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{-# OPTIONS --without-K --safe #-}
{-# OPTIONS --no-require-unique-meta-solutions #-}
open import Level using (Level)
module Functor.Instance.Underlying.SymmetricMonoidal.FinitelyCocomplete {o ℓ e : Level} where
import Categories.Morphism.Reasoning as ⇒-Reasoning
import Categories.Object.Coproduct as Coproduct
import Categories.Object.Initial as Initial
open import Categories.Functor using (Functor; _∘F_)
open import Categories.Functor.Properties using ([_]-resp-square; [_]-resp-∘)
open import Categories.Functor.Monoidal using (IsMonoidalFunctor)
open import Categories.Functor.Monoidal.Braided using (module Lax)
open import Categories.Functor.Monoidal.Properties using (idF-SymmetricMonoidal; ∘-SymmetricMonoidal)
open import Categories.Functor.Monoidal.Symmetric using (module Lax)
open import Categories.Category using (Category; _[_,_]; _[_≈_]; _[_∘_])
open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory; BraidedMonoidalCategory; MonoidalCategory)
open import Categories.Category.Product using (_⁂_)
open import Categories.Morphism using (_≅_)
open import Categories.Morphism.Notation using (_[_≅_])
open import Categories.NaturalTransformation.Core using (NaturalTransformation; ntHelper)
open import Categories.NaturalTransformation.NaturalIsomorphism using (NaturalIsomorphism; niHelper) renaming (refl to ≃-refl)
open import Categories.NaturalTransformation.NaturalIsomorphism.Monoidal.Symmetric using (module Lax)
open import Data.Product.Base using (_,_)
open import Category.Instance.FinitelyCocompletes {o} {ℓ} {e} using (FinitelyCocompletes)
open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory)
open import Category.Instance.SymMonCat {o} {ℓ} {e} using (SymMonCat)
open import Functor.Exact using (RightExactFunctor; idREF; ∘-RightExactFunctor)
open FinitelyCocompleteCategory using () renaming (symmetricMonoidalCategory to smc)
open SymmetricMonoidalCategory using (unit) renaming (braidedMonoidalCategory to bmc)
open BraidedMonoidalCategory using () renaming (monoidalCategory to mc)
private
variable
A B C : FinitelyCocompleteCategory o ℓ e
F₀ : FinitelyCocompleteCategory o ℓ e → SymmetricMonoidalCategory o ℓ e
F₀ C = smc C
{-# INJECTIVE_FOR_INFERENCE F₀ #-}
F₁ : RightExactFunctor A B → Lax.SymmetricMonoidalFunctor (F₀ A) (F₀ B)
F₁ {A} {B} F = record
{ F = F.F
; isBraidedMonoidal = record
{ isMonoidal = record
{ ε = ε-iso.from
; ⊗-homo = ⊗-homo
; associativity = assoc
; unitaryˡ = unitaryˡ
; unitaryʳ = unitaryʳ
}
; braiding-compat = braiding-compat
}
}
where
module F = RightExactFunctor F
module A = SymmetricMonoidalCategory (F₀ A)
module B = SymmetricMonoidalCategory (F₀ B)
module A′ = FinitelyCocompleteCategory A
module B′ = FinitelyCocompleteCategory B
ε-iso : B.U [ B.unit ≅ F.₀ A.unit ]
ε-iso = Initial.up-to-iso B.U B′.initial (record { ⊥ = F.₀ A′.⊥ ; ⊥-is-initial = F.F-resp-⊥ A′.⊥-is-initial })
module ε-iso = _≅_ ε-iso
+-iso : ∀ {X Y} → B.U [ F.₀ X B′.+ F.₀ Y ≅ F.₀ (X A′.+ Y) ]
+-iso = Coproduct.up-to-iso B.U B′.coproduct (Coproduct.IsCoproduct⇒Coproduct B.U (F.F-resp-+ (Coproduct.Coproduct⇒IsCoproduct A.U A′.coproduct)))
module +-iso {X Y} = _≅_ (+-iso {X} {Y})
module B-proofs where
open ⇒-Reasoning B.U
open B.HomReasoning
open B.Equiv
open B using (_∘_; _≈_)
open B′ using (_+₁_; []-congˡ; []-congʳ; []-cong₂)
open A′ using (_+_; i₁; i₂)
⊗-homo : NaturalTransformation (B.⊗ ∘F (F.F ⁂ F.F)) (F.F ∘F A.⊗)
⊗-homo = ntHelper record
{ η = λ { (X , Y) → +-iso.from {X} {Y} }
; commute = λ { {X , Y} {X′ , Y′} (f , g) →
B′.coproduct.∘-distribˡ-[]
○ B′.coproduct.[]-cong₂
(pullˡ B′.coproduct.inject₁ ○ [ F.F ]-resp-square (A.Equiv.sym A′.coproduct.inject₁))
(pullˡ B′.coproduct.inject₂ ○ [ F.F ]-resp-square (A.Equiv.sym A′.coproduct.inject₂))
○ sym B′.coproduct.∘-distribˡ-[] }
}
assoc
: {X Y Z : A.Obj}
→ F.₁ A′.+-assocˡ
∘ +-iso.from {X + Y} {Z}
∘ (+-iso.from {X} {Y} +₁ B.id {F.₀ Z})
≈ +-iso.from {X} {Y + Z}
∘ (B.id {F.₀ X} +₁ +-iso.from {Y} {Z})
∘ B′.+-assocˡ
assoc {X} {Y} {Z} = begin
F.₁ A′.+-assocˡ ∘ +-iso.from ∘ (+-iso.from +₁ B.id) ≈⟨ refl⟩∘⟨ B′.[]∘+₁ ⟩
F.₁ A′.+-assocˡ ∘ B′.[ F.₁ i₁ ∘ +-iso.from , F.₁ i₂ ∘ B.id ] ≈⟨ refl⟩∘⟨ []-congʳ B′.coproduct.∘-distribˡ-[] ⟩
F.₁ A′.+-assocˡ ∘ B′.[ B′.[ F.₁ i₁ ∘ F.₁ i₁ , F.₁ i₁ ∘ F.₁ i₂ ] , F.₁ i₂ ∘ B.id ] ≈⟨ B′.coproduct.∘-distribˡ-[] ⟩
B′.[ F.₁ A′.+-assocˡ ∘ B′.[ F.₁ i₁ ∘ F.₁ i₁ , F.₁ i₁ ∘ F.₁ i₂ ] , _ ] ≈⟨ []-congʳ B′.coproduct.∘-distribˡ-[] ⟩
B′.[ B′.[ F.₁ A′.+-assocˡ ∘ F.₁ i₁ ∘ F.₁ i₁ , F.₁ A′.+-assocˡ ∘ _ ] , _ ] ≈⟨ []-congʳ ([]-congʳ (pullˡ ([ F.F ]-resp-∘ A′.coproduct.inject₁))) ⟩
B′.[ B′.[ F.₁ A′.[ i₁ , i₂ A′.∘ i₁ ] ∘ F.₁ i₁ , F.₁ A′.+-assocˡ ∘ _ ] , _ ] ≈⟨ []-congʳ ([]-congʳ ([ F.F ]-resp-∘ A′.coproduct.inject₁)) ⟩
B′.[ B′.[ F.₁ i₁ , F.₁ A′.+-assocˡ ∘ F.₁ i₁ ∘ F.₁ i₂ ] , _ ] ≈⟨ []-congʳ ([]-congˡ (pullˡ ([ F.F ]-resp-∘ A′.coproduct.inject₁))) ⟩
B′.[ B′.[ F.₁ i₁ , F.₁ A′.[ i₁ , i₂ A′.∘ i₁ ] ∘ F.₁ i₂ ] , _ ] ≈⟨ []-congʳ ([]-congˡ ([ F.F ]-resp-∘ A′.coproduct.inject₂)) ⟩
B′.[ B′.[ F.₁ i₁ , F.₁ (i₂ A′.∘ i₁) ] , F.₁ A′.+-assocˡ ∘ F.₁ i₂ ∘ B.id ] ≈⟨ []-congˡ (pullˡ ([ F.F ]-resp-∘ A′.coproduct.inject₂)) ⟩
B′.[ B′.[ F.₁ i₁ , F.₁ (i₂ A′.∘ i₁) ] , F.₁ (i₂ A′.∘ i₂) ∘ B.id ] ≈⟨ []-cong₂ ([]-congˡ F.homomorphism) (B.identityʳ ○ F.homomorphism) ⟩
B′.[ B′.[ F.₁ i₁ , F.₁ i₂ B′.∘ F.₁ i₁ ] , F.₁ i₂ ∘ F.₁ i₂ ] ≈⟨ []-congʳ ([]-congˡ B′.coproduct.inject₁) ⟨
B′.[ B′.[ F.₁ i₁ , B′.[ F.₁ i₂ B′.∘ F.₁ i₁ , _ ] ∘ B′.i₁ ] , _ ] ≈⟨ []-congʳ ([]-cong₂ (sym B′.coproduct.inject₁) (pushˡ (sym B′.coproduct.inject₂))) ⟩
B′.[ B′.[ B′.[ F.₁ i₁ , _ ] ∘ B′.i₁ , B′.[ F.₁ i₁ , _ ] ∘ B′.i₂ ∘ B′.i₁ ] , _ ] ≈⟨ []-congʳ B′.coproduct.∘-distribˡ-[] ⟨
B′.[ B′.[ F.₁ i₁ , _ ] ∘ B′.[ B′.i₁ , B′.i₂ ∘ B′.i₁ ] , F.₁ i₂ ∘ F.₁ i₂ ] ≈⟨ []-congˡ B′.coproduct.inject₂ ⟨
B′.[ B′.[ F.₁ i₁ , _ ] ∘ B′.[ _ , _ ] , B′.[ _ , F.₁ i₂ ∘ F.₁ i₂ ] ∘ B′.i₂ ] ≈⟨ []-congˡ (pushˡ (sym B′.coproduct.inject₂)) ⟩
B′.[ B′.[ F.₁ i₁ , _ ] ∘ B′.[ _ , _ ] , B′.[ F.₁ i₁ , _ ] ∘ B′.i₂ ∘ B′.i₂ ] ≈⟨ B′.coproduct.∘-distribˡ-[] ⟨
B′.[ F.₁ i₁ , B′.[ F.₁ i₂ ∘ F.₁ i₁ , F.₁ i₂ ∘ F.₁ i₂ ] ] ∘ B′.+-assocˡ ≈⟨ []-cong₂ B.identityʳ (B′.coproduct.∘-distribˡ-[]) ⟩∘⟨refl ⟨
B′.[ F.₁ i₁ B′.∘ B′.id , F.₁ i₂ ∘ B′.[ F.₁ i₁ , F.₁ i₂ ] ] ∘ B′.+-assocˡ ≈⟨ pushˡ (sym B′.[]∘+₁) ⟩
+-iso.from ∘ (B.id +₁ +-iso.from) ∘ B′.+-assocˡ ∎
unitaryˡ
: {X : A.Obj}
→ F.₁ A′.[ A′.initial.! , A.id {X} ]
∘ B′.[ F.₁ i₁ , F.₁ i₂ ]
∘ B′.[ B′.i₁ ∘ B′.initial.! , B′.i₂ ∘ B.id ]
≈ B′.[ B′.initial.! , B.id ]
unitaryˡ {X} = begin
F.₁ A′.[ A′.initial.! , A.id ] ∘ B′.[ F.₁ i₁ , F.₁ i₂ ] ∘ B′.[ _ , B′.i₂ ∘ B.id ] ≈⟨ refl⟩∘⟨ B′.coproduct.∘-distribˡ-[] ⟩
_ ∘ B′.[ _ ∘ B′.i₁ ∘ B′.initial.! , B′.[ F.₁ i₁ , F.₁ i₂ ] ∘ B′.i₂ ∘ B.id ] ≈⟨ refl⟩∘⟨ []-cong₂ (sym (B′.¡-unique _)) (pullˡ B′.coproduct.inject₂) ⟩
F.₁ A′.[ A′.initial.! , A.id ] ∘ B′.[ B′.initial.! , F.₁ i₂ ∘ B.id ] ≈⟨ B′.coproduct.∘-distribˡ-[] ⟩
B′.[ _ ∘ B′.initial.! , F.₁ A′.[ A′.initial.! , A.id ] ∘ F.₁ i₂ ∘ B.id ] ≈⟨ []-cong₂ (sym (B′.¡-unique _)) (pullˡ ([ F.F ]-resp-∘ A′.coproduct.inject₂)) ⟩
B′.[ B′.initial.! , F.₁ A.id ∘ B.id ] ≈⟨ []-congˡ (elimˡ F.identity) ⟩
B′.[ B′.initial.! , B.id ] ∎
unitaryʳ
: {X : A.Obj}
→ F.₁ A′.[ A′.id {X} , A′.initial.! ]
∘ B′.[ F.₁ i₁ , F.₁ i₂ ]
∘ B′.[ B′.i₁ ∘ B.id , B′.i₂ ∘ B′.initial.! ]
≈ B′.[ B.id , B′.initial.! ]
unitaryʳ {X} = begin
F.₁ A′.[ A.id , A′.initial.! ] ∘ B′.[ F.₁ i₁ , F.₁ i₂ ] ∘ B′.[ B′.i₁ ∘ B.id , _ ] ≈⟨ refl⟩∘⟨ B′.coproduct.∘-distribˡ-[] ⟩
_ ∘ B′.[ B′.[ F.₁ i₁ , F.₁ i₂ ] ∘ B′.i₁ ∘ B.id , _ ∘ B′.i₂ ∘ B′.initial.! ] ≈⟨ refl⟩∘⟨ []-cong₂ (pullˡ B′.coproduct.inject₁) (sym (B′.¡-unique _)) ⟩
F.₁ A′.[ A.id , A′.initial.! ] ∘ B′.[ F.₁ i₁ ∘ B.id , B′.initial.! ] ≈⟨ B′.coproduct.∘-distribˡ-[] ⟩
B′.[ F.₁ A′.[ A.id , A′.initial.! ] ∘ F.₁ i₁ ∘ B.id , _ ∘ B′.initial.! ] ≈⟨ []-cong₂ (pullˡ ([ F.F ]-resp-∘ A′.coproduct.inject₁)) (sym (B′.¡-unique _)) ⟩
B′.[ F.₁ A.id ∘ B.id , B′.initial.! ] ≈⟨ []-congʳ (elimˡ F.identity) ⟩
B′.[ B.id , B′.initial.! ] ∎
braiding-compat
: {X Y : A.Obj}
→ F.₁ A′.[ i₂ {X} {Y} , i₁ ] ∘ B′.[ F.₁ i₁ , F.₁ i₂ ]
≈ B′.[ F.F₁ i₁ , F.F₁ i₂ ] ∘ B′.[ B′.i₂ , B′.i₁ ]
braiding-compat = begin
F.₁ A′.[ i₂ , i₁ ] ∘ B′.[ F.₁ i₁ , F.₁ i₂ ] ≈⟨ B′.coproduct.∘-distribˡ-[] ⟩
B′.[ F.₁ A′.[ i₂ , i₁ ] ∘ F.₁ i₁ , F.₁ A′.[ i₂ , i₁ ] ∘ F.₁ i₂ ] ≈⟨ []-cong₂ ([ F.F ]-resp-∘ A′.coproduct.inject₁) ([ F.F ]-resp-∘ A′.coproduct.inject₂) ⟩
B′.[ F.₁ i₂ , F.₁ i₁ ] ≈⟨ []-cong₂ B′.coproduct.inject₂ B′.coproduct.inject₁ ⟨
B′.[ B′.[ F.₁ i₁ , F.₁ i₂ ] ∘ B′.i₂ , B′.[ F.₁ i₁ , F.₁ i₂ ] ∘ B′.i₁ ] ≈⟨ B′.coproduct.∘-distribˡ-[] ⟨
B′.[ F.₁ i₁ , F.₁ i₂ ] ∘ B′.[ B′.i₂ , B′.i₁ ] ∎
open B-proofs
identity : Lax.SymmetricMonoidalNaturalIsomorphism (F₁ (Functor.Exact.idREF {o} {ℓ} {e} {A})) (idF-SymmetricMonoidal (F₀ A))
identity {A} = record
{ U = ≃-refl
; F⇒G-isMonoidal = record
{ ε-compat = ¡-unique₂ (id ∘ ¡) id
; ⊗-homo-compat = refl⟩∘⟨ sym ([]-cong₂ identityʳ identityʳ)
}
}
where
open FinitelyCocompleteCategory A
open HomReasoning
open Equiv
homomorphism
: {F : RightExactFunctor A B}
{G : RightExactFunctor B C}
→ Lax.SymmetricMonoidalNaturalIsomorphism
(F₁ {A} {C} (∘-RightExactFunctor G F))
(∘-SymmetricMonoidal (F₁ {B} {C} G) (F₁ {A} {B} F))
homomorphism {A} {B} {C} {F} {G} = record
{ U = ≃-refl
; F⇒G-isMonoidal = record
{ ε-compat = ¡-unique₂ (id ∘ ¡) (G.₁ B.¡ ∘ ¡)
; ⊗-homo-compat =
identityˡ
○ sym
([]-cong₂
([ G.F ]-resp-∘ B.coproducts.inject₁)
([ G.F ]-resp-∘ B.coproducts.inject₂))
○ sym ∘-distribˡ-[]
○ pushʳ (introʳ C.⊗.identity)
}
}
where
module A = FinitelyCocompleteCategory A
module B = FinitelyCocompleteCategory B
open FinitelyCocompleteCategory C
module C = SymmetricMonoidalCategory (F₀ C)
open HomReasoning
open Equiv
open ⇒-Reasoning U
module F = RightExactFunctor F
module G = RightExactFunctor G
module _ {F G : RightExactFunctor A B} where
module F = RightExactFunctor F
module G = RightExactFunctor G
F-resp-≈
: NaturalIsomorphism F.F G.F
→ Lax.SymmetricMonoidalNaturalIsomorphism (F₁ {A} {B} F) (F₁ {A} {B} G)
F-resp-≈ F≃G = record
{ U = F≃G
; F⇒G-isMonoidal = record
{ ε-compat = sym (¡-unique (⇒.η A.⊥ ∘ ¡))
; ⊗-homo-compat =
∘-distribˡ-[]
○ []-cong₂ (⇒.commute A.i₁) (⇒.commute A.i₂)
○ sym []∘+₁
}
}
where
module A = FinitelyCocompleteCategory A
open NaturalIsomorphism F≃G
open FinitelyCocompleteCategory B
open HomReasoning
open Equiv
Underlying : Functor FinitelyCocompletes SymMonCat
Underlying = record
{ F₀ = F₀
; F₁ = F₁
; identity = λ { {X} → identity {X} }
; homomorphism = λ { {X} {Y} {Z} {F} {G} → homomorphism {X} {Y} {Z} {F} {G} }
; F-resp-≈ = λ { {X} {Y} {F} {G} → F-resp-≈ {X} {Y} {F} {G} }
}
|
