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{-# OPTIONS --without-K --safe #-}
open import Level using (Level)
module Data.Monoid {c ℓ : Level} where
open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory)
open import Category.Instance.Setoids.SymmetricMonoidal {c} {ℓ} using (Setoids-×; ×-monoidal′)
open import Categories.Object.Monoid using (Monoid; Monoid⇒)
import Algebra.Bundles as Alg
open import Data.Setoid using (∣_∣)
open import Relation.Binary using (Setoid)
open import Function using (Func)
open import Data.Product using (curry′; uncurry′; _,_)
open Func
-- A monoid object in the (monoidal) category of setoids is just a monoid
open import Function.Construct.Constant using () renaming (function to Const)
open import Data.Setoid.Unit using (⊤ₛ)
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
fromMonoid : Alg.Monoid c ℓ → Monoid Setoids-×.monoidal
fromMonoid M = record
{ Carrier = setoid
; isMonoid = record
{ μ = record { to = uncurry′ _∙_ ; cong = uncurry′ ∙-cong }
; η = Const ⊤ₛ setoid ε
; assoc = λ { {(x , y) , z} → assoc x y z }
; identityˡ = λ { {_ , x} → sym (identityˡ x) }
; identityʳ = λ { {x , _} → sym (identityʳ x) }
}
}
where
open Alg.Monoid M
-- A morphism of monoids in the (monoidal) category of setoids is a monoid homomorphism
module _ (M N : Monoid Setoids-×.monoidal) where
module M = Alg.Monoid (toMonoid M)
module N = Alg.Monoid (toMonoid N)
open import Data.Product using (Σ; _,_)
open import Function using (_⟶ₛ_)
open import Algebra.Morphism using (IsMonoidHomomorphism)
open Monoid⇒
opaque
unfolding toMonoid
toMonoid⇒
: Monoid⇒ Setoids-×.monoidal M N
→ Σ (M.setoid ⟶ₛ N.setoid) (λ f
→ IsMonoidHomomorphism M.rawMonoid N.rawMonoid (to f))
toMonoid⇒ f = arr f , record
{ isMagmaHomomorphism = record
{ isRelHomomorphism = record { cong = cong (arr f) }
; homo = λ x y → preserves-μ f {x , y}
}
; ε-homo = preserves-η f
}
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