Added K monad

MonadK.agda

@@ -0,0 +1,40 @@
+open import Level
+open import Categories.Category.Core
+open import Categories.Category.Extensive.Bundle
+open import Function using (id)
+
+module MonadK {o ℓ e} (D : ExtensiveDistributiveCategory o ℓ e) where
+  open ExtensiveDistributiveCategory D renaming (U to C; id to idC)
+  open import UniformIterationAlgebras
+  open import UniformIterationAlgebra
+  open import Categories.FreeObjects.Free
+  open import Categories.Functor.Core
+  open import Categories.Adjoint
+  open import Categories.Adjoint.Properties
+  open import Categories.NaturalTransformation.Core renaming (id to idN)
+  open import Categories.Monad
+  open Equiv
+
+  record MonadK : Set (suc o ⊔ suc ℓ ⊔ suc e) where
+    forgetfulF : Functor (Uniform-Iteration-Algebras D) 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
+      }
+    
+    field
+      algebras : ∀ X → FreeObject {C = C} {D = Uniform-Iteration-Algebras D} forgetfulF X
+
+    freeF : Functor C (Uniform-Iteration-Algebras D)
+    freeF = FO⇒Functor forgetfulF algebras
+    
+    adjoint : freeF ⊣ forgetfulF
+    adjoint = FO⇒LAdj forgetfulF algebras
+
+    K : Monad C
+    K = adjoint⇒monad adjoint
+
+    -- TODO show that the category of K-Algebras is the category of uniform-iteration algebras