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{-# OPTIONS --without-K --safe #-}
{-# OPTIONS --lossy-unification #-}
open import Level using (Level; suc; _⊔_)
module Category.Instance.Properties.SymMonCat {o ℓ e : Level} where
import Categories.Category.Monoidal.Reasoning as ⊗-Reasoning
import Categories.Morphism.Reasoning as ⇒-Reasoning
import Categories.NaturalTransformation.NaturalIsomorphism.Monoidal.Symmetric as SMNI
import Categories.Functor.Monoidal.Symmetric {o} {o} {ℓ} {ℓ} {e} {e} as SMF
open import Category.Instance.SymMonCat {o} {ℓ} {e} using (SymMonCat)
open import Categories.Category using (Category; _[_≈_]; _[_∘_])
open import Categories.Object.Product.Core SymMonCat using (Product)
open import Categories.Object.Terminal SymMonCat using (Terminal)
open import Categories.Category.Instance.One using (One)
open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory)
open import Categories.Category.Cartesian SymMonCat using (Cartesian)
open import Categories.Category.Cartesian.Bundle using (CartesianCategory)
open import Categories.Functor.Monoidal.Properties using (idF-SymmetricMonoidal; ∘-SymmetricMonoidal)
open import Categories.Category.BinaryProducts SymMonCat using (BinaryProducts)
open import Categories.Functor.Monoidal.Construction.Product
using ()
renaming
( πˡ-SymmetricMonoidalFunctor to πˡ-SMF
; πʳ-SymmetricMonoidalFunctor to πʳ-SMF
; ※-SymmetricMonoidalFunctor to ※-SMF
)
open import Categories.Category.Monoidal.Construction.Product using (Product-SymmetricMonoidalCategory)
open import Categories.Category.Product.Properties using () renaming (project₁ to p₁; project₂ to p₂; unique to u)
open import Data.Product.Base using (_,_; proj₁; proj₂)
open SMF.Lax using (SymmetricMonoidalFunctor)
open SMNI.Lax using (SymmetricMonoidalNaturalIsomorphism; id; isEquivalence)
module Cone
{A B X : SymmetricMonoidalCategory o ℓ e}
{F : SymmetricMonoidalFunctor X A}
{G : SymmetricMonoidalFunctor X B} where
module A = SymmetricMonoidalCategory A
module B = SymmetricMonoidalCategory B
module X = SymmetricMonoidalCategory X
module F = SymmetricMonoidalFunctor X A F
module G = SymmetricMonoidalFunctor X B G
A×B : SymmetricMonoidalCategory o ℓ e
A×B = (Product-SymmetricMonoidalCategory A B)
πˡ : SymmetricMonoidalFunctor A×B A
πˡ = πˡ-SMF {o} {ℓ} {e} {o} {ℓ} {e} {A} {B}
πʳ : SymmetricMonoidalFunctor A×B B
πʳ = πʳ-SMF {o} {ℓ} {e} {o} {ℓ} {e} {A} {B}
module _ where
open Category A.U
open Equiv
open ⇒-Reasoning A.U
open ⊗-Reasoning A.monoidal
project₁ : SymMonCat [ SymMonCat [ πˡ ∘ ※-SMF F G ] ≈ F ]
project₁ = record
{ U = p₁ {o} {ℓ} {e} {o} {ℓ} {e} {o} {ℓ} {e} {A.U} {B.U} {X.U} {F.F} {G.F}
; F⇒G-isMonoidal = record
{ ε-compat = identityˡ ○ identityʳ
; ⊗-homo-compat = λ { {C} {D} → identityˡ ○ refl⟩∘⟨ sym A.⊗.identity }
}
}
module _ (H : SymmetricMonoidalFunctor X A×B) (eq₁ : SymMonCat [ SymMonCat [ πˡ ∘ H ] ≈ F ]) where
private
module H = SymmetricMonoidalFunctor X A×B H
open SymmetricMonoidalNaturalIsomorphism eq₁
ε-compat₁ : ⇐.η X.unit A.∘ F.ε A.≈ H.ε .proj₁
ε-compat₁ = refl⟩∘⟨ sym ε-compat ○ cancelˡ (iso.isoˡ X.unit) ○ identityʳ
⊗-homo-compat₁
: ∀ {C D}
→ ⇐.η (X.⊗.₀ (C , D)) ∘ F.⊗-homo.η (C , D)
≈ H.⊗-homo.η (C , D) .proj₁ ∘ A.⊗.₁ (⇐.η C , ⇐.η D)
⊗-homo-compat₁ {C} {D} =
insertʳ
(sym ⊗-distrib-over-∘
○ iso.isoʳ C ⟩⊗⟨ iso.isoʳ D
○ A.⊗.identity)
○ (pullʳ (sym ⊗-homo-compat)
○ cancelˡ (iso.isoˡ (X.⊗.₀ (C , D)))
○ identityʳ) ⟩∘⟨refl
module _ where
open Category B.U
open Equiv
open ⇒-Reasoning B.U
open ⊗-Reasoning B.monoidal
project₂ : SymMonCat [ SymMonCat [ πʳ ∘ ※-SMF F G ] ≈ G ]
project₂ = record
{ U = p₂ {o} {ℓ} {e} {o} {ℓ} {e} {o} {ℓ} {e} {A.U} {B.U} {X.U} {F.F} {G.F}
; F⇒G-isMonoidal = record
{ ε-compat = identityˡ ○ identityʳ
; ⊗-homo-compat = λ { {C} {D} → identityˡ ○ refl⟩∘⟨ sym B.⊗.identity }
}
}
module _ (H : SymmetricMonoidalFunctor X A×B) (eq₂ : SymMonCat [ SymMonCat [ πʳ ∘ H ] ≈ G ]) where
private
module H = SymmetricMonoidalFunctor X A×B H
open SymmetricMonoidalNaturalIsomorphism eq₂
ε-compat₂ : ⇐.η X.unit ∘ G.ε ≈ H.ε .proj₂
ε-compat₂ = refl⟩∘⟨ sym ε-compat ○ cancelˡ (iso.isoˡ X.unit) ○ identityʳ
⊗-homo-compat₂
: ∀ {C D}
→ ⇐.η (X.⊗.₀ (C , D)) ∘ G.⊗-homo.η (C , D)
≈ H.⊗-homo.η (C , D) .proj₂ ∘ B.⊗.₁ (⇐.η C , ⇐.η D)
⊗-homo-compat₂ {C} {D} =
insertʳ
(sym ⊗-distrib-over-∘
○ iso.isoʳ C ⟩⊗⟨ iso.isoʳ D
○ B.⊗.identity)
○ (pullʳ (sym ⊗-homo-compat)
○ cancelˡ (iso.isoˡ (X.⊗.₀ (C , D)))
○ identityʳ) ⟩∘⟨refl
unique
: (H : SymmetricMonoidalFunctor X A×B)
→ SymMonCat [ SymMonCat [ πˡ ∘ H ] ≈ F ]
→ SymMonCat [ SymMonCat [ πʳ ∘ H ] ≈ G ]
→ SymMonCat [ ※-SMF F G ≈ H ]
unique H eq₁ eq₂ = record
{ U = u {o} {ℓ} {e} {o} {ℓ} {e} {o} {ℓ} {e} {A.U} {B.U} {X.U} {F.F} {G.F} {H.F} eq₁.U eq₂.U
; F⇒G-isMonoidal = record
{ ε-compat = ε-compat₁ H eq₁ , ε-compat₂ H eq₂
; ⊗-homo-compat = ⊗-homo-compat₁ H eq₁ , ⊗-homo-compat₂ H eq₂
}
}
where
module H = SymmetricMonoidalFunctor X A×B H
module eq₁ = SymmetricMonoidalNaturalIsomorphism eq₁
module eq₂ = SymmetricMonoidalNaturalIsomorphism eq₂
prod-SymMonCat : ∀ {A B} → Product A B
prod-SymMonCat {A} {B} = record
{ A×B = Product-SymmetricMonoidalCategory A B
; π₁ = πˡ-SMF {o} {ℓ} {e} {o} {ℓ} {e} {A} {B}
; π₂ = πʳ-SMF {o} {ℓ} {e} {o} {ℓ} {e} {A} {B}
; ⟨_,_⟩ = ※-SMF
; project₁ = λ { {X} {f} {g} → Cone.project₁ {A} {B} {X} {f} {g} }
; project₂ = λ { {X} {f} {g} → Cone.project₂ {A} {B} {X} {f} {g} }
; unique = λ { {X} {h} {f} {g} eq₁ eq₂ → Cone.unique {A} {B} {X} {f} {g} h eq₁ eq₂ }
}
SymMonCat-BinaryProducts : BinaryProducts
SymMonCat-BinaryProducts = record { product = prod-SymMonCat }
SymMonCat-Terminal : Terminal
SymMonCat-Terminal = record
{ ⊤ = record
{ U = One
; monoidal = _
; symmetric = _
}
; ⊤-is-terminal = _
}
SymMonCat-Cartesian : Cartesian
SymMonCat-Cartesian = record
{ terminal = SymMonCat-Terminal
; products = SymMonCat-BinaryProducts
}
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