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Show that K is initial pre-Elgot

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Leon Vatthauer 2023-11-15 19:25:35 +01:00
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9
src/Monad/ElgotMonad.lagda.md
View file

@ -4,7 +4,9 @@
open import Level
open import Category.Instance.AmbientCategory using (Ambient)
open import Categories.Monad.Construction.Kleisli
open import Categories.Monad
open import Categories.Monad.Relative renaming (Monad to RMonad)
open import Categories.Functor
```
-->
@ -36,6 +38,7 @@ module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
```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
@ -46,8 +49,8 @@ module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
-- with the following associativity
field
assoc : ∀ {X Y Z} (f : Z ⇒ T₀ X + Z) (h : X ⇒ T₀ Y)
→ elgotalgebras._# (((μ.η _ ∘ T₁ h) +₁ idC) ∘ f) ≈ (μ.η _ ∘ T₁ h) ∘ (elgotalgebras._# {X}) f
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
@ -89,7 +92,6 @@ module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
### *Proposition 15*: (Strong) Elgot monads are (strong) pre-Elgot
```agda
-- elgot monads are pre-elgot
Elgot⇒PreElgot : ElgotMonad → PreElgotMonad
Elgot⇒PreElgot EM = record
@ -146,4 +148,3 @@ module Monad.ElgotMonad {o ℓ e} (ambient : Ambient o ℓ e) where
module T = Monad T
open T using (F; η; μ)
open Functor F renaming (F₀ to T₀; F₁ to T₁)
```

104
src/Monad/Instance/K/PreElgot.lagda.md
View file

@ -7,6 +7,8 @@ open import Categories.Object.Initial
open import Categories.NaturalTransformation
open import Categories.NaturalTransformation.Equivalence
open import Categories.Monad
open import Categories.Monad.Relative renaming (Monad to RMonad)
open import Categories.Monad.Construction.Kleisli
import Monad.Instance.K as MIK
```
-->
@ -22,6 +24,7 @@ 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 Monad.Instance.K.Strong ambient MK
open import Category.Construction.PreElgotMonads ambient
open import Category.Construction.ElgotAlgebras ambient
@ -34,16 +37,17 @@ open M C
# K is a pre-Elgot monad
```agda
open kleisliK using (extend)
-- TODO fix global declarations on Commutative.lagda.md
-- open Elgot-Algebra-on using (#-Compositionality)
_# = λ {A} {X} f → Uniform-Iteration-Algebra._# (algebras A) {X = X} f
-- TODO fix this import mess!!!
-- _# = λ {A} {X} f → Uniform-Iteration-Algebra._# (algebras A) {X = X} f
isPreElgot : IsPreElgot monadK
isPreElgot = record
{ elgotalgebras = λ {X} → elgot X
; assoc = λ f h → sym (extend-preserve h f)
; pres = λ f h → sym (extend-preserve h f)
}
where open kleisliK using (extend)
preElgot : PreElgotMonad
preElgot = record { T = monadK ; isPreElgot = isPreElgot }
@ -59,19 +63,105 @@ initialPreElgot = record
!′ {A} = record
{ α = ntHelper (record
{ η = η'
; commute = {!!}
; commute = commute
})
; α-η = {!!}
; α-μ = {!!}
; α-η = FreeObject.*-lift (freealgebras _) (T.η.η _)
; α-μ = α-μ
; preserves = λ {X} {B} f → Elgot-Algebra-Morphism.preserves (((freeElgot B) FreeObject.*) (T.η.η B))
}
where
open PreElgotMonad A using (T)
open RMonad (Monad⇒Kleisli C T) using (extend)
module T = Monad T
open PreElgotMonad preElgot using ()
open monadK using () renaming (η to ηK; μ to μK)
open Elgot-Algebra-on using (#-resp-≈)
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
commute : ∀ {X Y : Obj} (f : X ⇒ Y) → η' Y ∘ K.₁ f ≈ T.F.₁ f ∘ η' X
commute {X} {Y} f = begin
η' Y ∘ K.₁ f ≈⟨ *-uniq (T.F.₁ f ∘ T.η.η X) (record { h = η' Y ∘ K.₁ f ; preserves = pres₁ }) (begin
(η' Y ∘ K.₁ f) ∘ η ≈⟨ pullʳ (K₁η f) ⟩
η' Y ∘ ηK.η _ ∘ f ≈⟨ pullˡ (FreeObject.*-lift (freealgebras Y) (T.η.η Y)) ⟩
T.η.η Y ∘ f ≈⟨ NaturalTransformation.commute T.η f ⟩
T.F.₁ f ∘ T.η.η X ∎) ⟩
Elgot-Algebra-Morphism.h (_* {A = record { A = T.F.F₀ Y ; algebra = PreElgotMonad.elgotalgebras A }} (T.F.₁ f ∘ T.η.η _)) ≈⟨ sym (*-uniq (T.F.₁ f ∘ T.η.η X) (record { h = T.F.₁ f ∘ η' X ; preserves = pres₂ }) (begin
(T.F.₁ f ∘ η' X) ∘ η ≈⟨ pullʳ (FreeObject.*-lift (freealgebras X) (T.η.η X)) ⟩
T.F.₁ f ∘ T.η.η X ∎)) ⟩
T.F.₁ f ∘ η' X ∎
where
open FreeObject (freeElgot X)
_#K = λ {B} {C} f → Elgot-Algebra._# (FreeObject.FX (freeElgot C)) {B} f
_#T = λ {B} {C} f → PreElgotMonad.elgotalgebras._# A {B} {C} f
pres₁ : ∀ {Z} {g : Z ⇒ K.₀ X + Z} → (η' Y ∘ K.₁ f) ∘ g #K ≈ ((η' Y ∘ K.₁ f +₁ idC) ∘ g) #T
pres₁ {Z} {g} = begin
(η' Y ∘ K.₁ f) ∘ (g #K) ≈⟨ pullʳ (Elgot-Algebra-Morphism.preserves (((freeElgot X) FreeObject.*) (ηK.η Y ∘ f))) ⟩
η' Y ∘ (((K.₁ f +₁ idC) ∘ g) #K) ≈⟨ Elgot-Algebra-Morphism.preserves (((freeElgot Y) FreeObject.*) {A = record
{ A = T.F.F₀ Y
; algebra =
record
{ _# = λ {X = X₁} → A PreElgotMonad.elgotalgebras.#
; #-Fixpoint = PreElgotMonad.elgotalgebras.#-Fixpoint A
; #-Uniformity = PreElgotMonad.elgotalgebras.#-Uniformity A
; #-Folding = PreElgotMonad.elgotalgebras.#-Folding A
; #-resp-≈ = PreElgotMonad.elgotalgebras.#-resp-≈ A
}
}} (T.η.η Y)) ⟩
(((η' Y +₁ idC) ∘ (K.₁ f +₁ idC) ∘ g) #T) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
((η' Y ∘ K.₁ f +₁ idC) ∘ g) #T ∎
pres₂ : ∀ {Z} {g : Z ⇒ K.₀ X + Z} → (T.F.₁ f ∘ η' X) ∘ g #K ≈ ((T.F.₁ f ∘ η' X +₁ idC) ∘ g) #T
pres₂ {Z} {g} = begin
(T.F.₁ f ∘ η' X) ∘ g #K ≈⟨ pullʳ (Elgot-Algebra-Morphism.preserves (((freeElgot X) FreeObject.*) (T.η.η X))) ⟩
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) +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 ∎
α-μ : ∀ {X : Obj} → η' X ∘ μK.η X ≈ T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)
α-μ {X} = begin
η' X ∘ μK.η X ≈⟨ FreeObject.*-uniq (freeElgot (K.₀ X)) (η' X) (record { h = η' X ∘ μK.η X ; preserves = pres₁ }) (cancelʳ monadK.identityʳ) ⟩
Elgot-Algebra-Morphism.h (((freeElgot (K.₀ X)) FreeObject.*) {A = record
{ A = T.F.F₀ X
; algebra =
record
{ _# = λ Z → (A PreElgotMonad.elgotalgebras.#) Z
; #-Fixpoint = PreElgotMonad.elgotalgebras.#-Fixpoint A
; #-Uniformity = PreElgotMonad.elgotalgebras.#-Uniformity A
; #-Folding = PreElgotMonad.elgotalgebras.#-Folding A
; #-resp-≈ = PreElgotMonad.elgotalgebras.#-resp-≈ A
}
}} (η' X)) ≈⟨ sym (FreeObject.*-uniq (freeElgot (K.₀ X)) (η' X) (record { h = T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) ; preserves = pres₂ }) (begin
(T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X)) ∘ ηK.η (K.₀ X) ≈⟨ (refl⟩∘⟨ sym (commute (η' X))) ⟩∘⟨refl ⟩
(T.μ.η X ∘ η' _ ∘ K.₁ (η' X)) ∘ ηK.η (K.₀ X) ≈⟨ assoc ○ refl⟩∘⟨ (assoc ○ refl⟩∘⟨ sym (monadK.η.commute (η' X))) ⟩
T.μ.η X ∘ η' _ ∘ ηK.η (T.F.F₀ X) ∘ η' X ≈⟨ refl⟩∘⟨ (pullˡ (FreeObject.*-lift (freealgebras _) (T.η.η _))) ⟩
T.μ.η X ∘ T.η.η _ ∘ η' X ≈⟨ cancelˡ (Monad.identityʳ T) ⟩
η' X ∎)) ⟩
T.μ.η X ∘ T.F.₁ (η' X) ∘ η' (K.₀ X) ∎
where
_#K = λ {B} {C} f → Elgot-Algebra._# (FreeObject.FX (freeElgot C)) {B} f
_#T = λ {B} {C} f → PreElgotMonad.elgotalgebras._# A {B} {C} f
pres₁ : ∀ {Z} {g : Z ⇒ K.₀ (K.₀ X) + Z} → (η' X ∘ μK.η X) ∘ g #K ≈ ((η' X ∘ μK.η X +₁ idC) ∘ g) #T
pres₁ {Z} {g} = begin
(η' X ∘ μK.η X) ∘ (g #K) ≈⟨ pullʳ (Elgot-Algebra-Morphism.preserves (((freeElgot (K.₀ X)) FreeObject.*) idC)) ⟩
η' X ∘ ((μK.η X +₁ idC) ∘ g) #K ≈⟨ Elgot-Algebra-Morphism.preserves (((freeElgot X) FreeObject.*) (T.η.η X)) ⟩
(((η' X +₁ idC) ∘ (μK.η X +₁ idC) ∘ g) #T) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²)) ⟩
(((η' X ∘ μK.η X +₁ idC) ∘ g) #T) ∎
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 (Elgot-Algebra-Morphism.preserves (((freeElgot (K.₀ X)) FreeObject.*) (T.η.η (K.₀ 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.pres 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 μ-extend) ⟩∘⟨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 +₁ idC) ∘ (extend (T.η.η _ ∘ η' _) +₁ idC) ∘ ((η' _ +₁ idC)) ∘ g) #T) ≈⟨ #-resp-≈ (PreElgotMonad.elgotalgebras A) (pullˡ (+₁∘+₁ ○ +₁-cong₂ (μ-extend ⟩∘⟨ (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) ∎
where
μ-extend : extend idC ≈ T.μ.η X
μ-extend = begin
T.μ.η _ ∘ T.F.₁ idC ≈⟨ elimʳ T.F.identity ⟩
T.μ.η X ∎
!-unique′ : ∀ {A : PreElgotMonad} (f : PreElgotMonad-Morphism preElgot A) → PreElgotMonad-Morphism.α (!′ {A = A}) ≃ PreElgotMonad-Morphism.α f
!-unique′ {A} f {X} = sym (*-uniq (T.η.η X) (record
{ h = α.η X
; preserves = preserves _