bsc-leon-vatthauer/src/Monad/Instance/K.lagda.md

Summary

In this file I explore the monad K and its properties:

• Lemma 16 Definition of the monad
• Lemma 16 EilenbergMoore⇒UniformIterationAlgebras (use crude monadicity theorem)
• Proposition 19 K is strong
• Theorem 22 K is an equational lifting monad
• Proposition 23 The Kleisli category of K is enriched over pointed partial orders and strict monotone maps
• Proposition 25 K is copyable and weakly discardable
• Theorem 29 K is an initial pre-Elgot monad and an initial strong pre-Elgot monad

Code

module Monad.Instance.K {o ℓ e} (ambient : Ambient o ℓ e) where
  open Ambient ambient
  open import Category.Construction.UniformIterationAlgebras ambient
  open import Algebra.UniformIterationAlgebra ambient
  open Equiv

  -- TODO move this to a different file

  forgetfulF : Functor Uniform-Iteration-Algebras C
  forgetfulF = record
    { F₀ = λ X → Uniform-Iteration-Algebra.A X
    ; F₁ = λ f → Uniform-Iteration-Algebra-Morphism.h f
    ; identity = refl
    ; homomorphism = refl
    ; F-resp-≈ = id
    }

  -- typedef
  FreeUniformIterationAlgebra : Obj → Set (suc o ⊔ suc ℓ ⊔ suc e)
  FreeUniformIterationAlgebra X = FreeObject {C = C} {D = Uniform-Iteration-Algebras} forgetfulF X

Lemma 16: definition of monad K

  record MonadK : Set (suc o ⊔ suc ℓ ⊔ suc e) where
    field
      algebras : ∀ X → FreeUniformIterationAlgebra X

    freeF : Functor C Uniform-Iteration-Algebras
    freeF = FO⇒Functor forgetfulF algebras
    
    adjoint : freeF ⊣ forgetfulF
    adjoint = FO⇒LAdj forgetfulF algebras

    K : Monad C
    K = adjoint⇒monad adjoint

    -- EilenbergMoore⇒UniformIterationAlgebras : StrongEquivalence (EilenbergMoore K) (Uniform-Iteration-Algebras D)
    -- EilenbergMoore⇒UniformIterationAlgebras = {!   !}