From 4326ca8d5b9386a480303418fb3e57b9054213f0 Mon Sep 17 00:00:00 2001
From: Jacques Comeaux <jacquesrcomeaux@protonmail.com>
Date: Wed, 7 Jan 2026 10:56:40 -0600
Subject: Add monoidal preorders to monoids functor

---
 .../Cartesian/Instance/Preorders/Primitive.agda    | 68 ++++++++++++++++++++++
 1 file changed, 68 insertions(+)
 create mode 100644 Category/Cartesian/Instance/Preorders/Primitive.agda

(limited to 'Category/Cartesian/Instance')

diff --git a/Category/Cartesian/Instance/Preorders/Primitive.agda b/Category/Cartesian/Instance/Preorders/Primitive.agda
new file mode 100644
index 0000000..d366722
--- /dev/null
+++ b/Category/Cartesian/Instance/Preorders/Primitive.agda
@@ -0,0 +1,68 @@
+{-# OPTIONS --without-K --safe #-}
+
+module Category.Cartesian.Instance.Preorders.Primitive where
+
+open import Categories.Category.Cartesian using (Cartesian)
+open import Level using (Level; suc; _⊔_)
+open import Categories.Category.Cartesian.Bundle using (CartesianCategory)
+open import Category.Instance.Preorder.Primitive.Preorders using (Preorders)
+open import Preorder.Primitive using (Preorder; module Isomorphism)
+open import Preorder.Primitive.MonotoneMap using (MonotoneMap; _≃_; module ≃)
+open import Preorder.Primitive.Monoidal using (_×ₚ_)
+open import Data.Unit.Polymorphic using (⊤)
+open import Data.Product using (proj₁; proj₂; <_,_>; _,_)
+
+
+module _ (c ℓ : Level) where
+
+  private module _ (A B : Preorder c ℓ) where
+
+    π₁ : MonotoneMap (A ×ₚ B) A
+    π₁ = record
+        { map = proj₁
+        ; mono = proj₁
+        }
+
+    π₂ : MonotoneMap (A ×ₚ B) B
+    π₂ = record
+        { map = proj₂
+        ; mono = proj₂
+        }
+
+    pair
+        : {C : Preorder c ℓ}
+        → MonotoneMap C A
+        → MonotoneMap C B
+        → MonotoneMap C (A ×ₚ B)
+    pair f g = let open MonotoneMap in record
+        { map = < map f , map g >
+        ; mono = < mono f , mono g >
+        }
+
+  ⊤ₚ : Preorder c ℓ
+  ⊤ₚ = record
+      { Carrier = ⊤
+      ; _≲_ = λ _ _ → ⊤
+      }
+
+  Preorders-Cartesian : Cartesian (Preorders c ℓ)
+  Preorders-Cartesian = record
+      { terminal = record { ⊤ = ⊤ₚ }
+      ; products = record
+          { product = λ {A} {B} → record
+              { A×B = A ×ₚ B
+              ; π₁ = π₁ A B
+              ; π₂ = π₂ A B
+              ; ⟨_,_⟩ = pair A B
+              ; project₁ = λ {h = h} → ≃.refl {x = h}
+              ; project₂ = λ {i = i} → ≃.refl {x = i}
+              ; unique = λ π₁∘j≃h π₂∘j≃i x → let open Isomorphism._≅_ in record
+                    { from = to (π₁∘j≃h x) , to (π₂∘j≃i x)
+                    ; to = from (π₁∘j≃h x) , from (π₂∘j≃i x)
+                    }
+              }
+          }
+      }
+
+  Preorders-CC : CartesianCategory (suc (c ⊔ ℓ)) (c ⊔ ℓ) (c ⊔ ℓ)
+  Preorders-CC = record { cartesian = Preorders-Cartesian }
-- 
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