Add monoidal preorders to monoids functor

1 files changed, 54 insertions, 0 deletions

diff --git a/Functor/Cartesian/Instance/InducedSetoid.agda b/Functor/Cartesian/Instance/InducedSetoid.agda
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+++ b/Functor/Cartesian/Instance/InducedSetoid.agda
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+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Cartesian.Instance.InducedSetoid where
+
+-- The induced setoid functor is cartesian
+
+open import Categories.Category.BinaryProducts using (BinaryProducts)
+open import Categories.Category.Cartesian.Bundle using (CartesianCategory)
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Category.Monoidal.Instance.Setoids using (Setoids-CartesianCategory)
+open import Categories.Functor.Cartesian using (CartesianF)
+open import Categories.Object.Product using (IsProduct)
+open import Category.Cartesian.Instance.Preorders.Primitive using (Preorders-CC)
+open import Data.Product using (<_,_>; _,_)
+open import Function using (Func)
+open import Functor.Free.Instance.InducedSetoid using () renaming (InducedSetoid to IS)
+open import Level using (Level)
+open import Preorder.Primitive using (Preorder; module Isomorphism)
+
+module _ {c ℓ : Level} where
+
+  private
+    module Preorders-CC = CartesianCategory (Preorders-CC c ℓ)
+    module BP = BinaryProducts (Preorders-CC.products)
+
+    IS-resp-×
+        : {A B : Preorder c ℓ}
+        → IsProduct (Setoids c ℓ) (IS.₁ {B = A} (BP.π₁ {B = B})) (IS.₁ BP.π₂)
+    IS-resp-× {A} {B} = record
+        { ⟨_,_⟩ = λ {S} S⟶ₛIS₀A S⟶ₛIS₀B → let open Func in record
+            { to = < to S⟶ₛIS₀A , to S⟶ₛIS₀B >
+            ; cong = λ x≈y → let open Isomorphism._≅_ in record
+                { from = from (cong S⟶ₛIS₀A x≈y) , from (cong S⟶ₛIS₀B x≈y)
+                ; to = to (cong S⟶ₛIS₀A x≈y) , to (cong S⟶ₛIS₀B x≈y)
+                }
+            }
+        ; project₁ = Isomorphism.≅.refl A
+        ; project₂ = Isomorphism.≅.refl B
+        ; unique = λ {S} {h} {i} {j} π₁∙h≈i π₂∙h≈j {x} → let open Isomorphism._≅_ in record
+            { from = to (π₁∙h≈i {x}) , to (π₂∙h≈j {x})
+            ; to = from (π₁∙h≈i {x}) , from (π₂∙h≈j {x})
+            }
+        }
+
+  InducedSetoid : CartesianF (Preorders-CC c ℓ) (Setoids-CartesianCategory c ℓ)
+  InducedSetoid = record
+      { F = IS
+      ; isCartesian = record
+          { F-resp-⊤ = _
+          ; F-resp-× = IS-resp-×
+          }
+      }
+
+  module InducedSetoid = CartesianF InducedSetoid