2 files changed, 113 insertions, 1 deletions
diff --git a/Adjoint/Instance/List.agda b/Adjoint/Instance/List.agda
index f3bf061..1b65985 100644
--- a/Adjoint/Instance/List.agda
+++ b/Adjoint/Instance/List.agda
@@ -1,6 +1,6 @@
 {-# OPTIONS --without-K --safe #-}
 
-open import Level using (Level; _⊔_; suc; 0ℓ)
+open import Level using (Level; _⊔_; suc)
 
 module Adjoint.Instance.List {ℓ : Level} where
 
diff --git a/Adjoint/Instance/Multiset.agda b/Adjoint/Instance/Multiset.agda
new file mode 100644
index 0000000..c51baa9
--- /dev/null
+++ b/Adjoint/Instance/Multiset.agda
@@ -0,0 +1,112 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Level using (Level; _⊔_; suc; 0ℓ)
+
+module Adjoint.Instance.Multiset {ℓ : Level} where
+
+open import Category.Instance.Setoids.SymmetricMonoidal {ℓ} {ℓ} using (Setoids-×)
+
+private
+  module S = Setoids-×
+
+import Categories.Object.Monoid S.monoidal as MonoidObject
+import Data.List as L
+import Data.List.Relation.Binary.Permutation.Setoid as ↭
+import Functor.Forgetful.Instance.CMonoid S.symmetric as CMonoid
+import Functor.Forgetful.Instance.Monoid S.monoidal as Monoid
+import Object.Monoid.Commutative S.symmetric as CMonoidObject
+
+open import Categories.Adjoint using (_⊣_)
+open import Categories.Category using (Category)
+open import Categories.Functor using (Functor; id; _∘F_)
+open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
+open import Category.Construction.CMonoids using (CMonoids)
+open import Data.CMonoid using (toCMonoid; toCMonoid⇒)
+open import Data.Monoid using (toMonoid; toMonoid⇒)
+open import Data.Opaque.Multiset using ([-]ₛ; Multisetₛ; foldₛ; ++ₛ-homo; []ₛ-homo; fold-mapₛ; fold)
+open import Data.Product using (_,_; uncurry)
+open import Data.Setoid using (∣_∣)
+open import Function using (_⟶ₛ_; _⟨$⟩_)
+open import Functor.Free.Instance.CMonoid {ℓ} {ℓ} using (Multisetₘ; mapₘ; MultisetCMonoid) renaming (Free to Free′)
+open import Relation.Binary using (Setoid)
+
+open CMonoidObject using (CommutativeMonoid; CommutativeMonoid⇒)
+open CommutativeMonoid using (Carrier; monoid; identityʳ)
+open CommutativeMonoid⇒ using (arr; monoid⇒)
+open MonoidObject using (Monoid; Monoid⇒)
+open Monoid⇒ using (preserves-μ; preserves-η)
+
+CMon[S] : Category (suc ℓ) ℓ ℓ
+CMon[S] = CMonoids S.symmetric
+
+Free : Functor S.U CMon[S]
+Free = Free′
+
+Forget : Functor CMon[S] S.U
+Forget = Monoid.Forget ∘F CMonoid.Forget
+
+opaque
+  unfolding [-]ₛ
+  map-[-]ₛ
+      : {X Y : Setoid ℓ ℓ}
+        (f : X ⟶ₛ Y)
+        {x : ∣ X ∣}
+      → (open Setoid (Multisetₛ Y))
+      → [-]ₛ ⟨$⟩ (f ⟨$⟩ x)
+      ≈ arr (mapₘ f) ⟨$⟩ ([-]ₛ ⟨$⟩ x)
+  map-[-]ₛ {X} {Y} f {x} = Setoid.refl (Multisetₛ Y)
+
+unit : NaturalTransformation id (Forget ∘F Free)
+unit = ntHelper record
+    { η = λ X → [-]ₛ {ℓ} {ℓ} {X}
+    ; commute = map-[-]ₛ
+    }
+
+opaque
+  unfolding toMonoid MultisetCMonoid
+  foldₘ : (X : CommutativeMonoid) → CommutativeMonoid⇒ (Multisetₘ (Carrier X)) X
+  foldₘ X .monoid⇒ .Monoid⇒.arr = foldₛ (toCMonoid X)
+  foldₘ X .monoid⇒ .preserves-μ {xs , ys} = ++ₛ-homo (toCMonoid X) xs ys
+  foldₘ X .monoid⇒ .preserves-η {_} = []ₛ-homo (toCMonoid X)
+
+opaque
+  unfolding foldₘ toMonoid⇒
+  fold-mapₘ
+      : {X Y : CommutativeMonoid}
+        (f : CommutativeMonoid⇒ X Y)
+        {x : ∣ Multisetₛ (Carrier X) ∣}
+      → (open Setoid (Carrier Y))
+      → arr (foldₘ Y) ⟨$⟩ (arr (mapₘ (arr f)) ⟨$⟩ x)
+      ≈ arr f ⟨$⟩ (arr (foldₘ X) ⟨$⟩ x)
+  fold-mapₘ {X} {Y} f = uncurry (fold-mapₛ (toCMonoid X) (toCMonoid Y)) (toCMonoid⇒ X Y f)
+
+counit : NaturalTransformation (Free ∘F Forget) id
+counit = ntHelper record
+    { η = foldₘ
+    ; commute = fold-mapₘ
+    }
+
+opaque
+  unfolding foldₘ fold Multisetₛ
+  zig : (Aₛ  : Setoid ℓ ℓ)
+        {xs : ∣ Multisetₛ Aₛ ∣}
+      → (open Setoid (Multisetₛ Aₛ))
+      → arr (foldₘ (Multisetₘ Aₛ)) ⟨$⟩ (arr (mapₘ [-]ₛ) ⟨$⟩ xs) ≈ xs
+  zig Aₛ {L.[]} = ↭.↭-refl Aₛ
+  zig Aₛ {x L.∷ xs} = ↭.prep (Setoid.refl Aₛ) (zig Aₛ)
+
+opaque
+  unfolding foldₘ fold
+  zag : (M : CommutativeMonoid)
+        {x : ∣ Carrier M ∣}
+      → (open Setoid (Carrier M))
+      → arr (foldₘ M) ⟨$⟩ ([-]ₛ ⟨$⟩ x) ≈ x
+  zag M {x} = Setoid.sym (Carrier M) (identityʳ M {x , _})
+
+Multiset⊣ : Free ⊣ Forget
+Multiset⊣ = record
+    { unit = unit
+    ; counit = counit
+    ; zig = λ {X} → zig X
+    ; zag = λ {M} → zag M
+    }