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Work on initial pre-Elgot

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Leon Vatthauer 2023-11-15 16:00:08 +01:00
parent 08cf9b6f61
commit 60f60ce3bf
Signed by: leonv
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2 changed files with 64 additions and 7 deletions
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29
src/Category/Construction/PreElgotMonads.lagda.md
View file

@ -18,6 +18,7 @@ open import Categories.Category.Core
module Category.Construction.PreElgotMonads {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient
open import Monad.ElgotMonad ambient
open import Algebra.ElgotAlgebra ambient
open HomReasoning
open Equiv
open M C
@ -25,12 +26,14 @@ open MR C
module _ (P S : PreElgotMonad) where
private
open PreElgotMonad P using () renaming (T to TP)
open PreElgotMonad S using () renaming (T to TS)
open PreElgotMonad P using () renaming (T to TP; elgotalgebras to P-elgots)
open PreElgotMonad S using () renaming (T to TS; elgotalgebras to S-elgots)
module TP = Monad TP
module TS = Monad TS
open RMonad (Monad⇒Kleisli C TP) using () renaming (extend to extendP)
open RMonad (Monad⇒Kleisli C TS) using () renaming (extend to extendS)
_#P = λ {X} {A} f → P-elgots._# {X} {A} f
_#S = λ {X} {A} f → S-elgots._# {X} {A} f
record PreElgotMonad-Morphism : Set (o ⊔ ℓ ⊔ e) where
field
α : NaturalTransformation TP.F TS.F
@ -40,6 +43,7 @@ module _ (P S : PreElgotMonad) where
→ α.η X ∘ TP.η.η X ≈ TS.η.η X
α-μ : ∀ {X}
→ α.η X ∘ TP.μ.η X ≈ TS.μ.η X ∘ TS.F.₁ (α.η X) ∘ α.η (TP.F.₀ X)
preserves : ∀ {X A} (f : X ⇒ TP.F.₀ A + X) → α.η A ∘ f #P ≈ ((α.η A +₁ idC) ∘ f) #S
PreElgotMonads : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) (o ⊔ e)
PreElgotMonads = record
@ -57,6 +61,7 @@ PreElgotMonads = record
; ∘-resp-≈ = λ f≈h g≈i → ∘-resp-≈ f≈h g≈i
}
where
open Elgot-Algebra-on using (#-resp-≈)
id' : ∀ {A : PreElgotMonad} → PreElgotMonad-Morphism A A
id' {A} = record
{ α = ntHelper (record
@ -68,10 +73,16 @@ PreElgotMonads = record
T.μ.η _ ∘ T.F.₁ idC ∘ idC ≈⟨ refl⟩∘⟨ identityʳ ⟩
T.μ.η _ ∘ T.F.₁ idC ≈⟨ elimʳ T.F.identity ⟩
T.μ.η _ ≈⟨ sym identityˡ ⟩
idC ∘ T.μ.η _ ∎) }
idC ∘ T.μ.η _ ∎)
; preserves = λ f → begin
idC ∘ f # ≈⟨ identityˡ ⟩
f # ≈⟨ sym (#-resp-≈ elgotalgebras (elimˡ ([]-unique id-comm-sym id-comm-sym))) ⟩
((idC +₁ idC) ∘ f) # ∎
}
where
open PreElgotMonad A using (T)
open PreElgotMonad A using (T; elgotalgebras)
module T = Monad T
_# = λ {X} {A} f → elgotalgebras._# {X} {A} f
_∘'_ : ∀ {X Y Z : PreElgotMonad} → PreElgotMonad-Morphism Y Z → PreElgotMonad-Morphism X Y → PreElgotMonad-Morphism X Z
_∘'_ {X} {Y} {Z} f g = record
{ α = αf ∘ᵥ αg
@ -86,13 +97,21 @@ PreElgotMonads = record
TZ.μ.η A ∘ TZ.F.₁ (αf.η A) ∘ (TZ.F.₁ (αg.η A) ∘ αf.η (TX.F.₀ A)) ∘ αg.η (TX.F.₀ A) ≈⟨ refl⟩∘⟨ pullˡ (pullˡ (sym (Functor.homomorphism TZ.F))) ⟩
TZ.μ.η A ∘ (TZ.F.₁ (αf.η A ∘ αg.η A) ∘ αf.η (TX.F.₀ A)) ∘ αg.η (TX.F.₀ A) ≈⟨ refl⟩∘⟨ assoc ⟩
TZ.μ.η A ∘ TZ.F.₁ ((αf.η A ∘ αg.η A)) ∘ αf.η (TX.F.₀ A) ∘ αg.η (TX.F.₀ A) ∎
; preserves = λ {A} {B} h → begin
(αf.η B ∘ αg.η B) ∘ (h #X) ≈⟨ pullʳ (preserves g h) ⟩
αf.η B ∘ ((αg.η B +₁ idC) ∘ h) #Y ≈⟨ preserves f ((αg.η B +₁ idC) ∘ h) ⟩
(((αf.η B +₁ idC) ∘ (αg.η B +₁ idC) ∘ h) #Z) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras Z) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
(((αf.η B ∘ αg.η B +₁ idC) ∘ h) #Z) ∎
}
where
module TX = Monad (PreElgotMonad.T X)
module TY = Monad (PreElgotMonad.T Y)
module TZ = Monad (PreElgotMonad.T Z)
_#X = λ {A} {B} f → PreElgotMonad.elgotalgebras._# X {A} {B} f
_#Y = λ {A} {B} f → PreElgotMonad.elgotalgebras._# Y {A} {B} f
_#Z = λ {A} {B} f → PreElgotMonad.elgotalgebras._# Z {A} {B} f
open PreElgotMonad-Morphism using (α-η; α-μ)
open PreElgotMonad-Morphism using (α-η; α-μ; preserves)
open PreElgotMonad-Morphism f using () renaming (α to αf)
open PreElgotMonad-Morphism g using () renaming (α to αg)

40
src/Monad/Instance/K/PreElgot.lagda.md
View file

@ -2,6 +2,11 @@
```agda
open import Level
open import Category.Instance.AmbientCategory
open import Categories.FreeObjects.Free
open import Categories.Object.Initial
open import Categories.NaturalTransformation
open import Categories.NaturalTransformation.Equivalence
open import Categories.Monad
import Monad.Instance.K as MIK
```
-->
@ -17,7 +22,8 @@ open import Monad.ElgotMonad ambient
open import Monad.Instance.K ambient
open import Monad.Instance.K.Compositionality ambient MK
open import Monad.Instance.K.Commutative ambient MK
open import Categories.FreeObjects.Free
open import Category.Construction.PreElgotMonads ambient
open import Category.Construction.ElgotAlgebras ambient
open Equiv
open HomReasoning
@ -43,4 +49,36 @@ preElgot : PreElgotMonad
preElgot = record { T = monadK ; isPreElgot = isPreElgot }
-- initialPreElgot :
initialPreElgot : IsInitial PreElgotMonads preElgot
initialPreElgot = record
{ ! = !′
; !-unique = !-unique′
}
where
!′ : ∀ {A : PreElgotMonad} → PreElgotMonad-Morphism preElgot A
!′ {A} = record
{ α = ntHelper (record
{ η = η'
; commute = {!!}
})
; α-η = {!!}
; α-μ = {!!}
}
where
open PreElgotMonad A using (T)
module T = Monad T
open PreElgotMonad preElgot using ()
η' : ∀ (X : Obj) → K.₀ X ⇒ T.F.₀ X
η' X = Elgot-Algebra-Morphism.h (_* {A = record { A = T.F.F₀ X ; algebra = PreElgotMonad.elgotalgebras A }} (T.η.η X))
where open FreeObject (freeElgot X)
!-unique′ : ∀ {A : PreElgotMonad} (f : PreElgotMonad-Morphism preElgot A) → PreElgotMonad-Morphism.α !′ ≃ PreElgotMonad-Morphism.α f
!-unique′ {A} f {X} = sym (*-uniq (T.η.η X) (record
{ h = α.η X
; preserves = preserves _
}) α-η)
where
open PreElgotMonad-Morphism f using (α; α-η; preserves)
open PreElgotMonad A using (T)
module T = Monad T
open FreeObject (freeElgot X)
```