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{-# OPTIONS --without-K --safe #-}
open import Level using (Level; _⊔_; suc)
module SplitIdempotents.Monoids {c ℓ : Level} where
import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
open import Category.Instance.Setoids.SymmetricMonoidal {c} {ℓ} using (Setoids-×; ×-monoidal′)
open import Categories.Category using (Category)
open import Categories.Category.Construction.Monoids Setoids-×.monoidal using (Monoids)
open import Categories.Category.Instance.Setoids using (Setoids)
open import Categories.Category.Monoidal using (Monoidal)
open import Categories.Object.Monoid Setoids-×.monoidal using (Monoid; Monoid⇒)
open import Data.Product using (_,_)
open import Data.Setoid using (∣_∣)
open import Function using (Func; _⟶ₛ_; _⟨$⟩_; id)
open import Function.Construct.Identity using () renaming (function to Id)
open import Function.Construct.Setoid using (_∙_)
open import Morphism.SplitIdempotent Monoids using (IsSplitIdempotent)
open import Relation.Binary using (Setoid)
open import SplitIdempotents.Setoids using () renaming (Q to Q′)
open Category Monoids using (_∘_; _≈_) renaming (id to Id⇒)
open Func
open Monoidal Setoids-×.monoidal using (_⊗₀_; unit; _⊗₁_)
module _ {M : Monoid} (F : Monoid⇒ M M) where
private
module M = Monoid M
module F = Monoid⇒ F
X′ : Setoid c ℓ
X′ = Q′ F.arr
private
module S = Setoid M.Carrier
module X′ = Setoid X′
open ≈-Reasoning M.Carrier
opaque
unfolding ×-monoidal′
μ : X′ ⊗₀ X′ ⟶ₛ X′
μ = record
{ to = λ (x , y) → M.μ ⟨$⟩ (x , y)
; cong = λ { {A , B} {C , D} (FA≈FB , FC≈FD) → begin
F.arr ⟨$⟩ (M.μ ⟨$⟩ (A , B)) ≈⟨ F.preserves-μ ⟩
M.μ ⟨$⟩ (F.arr ⟨$⟩ A , F.arr ⟨$⟩ B) ≈⟨ cong M.μ (FA≈FB , FC≈FD) ⟩
M.μ ⟨$⟩ (F.arr ⟨$⟩ C , F.arr ⟨$⟩ D) ≈⟨ F.preserves-μ ⟨
F.arr ⟨$⟩ (M.μ ⟨$⟩ (C , D)) ∎
}
}
η : unit ⟶ₛ X′
η = record
{ to = to M.η
; cong = λ { {A} {B} eq → begin
F.arr ⟨$⟩ (M.η ⟨$⟩ A) ≈⟨ F.preserves-η ⟩
M.η ⟨$⟩ A ≈⟨ cong M.η eq ⟩
M.η ⟨$⟩ B ≈⟨ F.preserves-η ⟨
F.arr ⟨$⟩ (M.η ⟨$⟩ B) ∎
}
}
opaque
unfolding μ
assoc
: {x : ∣ (X′ ⊗₀ X′) ⊗₀ X′ ∣}
→ μ ∙ μ ⊗₁ Setoids-×.id ⟨$⟩ x
X′.≈ (μ ∙ Id X′ ⊗₁ μ ∙ Setoids-×.associator.from ⟨$⟩ x)
assoc {(x , y) , z} = begin
F.arr ⟨$⟩ (μ ⟨$⟩ (μ ⟨$⟩ (x , y) , z)) ≈⟨ F.preserves-μ ⟩
μ ⟨$⟩ (F.arr ⟨$⟩ (μ ⟨$⟩ (x , y)) , F.arr ⟨$⟩ z) ≈⟨ cong M.μ (F.preserves-μ , X′.refl) ⟩
μ ⟨$⟩ (μ ⟨$⟩ (F.arr ⟨$⟩ x , F.arr ⟨$⟩ y) , F.arr ⟨$⟩ z) ≈⟨ M.assoc ⟩
μ ⟨$⟩ (F.arr ⟨$⟩ x , μ ⟨$⟩ (F.arr ⟨$⟩ y , F.arr ⟨$⟩ z)) ≈⟨ cong M.μ (X′.refl , F.preserves-μ) ⟨
μ ⟨$⟩ (F.arr ⟨$⟩ x , F.arr ⟨$⟩ (μ ⟨$⟩ (y , z))) ≈⟨ F.preserves-μ ⟨
F.arr ⟨$⟩ (μ ⟨$⟩ (x , μ ⟨$⟩ (y , z))) ∎
opaque
unfolding μ
identityˡ
: {x : ∣ unit ⊗₀ X′ ∣}
→ (Setoids-×.unitorˡ.from ⟨$⟩ x)
X′.≈ (μ ∙ η ⊗₁ Id X′ ⟨$⟩ x)
identityˡ {_ , x} = begin
F.arr ⟨$⟩ x ≈⟨ M.identityˡ ⟩
μ ⟨$⟩ (η ⟨$⟩ _ , F.arr ⟨$⟩ x) ≈⟨ cong M.μ (F.preserves-η , X′.refl) ⟨
μ ⟨$⟩ (F.arr ⟨$⟩ (η ⟨$⟩ _) , F.arr ⟨$⟩ x) ≈⟨ F.preserves-μ ⟨
F.arr ⟨$⟩ (μ ⟨$⟩ (η ⟨$⟩ _ , x)) ∎
opaque
unfolding μ
identityʳ
: {x : ∣ X′ ⊗₀ unit ∣}
→ (Setoids-×.unitorʳ.from ⟨$⟩ x)
X′.≈ (μ ∙ Id _ ⊗₁ η ⟨$⟩ x)
identityʳ {x , _} = begin
F.arr ⟨$⟩ x ≈⟨ M.identityʳ ⟩
μ ⟨$⟩ (F.arr ⟨$⟩ x , η ⟨$⟩ _) ≈⟨ cong M.μ (X′.refl , F.preserves-η) ⟨
μ ⟨$⟩ (F.arr ⟨$⟩ x , F.arr ⟨$⟩ (η ⟨$⟩ _)) ≈⟨ F.preserves-μ ⟨
F.arr ⟨$⟩ (μ ⟨$⟩ (x , to M.η _)) ∎
Q : Monoid
Q = record
{ Carrier = X′
; isMonoid = record
{ μ = μ
; η = η
; assoc = assoc
; identityˡ = identityˡ
; identityʳ = identityʳ
}
}
M⇒Q : Monoid⇒ M Q
M⇒Q = record
{ arr = arr
; preserves-μ = preserves-μ
; preserves-η = S.refl
}
where
arr : M.Carrier ⟶ₛ X′
arr = record
{ to = id
; cong = cong F.arr
}
opaque
unfolding μ
preserves-μ : {x : ∣ M.Carrier ⊗₀ M.Carrier ∣} → (arr ∙ M.μ ⟨$⟩ x) X′.≈ (μ ∙ arr ⊗₁ arr ⟨$⟩ x)
preserves-μ = S.refl
Q⇒M : Monoid⇒ Q M
Q⇒M = record
{ arr = arr
; preserves-μ = preserves-μ
; preserves-η = F.preserves-η
}
where
arr : X′ ⟶ₛ M.Carrier
arr = record
{ to = to F.arr
; cong = id
}
opaque
unfolding μ
preserves-μ : {x : ∣ X′ ⊗₀ X′ ∣} → arr ∙ μ ⟨$⟩ x S.≈ M.μ ∙ arr ⊗₁ arr ⟨$⟩ x
preserves-μ = F.preserves-μ
module _ (idem : F ∘ F ≈ F) where
split : IsSplitIdempotent F
split = record
{ B = Q
; r = M⇒Q
; s = Q⇒M
; s∘r = S.refl
; r∘s = idem
}
|
