Add commutative monoid conversions

1 files changed, 58 insertions, 36 deletions

diff --git a/Data/Monoid.agda b/Data/Monoid.agda
index ae2ded9..dba3e79 100644
--- a/Data/Monoid.agda
+++ b/Data/Monoid.agda
@@ -4,7 +4,7 @@ open import Level using (Level)
 module Data.Monoid {c ℓ : Level} where
 
 open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory)
-open import Category.Instance.Setoids.SymmetricMonoidal {c} {ℓ} using (Setoids-×)
+open import Category.Instance.Setoids.SymmetricMonoidal {c} {ℓ} using (Setoids-×; ×-monoidal′)
 open import Categories.Object.Monoid using (Monoid; Monoid⇒)
 
 module Setoids-× = SymmetricMonoidalCategory Setoids-×
@@ -14,31 +14,50 @@ import Algebra.Bundles as Alg
 open import Data.Setoid using (∣_∣)
 open import Relation.Binary using (Setoid)
 open import Function using (Func)
-open import Data.Product using (curry′; _,_)
+open import Data.Product using (curry′; uncurry′; _,_)
 open Func
 
 -- A monoid object in the (monoidal) category of setoids is just a monoid
 
-toMonoid : Monoid Setoids-×.monoidal → Alg.Monoid c ℓ
-toMonoid M = record
-    { Carrier = Carrier
-    ; _≈_ = _≈_
-    ; _∙_ = curry′ (to μ)
-    ; ε = to η _
-    ; isMonoid = record
-        { isSemigroup = record
-            { isMagma = record
-                { isEquivalence = isEquivalence
-                ; ∙-cong = curry′ (cong μ)
-                }
-            ; assoc = λ x y z → assoc {(x , y) , z}
-            }
-        ; identity = (λ x → sym (identityˡ {_ , x}) ) , λ x → sym (identityʳ {x , _})
-        }
-    }
-  where
-    open Monoid M renaming (Carrier to A)
-    open Setoid A
+open import Function.Construct.Constant using () renaming (function to Const)
+open import Categories.Category.Instance.SingletonSet using () renaming (SingletonSetoid to ⊤ₛ)
+
+opaque
+  unfolding ×-monoidal′
+  toMonoid : Monoid Setoids-×.monoidal → Alg.Monoid c ℓ
+  toMonoid M = record
+      { Carrier = Carrier
+      ; _≈_ = _≈_
+      ; _∙_ = curry′ (to μ)
+      ; ε = to η _
+      ; isMonoid = record
+          { isSemigroup = record
+              { isMagma = record
+                  { isEquivalence = isEquivalence
+                  ; ∙-cong = curry′ (cong μ)
+                  }
+              ; assoc = λ x y z → assoc {(x , y) , z}
+              }
+          ; identity = (λ x → sym (identityˡ {_ , x}) ) , λ x → sym (identityʳ {x , _})
+          }
+      }
+    where
+      open Monoid M renaming (Carrier to A)
+      open Setoid A
+
+  fromMonoid : Alg.Monoid c ℓ → Monoid Setoids-×.monoidal
+  fromMonoid M = record
+      { Carrier = setoid
+      ; isMonoid = record
+          { μ = record { to = uncurry′ _∙_ ; cong = uncurry′ ∙-cong }
+          ; η = Const ⊤ₛ setoid ε
+          ; assoc = λ { {(x , y) , z} → assoc x y z }
+          ; identityˡ = λ { {_ , x} → sym (identityˡ x) }
+          ; identityʳ = λ { {x , _} → sym (identityʳ x) }
+          }
+      }
+    where
+      open Alg.Monoid M
 
 -- A morphism of monoids in the (monoidal) category of setoids is a monoid homomorphism
 
@@ -48,19 +67,22 @@ module  _ (M N : Monoid Setoids-×.monoidal) where
   module N = Alg.Monoid (toMonoid N)
 
   open import Data.Product using (Σ; _,_)
-  open import Function using (_⟶ₛ_; _⟨$⟩_)
+  open import Function using (_⟶ₛ_)
   open import Algebra.Morphism using (IsMonoidHomomorphism)
   open Monoid⇒
-  toMonoid⇒
-      : Monoid⇒ Setoids-×.monoidal M N
-      → Σ (M.setoid ⟶ₛ N.setoid) (λ f
-      → IsMonoidHomomorphism M.rawMonoid N.rawMonoid (to f))
-  toMonoid⇒ f = arr f , record
-      { isMagmaHomomorphism = record
-          { isRelHomomorphism = record
-              { cong = cong (arr f)
-              }
-          ; homo = λ x y → preserves-μ f {x , y}
-          }
-      ; ε-homo = preserves-η f
-      }
+
+  opaque
+
+    unfolding toMonoid
+
+    toMonoid⇒
+        : Monoid⇒ Setoids-×.monoidal M N
+        → Σ (M.setoid ⟶ₛ N.setoid) (λ f
+        → IsMonoidHomomorphism M.rawMonoid N.rawMonoid (to f))
+    toMonoid⇒ f = arr f , record
+        { isMagmaHomomorphism = record
+            { isRelHomomorphism = record { cong = cong (arr f) }
+            ; homo = λ x y → preserves-μ f {x , y}
+            }
+        ; ε-homo = preserves-η f
+        }