Add adjunction between free monoid and forget

2 files changed, 6 insertions, 68 deletions

diff --git a/Functor/Instance/FreeMonoid.agda b/Functor/Instance/FreeMonoid.agda
deleted file mode 100644
index bb26fd4..0000000
--- a/Functor/Instance/FreeMonoid.agda
+++ /dev/null
@@ -1,64 +0,0 @@
-{-# 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}
diff --git a/Functor/Instance/List.agda b/Functor/Instance/List.agda
index ceb73e1..a280218 100644
--- a/Functor/Instance/List.agda
+++ b/Functor/Instance/List.agda
@@ -18,7 +18,7 @@ open Functor
 open Setoid using (reflexive)
 open Func
 
-open import Data.Opaque.List as List hiding (List)
+open import Data.Opaque.List as L hiding (List)
 
 private
   variable
@@ -29,7 +29,7 @@ open import Function.Construct.Setoid using (_∙_)
 
 opaque
 
-  unfolding List.List
+  unfolding L.List
 
   map-id
       : (xs : ∣ Listₛ A ∣)
@@ -58,8 +58,10 @@ opaque
 -- which applies the same function to every element of a list
 
 List : Functor (Setoids c ℓ) (Setoids c (c ⊔ ℓ))
-List .F₀ = List.Listₛ
-List .F₁ = List.mapₛ
+List .F₀ = Listₛ
+List .F₁ = mapₛ
 List .identity {_} {xs} = map-id xs
 List .homomorphism {f = f} {g} {xs} = List-homo f g xs
 List .F-resp-≈ {f = f} {g} f≈g = List-resp-≈ f g f≈g
+
+module List = Functor List