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Added distributivity
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4 changed files with 92 additions and 13 deletions
20
Distributive/Bundle.agda
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20
Distributive/Bundle.agda
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@ -0,0 +1,20 @@
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{-# OPTIONS --without-K --safe #-}
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open import Categories.Category
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open import Categories.Category.Cartesian
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open import Categories.Category.BinaryProducts
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open import Categories.Category.Cocartesian
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open import Distributive.Core using (Distributive)
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import Categories.Morphism as M
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module Distributive.Bundle where
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open import Level
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record DistributiveCategory o ℓ e : Set (suc (o ⊔ ℓ ⊔ e)) where
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field
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U : Category o ℓ e
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distributive : Distributive U
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open Category U public
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open Distributive distributive public
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28
Distributive/Core.agda
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28
Distributive/Core.agda
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{-# OPTIONS --without-K --safe #-}
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open import Categories.Category
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open import Categories.Category.Cartesian
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open import Categories.Category.BinaryProducts
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open import Categories.Category.Cocartesian
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import Categories.Morphism as M
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module Distributive.Core {o ℓ e} (𝒞 : Category o ℓ e) where
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open import Level
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open Category 𝒞
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open M 𝒞
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record Distributive : Set (levelOfTerm 𝒞) where
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field
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cartesian : Cartesian 𝒞
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cocartesian : Cocartesian 𝒞
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open Cartesian cartesian
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open BinaryProducts products
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open Cocartesian cocartesian
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distribute : ∀ {A B C} → (A × B + A × C) ⇒ (A × (B + C))
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distribute = [ id ⁂ i₁ , id ⁂ i₂ ]
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field
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iso : ∀ {A B C} → IsIso (distribute {A} {B} {C})
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@ -8,19 +8,26 @@ open import Categories.Functor.Algebra
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open import Categories.Object.Product
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open import Categories.Object.Coproduct
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open import Categories.Category
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open import Distributive.Bundle
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open import Distributive.Core
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open import Categories.Category.Cartesian
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open import Categories.Category.BinaryProducts
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open import Categories.Category.Cocartesian
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private
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variable
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o ℓ e : Level
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module _ (C𝒞 : CocartesianCategory o ℓ e) where
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open CocartesianCategory C𝒞 renaming (U to 𝒞; id to idC)
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module _ (D : DistributiveCategory o ℓ e) where
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open DistributiveCategory D renaming (U to C; id to idC)
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open Cocartesian cocartesian
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open Cartesian cartesian
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--*
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-- F-guarded Elgot Algebra
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--*
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module _ {F : Endofunctor 𝒞} (FA : F-Algebra F) where
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module _ {F : Endofunctor C} (FA : F-Algebra F) where
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record Guarded-Elgot-Algebra : Set (o ⊔ ℓ ⊔ e) where
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open Functor F public
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open F-Algebra FA public
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@ -116,8 +123,8 @@ module _ (C𝒞 : CocartesianCategory o ℓ e) where
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private
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-- identity functor on 𝒞
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Id : Functor 𝒞 𝒞
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Id = idF {C = 𝒞}
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Id : Functor C C
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Id = idF {C = C}
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-- identity algebra
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Id-Algebra : Obj → F-Algebra Id
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@ -2,16 +2,20 @@ open import Level renaming (suc to ℓ-suc)
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open import Function using (_$_) renaming (id to idf; _∘_ to _∘ᶠ_)
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open import Data.Product using (_,_) renaming (_×_ to _∧_)
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open import Categories.Category.Cocartesian.Bundle using (CocartesianCategory)
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open import Categories.Category.Cocartesian
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open import Categories.Category.Cocartesian.Bundle
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open import Categories.Category.Cartesian
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open import Categories.Functor renaming (id to idF)
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open import Categories.Functor.Algebra
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open import Categories.Object.Terminal
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open import Categories.Object.Product
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open import Categories.Object.Exponential
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open import Categories.Object.Coproduct
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open import Categories.Category.BinaryProducts
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open import Categories.Category
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open import ElgotAlgebra
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open import Distributive.Bundle
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open import Distributive.Core
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module ElgotAlgebras where
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@ -19,15 +23,22 @@ private
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variable
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o ℓ e : Level
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module _ (CC : CocartesianCategory o ℓ e) where
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open CocartesianCategory CC renaming (U to C; id to idC)
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module _ (D : DistributiveCategory o ℓ e) where
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open DistributiveCategory D renaming (U to C; id to idC)
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open Cocartesian cocartesian
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CC : CocartesianCategory o ℓ e
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CC = record
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{ U = C
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; cocartesian = cocartesian
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}
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--*
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-- let's define the category of elgot-algebras
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--*
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-- iteration preversing morphism between two elgot-algebras
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module _ (E₁ E₂ : Elgot-Algebra CC) where
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module _ (E₁ E₂ : Elgot-Algebra D) where
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open Elgot-Algebra E₁ renaming (_# to _#₁)
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open Elgot-Algebra E₂ renaming (_# to _#₂; A to B)
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record Elgot-Algebra-Morphism : Set (o ⊔ ℓ ⊔ e) where
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@ -38,7 +49,7 @@ module _ (CC : CocartesianCategory o ℓ e) where
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-- the category of elgot algebras for a given (cocartesian-)category
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Elgot-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
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Elgot-Algebras = record
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{ Obj = Elgot-Algebra CC
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{ Obj = Elgot-Algebra D
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; _⇒_ = Elgot-Algebra-Morphism
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; _≈_ = λ f g → Elgot-Algebra-Morphism.h f ≈ Elgot-Algebra-Morphism.h g
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; id = λ {EB} → let open Elgot-Algebra EB in
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@ -105,7 +116,7 @@ module _ (CC : CocartesianCategory o ℓ e) where
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open Equiv
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-- if the carriers of the algebra form a product, so do the algebras
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A×B-Helper : ∀ {EA EB : Elgot-Algebra CC} → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Elgot-Algebra CC
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A×B-Helper : ∀ {EA EB : Elgot-Algebra D} → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Elgot-Algebra D
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A×B-Helper {EA} {EB} p = record
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{ A = A×B
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; _# = λ {X : Obj} (h : X ⇒ A×B + X) → ⟨ ((π₁ +₁ idC) ∘ h)#ᵃ , ((π₂ +₁ idC) ∘ h)#ᵇ ⟩
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@ -202,7 +213,7 @@ module _ (CC : CocartesianCategory o ℓ e) where
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(([ (π₂ +₁ idC) ∘ (idC +₁ i₁) ∘ f , (π₂ +₁ idC) ∘ i₂ ∘ h ])#ᵇ) ≈⟨ sym (#ᵇ-resp-≈ ∘[]) ⟩
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((π₂ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ ∎
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Product-Elgot-Algebras : ∀ (EA EB : Elgot-Algebra CC) → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Product Elgot-Algebras EA EB
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Product-Elgot-Algebras : ∀ (EA EB : Elgot-Algebra D) → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Product Elgot-Algebras EA EB
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Product-Elgot-Algebras EA EB p = record
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{ A×B = A×B-Helper {EA} {EB} p
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; π₁ = record { h = π₁ ; preserves = λ {X} {f} → project₁ }
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@ -241,3 +252,16 @@ module _ (CC : CocartesianCategory o ℓ e) where
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open Cartesian CaC using (terminal; products)
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open BinaryProducts products using (product)
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open Equiv
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-- if the carriers of the algebra form a exponential, so do the algebras
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B^A-Helper : ∀ {EA EB : Elgot-Algebra D} → Exponential C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Elgot-Algebra D
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B^A-Helper {EA} {EB} e = record
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{ A = B^A
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; _# = λ {X} f → λg {! !} {! !}
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; #-Fixpoint = {! !}
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; #-Uniformity = {! !}
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; #-Folding = {! !}
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; #-resp-≈ = {! !}
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}
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where
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open Exponential e
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