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{-# OPTIONS --without-K --safe #-}
{-# OPTIONS --hidden-argument-puns #-}
{-# OPTIONS --lossy-unification #-}
module Functor.Instance.Nat.System.Looped where
open import Level using (suc; 0ℓ)
import Data.System.Monoidal as System-⊗
import Functor.Instance.Nat.System as Unlooped
import Categories.Morphism as Morphism
import Functor.Free.Instance.SymmetricMonoidalPreorder.Strong as SymmetricMonoidalPreorder
open import Category.Instance.Setoids.SymmetricMonoidal using (Setoids-×)
open import Categories.Category.Instance.Cats using (Cats)
open import Categories.Category.Instance.Monoidals using (StrongMonoidals)
open import Categories.Category.Instance.Nat using (Nat)
open import Categories.Functor using (Functor; _∘F_) renaming (id to idF)
open import Categories.Functor.Monoidal using (StrongMonoidalFunctor)
open import Categories.Functor.Monoidal.Properties using (idF-StrongMonoidal; ∘-StrongMonoidal)
open import Categories.Functor.Monoidal.Symmetric using () renaming (module Strong to Strong₃)
open import Categories.Functor.Monoidal.Symmetric.Properties using (idF-StrongSymmetricMonoidal; ∘-StrongSymmetricMonoidal)
open import Categories.NaturalTransformation.NaturalIsomorphism using (_≃_; NaturalIsomorphism)
open import Categories.NaturalTransformation.NaturalIsomorphism.Monoidal using () renaming (module Strong to Strong₂)
open import Categories.NaturalTransformation.NaturalIsomorphism.Monoidal.Symmetric using () renaming (module Strong to Strong₄)
open import Category.Construction.CMonoids (Setoids-×.symmetric {suc 0ℓ} {0ℓ}) using (CMonoids)
open import Category.Instance.SymMonCat using () renaming (module Strong to Strong₁)
open import Data.Circuit.Value using (monoid)
open import Data.Fin using (Fin)
open import Data.Nat using (ℕ)
open import Data.System using (System; _≤_; _≈_; Systems[_]; ≤-refl; ≤-trans; discrete)
open import Data.System.Looped.Monoidal using (Systems-MC; Systems-SMC)
open import Data.Values monoid using (module ≋; module Algebra; Values; ≋-isEquiv)
open import Function using (Func; _⟶ₛ_; _⟨$⟩_; _∘_; id)
open import Function.Construct.Setoid using (_∙_)
open import Functor.Free.Instance.InducedCMonoid using (InducedCMonoid)
open import Functor.Instance.Nat.Pull using (Pull)
open import Functor.Instance.Nat.Push using (Push)
open import Object.Monoid.Commutative (Setoids-×.symmetric {0ℓ} {0ℓ}) using (CommutativeMonoid; CommutativeMonoid⇒)
open import Relation.Binary using (Setoid)
open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
open Functor
open Strong₁ using (SymMonCat)
open Strong₂ using (MonoidalNaturalIsomorphism)
open Strong₃ using (SymmetricMonoidalFunctor)
open Strong₄ using (SymmetricMonoidalNaturalIsomorphism)
open Algebra using (Valuesₘ)
private
variable A B C : ℕ
opaque
unfolding ≋-isEquiv
Sys₁ : (Fin A → Fin B) → Functor Systems[ Valuesₘ A ] Systems[ Valuesₘ B ]
Sys₁ f = record { Functor (Unlooped.NatCat.Sys.₁ f) }
Sys-identity : Sys₁ {A} id ≃ idF
Sys-identity {A} = record
{ F⇒G = record { NI.⇒ }
; F⇐G = record { NI.⇐ }
; iso = λ X → record { NI.iso X }
}
where
module NI = NaturalIsomorphism (Unlooped.NatCat.Sys.identity {A})
Sys-homo
: (f : Fin A → Fin B)
(g : Fin B → Fin C)
→ Sys₁ (g ∘ f) ≃ Sys₁ g ∘F Sys₁ f
Sys-homo {A} f g = record
{ F⇒G = record { NI.⇒ }
; F⇐G = record { NI.⇐ }
; iso = λ X → record { NI.iso X }
}
where
module NI = NaturalIsomorphism (Unlooped.NatCat.Sys.homomorphism {f = f} {g})
Sys-resp-≈ : {f g : Fin A → Fin B} → f ≗ g → Sys₁ f ≃ Sys₁ g
Sys-resp-≈ f≗g = record
{ F⇒G = record { NI.⇒ }
; F⇐G = record { NI.⇐ }
; iso = λ X → record { NI.iso X }
}
where
module NI = NaturalIsomorphism (Unlooped.NatCat.Sys.F-resp-≈ f≗g)
module NatCat where
Sys : Functor Nat (Cats (suc 0ℓ) 0ℓ 0ℓ)
Sys .F₀ = λ n → Systems[ Valuesₘ n ]
Sys .F₁ = Sys₁
Sys .identity = Sys-identity
Sys .homomorphism = Sys-homo _ _
Sys .F-resp-≈ = Sys-resp-≈
module Sys = Functor Sys
module NatMC where
module _ (f : Fin A → Fin B) where
-- module A = System-⊗ A A
-- module B = System-⊗ B B
module MF = StrongMonoidalFunctor (Unlooped.NatMC.Sys.₁ f)
open Morphism using (_≅_; Iso)
opaque
unfolding Sys₁ ≋-isEquiv
Sys-MC₁ : StrongMonoidalFunctor (Systems-MC (Valuesₘ A)) (Systems-MC (Valuesₘ B))
Sys-MC₁ = record
{ F = Sys₁ f
; isStrongMonoidal = record
{ ε = record
{ _≅_ MF.ε
; iso = record { Iso MF.ε.iso }
}
; ⊗-homo = record
{ F⇒G = record { MF.⊗-homo.⇒ }
; F⇐G = record { MF.⊗-homo.⇐ }
; iso = λ X → record { MF.⊗-homo.iso X }
}
; associativity = λ {X Y Z} → MF.associativity {X} {Y} {Z}
; unitaryˡ = λ {X} → MF.unitaryˡ {X}
; unitaryʳ = λ {X} → MF.unitaryʳ {X}
}
}
opaque
unfolding Sys-MC₁
Sys-MC-identity : MonoidalNaturalIsomorphism (Sys-MC₁ id) (idF-StrongMonoidal (Systems-MC (Valuesₘ A)))
Sys-MC-identity {A} = record
{ U = record
{ F⇒G = record { ⇒ }
; F⇐G = record { ⇐ }
; iso = λ X → record { iso X}
}
; F⇒G-isMonoidal = record
{ ε-compat = ε-compat
; ⊗-homo-compat = λ {X Y} → ⊗-homo-compat {X} {Y}
}
}
where
open MonoidalNaturalIsomorphism (Unlooped.NatMC.Sys.identity {A})
Sys-MC-homomorphism
: {g : Fin B → Fin C}
{f : Fin A → Fin B}
→ MonoidalNaturalIsomorphism (Sys-MC₁ (g ∘ f)) (∘-StrongMonoidal (Sys-MC₁ g) (Sys-MC₁ f))
Sys-MC-homomorphism {g} {f} = record
{ U = record
{ F⇒G = record { ⇒ }
; F⇐G = record { ⇐ }
; iso = λ X → record { iso X}
}
; F⇒G-isMonoidal = record
{ ε-compat = ε-compat
; ⊗-homo-compat = λ {X Y} → ⊗-homo-compat {X} {Y}
}
}
where
open MonoidalNaturalIsomorphism (Unlooped.NatMC.Sys.homomorphism {f = f} {g})
Sys-MC-resp-≈
: {f g : Fin A → Fin B}
→ f ≗ g
→ MonoidalNaturalIsomorphism (Sys-MC₁ f) (Sys-MC₁ g)
Sys-MC-resp-≈ f≗g = record
{ U = record
{ F⇒G = record { ⇒ }
; F⇐G = record { ⇐ }
; iso = λ X → record { iso X}
}
; F⇒G-isMonoidal = record
{ ε-compat = ε-compat
; ⊗-homo-compat = λ {X Y} → ⊗-homo-compat {X} {Y}
}
}
where
open MonoidalNaturalIsomorphism (Unlooped.NatMC.Sys.F-resp-≈ f≗g)
Sys : Functor Nat (StrongMonoidals (suc 0ℓ) 0ℓ 0ℓ)
Sys .F₀ = λ n → Systems-MC (Valuesₘ n)
Sys .F₁ = Sys-MC₁
Sys .identity = Sys-MC-identity
Sys .homomorphism = Sys-MC-homomorphism
Sys .F-resp-≈ = Sys-MC-resp-≈
module Sys = Functor Sys
module NatSMC where
module _ (f : Fin A → Fin B) where
F-MF : StrongMonoidalFunctor (Systems-MC (Valuesₘ A)) (Systems-MC (Valuesₘ B))
F-MF = NatMC.Sys.₁ f
module F-MF = StrongMonoidalFunctor F-MF
module SMF = SymmetricMonoidalFunctor (Unlooped.NatSMC.Sys.₁ f)
opaque
unfolding NatMC.Sys-MC₁
Sys-SMC₁ : SymmetricMonoidalFunctor (Systems-SMC (Valuesₘ A)) (Systems-SMC (Valuesₘ B))
Sys-SMC₁ = record
{ F-MF
; isBraidedMonoidal = record
{ F-MF
; braiding-compat = λ {X Y} → SMF.braiding-compat {X} {Y}
}
}
opaque
unfolding Sys-SMC₁
Sys-SMC-identity : SymmetricMonoidalNaturalIsomorphism (Sys-SMC₁ id) (idF-StrongSymmetricMonoidal (Systems-SMC (Valuesₘ A)))
Sys-SMC-identity = record { MonoidalNaturalIsomorphism NatMC.Sys.identity }
Sys-SMC-homomorphism
: {g : Fin B → Fin C}
{f : Fin A → Fin B}
→ SymmetricMonoidalNaturalIsomorphism (Sys-SMC₁ (g ∘ f)) (∘-StrongSymmetricMonoidal (Sys-SMC₁ g) (Sys-SMC₁ f))
Sys-SMC-homomorphism = record { MonoidalNaturalIsomorphism NatMC.Sys.homomorphism }
Sys-SMC-resp-≈
: {f g : Fin A → Fin B}
→ f ≗ g
→ SymmetricMonoidalNaturalIsomorphism (Sys-SMC₁ f) (Sys-SMC₁ g)
Sys-SMC-resp-≈ f≗g = record { MonoidalNaturalIsomorphism (NatMC.Sys.F-resp-≈ f≗g) }
Sys : Functor Nat (SymMonCat {suc 0ℓ} {0ℓ} {0ℓ})
Sys .F₀ = λ n → Systems-SMC (Valuesₘ n)
Sys .F₁ = Sys-SMC₁
Sys .identity = Sys-SMC-identity
Sys .homomorphism = Sys-SMC-homomorphism
Sys .F-resp-≈ = Sys-SMC-resp-≈
module Sys = Functor Sys
module NatCMon where
Sys : Functor Nat CMonoids
Sys = InducedCMonoid ∘F SymmetricMonoidalPreorder.Free ∘F NatSMC.Sys
module Sys = Functor Sys
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