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show that K is strong pre-Elgot
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4 changed files with 89 additions and 50 deletions
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@ -17,7 +17,7 @@ open import Categories.Category.Core
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```agda
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module Category.Construction.PreElgotMonads {o ℓ e} (ambient : Ambient o ℓ e) where
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open Ambient ambient
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open import Monad.ElgotMonad ambient
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open import Monad.PreElgot ambient
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open import Algebra.ElgotAlgebra ambient
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open HomReasoning
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open Equiv
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@ -1,7 +1,5 @@
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<!--
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```agda
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{-# OPTIONS --allow-unsolved-metas #-}
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open import Level
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open import Category.Instance.AmbientCategory using (Ambient)
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open import Categories.Monad.Construction.Kleisli
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@ -11,54 +9,16 @@ open import Categories.Functor
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```
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-->
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## Summary
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This file introduces Elgot Monads.
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TODO: Probably only Pre-Elgot is needed
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- [X] *Definition 13* Pre-Elgot Monads
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- [ ] *Definition 13* strong pre-Elgot
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- [X] *Definition 14* Elgot Monads
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- [ ] *Definition 14* strong Elgot
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- [ ] *Proposition 15* (Strong) Elgot monads are (strong) pre-Elgot
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## Code
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```agda
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module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
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module Misc.Monad.Elgot {o ℓ e} (ambient : Ambient o ℓ e) where
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open Ambient ambient
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open HomReasoning
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open MR C
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open Equiv
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open import Algebra.ElgotAlgebra ambient
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open import Monad.PreElgot ambient
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```
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### *Definition 13*: Pre-Elgot Monads
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```agda
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record IsPreElgot (T : Monad C) : Set (o ⊔ ℓ ⊔ e) where
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open Monad T
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open RMonad (Monad⇒Kleisli C T) using (extend)
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open Functor F renaming (F₀ to T₀; F₁ to T₁)
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-- every TX needs to be equipped with an elgot algebra structure
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field
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elgotalgebras : ∀ {X} → Elgot-Algebra-on (T₀ X)
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module elgotalgebras {X} = Elgot-Algebra-on (elgotalgebras {X})
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-- with the following associativity
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field
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pres : ∀ {X Y Z} (f : Z ⇒ T₀ X + Z) (h : X ⇒ T₀ Y)
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→ elgotalgebras._# ((extend h +₁ idC) ∘ f) ≈ extend h ∘ (elgotalgebras._# {X}) f
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record PreElgotMonad : Set (o ⊔ ℓ ⊔ e) where
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field
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T : Monad C
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isPreElgot : IsPreElgot T
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open IsPreElgot isPreElgot public
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```
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### *Definition 14*: Elgot Monads
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```agda
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@ -90,7 +50,7 @@ module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
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open IsElgot isElgot public
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```
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### *Proposition 15*: (Strong) Elgot monads are (strong) pre-Elgot
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### *Proposition 15*: Elgot monads are pre-Elgot
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-- elgot monads are pre-elgot
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Elgot⇒PreElgot : ElgotMonad → PreElgotMonad
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@ -147,4 +107,4 @@ module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
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open ElgotMonad EM
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module T = Monad T
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open T using (F; η; μ)
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open Functor F renaming (F₀ to T₀; F₁ to T₁)
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open Functor F renaming (F₀ to T₀; F₁ to T₁)
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@ -20,7 +20,7 @@ open MIK ambient
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open MonadK MK
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open import Algebra.ElgotAlgebra ambient
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open import Algebra.UniformIterationAlgebra ambient
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open import Monad.ElgotMonad ambient
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open import Monad.PreElgot ambient
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open import Monad.Instance.K ambient
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open import Monad.Instance.K.Elgot ambient MK
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open import Monad.Instance.K.Commutative ambient MK
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@ -40,13 +40,19 @@ open M C
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isPreElgot : IsPreElgot monadK
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isPreElgot = record
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{ elgotalgebras = λ {X} → elgot X
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; pres = λ f h → sym (extend-preserve h f)
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; extend-preserves = λ f h → sym (extend-preserve h f)
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}
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where open kleisliK using (extend)
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preElgot : PreElgotMonad
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preElgot = record { T = monadK ; isPreElgot = isPreElgot }
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strongPreElgot : IsStrongPreElgot KStrong
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strongPreElgot = record
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{ preElgot = isPreElgot
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; strengthen-preserves = τ-comm
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}
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initialPreElgot : IsInitial PreElgotMonads preElgot
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initialPreElgot = record
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{ ! = !′
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@ -111,7 +117,7 @@ initialPreElgot = record
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pres₂ {Z} {g} = begin
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(T.F.₁ f ∘ η' X) ∘ g #K ≈⟨ pullʳ (η'-preserves g) ⟩
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T.F.₁ f ∘ ((η' X +₁ idC) ∘ g) #T ≈⟨ (sym (F₁⇒extend T f)) ⟩∘⟨refl ⟩
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extend (T.η.η Y ∘ f) ∘ ((η' X +₁ idC) ∘ g) #T ≈⟨ sym (PreElgotMonad.pres A ((η' X +₁ idC) ∘ g) (T.η.η Y ∘ f)) ⟩
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extend (T.η.η Y ∘ f) ∘ ((η' X +₁ idC) ∘ g) #T ≈⟨ sym (PreElgotMonad.extend-preserves A ((η' X +₁ idC) ∘ g) (T.η.η Y ∘ f)) ⟩
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(((extend (T.η.η Y ∘ f) +₁ idC) ∘ (η' X +₁ idC) ∘ g) #T) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ ((F₁⇒extend T f) ⟩∘⟨refl) identity²)) ⟩
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((T.F.₁ f ∘ η' X +₁ idC) ∘ g) #T ∎
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comm₁ : (η' Y ∘ K.₁ f) ∘ _ ≈ T.F.₁ f ∘ T.η.η X
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@ -145,9 +151,9 @@ initialPreElgot = record
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pres₂ : ∀ {Z} {g : Z ⇒ K.₀ (K.₀ X) + Z} → (T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)) ∘ g #K ≈ ((T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) +₁ idC) ∘ g) #T
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pres₂ {Z} {g} = begin
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(T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)) ∘ (g #K) ≈⟨ pullʳ (pullʳ (η'-preserves g)) ⟩
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T.μ.η X ∘ T.F.₁ (η' X) ∘ (((η' (K.₀ X) +₁ idC) ∘ g) #T) ≈⟨ refl⟩∘⟨ ((sym (F₁⇒extend T (η' X))) ⟩∘⟨refl ○ sym (PreElgotMonad.pres A ((η' (K.₀ X) +₁ idC) ∘ g) (T.η.η (T.F.F₀ X) ∘ η' X)) )⟩
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T.μ.η X ∘ T.F.₁ (η' X) ∘ (((η' (K.₀ X) +₁ idC) ∘ g) #T) ≈⟨ refl⟩∘⟨ ((sym (F₁⇒extend T (η' X))) ⟩∘⟨refl ○ sym (PreElgotMonad.extend-preserves A ((η' (K.₀ X) +₁ idC) ∘ g) (T.η.η (T.F.F₀ X) ∘ η' X)) )⟩
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T.μ.η X ∘ ((extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T ≈⟨ (sym (elimʳ T.F.identity)) ⟩∘⟨refl ⟩
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extend idC ∘ ((extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T ≈⟨ sym (PreElgotMonad.pres A ((extend (T.η.η (T.F.F₀ X) ∘ η' X) +₁ idC) ∘ (η' (K.₀ X) +₁ idC) ∘ g) idC) ⟩
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extend idC ∘ ((extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T ≈⟨ sym (PreElgotMonad.extend-preserves A ((extend (T.η.η (T.F.F₀ X) ∘ η' X) +₁ idC) ∘ (η' (K.₀ X) +₁ idC) ∘ g) idC) ⟩
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(((extend idC +₁ idC) ∘ (extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ ((elimʳ T.F.identity) ⟩∘⟨ (F₁⇒extend T (η' X))) identity²)) ⟩
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(((T.μ.η X ∘ T.F.₁ (η' X) +₁ idC) ∘ (η' _ +₁ idC) ∘ g) #T) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ assoc identity²)) ⟩
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(((T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) +₁ idC) ∘ g) #T) ∎
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73
src/Monad/PreElgot.lagda.md
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73
src/Monad/PreElgot.lagda.md
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@ -0,0 +1,73 @@
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<!--
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```agda
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{-# OPTIONS --allow-unsolved-metas #-}
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open import Level
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open import Category.Instance.AmbientCategory using (Ambient)
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open import Categories.Monad.Construction.Kleisli
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open import Categories.Monad
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open import Categories.Monad.Strong
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open import Categories.Monad.Relative renaming (Monad to RMonad)
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open import Categories.Functor
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open import Data.Product using (_,_)
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```
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-->
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```agda
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module Monad.PreElgot {o ℓ e} (ambient : Ambient o ℓ e) where
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open Ambient ambient
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open HomReasoning
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open MR C
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open Equiv
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open import Algebra.ElgotAlgebra ambient
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```
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# (strong) pre-Elgot monads
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```agda
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record IsPreElgot (T : Monad C) : Set (o ⊔ ℓ ⊔ e) where
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open Monad T
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open RMonad (Monad⇒Kleisli C T) using (extend)
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open Functor F renaming (F₀ to T₀; F₁ to T₁)
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-- every TX needs to be equipped with an elgot algebra structure
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field
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elgotalgebras : ∀ {X} → Elgot-Algebra-on (T₀ X)
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module elgotalgebras {X} = Elgot-Algebra-on (elgotalgebras {X})
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-- where kleisli lifting preserves iteration
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field
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extend-preserves : ∀ {X Y Z} (f : Z ⇒ T₀ X + Z) (h : X ⇒ T₀ Y)
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→ elgotalgebras._# ((extend h +₁ idC) ∘ f) ≈ extend h ∘ elgotalgebras._# {X} f
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record PreElgotMonad : Set (o ⊔ ℓ ⊔ e) where
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field
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T : Monad C
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isPreElgot : IsPreElgot T
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open IsPreElgot isPreElgot public
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record IsStrongPreElgot (SM : StrongMonad monoidal) : Set (o ⊔ ℓ ⊔ e) where
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open StrongMonad SM using (M; strengthen)
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open Monad M using (F)
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-- M is pre-Elgot
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field
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preElgot : IsPreElgot M
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open IsPreElgot preElgot public
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-- and strength is iteration preserving
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field
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strengthen-preserves : ∀ {X Y Z} (f : Z ⇒ F.₀ Y + Z)
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→ strengthen.η (X , Y) ∘ (idC ⁂ elgotalgebras._# f) ≈ elgotalgebras._# ((strengthen.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ f))
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record StrongPreElgotMonad : Set (o ⊔ ℓ ⊔ e) where
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field
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SM : StrongMonad monoidal
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isStrongPreElgot : IsStrongPreElgot SM
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open IsStrongPreElgot isStrongPreElgot public
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```
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