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{-# OPTIONS --without-K --safe #-}
open import Data.Nat using (ℕ)
open import Level using (Level; suc; 0ℓ)
module Data.System.Monoidal {ℓ : Level} (n : ℕ) where
open import Data.System {ℓ} using (System; Systems; _≤_; ≤-refl; ≤-trans; _≈_; discrete)
open import Categories.Category.Monoidal using (Monoidal)
open import Categories.Category.Monoidal.Symmetric using (Symmetric)
open import Categories.Category.Monoidal.Bundle using (MonoidalCategory; SymmetricMonoidalCategory)
open import Categories.Functor using (Functor)
open import Categories.Functor.Bifunctor using (Bifunctor; flip-bifunctor)
open import Categories.Morphism (Systems n) using (_≅_; Iso)
open import Categories.NaturalTransformation.NaturalIsomorphism using (_≃_; niHelper)
open import Data.Circuit.Value using (Monoid)
open import Data.Product using (_,_; _×_; uncurry′)
open import Data.Product.Function.NonDependent.Setoid using (_×-function_; proj₁ₛ; proj₂ₛ; swapₛ)
open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
open import Data.Setoid using (_⇒ₛ_; _×-⇒_; assocₛ⇒; assocₛ⇐)
open import Data.System.Values Monoid using (Values; _⊕_; ⊕-cong; ⊕-identityˡ; ⊕-identityʳ; ⊕-assoc; ⊕-comm; module ≋)
open import Function using (Func; _⟶ₛ_)
open import Function.Construct.Setoid using (_∙_)
open import Relation.Binary using (Setoid)
open _≤_
open Setoid
module _ where
open Func
δₛ : Values n ⟶ₛ Values n ×ₛ Values n
δₛ .to v = v , v
δₛ .cong v≋w = v≋w , v≋w
⊕ₛ : Values n ×ₛ Values n ⟶ₛ Values n
⊕ₛ .to (v , w) = v ⊕ w
⊕ₛ .cong (v₁≋v₂ , w₁≋w₂) = ⊕-cong v₁≋v₂ w₁≋w₂
_⊗₀_ : System n → System n → System n
_⊗₀_ X Y = let open System in record
{ S = S X ×ₛ S Y
; fₛ = fₛ X ×-⇒ fₛ Y ∙ δₛ
; fₒ = ⊕ₛ ∙ fₒ X ×-function fₒ Y
}
_⊗₁_
: {A A′ B B′ : System n}
(f : A ≤ A′)
(g : B ≤ B′)
→ A ⊗₀ B ≤ A′ ⊗₀ B′
_⊗₁_ f g .⇒S = ⇒S f ×-function ⇒S g
_⊗₁_ f g .≗-fₛ i (s₁ , s₂) = ≗-fₛ f i s₁ , ≗-fₛ g i s₂
_⊗₁_ f g .≗-fₒ (s₁ , s₂) = ⊕-cong (≗-fₒ f s₁) (≗-fₒ g s₂)
module _ where
open Functor
open System
⊗ : Bifunctor (Systems n) (Systems n) (Systems n)
⊗ .F₀ = uncurry′ _⊗₀_
⊗ .F₁ = uncurry′ _⊗₁_
⊗ .identity {X , Y} = refl (S X) , refl (S Y)
⊗ .homomorphism {_} {_} {X″ , Y″} = refl (S X″) , refl (S Y″)
⊗ .F-resp-≈ (f≈f′ , g≈g′) = f≈f′ , g≈g′
module Unitors {X : System n} where
open System X
⊗-discreteˡ-≤ : discrete n ⊗₀ X ≤ X
⊗-discreteˡ-≤ .⇒S = proj₂ₛ
⊗-discreteˡ-≤ .≗-fₛ i s = S.refl
⊗-discreteˡ-≤ .≗-fₒ (_ , s) = ⊕-identityˡ (fₒ′ s)
⊗-discreteˡ-≥ : X ≤ discrete n ⊗₀ X
⊗-discreteˡ-≥ .⇒S = record { to = λ s → _ , s ; cong = λ s≈s′ → _ , s≈s′ }
⊗-discreteˡ-≥ .≗-fₛ i s = _ , S.refl
⊗-discreteˡ-≥ .≗-fₒ s = ≋.sym (⊕-identityˡ (fₒ′ s))
⊗-discreteʳ-≤ : X ⊗₀ discrete n ≤ X
⊗-discreteʳ-≤ .⇒S = proj₁ₛ
⊗-discreteʳ-≤ .≗-fₛ i s = S.refl
⊗-discreteʳ-≤ .≗-fₒ (s , _) = ⊕-identityʳ (fₒ′ s)
⊗-discreteʳ-≥ : X ≤ X ⊗₀ discrete n
⊗-discreteʳ-≥ .⇒S = record { to = λ s → s , _ ; cong = λ s≈s′ → s≈s′ , _ }
⊗-discreteʳ-≥ .≗-fₛ i s = S.refl , _
⊗-discreteʳ-≥ .≗-fₒ s = ≋.sym (⊕-identityʳ (fₒ′ s))
open _≅_
open Iso
unitorˡ : discrete n ⊗₀ X ≅ X
unitorˡ .from = ⊗-discreteˡ-≤
unitorˡ .to = ⊗-discreteˡ-≥
unitorˡ .iso .isoˡ = _ , S.refl
unitorˡ .iso .isoʳ = S.refl
unitorʳ : X ⊗₀ discrete n ≅ X
unitorʳ .from = ⊗-discreteʳ-≤
unitorʳ .to = ⊗-discreteʳ-≥
unitorʳ .iso .isoˡ = S.refl , _
unitorʳ .iso .isoʳ = S.refl
open Unitors using (unitorˡ; unitorʳ) public
module Associator {X Y Z : System n} where
module X = System X
module Y = System Y
module Z = System Z
assoc-≤ : (X ⊗₀ Y) ⊗₀ Z ≤ X ⊗₀ (Y ⊗₀ Z)
assoc-≤ .⇒S = assocₛ⇒
assoc-≤ .≗-fₛ i ((s₁ , s₂) , s₃) = X.S.refl , Y.S.refl , Z.S.refl
assoc-≤ .≗-fₒ ((s₁ , s₂) , s₃) = ⊕-assoc (X.fₒ′ s₁) (Y.fₒ′ s₂) (Z.fₒ′ s₃)
assoc-≥ : X ⊗₀ (Y ⊗₀ Z) ≤ (X ⊗₀ Y) ⊗₀ Z
assoc-≥ .⇒S = assocₛ⇐
assoc-≥ .≗-fₛ i (s₁ , (s₂ , s₃)) = (X.S.refl , Y.S.refl) , Z.S.refl
assoc-≥ .≗-fₒ (s₁ , (s₂ , s₃)) = ≋.sym (⊕-assoc (X.fₒ′ s₁) (Y.fₒ′ s₂) (Z.fₒ′ s₃) )
open _≅_
open Iso
associator : (X ⊗₀ Y) ⊗₀ Z ≅ X ⊗₀ (Y ⊗₀ Z)
associator .from = assoc-≤
associator .to = assoc-≥
associator .iso .isoˡ = (X.S.refl , Y.S.refl) , Z.S.refl
associator .iso .isoʳ = X.S.refl , Y.S.refl , Z.S.refl
open Associator using (associator) public
Systems-Monoidal : Monoidal (Systems n)
Systems-Monoidal = let open System in record
{ ⊗ = ⊗
; unit = discrete n
; unitorˡ = unitorˡ
; unitorʳ = unitorʳ
; associator = associator
; unitorˡ-commute-from = λ {_} {Y} → refl (S Y)
; unitorˡ-commute-to = λ {_} {Y} → _ , refl (S Y)
; unitorʳ-commute-from = λ {_} {Y} → refl (S Y)
; unitorʳ-commute-to = λ {_} {Y} → refl (S Y) , _
; assoc-commute-from = λ {_} {X′} {_} {_} {Y′} {_} {_} {Z′} → refl (S X′) , refl (S Y′) , refl (S Z′)
; assoc-commute-to = λ {_} {X′} {_} {_} {Y′} {_} {_} {Z′} → (refl (S X′) , refl (S Y′)) , refl (S Z′)
; triangle = λ {X} {Y} → refl (S X) , refl (S Y)
; pentagon = λ {W} {X} {Y} {Z} → refl (S W) , refl (S X) , refl (S Y) , refl (S Z)
}
open System
⊗-swap-≤ : {X Y : System n} → Y ⊗₀ X ≤ X ⊗₀ Y
⊗-swap-≤ .⇒S = swapₛ
⊗-swap-≤ {X} {Y} .≗-fₛ i (s₁ , s₂) = refl (S X) , refl (S Y)
⊗-swap-≤ {X} {Y} .≗-fₒ (s₁ , s₂) = ⊕-comm (fₒ′ Y s₁) (fₒ′ X s₂)
braiding : ⊗ ≃ flip-bifunctor ⊗
braiding = niHelper record
{ η = λ (X , Y) → ⊗-swap-≤
; η⁻¹ = λ (X , Y) → ⊗-swap-≤
; commute = λ { {X , Y} {X′ , Y′} (f , g) → refl (S Y′) , refl (S X′) }
; iso = λ (X , Y) → record
{ isoˡ = refl (S X) , refl (S Y)
; isoʳ = refl (S Y) , refl (S X)
}
}
Systems-Symmetric : Symmetric Systems-Monoidal
Systems-Symmetric = record
{ braided = record
{ braiding = braiding
; hexagon₁ = λ {X} {Y} {Z} → refl (S Y) , refl (S Z) , refl (S X)
; hexagon₂ = λ {X} {Y} {Z} → (refl (S Z) , refl (S X)) , refl (S Y)
}
; commutative = λ {X} {Y} → refl (S Y) , refl (S X)
}
Systems-MC : MonoidalCategory (suc 0ℓ) ℓ 0ℓ
Systems-MC = record { monoidal = Systems-Monoidal }
Systems-SMC : SymmetricMonoidalCategory (suc 0ℓ) ℓ 0ℓ
Systems-SMC = record { symmetric = Systems-Symmetric }
|
