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https://git8.cs.fau.de/theses/bsc-leon-vatthauer.git
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77 lines
2.6 KiB
Markdown
77 lines
2.6 KiB
Markdown
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<!--
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```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.Equivalence using (StrongEquivalence)
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open import Categories.Category.Extensive.Bundle
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open import Function using (id)
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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.Adjoint.Monadic.Crude
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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 import Categories.Category.Construction.EilenbergMoore
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open import Categories.Category.Slice
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```
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-->
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## Summary
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In this file I explore the monad ***K*** and its properties:
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- [X] *Lemma 16* Definition of the monad
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- [ ] *Lemma 16* EilenbergMoore⇒UniformIterationAlgebras (use [crude monadicity theorem](https://agda.github.io/agda-categories/Categories.Adjoint.Monadic.Crude.html))
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- [ ] *Proposition 19* ***K*** is strong
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- [ ] *Theorem 22* ***K*** is an equational lifting monad
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- [ ] *Proposition 23* The Kleisli category of ***K*** is enriched over pointed partial orders and strict monotone maps
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- [ ] *Proposition 25* ***K*** is copyable and weakly discardable
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- [ ] *Theorem 29* ***K*** is an initial pre-Elgot monad and an initial strong pre-Elgot monad
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## Code
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```agda
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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 Equiv
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-- TODO move this to a different file
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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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-- typedef
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FreeUniformIterationAlgebra : Obj → Set (suc o ⊔ suc ℓ ⊔ suc e)
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FreeUniformIterationAlgebra X = FreeObject {C = C} {D = Uniform-Iteration-Algebras D} forgetfulF X
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```
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### *Lemma 16*: definition of monad ***K***
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```agda
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record MonadK : Set (suc o ⊔ suc ℓ ⊔ suc e) where
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field
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algebras : ∀ X → FreeUniformIterationAlgebra 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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-- EilenbergMoore⇒UniformIterationAlgebras : StrongEquivalence (EilenbergMoore K) (Uniform-Iteration-Algebras D)
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-- EilenbergMoore⇒UniformIterationAlgebras = {! !}
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```
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