bsc-leon-vatthauer/src/ElgotAlgebras.lagda.md

Summary

This file introduces the category of unguarded elgot algebras

• Definition 7 Category of elgot algebras
• Lemma 11 Products of elgot algebras
• Lemma 11 Exponentials of elgot algebras

Code

module ElgotAlgebras where

private
  variable
    o ℓ e : Level

module _ (D : ExtensiveDistributiveCategory o ℓ e) where
  open ExtensiveDistributiveCategory D renaming (U to C; id to idC)
  open Cocartesian (Extensive.cocartesian extensive)
  open Cartesian (ExtensiveDistributiveCategory.cartesian D)
  open BinaryProducts products
  open M C
  open MR C
  open HomReasoning
  open Equiv

Definition 7: Category of elgot algebras

  -- iteration preversing morphism between two elgot-algebras
  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
      field
        h : A ⇒ B
        preserves : ∀ {X} {f : X ⇒ A + X} → h ∘ (f #₁) ≈ ((h +₁ idC) ∘ f)#₂

  -- the category of elgot algebras for a given category
  Elgot-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
  Elgot-Algebras = record
    { 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 
    record { h = idC; preserves = λ {X : Obj} {f : X ⇒ A + X} → begin
      idC ∘ f #            ≈⟨ identityˡ ⟩ 
      f #                  ≈⟨ #-resp-≈ (introˡ (coproduct.unique id-comm-sym id-comm-sym)) ⟩
      ((idC +₁ idC) ∘ f) # ∎ }
    ; _∘_       = λ {EA} {EB} {EC} f g → let 
      open Elgot-Algebra-Morphism f renaming (h to hᶠ; preserves to preservesᶠ)
      open Elgot-Algebra-Morphism g renaming (h to hᵍ; preserves to preservesᵍ)
      open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ)
      open Elgot-Algebra EB using () renaming (_# to _#ᵇ; A to B)
      open Elgot-Algebra EC using () renaming (_# to _#ᶜ; A to C; #-resp-≈ to #ᶜ-resp-≈)
      in record { h = hᶠ ∘ hᵍ; preserves = λ {X} {f : X ⇒ A + X} → begin 
        (hᶠ ∘ hᵍ) ∘ (f #ᵃ)                     ≈⟨ pullʳ preservesᵍ ⟩
        (hᶠ ∘ (((hᵍ +₁ idC) ∘ f) #ᵇ))          ≈⟨ preservesᶠ ⟩ 
        (((hᶠ +₁ idC) ∘ (hᵍ +₁ idC) ∘ f) #ᶜ)   ≈⟨ #ᶜ-resp-≈ (pullˡ (trans +₁∘+₁ (+₁-cong₂ refl (identity²)))) ⟩ 
        ((hᶠ ∘ hᵍ +₁ idC) ∘ f) #ᶜ              ∎ }
    ; identityˡ = identityˡ
    ; identityʳ = identityʳ
    ; identity² = identity²
    ; assoc     = assoc
    ; sym-assoc = sym-assoc
    ; equiv     = record
      { refl  = refl
      ; sym   = sym
      ; trans = trans
      }
    ; ∘-resp-≈  = ∘-resp-≈
    }
    where open Elgot-Algebra-Morphism

Lemma 11: Products of elgot algebras

  -- if the carrier contains a terminal, so does elgot-algebras
  Terminal-Elgot-Algebras : Terminal C → Terminal Elgot-Algebras
  Terminal-Elgot-Algebras T = record 
    { ⊤ = record
      { A = ⊤
      ; algebra = record
        { _# = λ x → !
        ; #-Fixpoint = λ {_ f} → !-unique ([ idC , ! ] ∘ f)
        ; #-Uniformity = λ {_ _ _ _ h} _ → !-unique (! ∘ h)
        ; #-Folding = refl
        ; #-resp-≈ = λ _ → refl
        }
      } 
    ; ⊤-is-terminal = record 
      { ! = λ {A} → record { h = ! ; preserves = λ {X} {f} → sym (!-unique (! ∘ (A Elgot-Algebra.#) f))   } 
      ; !-unique = λ {A} f → !-unique (Elgot-Algebra-Morphism.h f) 
      } 
    }
    where 
    open Terminal T

  -- if the carriers of the algebra form a product, so do the algebras
  A×B-Helper : ∀ {EA EB : Elgot-Algebra D} → Elgot-Algebra D
  A×B-Helper {EA} {EB}  = record
    { A = A × B
    ; algebra = record
      { _# = λ {X : Obj} (h : X ⇒ A×B + X) → ⟨ ((π₁ +₁ idC) ∘ h)#ᵃ , ((π₂ +₁ idC) ∘ h)#ᵇ ⟩
      ; #-Fixpoint = λ {X} {f} → begin
        ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩                                                             ≈⟨ ⟨⟩-cong₂ #ᵃ-Fixpoint #ᵇ-Fixpoint ⟩
        ⟨ [ idC , ((π₁ +₁ idC) ∘ f)#ᵃ ] ∘ ((π₁ +₁ idC) ∘ f) , [ idC , ((π₂ +₁ idC) ∘ f)#ᵇ ] ∘ ((π₂ +₁ idC) ∘ f) ⟩ ≈⟨ ⟨⟩-cong₂ (pullˡ []∘+₁) (pullˡ []∘+₁) ⟩
        ⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] ∘ f , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ∘ f ⟩           ≈˘⟨ ⟨⟩∘ ⟩
        (⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ∘ f)             ≈⟨ ∘-resp-≈ˡ (unique′ 
          (begin 
            π₁ ∘ ⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ≈⟨ project₁ ⟩ 
            [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ]                                                     ≈⟨ []-cong₂ id-comm-sym (trans identityʳ (sym project₁)) ⟩
            [ π₁ ∘ idC , π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ]                            ≈˘⟨ ∘[] ⟩
            π₁ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ]                                 ∎) 
          (begin 
            π₂ ∘ ⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ≈⟨ project₂ ⟩ 
            [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ]                                                     ≈⟨ []-cong₂ id-comm-sym (trans identityʳ (sym project₂)) ⟩
            [ π₂ ∘ idC , π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ]                            ≈˘⟨ ∘[] ⟩
            π₂ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ]                                 ∎)
        )⟩
        ([ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∘ f)                                             ∎
      ; #-Uniformity = λ {X Y f g h} uni → unique′ 
        (begin 
          π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ project₁ ⟩ 
            (((π₁ +₁ idC) ∘ f)#ᵃ)                            ≈⟨ #ᵃ-Uniformity 
              (begin 
              -- TODO factor these out or adjust +₁ swap... maybe call it +₁-id-comm
                (idC +₁ h) ∘ (π₁ +₁ idC) ∘ f                     ≈⟨ pullˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm)) ⟩ 
                (π₁ ∘ idC +₁ idC ∘ h) ∘ f                        ≈˘⟨ pullˡ +₁∘+₁ ⟩
                (π₁ +₁ idC) ∘ (idC +₁ h) ∘ f                     ≈⟨ pushʳ uni ⟩
                ((π₁ +₁ idC) ∘ g) ∘ h                            ∎)⟩
        (((π₁ +₁ idC) ∘ g)#ᵃ ∘ h)                                  ≈˘⟨ pullˡ project₁ ⟩
          π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩ ∘ h     ∎) 
        (begin 
          π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ project₂ ⟩ 
          ((π₂ +₁ idC) ∘ f)#ᵇ                                ≈⟨ #ᵇ-Uniformity 
            (begin 
              (idC +₁ h) ∘ (π₂ +₁ idC) ∘ f   ≈⟨ pullˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm))⟩
              ((π₂ ∘ idC +₁ idC ∘ h) ∘ f)    ≈˘⟨ pullˡ +₁∘+₁ ⟩
              (π₂ +₁ idC) ∘ ((idC +₁ h)) ∘ f ≈⟨ pushʳ uni ⟩
              ((π₂ +₁ idC) ∘ g) ∘ h          ∎)⟩
          ((π₂ +₁ idC) ∘ g)#ᵇ ∘ h                                    ≈˘⟨ pullˡ project₂ ⟩
          π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩ ∘ h     ∎)
      ; #-Folding = λ {X} {Y} {f} {h} → ⟨⟩-cong₂ (foldingˡ {X} {Y}) (foldingʳ {X} {Y})
      ; #-resp-≈ = λ fg → ⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ʳ fg)) (#ᵇ-resp-≈ (∘-resp-≈ʳ fg)) 
      }
    }
    where 
    open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
    open Elgot-Algebra EB using () renaming (A to B; _# to _#ᵇ; #-Fixpoint to #ᵇ-Fixpoint; #-Uniformity to #ᵇ-Uniformity; #-Folding to #ᵇ-Folding; #-resp-≈ to #ᵇ-resp-≈)
    +₁-id-swap : ∀ {X Y C} {f : X ⇒ (A × B) + X} {h : Y ⇒ X + Y} (π : A × B ⇒ C) → [ (idC +₁ i₁) ∘ ((π +₁ idC) ∘ f) , i₂ ∘ h ] ≈ (π +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ]
    +₁-id-swap {X} {Y} {C} {f} {h} π = begin ([ (idC +₁ i₁) ∘ ((π +₁ idC) ∘ f) , i₂ ∘ h ] )                      ≈⟨ ([]-congʳ sym-assoc) ⟩ 
      ([ ((idC +₁ i₁) ∘ (π +₁ idC)) ∘ f , i₂ ∘ h ] )                      ≈⟨ []-cong₂ (∘-resp-≈ˡ (trans +₁∘+₁ (+₁-cong₂ id-comm-sym id-comm))) (∘-resp-≈ˡ (sym identityʳ)) ⟩ 
      (([ (π ∘ idC +₁ idC ∘ i₁) ∘ f ,  (i₂ ∘ idC) ∘ h ]))                 ≈˘⟨ []-cong₂ (pullˡ +₁∘+₁) (pullˡ +₁∘i₂) ⟩ 
      (([ (π +₁ idC) ∘ (idC +₁ i₁) ∘ f , (π +₁ idC) ∘ i₂ ∘ h ]))          ≈˘⟨ ∘[] ⟩ 
      ((π +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])                         ∎
    foldingˡ : ∀ {X} {Y} {f} {h} → (((π₁ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ) ≈ ((π₁ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ
    foldingˡ {X} {Y} {f} {h} = begin 
      ((π₁ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ ≈⟨ #ᵃ-resp-≈ (trans +₁∘+₁ (+₁-cong₂ project₁ identityˡ)) ⟩ 
      ((((π₁ +₁ idC) ∘ f)#ᵃ +₁ h)#ᵃ)                                         ≈⟨ #ᵃ-Folding ⟩ 
      ([ (idC +₁ i₁) ∘ ((π₁ +₁ idC) ∘ f) , i₂ ∘ h ] #ᵃ)                      ≈⟨ #ᵃ-resp-≈ (+₁-id-swap π₁)⟩
      ((π₁ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ                         ∎
    foldingʳ : ∀ {X} {Y} {f} {h} → ((π₂ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈ ((π₂ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ
    foldingʳ {X} {Y} {f} {h} = begin
      ((π₂ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈⟨ #ᵇ-resp-≈ (trans +₁∘+₁ (+₁-cong₂ project₂ identityˡ)) ⟩
      ((((π₂ +₁ idC) ∘ f)#ᵇ +₁ h)#ᵇ)                                         ≈⟨ #ᵇ-Folding ⟩
      [ (idC +₁ i₁) ∘ ((π₂ +₁ idC) ∘ f) , i₂ ∘ h ] #ᵇ                        ≈⟨ #ᵇ-resp-≈ (+₁-id-swap π₂) ⟩
      ((π₂ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ                         ∎

  Product-Elgot-Algebras : ∀ (EA EB : Elgot-Algebra D) → Product Elgot-Algebras EA EB
  Product-Elgot-Algebras EA EB = record
    { A×B = A×B-Helper {EA} {EB}
    ; π₁ = record { h = π₁ ; preserves = λ {X} {f} → project₁ }
    ; π₂ = record { h = π₂ ; preserves = λ {X} {f} → project₂ }
    ; ⟨_,_⟩ = λ {E} f g → let 
      open Elgot-Algebra-Morphism f renaming (h to f′; preserves to preservesᶠ)
      open Elgot-Algebra-Morphism g renaming (h to g′; preserves to preservesᵍ)
      open Elgot-Algebra E renaming (_# to _#ᵉ) in record { h = ⟨ f′ , g′ ⟩ ; preserves = λ {X} {h} → 
      begin 
        ⟨ f′ , g′ ⟩ ∘ (h #ᵉ)                                                                              ≈⟨ ⟨⟩∘ ⟩ 
        ⟨ f′ ∘ (h #ᵉ) , g′ ∘ (h #ᵉ) ⟩                                                                     ≈⟨ ⟨⟩-cong₂ preservesᶠ preservesᵍ ⟩
        ⟨ ((f′ +₁ idC) ∘ h) #ᵃ , ((g′ +₁ idC) ∘ h) #ᵇ ⟩                                                   ≈˘⟨ ⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ project₁ identity²))) (#ᵇ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ project₂ identity²))) ⟩
        ⟨ ((π₁ ∘ ⟨ f′ , g′ ⟩ +₁ idC ∘ idC) ∘ h) #ᵃ , ((π₂ ∘ ⟨ f′ , g′ ⟩ +₁ idC ∘ idC) ∘ h) #ᵇ ⟩           ≈˘⟨ ⟨⟩-cong₂ (#ᵃ-resp-≈ (pullˡ +₁∘+₁)) (#ᵇ-resp-≈ (pullˡ +₁∘+₁)) ⟩
        ⟨ ((π₁ +₁ idC) ∘ (⟨ f′ , g′ ⟩ +₁ idC) ∘ h) #ᵃ , ((π₂ +₁ idC) ∘ (⟨ f′ , g′ ⟩ +₁ idC) ∘ h) #ᵇ ⟩     ∎ }
    ; project₁ = project₁
    ; project₂ = project₂
    ; unique = unique
    }
    where
    open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
    open Elgot-Algebra EB using () renaming (A to B; _# to _#ᵇ; #-Fixpoint to #ᵇ-Fixpoint; #-Uniformity to #ᵇ-Uniformity; #-Folding to #ᵇ-Folding; #-resp-≈ to #ᵇ-resp-≈)
    open Elgot-Algebra (A×B-Helper {EA} {EB}) using () renaming (_# to _#ᵖ) 


  -- if the carrier is cartesian, so is the category of algebras
  Cartesian-Elgot-Algebras : Cartesian Elgot-Algebras
  Cartesian-Elgot-Algebras = record 
    { terminal = Terminal-Elgot-Algebras terminal
    ; products = record { product = λ {EA EB} → Product-Elgot-Algebras EA EB }
    }

Lemma 11: Exponentials of elgot algebras

  -- if the carriers of the algebra form a exponential, so do the algebras
  B^A-Helper : ∀ {EA : Elgot-Algebra D} {X : Obj} → Exponential C X (Elgot-Algebra.A EA) → Elgot-Algebra D
  B^A-Helper {EA} {X} exp = record
    { A = A^X
    ; algebra = record
      { _# = λ {Z} f → λg product (((((eval +₁ idC) ∘ (Categories.Object.Product.repack C product product' +₁ idC)) ∘ dstl) ∘ (f ⁂ idC)) #ᵃ) 
      ; #-Fixpoint = {!   !}
      ; #-Uniformity = {!   !}
      ; #-Folding = {!   !}
      ; #-resp-≈ = {!   !}
      }
    }
    where
    open Exponential exp renaming (B^A to A^X; product to product')
    open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
    dstr = λ {X Y Z} → IsIso.inv (isIsoˡ {X} {Y} {Z})
    dstl = λ {X Y Z} → IsIso.inv (isIsoʳ {X} {Y} {Z})