ} using (++)
+open import Relation.Binary using (Setoid)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
+
+module Setoids-× = SymmetricMonoidalCategory Setoids-×
+module ++ = NaturalTransformation ++
+module ⊤⇒[] = NaturalTransformation ⊤⇒[]
+
+open Functor
+open MonoidObject Setoids-×.monoidal using (Monoid; IsMonoid; Monoid⇒)
+open IsMonoid
+
+-- the functor sending a setoid A to the monoid List A
+
+module _ (X : Setoid c ℓ) where
+
+  open Setoid (List.₀ X)
+
+  opaque
+
+    unfolding []ₛ
+
+    ++ₛ-assoc
+        : (x y z : ∣ Listₛ X ∣)
+        → ++ₛ ⟨$⟩ (++ₛ ⟨$⟩ (x , y) , z)
+        ≈ ++ₛ ⟨$⟩ (x , ++ₛ ⟨$⟩ (y , z))
+    ++ₛ-assoc x y z = reflexive (++-assoc x y z)
+
+    ++ₛ-identityˡ
+        : (x : ∣ Listₛ X ∣)
+        → x ≈ ++ₛ ⟨$⟩ ([]ₛ ⟨$⟩ _ , x)
+    ++ₛ-identityˡ x = reflexive (++-identityˡ x)
+
+    ++ₛ-identityʳ
+        : (x : ∣ Listₛ X ∣)
+        → x ≈ ++ₛ ⟨$⟩ (x , []ₛ ⟨$⟩ _)
+    ++ₛ-identityʳ x = sym (reflexive (++-identityʳ x))
+
+  ListMonoid : IsMonoid (List.₀ X)
+  ListMonoid .μ = ++.η X
+  ListMonoid .η = ⊤⇒[].η X
+  ListMonoid .assoc {(x , y) , z} = ++ₛ-assoc x y z
+  ListMonoid .identityˡ {bro , x} = ++ₛ-identityˡ x
+  ListMonoid .identityʳ {x , _} = ++ₛ-identityʳ x
+
+Listₘ : Setoid c ℓ → Monoid
+Listₘ X = record { isMonoid = ListMonoid X }
+
+mapₘ
+    : {Aₛ Bₛ : Setoid c ℓ}
+      (f : Aₛ ⟶ₛ Bₛ)
+    → Monoid⇒ (Listₘ Aₛ) (Listₘ Bₛ)
+mapₘ f = record
+    { arr = List.₁ f
+    ; preserves-μ = λ {x,y} → ++.sym-commute f {x,y}
+    ; preserves-η = ⊤⇒[].sym-commute f
+    }
+
+Free : Functor (Setoids c ℓ) (Monoids Setoids-×.monoidal)
+Free .F₀ = Listₘ
+Free .F₁ = mapₘ
+Free .identity {X} = List.identity {X}
+Free .homomorphism {X} {Y} {Z} {f} {g} = List.homomorphism {X} {Y} {Z} {f} {g}
+Free .F-resp-≈ {A} {B} {f} {g} = List.F-resp-≈ {A} {B} {f} {g}