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{-# OPTIONS --without-K --safe #-}
open import Categories.Category using (Category)
open import Categories.Category.Cocartesian using (Cocartesian)
open import Categories.Category.Cocartesian.Bundle using (CocartesianCategory)
open import Categories.Category.Construction.Monoids using (Monoids)
open import Categories.Category.Monoidal.Bundle using (MonoidalCategory)
open import Categories.Functor using (Functor) renaming (_∘F_ to _∙_)
open import Level using (Level; _⊔_)
-- A functor from a cocartesian category 𝒞 to Monoids[S]
-- can be turned into a monoidal functor from 𝒞 to S
module Functor.Monoidal.Construction.FamilyOfMonoids
{o o′ ℓ ℓ′ e e′ : Level}
{𝒞 : Category o ℓ e}
(𝒞-+ : Cocartesian 𝒞)
{S : MonoidalCategory o′ ℓ′ e′}
(let module S = MonoidalCategory S)
(M : Functor 𝒞 (Monoids S.monoidal))
where
import Categories.Category.Monoidal.Reasoning as ⊗-Reasoning
import Categories.Category.Monoidal.Utilities as ⊗-Util
import Categories.Morphism.Reasoning as ⇒-Reasoning
import Categories.Object.Monoid as MonoidObject
open import Categories.Category using (module Definitions)
open import Categories.Category.Cocartesian using (module CocartesianMonoidal)
open import Categories.Category.Product using (_⁂_)
open import Categories.Functor.Monoidal using (MonoidalFunctor; IsMonoidalFunctor)
open import Categories.Functor.Properties using ([_]-resp-square; [_]-resp-∘)
open import Categories.Morphism using (_≅_)
open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
open import Data.Product using (_,_)
open import Functor.Forgetful.Instance.Monoid S using (Forget)
private
G : Functor 𝒞 S.U
G = Forget ∙ M
module 𝒞 = CocartesianCategory (record { cocartesian = 𝒞-+ })
module 𝒞-M = CocartesianMonoidal 𝒞 𝒞-+
𝒞-MC : MonoidalCategory o ℓ e
𝒞-MC = record { monoidal = 𝒞-M.+-monoidal }
module +-assoc {n} {m} {o} = _≅_ (𝒞.+-assoc {n} {m} {o})
module +-λ {n} = _≅_ (𝒞-M.⊥+A≅A {n})
module +-ρ {n} = _≅_ (𝒞-M.A+⊥≅A {n})
module G = Functor G
module M = Functor M
open MonoidObject S.monoidal using (Monoid; Monoid⇒)
open Monoid renaming (assoc to μ-assoc; identityˡ to μ-identityˡ; identityʳ to μ-identityʳ)
open Monoid⇒
open 𝒞 using (-+-; _+_; _+₁_; i₁; i₂; inject₁; inject₂)
module _ where
open Category 𝒞
open ⇒-Reasoning 𝒞
open ⊗-Reasoning 𝒞-M.+-monoidal
module _ {n m o : Obj} where
private
+-α : (n + m) + o 𝒞.⇒ n + (m + o)
+-α = +-assoc.to {n} {m} {o}
+-α∘i₂ : +-α ∘ i₂ ≈ i₂ ∘ i₂
+-α∘i₂ = inject₂
+-α∘i₁∘i₁ : (+-α ∘ i₁) ∘ i₁ ≈ i₁
+-α∘i₁∘i₁ = inject₁ ⟩∘⟨refl ○ inject₁
+-α∘i₁∘i₂ : (+-α ∘ i₁) ∘ i₂ ≈ i₂ ∘ i₁
+-α∘i₁∘i₂ = inject₁ ⟩∘⟨refl ○ inject₂
module _ {n : Obj} where
+-ρ∘i₁ : +-ρ.from {n} ∘ i₁ ≈ id
+-ρ∘i₁ = inject₁
+-λ∘i₂ : +-λ.from {n} ∘ i₂ ≈ id
+-λ∘i₂ = inject₂
open S
open ⇒-Reasoning U
open ⊗-Reasoning monoidal
open ⊗-Util.Shorthands monoidal
⊲ : {A : 𝒞.Obj} → G.₀ A ⊗₀ G.₀ A ⇒ G.₀ A
⊲ {A} = μ (M.₀ A)
⇒⊲ : {A B : 𝒞.Obj} (f : A 𝒞.⇒ B) → G.₁ f ∘ ⊲ ≈ ⊲ ∘ G.₁ f ⊗₁ G.₁ f
⇒⊲ f = preserves-μ (M.₁ f)
ε : {A : 𝒞.Obj} → unit ⇒ G.₀ A
ε {A} = η (M.₀ A)
⇒ε : {A B : 𝒞.Obj} (f : A 𝒞.⇒ B) → G.₁ f ∘ ε ≈ ε
⇒ε f = preserves-η (M.₁ f)
⊲-⊗ : {n m : 𝒞.Obj} → G.₀ n ⊗₀ G.₀ m ⇒ G.₀ (n + m)
⊲-⊗ = ⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂
module _ {n n′ m m′ : 𝒞.Obj} (f : n 𝒞.⇒ n′) (g : m 𝒞.⇒ m′) where
open Definitions S.U using (CommutativeSquare)
left₁ : CommutativeSquare (G.₁ i₁) (G.₁ f) (G.₁ (f +₁ g)) (G.₁ i₁)
left₁ = [ G ]-resp-square inject₁
left₂ : CommutativeSquare (G.₁ i₂) (G.₁ g) (G.₁ (f +₁ g)) (G.₁ i₂)
left₂ = [ G ]-resp-square inject₂
right : CommutativeSquare ⊲ (G.₁ (f +₁ g) ⊗₁ G.₁ (f +₁ g)) (G.₁ (f +₁ g)) ⊲
right = ⇒⊲ (f +₁ g)
⊲-⊗-commute :
CommutativeSquare
(⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂)
(G.₁ f ⊗₁ G.₁ g)
(G.₁ (f +₁ g))
(⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂)
⊲-⊗-commute = glue′ right (parallel left₁ left₂)
⊲-⊗-homo : NaturalTransformation (⊗ ∙ (G ⁂ G)) (G ∙ -+-)
⊲-⊗-homo = ntHelper record
{ η = λ (n , m) → ⊲-⊗ {n} {m}
; commute = λ (f , g) → Equiv.sym (⊲-⊗-commute f g)
}
⊲-⊗-α
: {n m o : 𝒞.Obj}
→ G.₁ (+-assoc.to {n} {m} {o})
∘ (μ (M.₀ ((n + m) + o)) ∘ G.₁ i₁ ⊗₁ G.₁ i₂)
∘ (μ (M.₀ (n + m)) ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ⊗₁ id
≈ (μ (M.₀ (n + m + o)) ∘ G.₁ i₁ ⊗₁ G.₁ i₂)
∘ id ⊗₁ (μ (M.₀ (m + o)) ∘ G.₁ i₁ ⊗₁ G.₁ i₂)
∘ α⇒
⊲-⊗-α {n} {m} {o} = begin
G.₁ +-α ∘ (⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ∘ (⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ⊗₁ id ≈⟨ refl⟩∘⟨ pullʳ merge₁ʳ ⟩
G.₁ +-α ∘ ⊲ ∘ (G.₁ i₁ ∘ ⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ⊗₁ G.₁ i₂ ≈⟨ extendʳ (⇒⊲ +-α) ⟩
⊲ ∘ G.₁ +-α ⊗₁ G.₁ +-α ∘ (G.₁ i₁ ∘ ⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ⊗₁ G.₁ i₂ ≈⟨ refl⟩∘⟨ ⊗-distrib-over-∘ ⟨
⊲ ∘ (G.₁ +-α ∘ G.₁ i₁ ∘ ⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ⊗₁ (G.₁ +-α ∘ G.₁ i₂) ≈⟨ refl⟩∘⟨ pullˡ (Equiv.sym G.homomorphism) ⟩⊗⟨ [ G ]-resp-square +-α∘i₂ ⟩
⊲ ∘ (G.₁ (+-α 𝒞.∘ i₁) ∘ ⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂) ≈⟨ refl⟩∘⟨ extendʳ (⇒⊲ (+-α 𝒞.∘ i₁)) ⟩⊗⟨refl ⟩
⊲ ∘ (⊲ ∘ G.₁ (+-α 𝒞.∘ i₁) ⊗₁ G.₁ (+-α 𝒞.∘ i₁) ∘ _) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ ⊗-distrib-over-∘) ⟩⊗⟨refl ⟨
⊲ ∘ (⊲ ∘ _ ⊗₁ (G.₁ (+-α 𝒞.∘ i₁) ∘ G.₁ i₂)) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ [ G ]-resp-∘ +-α∘i₁∘i₁ ⟩⊗⟨ [ G ]-resp-square +-α∘i₁∘i₂) ⟩⊗⟨refl ⟩
⊲ ∘ (⊲ ∘ G.₁ i₁ ⊗₁ (G.₁ i₂ ∘ G.₁ i₁)) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂) ≈⟨ refl⟩∘⟨ split₁ˡ ⟩
⊲ ∘ ⊲ ⊗₁ id ∘ (G.₁ i₁ ⊗₁ (G.₁ i₂ ∘ G.₁ i₁)) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂) ≈⟨ extendʳ (μ-assoc (M.₀ (n + (m + o)))) ⟩
⊲ ∘ (id ⊗₁ ⊲ ∘ α⇒) ∘ (G.₁ i₁ ⊗₁ (G.₁ i₂ ∘ G.₁ i₁)) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂) ≈⟨ refl⟩∘⟨ assoc ⟩
⊲ ∘ id ⊗₁ ⊲ ∘ α⇒ ∘ (G.₁ i₁ ⊗₁ (G.₁ i₂ ∘ G.₁ i₁)) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ assoc-commute-from ⟩
⊲ ∘ id ⊗₁ ⊲ ∘ G.₁ i₁ ⊗₁ ((G.₁ i₂ ∘ G.₁ i₁) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂)) ∘ α⇒ ≈⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩
⊲ ∘ G.₁ i₁ ⊗₁ (⊲ ∘ (G.₁ i₂ ∘ G.₁ i₁) ⊗₁ (G.₁ i₂ ∘ G.₁ i₂)) ∘ α⇒ ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ ⊗-distrib-over-∘) ⟩∘⟨refl ⟩
⊲ ∘ G.₁ i₁ ⊗₁ (⊲ ∘ G.₁ i₂ ⊗₁ G.₁ i₂ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ∘ α⇒ ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (extendʳ (⇒⊲ i₂)) ⟩∘⟨refl ⟨
⊲ ∘ G.₁ i₁ ⊗₁ (G.₁ i₂ ∘ ⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ∘ α⇒ ≈⟨ pushʳ (pushˡ split₂ʳ) ⟩
(⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ∘ id ⊗₁ (⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ∘ α⇒ ∎
where
+-α : (n + m) + o 𝒞.⇒ n + (m + o)
+-α = +-assoc.to {n} {m} {o}
module _ {A B : 𝒞.Obj} (f : A 𝒞.⇒ B) where
⊲-εʳ : ⊲ ∘ G.₁ f ⊗₁ ε ≈ G.₁ f ∘ ρ⇒
⊲-εʳ = begin
⊲ ∘ G.₁ f ⊗₁ ε ≈⟨ refl⟩∘⟨ serialize₂₁ ⟩
⊲ ∘ id ⊗₁ ε ∘ G.₁ f ⊗₁ id ≈⟨ pullˡ (Equiv.sym (μ-identityʳ (M.₀ B))) ⟩
ρ⇒ ∘ G.₁ f ⊗₁ id ≈⟨ unitorʳ-commute-from ⟩
G.₁ f ∘ ρ⇒ ∎
⊲-εˡ : ⊲ ∘ ε ⊗₁ G.₁ f ≈ G.₁ f ∘ λ⇒
⊲-εˡ = begin
⊲ ∘ ε ⊗₁ G.₁ f ≈⟨ refl⟩∘⟨ serialize₁₂ ⟩
⊲ ∘ ε ⊗₁ id ∘ id ⊗₁ G.₁ f ≈⟨ pullˡ (Equiv.sym (μ-identityˡ (M.₀ B))) ⟩
λ⇒ ∘ id ⊗₁ G.₁ f ≈⟨ unitorˡ-commute-from ⟩
G.₁ f ∘ λ⇒ ∎
module _ {n : 𝒞.Obj} where
⊲-⊗-λ : G.₁ (+-λ.from {n}) ∘ ⊲-⊗ ∘ ε ⊗₁ id ≈ λ⇒
⊲-⊗-λ = begin
G.₁ +-λ.from ∘ (⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ∘ ε ⊗₁ id ≈⟨ refl⟩∘⟨ pullʳ merge₁ʳ ⟩
G.₁ +-λ.from ∘ ⊲ ∘ (G.₁ i₁ ∘ ε) ⊗₁ G.₁ i₂ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ⇒ε i₁ ⟩⊗⟨refl ⟩
G.₁ +-λ.from ∘ ⊲ ∘ ε ⊗₁ G.₁ i₂ ≈⟨ refl⟩∘⟨ ⊲-εˡ i₂ ⟩
G.₁ +-λ.from ∘ G.₁ i₂ ∘ λ⇒ ≈⟨ cancelˡ ([ G ]-resp-∘ +-λ∘i₂ ○ G.identity) ⟩
λ⇒ ∎
⊲-⊗-ρ : G.₁ (+-ρ.from {n}) ∘ ⊲-⊗ ∘ id ⊗₁ ε ≈ ρ⇒
⊲-⊗-ρ = begin
G.₁ +-ρ.from ∘ (⊲ ∘ G.₁ i₁ ⊗₁ G.₁ i₂) ∘ id ⊗₁ ε ≈⟨ refl⟩∘⟨ pullʳ merge₂ʳ ⟩
G.₁ +-ρ.from ∘ ⊲ ∘ G.₁ i₁ ⊗₁ (G.₁ i₂ ∘ ε) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ ⇒ε i₂ ⟩
G.₁ +-ρ.from ∘ ⊲ ∘ G.₁ i₁ ⊗₁ ε ≈⟨ refl⟩∘⟨ ⊲-εʳ i₁ ⟩
G.₁ +-ρ.from ∘ G.₁ i₁ ∘ ρ⇒ ≈⟨ cancelˡ ([ G ]-resp-∘ +-ρ∘i₁ ○ G.identity) ⟩
ρ⇒ ∎
F,⊗,ε : MonoidalFunctor 𝒞-MC S
F,⊗,ε = record
{ F = G
; isMonoidal = record
{ ε = ε
; ⊗-homo = ⊲-⊗-homo
; associativity = ⊲-⊗-α
; unitaryˡ = ⊲-⊗-λ
; unitaryʳ = ⊲-⊗-ρ
}
}
module F,⊗,ε = MonoidalFunctor F,⊗,ε
|
