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{-# OPTIONS --without-K --safe #-}
open import Level using (Level)
module Data.Monoid {c ℓ : Level} where
import Algebra.Bundles as Alg
open import Algebra.Morphism.Bundles using (MonoidHomomorphism)
open import Categories.Object.Monoid using (Monoid; Monoid⇒)
open import Category.Instance.Setoids.SymmetricMonoidal {c} {ℓ} using (Setoids-×; ×-monoidal′)
open import Data.Product using (curry′; uncurry′; _,_; Σ)
open import Data.Setoid using (∣_∣)
open import Data.Setoid.Unit using (⊤ₛ)
open import Function using (Func; _⟶ₛ_; _⟨$⟩_)
open import Function.Construct.Constant using () renaming (function to Const)
open import Function.Construct.Identity using () renaming (function to Id)
open import Relation.Binary using (Setoid)
open Func
-- A monoid object in the (monoidal) category of setoids is just a monoid
opaque
unfolding ×-monoidal′
toMonoid : Monoid Setoids-×.monoidal → Alg.Monoid c ℓ
toMonoid M = record
{ Carrier = Carrier
; _≈_ = _≈_
; _∙_ = curry′ (to μ)
; ε = to η _
; isMonoid = record
{ isSemigroup = record
{ isMagma = record
{ isEquivalence = isEquivalence
; ∙-cong = curry′ (cong μ)
}
; assoc = λ x y z → assoc {(x , y) , z}
}
; identity = (λ x → sym (identityˡ {_ , x}) ) , λ x → sym (identityʳ {x , _})
}
}
where
open Monoid M renaming (Carrier to A)
open Setoid A
module FromMonoid (M : Alg.Monoid c ℓ) where
open Alg.Monoid M
open Setoids-× using (_⊗₁_; _⊗₀_; _∘_; unit; module unitorˡ; module unitorʳ; module associator)
opaque
unfolding ×-monoidal′
μ : setoid ⊗₀ setoid ⟶ₛ setoid
μ .to = uncurry′ _∙_
μ .cong = uncurry′ ∙-cong
η : unit ⟶ₛ setoid
η = Const ⊤ₛ setoid ε
opaque
unfolding μ
μ-assoc
: {x : ∣ (setoid ⊗₀ setoid) ⊗₀ setoid ∣}
→ μ ∘ μ ⊗₁ Id setoid ⟨$⟩ x
≈ μ ∘ Id setoid ⊗₁ μ ∘ associator.from ⟨$⟩ x
μ-assoc {(x , y) , z} = assoc x y z
μ-identityˡ
: {x : ∣ unit ⊗₀ setoid ∣}
→ unitorˡ.from ⟨$⟩ x
≈ μ ∘ η ⊗₁ Id setoid ⟨$⟩ x
μ-identityˡ {_ , x} = sym (identityˡ x)
μ-identityʳ
: {x : ∣ setoid ⊗₀ unit ∣}
→ unitorʳ.from ⟨$⟩ x
≈ μ ∘ Id setoid ⊗₁ η ⟨$⟩ x
μ-identityʳ {x , _} = sym (identityʳ x)
fromMonoid : Monoid Setoids-×.monoidal
fromMonoid = record
{ Carrier = setoid
; isMonoid = record
{ μ = μ
; η = η
; assoc = μ-assoc
; identityˡ = μ-identityˡ
; identityʳ = μ-identityʳ
}
}
open FromMonoid using (fromMonoid) public
-- A morphism of monoids in the (monoidal) category of setoids is a monoid homomorphism
module _ (M N : Monoid Setoids-×.monoidal) where
private
module M = Alg.Monoid (toMonoid M)
module N = Alg.Monoid (toMonoid N)
open Monoid⇒
opaque
unfolding toMonoid
toMonoid⇒
: Monoid⇒ Setoids-×.monoidal M N
→ MonoidHomomorphism M.rawMonoid N.rawMonoid
toMonoid⇒ f = record
{ ⟦_⟧ = to (arr f)
; isMonoidHomomorphism = record
{ isMagmaHomomorphism = record
{ isRelHomomorphism = record { cong = cong (arr f) }
; homo = λ x y → preserves-μ f {x , y}
}
; ε-homo = preserves-η f
}
}
module _ (M N : Alg.Monoid c ℓ) where
private
module M = Alg.Monoid M
module N = Alg.Monoid N
open MonoidHomomorphism
opaque
unfolding FromMonoid.μ
fromMonoid⇒
: MonoidHomomorphism M.rawMonoid N.rawMonoid
→ Monoid⇒ Setoids-×.monoidal (fromMonoid M) (fromMonoid N)
fromMonoid⇒ f = record
{ arr = record { cong = ⟦⟧-cong f }
; preserves-μ = λ { {x , y} → homo f x y }
; preserves-η = ε-homo f
}
|
