From 8ec259bb32b4339b27560f5ea13afa81b9b8febc Mon Sep 17 00:00:00 2001
From: Jacques Comeaux <jacquesrcomeaux@protonmail.com>
Date: Wed, 7 Jan 2026 10:12:13 -0600
Subject: Differentiate lax and strong monoidal monotones

---
 Preorder/Primitive/MonotoneMap/Monoidal/Lax.agda   | 41 +++++++++++++++++++++
 .../Primitive/MonotoneMap/Monoidal/Strong.agda     | 43 ++++++++++++++++++++++
 2 files changed, 84 insertions(+)
 create mode 100644 Preorder/Primitive/MonotoneMap/Monoidal/Lax.agda
 create mode 100644 Preorder/Primitive/MonotoneMap/Monoidal/Strong.agda

(limited to 'Preorder/Primitive/MonotoneMap/Monoidal')

diff --git a/Preorder/Primitive/MonotoneMap/Monoidal/Lax.agda b/Preorder/Primitive/MonotoneMap/Monoidal/Lax.agda
new file mode 100644
index 0000000..703f3be
--- /dev/null
+++ b/Preorder/Primitive/MonotoneMap/Monoidal/Lax.agda
@@ -0,0 +1,41 @@
+{-# OPTIONS --without-K --safe #-}
+
+module Preorder.Primitive.MonotoneMap.Monoidal.Lax where
+
+open import Level using (Level; _⊔_)
+open import Preorder.Primitive.Monoidal using (MonoidalPreorder; SymmetricMonoidalPreorder)
+open import Preorder.Primitive.MonotoneMap using (MonotoneMap)
+
+record MonoidalMonotone
+    {c₁ c₂ ℓ₁ ℓ₂ : Level}
+    (P : MonoidalPreorder c₁ ℓ₁)
+    (Q : MonoidalPreorder c₂ ℓ₂)
+  : Set (c₁ ⊔ c₂ ⊔ ℓ₁ ⊔ ℓ₂) where
+
+  private
+    module P = MonoidalPreorder P
+    module Q = MonoidalPreorder Q
+
+  field
+    F : MonotoneMap P.U Q.U
+
+  open MonotoneMap F public
+
+  field
+    ε : Q.unit Q.≲ map P.unit
+    ⊗-homo : (p₁ p₂ : P.Carrier) → map p₁ Q.⊗ map p₂ Q.≲ map (p₁ P.⊗ p₂)
+
+record SymmetricMonoidalMonotone
+    {c₁ c₂ ℓ₁ ℓ₂ : Level}
+    (P : SymmetricMonoidalPreorder c₁ ℓ₁)
+    (Q : SymmetricMonoidalPreorder c₂ ℓ₂)
+  : Set (c₁ ⊔ c₂ ⊔ ℓ₁ ⊔ ℓ₂) where
+
+  private
+    module P = SymmetricMonoidalPreorder P
+    module Q = SymmetricMonoidalPreorder Q
+
+  field
+    monoidalMonotone : MonoidalMonotone P.monoidalPreorder Q.monoidalPreorder
+
+  open MonoidalMonotone monoidalMonotone public
diff --git a/Preorder/Primitive/MonotoneMap/Monoidal/Strong.agda b/Preorder/Primitive/MonotoneMap/Monoidal/Strong.agda
new file mode 100644
index 0000000..f613b14
--- /dev/null
+++ b/Preorder/Primitive/MonotoneMap/Monoidal/Strong.agda
@@ -0,0 +1,43 @@
+{-# OPTIONS --without-K --safe #-}
+
+module Preorder.Primitive.MonotoneMap.Monoidal.Strong where
+
+open import Level using (Level; _⊔_)
+open import Preorder.Primitive using (module Isomorphism)
+open import Preorder.Primitive.Monoidal using (MonoidalPreorder; SymmetricMonoidalPreorder)
+open import Preorder.Primitive.MonotoneMap using (MonotoneMap)
+
+record MonoidalMonotone
+    {c₁ c₂ ℓ₁ ℓ₂ : Level}
+    (P : MonoidalPreorder c₁ ℓ₁)
+    (Q : MonoidalPreorder c₂ ℓ₂)
+  : Set (c₁ ⊔ c₂ ⊔ ℓ₁ ⊔ ℓ₂) where
+
+  private
+    module P = MonoidalPreorder P
+    module Q = MonoidalPreorder Q
+
+  field
+    F : MonotoneMap P.U Q.U
+
+  open MonotoneMap F public
+  open Isomorphism Q.U using (_≅_)
+
+  field
+    ε : Q.unit ≅ map P.unit
+    ⊗-homo : (p₁ p₂ : P.Carrier) → map p₁ Q.⊗ map p₂ ≅ map (p₁ P.⊗ p₂)
+
+record SymmetricMonoidalMonotone
+    {c₁ c₂ ℓ₁ ℓ₂ : Level}
+    (P : SymmetricMonoidalPreorder c₁ ℓ₁)
+    (Q : SymmetricMonoidalPreorder c₂ ℓ₂)
+  : Set (c₁ ⊔ c₂ ⊔ ℓ₁ ⊔ ℓ₂) where
+
+  private
+    module P = SymmetricMonoidalPreorder P
+    module Q = SymmetricMonoidalPreorder Q
+
+  field
+    monoidalMonotone : MonoidalMonotone P.monoidalPreorder Q.monoidalPreorder
+
+  open MonoidalMonotone monoidalMonotone public
-- 
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