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60 lines
3 KiB
Agda
60 lines
3 KiB
Agda
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open import Level
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open import Categories.Category.Core
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open import Categories.Category.Extensive.Bundle
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open import Categories.Category.Extensive
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open import Categories.Category.Cocartesian
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import Categories.Morphism.Reasoning as MR
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open import UniformIterationAlgebra
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module UniformIterationAlgebras {o ℓ e} (D : ExtensiveDistributiveCategory o ℓ e) where
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open ExtensiveDistributiveCategory D renaming (U to C; id to idC)
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open Cocartesian (Extensive.cocartesian extensive)
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open HomReasoning
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open MR C
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open Equiv
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-- iteration preversing morphism between two elgot-algebras
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module _ (E₁ E₂ : Uniform-Iteration-Algebra D) where
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open Uniform-Iteration-Algebra E₁ renaming (_# to _#₁)
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open Uniform-Iteration-Algebra E₂ renaming (_# to _#₂; A to B)
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record Uniform-Iteration-Algebra-Morphism : Set (o ⊔ ℓ ⊔ e) where
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field
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h : A ⇒ B
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preserves : ∀ {X} {f : X ⇒ A + X} → h ∘ (f #₁) ≈ ((h +₁ idC) ∘ f)#₂
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-- the category of uniform-iteration algebras for a given category
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Uniform-Iteration-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
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Uniform-Iteration-Algebras = record
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{ Obj = Uniform-Iteration-Algebra D
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; _⇒_ = Uniform-Iteration-Algebra-Morphism
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; _≈_ = λ f g → Uniform-Iteration-Algebra-Morphism.h f ≈ Uniform-Iteration-Algebra-Morphism.h g
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; id = λ {EB} → let open Uniform-Iteration-Algebra EB in
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record { h = idC; preserves = λ {X : Obj} {f : X ⇒ A + X} → begin
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idC ∘ f # ≈⟨ identityˡ ⟩
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f # ≈⟨ #-resp-≈ (introˡ (coproduct.unique id-comm-sym id-comm-sym)) ⟩
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((idC +₁ idC) ∘ f) # ∎ }
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; _∘_ = λ {EA} {EB} {EC} f g → let
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open Uniform-Iteration-Algebra-Morphism f renaming (h to hᶠ; preserves to preservesᶠ)
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open Uniform-Iteration-Algebra-Morphism g renaming (h to hᵍ; preserves to preservesᵍ)
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open Uniform-Iteration-Algebra EA using (A) renaming (_# to _#ᵃ)
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open Uniform-Iteration-Algebra EB using () renaming (_# to _#ᵇ; A to B)
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open Uniform-Iteration-Algebra EC using () renaming (_# to _#ᶜ; A to C; #-resp-≈ to #ᶜ-resp-≈)
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in record { h = hᶠ ∘ hᵍ; preserves = λ {X} {f : X ⇒ A + X} → begin
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(hᶠ ∘ hᵍ) ∘ (f #ᵃ) ≈⟨ pullʳ preservesᵍ ⟩
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(hᶠ ∘ (((hᵍ +₁ idC) ∘ f) #ᵇ)) ≈⟨ preservesᶠ ⟩
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(((hᶠ +₁ idC) ∘ (hᵍ +₁ idC) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ (pullˡ (trans +₁∘+₁ (+₁-cong₂ refl (identity²)))) ⟩
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((hᶠ ∘ hᵍ +₁ idC) ∘ f) #ᶜ ∎ }
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; identityˡ = identityˡ
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; identityʳ = identityʳ
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; identity² = identity²
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; assoc = assoc
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; sym-assoc = sym-assoc
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; equiv = record
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{ refl = refl
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; sym = sym
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; trans = trans
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}
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; ∘-resp-≈ = ∘-resp-≈
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}
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where open Uniform-Iteration-Algebra-Morphism
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