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40 lines
1.4 KiB
Agda
40 lines
1.4 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 Function using (id)
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module MonadK {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 import UniformIterationAlgebras
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open import UniformIterationAlgebra
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open import Categories.FreeObjects.Free
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open import Categories.Functor.Core
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open import Categories.Adjoint
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open import Categories.Adjoint.Properties
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open import Categories.NaturalTransformation.Core renaming (id to idN)
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open import Categories.Monad
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open Equiv
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record MonadK : Set (suc o ⊔ suc ℓ ⊔ suc e) where
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forgetfulF : Functor (Uniform-Iteration-Algebras D) C
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forgetfulF = record
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{ F₀ = λ X → Uniform-Iteration-Algebra.A X
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; F₁ = λ f → Uniform-Iteration-Algebra-Morphism.h f
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; identity = refl
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; homomorphism = refl
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; F-resp-≈ = id
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}
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field
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algebras : ∀ X → FreeObject {C = C} {D = Uniform-Iteration-Algebras D} forgetfulF X
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freeF : Functor C (Uniform-Iteration-Algebras D)
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freeF = FO⇒Functor forgetfulF algebras
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adjoint : freeF ⊣ forgetfulF
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adjoint = FO⇒LAdj forgetfulF algebras
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K : Monad C
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K = adjoint⇒monad adjoint
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-- TODO show that the category of K-Algebras is the category of uniform-iteration algebras
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