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

<!--
```agda
open import Level
open import Categories.FreeObjects.Free
open import Categories.Category.Product renaming (Product to CProduct; _⁂_ to _×C_)
open import Categories.Category
open import Categories.Functor.Core
open import Categories.Adjoint
open import Categories.Adjoint.Properties
open import Categories.Monad
open import Categories.Monad.Strong
open import Category.Instance.AmbientCategory using (Ambient)
open import Categories.NaturalTransformation
-- open import Data.Product using (_,_; Σ; Σ-syntax)
```
-->

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

- [X] *Lemma 16* Definition of the monad
- [ ] *Lemma 16* EilenbergMoore⇒UniformIterationAlgebras (use [crude monadicity theorem](https://agda.github.io/agda-categories/Categories.Adjoint.Monadic.Crude.html))
- [ ] *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

```agda
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
  open import Algebra.Properties ambient using (FreeUniformIterationAlgebra; uniformForgetfulF; IsStableFreeUniformIterationAlgebra)

  open Equiv

```

### *Lemma 16*: definition of monad ***K***

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

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

    K : Monad C
    K = adjoint⇒monad adjoint

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

### *Proposition 19* If the algebras are stable then K is strong

```agda
  record MonadKStrong : Set (suc o ⊔ suc ℓ ⊔ suc e) where
    field
      algebras : ∀ X → FreeUniformIterationAlgebra X
      stable : ∀ X → IsStableFreeUniformIterationAlgebra (algebras X)
    
    K : Monad C
    K = MonadK.K (record { algebras = algebras })

    open Monad K using (F)
    open Functor F using () renaming (F₀ to K₀; F₁ to K₁)

    KStrong : StrongMonad {C = C} monoidal
    KStrong = record 
      { M = K 
      ; strength = record
        { strengthen = ntHelper (record { η = τ ; commute = λ f → {!   !} }) 
        ; identityˡ = {!   !}
        ; η-comm = {!   !}
        ; μ-η-comm = {!   !}
        ; strength-assoc = {!   !}
        }
      }
      where
        open import Agda.Builtin.Sigma
        open IsStableFreeUniformIterationAlgebra using (♯-law; ♯-preserving)

        module _ (P : Category.Obj (CProduct C C)) where
          η = λ Z → FreeObject.η (algebras Z)
          [_,_,_]♯ = λ {A} X Y f → IsStableFreeUniformIterationAlgebra.[_,_]♯ {Y = X} (stable X) {X = A} Y f


          X = fst P
          Y = snd P
          τ : X × K₀ Y ⇒ K₀ (X × Y)
          τ = [ Y , FreeObject.FX (algebras (X × Y)) , η (X × Y) ]♯

          τ-η : τ ∘ (idC ⁂ η Y) ≈ η (X × Y)
          τ-η = sym (♯-law (stable Y) (η (X × Y)))

        [_,_]# : ∀ (A : Uniform-Iteration-Algebra ambient) {X} → (X ⇒ ((Uniform-Iteration-Algebra.A A) + X)) → (X ⇒ Uniform-Iteration-Algebra.A A)
        [ A , f ]# = Uniform-Iteration-Algebra._# A f

        τ-comm : ∀ {X Y Z : Obj} (h : Z ⇒ K₀ Y + Z) → τ (X , Y) ∘ (idC ⁂ [ FreeObject.FX (algebras Y) , h ]#) ≈ [ FreeObject.FX (algebras (X × Y)) , (τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h) ]#
        τ-comm {X} {Y} {Z} h = ♯-preserving (stable Y) (η (X , Y) (X × Y)) h
```