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

<!--
```agda
open import Level
open import Categories.FreeObjects.Free
open import Categories.Functor.Core
open import Categories.Adjoint
open import Categories.Adjoint.Properties
open import Categories.Monad
open import Category.Instance.AmbientCategory using (Ambient)
```
-->

## 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.Properties ambient using (FreeUniformIterationAlgebra; uniformForgetfulF)

  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 = {!   !}
```