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show that K is strong pre-Elgot

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Leon Vatthauer 2023-11-18 12:27:53 +01:00
parent 7b39c02b12
commit 1f4e5460d7
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2
src/Category/Construction/PreElgotMonads.lagda.md
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@ -17,7 +17,7 @@ open import Categories.Category.Core
```agda
module Category.Construction.PreElgotMonads {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient
open import Monad.ElgotMonad ambient
open import Monad.PreElgot ambient
open import Algebra.ElgotAlgebra ambient
open HomReasoning
open Equiv

46
src/Monad/ElgotMonad.lagda.md → src/Misc/Monad/Elgot.lagda.md
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@ -1,7 +1,5 @@
<!--
```agda
{-# OPTIONS --allow-unsolved-metas #-}
open import Level
open import Category.Instance.AmbientCategory using (Ambient)
open import Categories.Monad.Construction.Kleisli
@ -11,54 +9,16 @@ open import Categories.Functor
```
-->
## Summary
This file introduces Elgot Monads.
TODO: Probably only Pre-Elgot is needed
- [X] *Definition 13* Pre-Elgot Monads
- [ ] *Definition 13* strong pre-Elgot
- [X] *Definition 14* Elgot Monads
- [ ] *Definition 14* strong Elgot
- [ ] *Proposition 15* (Strong) Elgot monads are (strong) pre-Elgot
## Code
```agda
module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
module Misc.Monad.Elgot {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient
open HomReasoning
open MR C
open Equiv
open import Algebra.ElgotAlgebra ambient
open import Monad.PreElgot ambient
```
### *Definition 13*: Pre-Elgot Monads
```agda
record IsPreElgot (T : Monad C) : Set (o ⊔ ℓ ⊔ e) where
open Monad T
open RMonad (Monad⇒Kleisli C T) using (extend)
open Functor F renaming (F₀ to T₀; F₁ to T₁)
-- every TX needs to be equipped with an elgot algebra structure
field
elgotalgebras : ∀ {X} → Elgot-Algebra-on (T₀ X)
module elgotalgebras {X} = Elgot-Algebra-on (elgotalgebras {X})
-- with the following associativity
field
pres : ∀ {X Y Z} (f : Z ⇒ T₀ X + Z) (h : X ⇒ T₀ Y)
→ elgotalgebras._# ((extend h +₁ idC) ∘ f) ≈ extend h ∘ (elgotalgebras._# {X}) f
record PreElgotMonad : Set (o ⊔ ℓ ⊔ e) where
field
T : Monad C
isPreElgot : IsPreElgot T
open IsPreElgot isPreElgot public
```
### *Definition 14*: Elgot Monads
```agda
@ -90,7 +50,7 @@ module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
open IsElgot isElgot public
```
### *Proposition 15*: (Strong) Elgot monads are (strong) pre-Elgot
### *Proposition 15*: Elgot monads are pre-Elgot
-- elgot monads are pre-elgot
Elgot⇒PreElgot : ElgotMonad → PreElgotMonad

16
src/Monad/Instance/K/PreElgot.lagda.md
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@ -20,7 +20,7 @@ open MIK ambient
open MonadK MK
open import Algebra.ElgotAlgebra ambient
open import Algebra.UniformIterationAlgebra ambient
open import Monad.ElgotMonad ambient
open import Monad.PreElgot ambient
open import Monad.Instance.K ambient
open import Monad.Instance.K.Elgot ambient MK
open import Monad.Instance.K.Commutative ambient MK
@ -40,13 +40,19 @@ open M C
isPreElgot : IsPreElgot monadK
isPreElgot = record
{ elgotalgebras = λ {X} → elgot X
; pres = λ f h → sym (extend-preserve h f)
; extend-preserves = λ f h → sym (extend-preserve h f)
}
where open kleisliK using (extend)
preElgot : PreElgotMonad
preElgot = record { T = monadK ; isPreElgot = isPreElgot }
strongPreElgot : IsStrongPreElgot KStrong
strongPreElgot = record
{ preElgot = isPreElgot
; strengthen-preserves = τ-comm
}
initialPreElgot : IsInitial PreElgotMonads preElgot
initialPreElgot = record
{ ! = !′
@ -111,7 +117,7 @@ initialPreElgot = record
pres₂ {Z} {g} = begin
(T.F.₁ f ∘ η' X) ∘ g #K ≈⟨ pullʳ (η'-preserves g) ⟩
T.F.₁ f ∘ ((η' X +₁ idC) ∘ g) #T ≈⟨ (sym (F₁⇒extend T f)) ⟩∘⟨refl ⟩
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)) ⟩
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)) ⟩
(((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)) ⟩
((T.F.₁ f ∘ η' X +₁ idC) ∘ g) #T ∎
comm₁ : (η' Y ∘ K.₁ f) ∘ _ ≈ T.F.₁ f ∘ T.η.η X
@ -145,9 +151,9 @@ initialPreElgot = record
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
pres₂ {Z} {g} = begin
(T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)) ∘ (g #K) ≈⟨ pullʳ (pullʳ (η'-preserves g)) ⟩
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)) )⟩
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)) )⟩
T.μ.η X ∘ ((extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T ≈⟨ (sym (elimʳ T.F.identity)) ⟩∘⟨refl ⟩
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) ⟩
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) ⟩
(((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)) ⟩
(((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)) ⟩
(((T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) +₁ idC) ∘ g) #T) ∎

73
src/Monad/PreElgot.lagda.md Normal file
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@ -0,0 +1,73 @@
<!--
```agda
{-# OPTIONS --allow-unsolved-metas #-}
open import Level
open import Category.Instance.AmbientCategory using (Ambient)
open import Categories.Monad.Construction.Kleisli
open import Categories.Monad
open import Categories.Monad.Strong
open import Categories.Monad.Relative renaming (Monad to RMonad)
open import Categories.Functor
open import Data.Product using (_,_)
```
-->
```agda
module Monad.PreElgot {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient
open HomReasoning
open MR C
open Equiv
open import Algebra.ElgotAlgebra ambient
```
# (strong) pre-Elgot monads
```agda
record IsPreElgot (T : Monad C) : Set (o ⊔ ℓ ⊔ e) where
open Monad T
open RMonad (Monad⇒Kleisli C T) using (extend)
open Functor F renaming (F₀ to T₀; F₁ to T₁)
-- every TX needs to be equipped with an elgot algebra structure
field
elgotalgebras : ∀ {X} → Elgot-Algebra-on (T₀ X)
module elgotalgebras {X} = Elgot-Algebra-on (elgotalgebras {X})
-- where kleisli lifting preserves iteration
field
extend-preserves : ∀ {X Y Z} (f : Z ⇒ T₀ X + Z) (h : X ⇒ T₀ Y)
→ elgotalgebras._# ((extend h +₁ idC) ∘ f) ≈ extend h ∘ elgotalgebras._# {X} f
record PreElgotMonad : Set (o ⊔ ℓ ⊔ e) where
field
T : Monad C
isPreElgot : IsPreElgot T
open IsPreElgot isPreElgot public
record IsStrongPreElgot (SM : StrongMonad monoidal) : Set (o ⊔ ℓ ⊔ e) where
open StrongMonad SM using (M; strengthen)
open Monad M using (F)
-- M is pre-Elgot
field
preElgot : IsPreElgot M
open IsPreElgot preElgot public
-- and strength is iteration preserving
field
strengthen-preserves : ∀ {X Y Z} (f : Z ⇒ F.₀ Y + Z)
→ strengthen.η (X , Y) ∘ (idC ⁂ elgotalgebras._# f) ≈ elgotalgebras._# ((strengthen.η (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ f))
record StrongPreElgotMonad : Set (o ⊔ ℓ ⊔ e) where
field
SM : StrongMonad monoidal
isStrongPreElgot : IsStrongPreElgot SM
open IsStrongPreElgot isStrongPreElgot public
```