bsc-leon-vatthauer/src/Algebra/Properties.lagda.md

module Algebra.Properties {o ℓ e} (ambient : Ambient o ℓ e) where
  open Ambient ambient
  open import Algebra.ElgotAlgebra ambient
  open import Algebra.UniformIterationAlgebra ambient
  open import Category.Construction.ElgotAlgebras ambient
  open import Category.Construction.UniformIterationAlgebras ambient

  open Equiv

Properties of algebras

This file contains some typedefs and records concerning different algebras.

Free Uniform-Iteration Algebras

  uniformForgetfulF : Functor Uniform-Iteration-Algebras C
  uniformForgetfulF = record
    { F₀ = λ X → Uniform-Iteration-Algebra.A X
    ; F₁ = λ f → Uniform-Iteration-Algebra-Morphism.h f
    ; identity = refl
    ; homomorphism = refl
    ; F-resp-≈ = λ x → x
    }

  -- typedef
  FreeUniformIterationAlgebra : Obj → Set (suc o ⊔ suc ℓ ⊔ suc e)
  FreeUniformIterationAlgebra X = FreeObject {C = C} {D = Uniform-Iteration-Algebras} uniformForgetfulF X

Stable Free Uniform-Iteration Algebras

  record IsStableFreeUniformIterationAlgebra {Y : Obj} (freeAlgebra : FreeUniformIterationAlgebra Y) : Set (suc o ⊔ suc ℓ ⊔ suc e) where 
    open FreeObject freeAlgebra using (η) renaming (FX to FY)
    open Uniform-Iteration-Algebra FY using (_#)

    field
        -- TODO awkward notation...
        [_,_]♯ : ∀ {X : Obj} (A : Uniform-Iteration-Algebra) (f : X × Y ⇒ Uniform-Iteration-Algebra.A A) → X × Uniform-Iteration-Algebra.A FY ⇒ Uniform-Iteration-Algebra.A A
        ♯-law : ∀ {X : Obj} {A : Uniform-Iteration-Algebra} (f : X × Y ⇒ Uniform-Iteration-Algebra.A A) → f ≈ [ A , f ]♯ ∘ (idC ⁂ η)
        ♯-preserving : ∀ {X : Obj} {B : Uniform-Iteration-Algebra} (f : X × Y ⇒ Uniform-Iteration-Algebra.A B) {Z : Obj} (h : Z ⇒ Uniform-Iteration-Algebra.A FY + Z) → [ B , f ]♯ ∘ (idC ⁂ h #) ≈ Uniform-Iteration-Algebra._# B (([ B , f ]♯ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))
        ♯-unique : ∀ {X : Obj} {B : Uniform-Iteration-Algebra} (f : X × Y ⇒ Uniform-Iteration-Algebra.A B) (g : X × Uniform-Iteration-Algebra.A FY ⇒ Uniform-Iteration-Algebra.A B)
          → f ≈ g ∘ (idC ⁂ η)
          → (∀ {Z : Obj} (h : Z ⇒ Uniform-Iteration-Algebra.A FY + Z) → g ∘ (idC ⁂ h #) ≈ Uniform-Iteration-Algebra._# B ((g +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)))
          → g ≈ [ B , f ]♯

  record StableFreeUniformIterationAlgebra : Set (suc o ⊔ suc ℓ ⊔ suc e) where 
    field
      Y : Obj
      freeAlgebra : FreeUniformIterationAlgebra Y
      isStableFreeUniformIterationAlgebra : IsStableFreeUniformIterationAlgebra freeAlgebra
    
    open IsStableFreeUniformIterationAlgebra isStableFreeUniformIterationAlgebra public

Free Elgot Algebras

  elgotForgetfulF : Functor Elgot-Algebras C
  elgotForgetfulF = record
    { F₀ = λ X → Elgot-Algebra.A X
    ; F₁ = λ f → Elgot-Algebra-Morphism.h f
    ; identity = refl
    ; homomorphism = refl
    ; F-resp-≈ = λ x → x
    }

  -- typedef
  FreeElgotAlgebra : Obj → Set (suc o ⊔ suc ℓ ⊔ suc e)
  FreeElgotAlgebra X = FreeObject {C = C} {D = Elgot-Algebras} elgotForgetfulF X

Stable Free Elgot Algebras

-- TODO this is duplicated from uniform iteration and since free elgots coincide with free uniform iterations this can later be removed.

  record IsStableFreeElgotAlgebra {Y : Obj} (freeElgot : FreeElgotAlgebra Y) : Set (suc o ⊔ suc ℓ ⊔ suc e) where 
    open FreeObject freeElgot using (η) renaming (FX to FY)
    open Elgot-Algebra FY using (_#)

    field
        -- TODO awkward notation...
        [_,_]♯ : ∀ {X : Obj} (A : Elgot-Algebra) (f : X × Y ⇒ Elgot-Algebra.A A) → X × Elgot-Algebra.A FY ⇒ Elgot-Algebra.A A
        ♯-law : ∀ {X : Obj} {A : Elgot-Algebra} (f : X × Y ⇒ Elgot-Algebra.A A) → f ≈ [ A , f ]♯ ∘ (idC ⁂ η)
        ♯-preserving : ∀ {X : Obj} {B : Elgot-Algebra} (f : X × Y ⇒ Elgot-Algebra.A B) {Z : Obj} (h : Z ⇒ Elgot-Algebra.A FY + Z) → [ B , f ]♯ ∘ (idC ⁂ h #) ≈ Elgot-Algebra._# B (([ B , f ]♯ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))
        ♯-unique : ∀ {X : Obj} {B : Elgot-Algebra} (f : X × Y ⇒ Elgot-Algebra.A B) (g : X × Elgot-Algebra.A FY ⇒ Elgot-Algebra.A B)
          → f ≈ g ∘ (idC ⁂ η)
          → (∀ {Z : Obj} (h : Z ⇒ Elgot-Algebra.A FY + Z) → g ∘ (idC ⁂ h #) ≈ Elgot-Algebra._# B ((g +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)))
          → g ≈ [ B , f ]♯

  record StableFreeElgotAlgebra : Set (suc o ⊔ suc ℓ ⊔ suc e) where 
    field
      Y : Obj
      freeElgot : FreeElgotAlgebra Y
      isStableFreeElgotAlgebra : IsStableFreeElgotAlgebra freeElgot
    
    open IsStableFreeElgotAlgebra isStableFreeElgotAlgebra public

Free Elgot to Free Uniform Iteration

  elgot-to-uniformF : Functor Elgot-Algebras Uniform-Iteration-Algebras
  elgot-to-uniformF = record
    { F₀ = λ X → record { A = Elgot-Algebra.A X ; algebra = record
      { _# = λ f → (X Elgot-Algebra.#) f
      ; #-Fixpoint = Elgot-Algebra.#-Fixpoint X 
      ; #-Uniformity = Elgot-Algebra.#-Uniformity X 
      ; #-resp-≈ = Elgot-Algebra.#-resp-≈ X 
      }}
    ; F₁ = λ f → record { h = Elgot-Algebra-Morphism.h f ; preserves = Elgot-Algebra-Morphism.preserves f }
    ; identity = refl
    ; homomorphism = refl
    ; F-resp-≈ = λ x → x
    }

FreeElgots⇒FreeUniformIterations : (∀ X → FreeElgotAlgebra X) → (∀ X → FreeUniformIterationAlgebra X) FreeElgots⇒FreeUniformIterations free-elgots X = record { FX = F₀ (FreeObject.FX (free-elgots X)) ; η = FreeObject.η (free-elgots X) ; * = λ {FY} f → F₁ (FreeObject.* (free-elgots X) {A = record { A = Uniform-Iteration-Algebra.A FY ; algebra = record { _# = FY Uniform-Iteration-Algebra.# ; #-Fixpoint = Uniform-Iteration-Algebra.#-Fixpoint FY ; #-Uniformity = Uniform-Iteration-Algebra.#-Uniformity FY ; #-Folding = {! !} ; #-resp-≈ = Uniform-Iteration-Algebra.#-resp-≈ FY } }} f) -- FreeObject.FX (free-elgots (Uniform-Iteration-Algebra.A FY)) ; *-lift = {! !} ; *-uniq = {! !} } where open Functor elgot-to-uniformF open Functor (FO⇒Functor elgotForgetfulF free-elgots) using () renaming (F₀ to FO₀; F₁ to FO₁)