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29 lines
1.1 KiB
Agda
29 lines
1.1 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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module UniformIterationAlgebra {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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record Uniform-Iteration-Algebra-on (A : Obj) : Set (o ⊔ ℓ ⊔ e) where
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-- iteration operator
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field
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_# : ∀ {X} → (X ⇒ A + X) → (X ⇒ A)
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-- _# properties
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field
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#-Fixpoint : ∀ {X} {f : X ⇒ A + X }
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→ f # ≈ [ idC , f # ] ∘ f
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#-Uniformity : ∀ {X Y} {f : X ⇒ A + X} {g : Y ⇒ A + Y} {h : X ⇒ Y}
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→ (idC +₁ h) ∘ f ≈ g ∘ h
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→ f # ≈ g # ∘ h
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#-resp-≈ : ∀ {X} {f g : X ⇒ A + X} → f ≈ g → (f #) ≈ (g #)
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record Uniform-Iteration-Algebra : Set (o ⊔ ℓ ⊔ e) where
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field
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A : Obj
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algebra : Uniform-Iteration-Algebra-on A
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open Uniform-Iteration-Algebra-on algebra public
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