Added distributivity

Distributive/Bundle.agda

@@ -0,0 +1,20 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Categories.Category
+open import Categories.Category.Cartesian
+open import Categories.Category.BinaryProducts
+open import Categories.Category.Cocartesian
+open import Distributive.Core using (Distributive)
+import Categories.Morphism as M
+
+module Distributive.Bundle where
+
+open import Level
+
+record DistributiveCategory o ℓ e : Set (suc (o ⊔ ℓ ⊔ e)) where
+    field
+        U : Category o ℓ e
+        distributive : Distributive U
+
+    open Category U public
+    open Distributive distributive public

Distributive/Core.agda

@@ -0,0 +1,28 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Categories.Category
+open import Categories.Category.Cartesian
+open import Categories.Category.BinaryProducts
+open import Categories.Category.Cocartesian
+import Categories.Morphism as M
+
+module Distributive.Core {o ℓ e} (𝒞 : Category o ℓ e) where
+
+open import Level
+open Category 𝒞
+open M 𝒞
+
+record Distributive : Set (levelOfTerm 𝒞) where
+    field
+        cartesian : Cartesian 𝒞
+        cocartesian : Cocartesian 𝒞
+    
+    open Cartesian cartesian
+    open BinaryProducts products
+    open Cocartesian cocartesian
+
+    distribute : ∀ {A B C} → (A × B + A × C) ⇒ (A × (B + C))
+    distribute = [ id ⁂ i₁ , id ⁂ i₂ ]
+
+    field
+        iso : ∀ {A B C} → IsIso (distribute {A} {B} {C})

ElgotAlgebra.agda

@@ -8,19 +8,26 @@ open import Categories.Functor.Algebra
 open import Categories.Object.Product
 open import Categories.Object.Coproduct
 open import Categories.Category
+open import Distributive.Bundle
+open import Distributive.Core
+open import Categories.Category.Cartesian
+open import Categories.Category.BinaryProducts
+open import Categories.Category.Cocartesian
 
 private
     variable
         o ℓ e : Level
 
 
-module _ (C𝒞 : CocartesianCategory o ℓ e) where
+module _ (D : DistributiveCategory o ℓ e) where
-    open CocartesianCategory C𝒞 renaming (U to 𝒞; id to idC)
+    open DistributiveCategory D renaming (U to C; id to idC)
+    open Cocartesian cocartesian
+    open Cartesian cartesian
     
     --*
     -- F-guarded Elgot Algebra
     --*
-    module _ {F : Endofunctor 𝒞} (FA : F-Algebra F) where
+    module _ {F : Endofunctor C} (FA : F-Algebra F) where
         record Guarded-Elgot-Algebra : Set (o ⊔ ℓ ⊔ e) where
             open Functor F public
             open F-Algebra FA public
@@ -116,8 +123,8 @@ module _ (C𝒞 : CocartesianCategory o ℓ e) where
 
     private
         -- identity functor on 𝒞
-        Id : Functor 𝒞 𝒞
+        Id : Functor C C
-        Id = idF {C = 𝒞}
+        Id = idF {C = C}
 
         -- identity algebra
         Id-Algebra : Obj → F-Algebra Id

ElgotAlgebras.agda

@@ -2,16 +2,20 @@ open import Level renaming (suc to ℓ-suc)
 open import Function using (_$_) renaming (id to idf; _∘_ to _∘ᶠ_)
 open import Data.Product using (_,_) renaming (_×_ to _∧_)
 
-open import Categories.Category.Cocartesian.Bundle using (CocartesianCategory)
+open import Categories.Category.Cocartesian
+open import Categories.Category.Cocartesian.Bundle
 open import Categories.Category.Cartesian
 open import Categories.Functor renaming (id to idF)
 open import Categories.Functor.Algebra
 open import Categories.Object.Terminal
 open import Categories.Object.Product
+open import Categories.Object.Exponential
 open import Categories.Object.Coproduct
 open import Categories.Category.BinaryProducts
 open import Categories.Category
 open import ElgotAlgebra
+open import Distributive.Bundle
+open import Distributive.Core
 
 module ElgotAlgebras where
 
@@ -19,15 +23,22 @@ private
     variable
         o ℓ e : Level
 
-module _ (CC : CocartesianCategory o ℓ e) where
+module _ (D : DistributiveCategory o ℓ e) where
-    open CocartesianCategory CC renaming (U to C; id to idC)
+    open DistributiveCategory D renaming (U to C; id to idC)
+    open Cocartesian cocartesian
+
+    CC : CocartesianCategory o ℓ e
+    CC = record
+        { U = C
+        ; cocartesian = cocartesian
+        }
 
     --*
     -- let's define the category of elgot-algebras
     --*
 
     -- iteration preversing morphism between two elgot-algebras
-    module _ (E₁ E₂ : Elgot-Algebra CC) where
+    module _ (E₁ E₂ : Elgot-Algebra D) where
         open Elgot-Algebra E₁ renaming (_# to _#₁)
         open Elgot-Algebra E₂ renaming (_# to _#₂; A to B)
         record Elgot-Algebra-Morphism : Set (o ⊔ ℓ ⊔ e) where
@@ -38,7 +49,7 @@ module _ (CC : CocartesianCategory o ℓ e) where
     -- the category of elgot algebras for a given (cocartesian-)category
     Elgot-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
     Elgot-Algebras = record
-        { Obj       = Elgot-Algebra CC
+        { Obj       = Elgot-Algebra D
         ; _⇒_       = Elgot-Algebra-Morphism
         ; _≈_       = λ f g → Elgot-Algebra-Morphism.h f ≈ Elgot-Algebra-Morphism.h g
         ; id        = λ {EB} → let open Elgot-Algebra EB in 
@@ -105,7 +116,7 @@ module _ (CC : CocartesianCategory o ℓ e) where
             open Equiv
 
     -- if the carriers of the algebra form a product, so do the algebras
-    A×B-Helper : ∀ {EA EB : Elgot-Algebra CC} → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Elgot-Algebra CC
+    A×B-Helper : ∀ {EA EB : Elgot-Algebra D} → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Elgot-Algebra D
     A×B-Helper {EA} {EB} p = record
         { A = A×B
         ; _# = λ {X : Obj} (h : X ⇒ A×B + X) → ⟨ ((π₁ +₁ idC) ∘ h)#ᵃ , ((π₂ +₁ idC) ∘ h)#ᵇ ⟩
@@ -202,7 +213,7 @@ module _ (CC : CocartesianCategory o ℓ e) where
                 (([ (π₂ +₁ idC) ∘ (idC +₁ i₁) ∘ f , (π₂ +₁ idC) ∘ i₂ ∘ h ])#ᵇ)         ≈⟨ sym (#ᵇ-resp-≈ ∘[]) ⟩
                 ((π₂ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ                         ∎
 
-    Product-Elgot-Algebras : ∀ (EA EB : Elgot-Algebra CC) → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Product Elgot-Algebras EA EB
+    Product-Elgot-Algebras : ∀ (EA EB : Elgot-Algebra D) → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Product Elgot-Algebras EA EB
     Product-Elgot-Algebras EA EB p = record
         { A×B = A×B-Helper {EA} {EB} p
         ; π₁ = record { h = π₁ ; preserves = λ {X} {f} → project₁ }
@@ -241,3 +252,16 @@ module _ (CC : CocartesianCategory o ℓ e) where
             open Cartesian CaC using (terminal; products)
             open BinaryProducts products using (product)
             open Equiv
+
+    -- if the carriers of the algebra form a exponential, so do the algebras
+    B^A-Helper : ∀ {EA EB : Elgot-Algebra D} → Exponential C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Elgot-Algebra D
+    B^A-Helper {EA} {EB} e = record
+        { A = B^A
+        ; _# = λ {X} f → λg {!   !} {!   !}
+        ; #-Fixpoint = {!   !}
+        ; #-Uniformity = {!   !}
+        ; #-Folding = {!   !}
+        ; #-resp-≈ = {!   !}
+        }
+        where
+            open Exponential e