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Proof that KX is freeElgot (still assuming compositionality)

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Leon Vatthauer 2023-11-15 09:41:27 +01:00
parent 13450c1d23
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src/Monad/Instance/K/Compositionality.lagda.md
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@ -15,7 +15,12 @@ module Monad.Instance.K.Compositionality {o ℓ e} (ambient : Ambient o ℓ e) (
open MonadK MK
open import Algebra.UniformIterationAlgebra ambient
open import Category.Construction.UniformIterationAlgebras ambient
open import Category.Construction.ElgotAlgebras ambient
open import Algebra.ElgotAlgebra ambient
open import Algebra.Properties ambient
open import Categories.Functor.Core
open Functor using (F₀; F₁)
elgot : ∀ (A : Obj) → Elgot-Algebra-on (K.₀ A)
elgot A = record
@ -27,4 +32,19 @@ module Monad.Instance.K.Compositionality {o ℓ e} (ambient : Ambient o ℓ e) (
}
where
open Uniform-Iteration-Algebra (algebras A) using (_#; #-Fixpoint; #-Uniformity; #-resp-≈)
freeElgot : ∀ (A : Obj) → FreeElgotAlgebra A
freeElgot A = record
{ FX = record { A = K.₀ A ; algebra = elgot A }
; η = η (freealgebras A)
; _* = λ {X} f → record
{ h = Uniform-Iteration-Algebra-Morphism.h (_* (freealgebras A) {A = F₀ elgot-to-uniformF X} f)
; preserves = λ {Y} {g} → Uniform-Iteration-Algebra-Morphism.preserves (((freealgebras A) *) f)
}
; *-lift = *-lift (freealgebras A)
; *-uniq = λ {X} f g eq → *-uniq (freealgebras A) f (F₁ elgot-to-uniformF g) eq
}
where
open FreeObject using (η; _*; *-lift; *-uniq)
```
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