From 08982c82069c66f768c17979b7aa8d0a62d65cdb Mon Sep 17 00:00:00 2001
From: Jacques Comeaux <jacquesrcomeaux@protonmail.com>
Date: Mon, 13 Jul 2026 20:43:51 -0700
Subject: Add cartesian structure to category of semimodules

---
 Category/Cartesian/Instance/Semimodules.agda | 101 +++++++++++++++++++++++++++
 1 file changed, 101 insertions(+)
 create mode 100644 Category/Cartesian/Instance/Semimodules.agda

(limited to 'Category/Cartesian')

diff --git a/Category/Cartesian/Instance/Semimodules.agda b/Category/Cartesian/Instance/Semimodules.agda
new file mode 100644
index 0000000..8300c07
--- /dev/null
+++ b/Category/Cartesian/Instance/Semimodules.agda
@@ -0,0 +1,101 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Algebra using (CommutativeSemiring)
+open import Level using (Level; suc; _⊔_)
+
+module Category.Cartesian.Instance.Semimodules {c ℓ m ℓm : Level} (R : CommutativeSemiring c ℓ) where
+
+import Algebra.Module.Construct.Zero as 𝟎
+import Algebra.Module.Construct.DirectProduct as ×
+
+open import Algebra.Module using (Semimodule)
+open import Algebra.Morphism.Construct.DirectProduct using () renaming (module Monoid-Export to Monoid-×)
+open import Categories.Category.BinaryProducts using (BinaryProducts)
+open import Categories.Category.Cartesian using (Cartesian)
+open import Categories.Category.Cartesian.Bundle using (CartesianCategory)
+open import Categories.Object.Terminal using (Terminal)
+open import Category.Instance.Semimodules {c} {ℓ} {m} {ℓm} R using (Semimodules; SemimoduleHomomorphism)
+open import Data.Product using (_,_; proj₁; proj₂; <_,_>)
+open import Data.Unit.Polymorphic using (tt)
+
+open Semimodule using (≈ᴹ-refl; ≈ᴹ-sym)
+
+terminal : Terminal Semimodules
+terminal = record
+    { ⊤ = 𝟎.semimodule
+    ; ⊤-is-terminal = record
+        { ! = record { ⟦_⟧ = λ _ → tt }
+        ; !-unique = λ _ _ → tt
+        }
+    }
+
+module _ (A B : Semimodule R m ℓm) where
+
+  π₁ : SemimoduleHomomorphism (×.semimodule A B) A
+  π₁ = record
+      { ⟦_⟧ = proj₁
+      ; isSemimoduleHomomorphism = record
+          { isBisemimoduleHomomorphism = record
+              { +ᴹ-isMonoidHomomorphism = Monoid-×.proj₁ {refl = ≈ᴹ-refl A}
+              ; *ₗ-homo = λ _ _ → ≈ᴹ-refl A
+              ; *ᵣ-homo = λ _ _ → ≈ᴹ-refl A
+              }
+          }
+      }
+
+  π₂ : SemimoduleHomomorphism (×.semimodule A B) B
+  π₂ = record
+      { ⟦_⟧ = proj₂
+      ; isSemimoduleHomomorphism = record
+          { isBisemimoduleHomomorphism = record
+              { +ᴹ-isMonoidHomomorphism = Monoid-×.proj₂ {refl = ≈ᴹ-refl B}
+              ; *ₗ-homo = λ _ _ → ≈ᴹ-refl B
+              ; *ᵣ-homo = λ _ _ → ≈ᴹ-refl B
+              }
+          }
+      }
+
+module _ {A B C : Semimodule R m ℓm} where
+
+  open SemimoduleHomomorphism
+
+  ⟨_,_⟩  : SemimoduleHomomorphism C A
+      → SemimoduleHomomorphism C B
+      → SemimoduleHomomorphism C (×.semimodule A B)
+  ⟨_,_⟩ f g = record
+      { ⟦_⟧ = < ⟦ f ⟧ , ⟦ g ⟧ >
+      ; isSemimoduleHomomorphism = record
+          { isBisemimoduleHomomorphism = record
+              { +ᴹ-isMonoidHomomorphism = Monoid-×.< +ᴹ-isMonoidHomomorphism f , +ᴹ-isMonoidHomomorphism g >
+              ; *ₗ-homo = λ r x → *ₗ-homo f r x , *ₗ-homo g r x
+              ; *ᵣ-homo = λ r x → *ᵣ-homo f r x , *ᵣ-homo g r x
+              }
+          }
+      }
+
+products : BinaryProducts Semimodules
+products = record
+    { product = λ {A B} → record
+        { A×B = ×.semimodule A B
+        ; π₁ = π₁ A B
+        ; π₂ = π₂ A B
+        ; ⟨_,_⟩ = ⟨_,_⟩
+        ; project₁ = λ _ → ≈ᴹ-refl A
+        ; project₂ = λ _ → ≈ᴹ-refl B
+        ; unique = λ eq₁ eq₂ x → ≈ᴹ-sym A (eq₁ x) , ≈ᴹ-sym B (eq₂ x)
+        }
+    }
+
+Semimodules-Cartesian : Cartesian Semimodules
+Semimodules-Cartesian = record
+    { terminal = terminal
+    ; products = products
+    }
+
+Semimodules-CC : CartesianCategory (c ⊔ ℓ ⊔ suc (m ⊔ ℓm)) (c ⊔ m ⊔ ℓm) (m ⊔ ℓm)
+Semimodules-CC = record
+    { U = Semimodules
+    ; cartesian = Semimodules-Cartesian
+    }
+
+module Semimodules-CC = CartesianCategory Semimodules-CC
-- 
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