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{-# OPTIONS --without-K --safe #-}
open import Level using (Level; suc; _⊔_)
module Category.Equivalence.Instance.Monoids (c ℓ : Level) where
open import Category.Instance.Monoids c ℓ using (Monoids; MonoidHomomorphism; mk-⇒; _≗_)
import Algebra as Alg
import Algebra.Morphism.Bundles as Raw
open import Category.Instance.Setoids.SymmetricMonoidal {c} {ℓ} using (Setoids-×; ×-monoidal′)
open import Categories.Category using (module Category; _[_∘_])
open import Categories.Category.Construction.Monoids Setoids-×.monoidal using () renaming (Monoids to Monoids[Setoids])
open import Categories.Category.Equivalence using (StrongEquivalence)
open import Categories.Functor using (Functor; _∘F_) renaming (id to Id)
open import Categories.NaturalTransformation.NaturalIsomorphism using (niHelper; _≃_)
open import Categories.Object.Monoid Setoids-×.monoidal using (Monoid; Monoid⇒)
open import Data.Monoid using (fromMonoid; toMonoid; fromMonoid⇒; toMonoid⇒)
open import Function using (Func; _⟨$⟩_; _∘_) renaming (id to idf)
open import Function.Construct.Identity using () renaming (function to IdFunc)
open import Relation.Binary using (Setoid)
open Category using (id)
open Func
open MonoidHomomorphism using (rawMonoidHomomorphism; ⟦_⟧)
open Monoid⇒
abstract opaque
unfolding fromMonoid⇒
identity⇒
: (A : Alg.Monoid c ℓ)
(open Alg.Monoid A)
{x : Carrier}
→ arr (fromMonoid⇒ A A (rawMonoidHomomorphism (id Monoids {A}))) ⟨$⟩ x ≈ x
identity⇒ = Alg.Monoid.refl
homo⇒
: (X Y Z : Alg.Monoid c ℓ)
{f : MonoidHomomorphism X Y}
{g : MonoidHomomorphism Y Z}
{x : Alg.Monoid.Carrier X}
(open Alg.Monoid Z)
→ arr (fromMonoid⇒ X Z (rawMonoidHomomorphism (Monoids [ g ∘ f ]))) ⟨$⟩ x
≈ arr (fromMonoid⇒ Y Z (rawMonoidHomomorphism g)) ⟨$⟩ (arr (fromMonoid⇒ X Y (rawMonoidHomomorphism f)) ⟨$⟩ x)
homo⇒ X Y Z = Alg.Monoid.refl Z
resp⇒
: (A B : Alg.Monoid c ℓ)
{f g : MonoidHomomorphism A B}
→ f ≗ g
→ {x : Alg.Monoid.Carrier A}
(open Alg.Monoid B)
→ arr (fromMonoid⇒ A B (rawMonoidHomomorphism f)) ⟨$⟩ x
≈ arr (fromMonoid⇒ A B (rawMonoidHomomorphism g)) ⟨$⟩ x
resp⇒ A B f≗g {x} = f≗g x
From : Functor Monoids Monoids[Setoids]
From = record
{ F₀ = fromMonoid
; F₁ = λ {M N} f → fromMonoid⇒ M N (rawMonoidHomomorphism f)
; identity = λ {A} → identity⇒ A
; homomorphism = λ {X Y Z} → homo⇒ X Y Z
; F-resp-≈ = λ {A B} → resp⇒ A B
}
opaque
unfolding toMonoid⇒
identity⇐
: {A : Monoid}
→ mk-⇒ (toMonoid⇒ A A (id Monoids[Setoids])) ≗ id Monoids {toMonoid A}
identity⇐ {A} _ = Alg.Monoid.refl (toMonoid A)
homo⇐
: {X Y Z : Monoid}
{f : Monoid⇒ X Y}
{g : Monoid⇒ Y Z}
→ mk-⇒ (toMonoid⇒ X Z (Monoids[Setoids] [ g ∘ f ]))
≗ _[_∘_] Monoids {toMonoid X} {toMonoid Y} {toMonoid Z} (mk-⇒ (toMonoid⇒ Y Z g)) (mk-⇒ (toMonoid⇒ X Y f))
homo⇐ {X} {Y} {Z} {f} {g} x = Alg.Monoid.refl (toMonoid Z)
resp⇐
: {A B : Monoid}
{f g : Monoid⇒ A B}
→ (∀ {x} (open Setoid (Monoid.Carrier B)) → arr f ⟨$⟩ x ≈ arr g ⟨$⟩ x)
→ _≗_ {toMonoid A} {toMonoid B} (mk-⇒ (toMonoid⇒ A B f)) (mk-⇒ (toMonoid⇒ A B g))
resp⇐ {A} {B} {f} {g} f≈g x = f≈g {x}
To : Functor Monoids[Setoids] Monoids
To = record
{ F₀ = toMonoid
; F₁ = λ {M N} f → mk-⇒ (toMonoid⇒ M N f)
; identity = identity⇐
; homomorphism = homo⇐
; F-resp-≈ = resp⇐
}
opaque
unfolding toMonoid Data.Monoid.FromMonoid.μ
from∘to⇒ : (M : Monoid) → Monoid⇒ (fromMonoid (toMonoid M)) M
from∘to⇒ M = let open Alg.Monoid (toMonoid M) in record
{ arr = IdFunc setoid
; preserves-μ = refl
; preserves-η = refl
}
from∘to⇐ : (M : Monoid) → Monoid⇒ M (fromMonoid (toMonoid M))
from∘to⇐ M = let open Alg.Monoid (toMonoid M) in record
{ arr = IdFunc setoid
; preserves-μ = refl
; preserves-η = refl
}
from∘to-isoˡ
: (M : Monoid)
(open Alg.Monoid (toMonoid M))
→ {x : Alg.Monoid.Carrier (toMonoid M)}
→ arr (from∘to⇐ M) ⟨$⟩ (arr (from∘to⇒ M) ⟨$⟩ x) ≈ x
from∘to-isoˡ M = Setoid.refl (Monoid.Carrier M)
from∘to-isoʳ
: (M : Monoid)
(open Setoid (Monoid.Carrier M))
{x : Setoid.Carrier (Monoid.Carrier M)}
→ arr (from∘to⇒ M) ⟨$⟩ (arr (from∘to⇐ M) ⟨$⟩ x) ≈ x
from∘to-isoʳ M = Setoid.refl (Monoid.Carrier M)
opaque
unfolding fromMonoid⇒ toMonoid⇒
from∘to⇒-commute
: {M N : Monoid}
(f : Monoid⇒ M N)
{x : Alg.Monoid.Carrier (toMonoid M)}
(open Setoid (Monoid.Carrier N))
→ arr (from∘to⇒ N) ⟨$⟩ (arr (fromMonoid⇒ (toMonoid M) (toMonoid N) (toMonoid⇒ M N f)) ⟨$⟩ x)
≈ arr f ⟨$⟩ (arr (from∘to⇒ M) ⟨$⟩ x)
from∘to⇒-commute {M} {N} f = Setoid.refl (Monoid.Carrier N)
From∘To : From ∘F To ≃ Id
From∘To = niHelper record
{ η = from∘to⇒
; η⁻¹ = from∘to⇐
; commute = from∘to⇒-commute
; iso = λ M → record
{ isoˡ = from∘to-isoˡ M
; isoʳ = from∘to-isoʳ M
}
}
opaque
unfolding toMonoid Data.Monoid.FromMonoid.μ
to∘from⇒ : (M : Alg.Monoid c ℓ) → MonoidHomomorphism (toMonoid (fromMonoid M)) M
to∘from⇒ M = let open Alg.Monoid M in mk-⇒ record
{ ⟦_⟧ = idf
; isMonoidHomomorphism = record
{ isMagmaHomomorphism = record
{ isRelHomomorphism = record
{ cong = idf
}
; homo = λ _ _ → refl
}
; ε-homo = refl
}
}
to∘from⇐ : (M : Alg.Monoid c ℓ) → MonoidHomomorphism M (toMonoid (fromMonoid M))
to∘from⇐ M = let open Alg.Monoid M in mk-⇒ record
{ ⟦_⟧ = idf
; isMonoidHomomorphism = record
{ isMagmaHomomorphism = record
{ isRelHomomorphism = record
{ cong = idf
}
; homo = λ _ _ → refl
}
; ε-homo = refl
}
}
to∘from-isoˡ
: (M : Alg.Monoid c ℓ)
(open Alg.Monoid (toMonoid (fromMonoid M)))
→ Monoids [ to∘from⇐ M ∘ to∘from⇒ M ] ≗ id Monoids {toMonoid (fromMonoid M)}
to∘from-isoˡ M _ = Alg.Monoid.refl M
to∘from-isoʳ
: (M : Alg.Monoid c ℓ)
(open Alg.Monoid M)
→ Monoids [ to∘from⇒ M ∘ to∘from⇐ M ] ≗ id Monoids {M}
to∘from-isoʳ M _ = Alg.Monoid.refl M
opaque
unfolding toMonoid⇒ fromMonoid⇒
to∘from⇒-commute
: {M N : Alg.Monoid c ℓ}
(f : MonoidHomomorphism M N)
(open Alg.Monoid N)
→ Monoids [ to∘from⇒ N ∘ mk-⇒ (toMonoid⇒ (fromMonoid M) (fromMonoid N) (fromMonoid⇒ M N (rawMonoidHomomorphism f))) ]
≗ Monoids [ f ∘ to∘from⇒ M ]
to∘from⇒-commute {_} {N} _ _ = Alg.Monoid.refl N
To∘From : To ∘F From ≃ Id
To∘From = niHelper record
{ η = to∘from⇒
; η⁻¹ = to∘from⇐
; commute = to∘from⇒-commute
; iso = λ M → record
{ isoˡ = to∘from-isoˡ M
; isoʳ = to∘from-isoʳ M
}
}
-- The category of monoids is equivalent to the category of monoid objects in Setoids
Monoids≈Monoids[Setoids] : StrongEquivalence Monoids Monoids[Setoids]
Monoids≈Monoids[Setoids] = record
{ F = From
; G = To
; weak-inverse = record
{ F∘G≈id = From∘To
; G∘F≈id = To∘From
}
}
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