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{-# OPTIONS --without-K --safe #-}
open import Categories.Category using (Category)
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)
module Functor.Monoidal.Construction.ListOf
{o o′ ℓ ℓ′ e e′ : Level}
{𝒞 : CocartesianCategory o ℓ e}
{S : MonoidalCategory o′ ℓ′ e′}
(let module 𝒞 = CocartesianCategory 𝒞)
(let module S = MonoidalCategory S)
(G : Functor 𝒞.U S.U)
(M : Functor S.U (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.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 (_,_)
module G = Functor G
module M = Functor M
module 𝒞-M = CocartesianMonoidal 𝒞.U 𝒞.cocartesian
open 𝒞 using (⊥; -+-; _+_; _+₁_; i₁; i₂; inject₁; inject₂)
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 _ where
open 𝒞
open ⇒-Reasoning U
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 Functor
open MonoidalFunctor
open MonoidObject S.monoidal using (Monoid; Monoid⇒)
open Monoid renaming (assoc to μ-assoc; identityˡ to μ-identityˡ; identityʳ to μ-identityʳ)
open Monoid⇒
Forget : Functor (Monoids monoidal) U
Forget .F₀ = Carrier
Forget .F₁ = arr
Forget .identity = Equiv.refl
Forget .homomorphism = Equiv.refl
Forget .F-resp-≈ x = x
𝒞-MC : MonoidalCategory o ℓ e
𝒞-MC = record { monoidal = 𝒞-M.+-monoidal }
List : Functor U U
List = Forget ∙ M
module List = Functor List
List∘G : Functor 𝒞.U U
List∘G = List ∙ G
module LG = Functor List∘G
[] : {A : Obj} → unit ⇒ List.₀ A
[] {A} = η (M.₀ A)
++ : {A : Obj} → List.₀ A ⊗₀ List.₀ A ⇒ List.₀ A
++ {A} = μ (M.₀ A)
++⇒ : {A B : Obj} (f : A ⇒ B) → List.₁ f ∘ ++ ≈ ++ ∘ List.₁ f ⊗₁ List.₁ f
++⇒ f = preserves-μ (M.₁ f)
[]⇒ : {A B : Obj} (f : A ⇒ B) → List.₁ f ∘ [] ≈ []
[]⇒ f = preserves-η (M.₁ f)
++-⊗ : {n m : 𝒞.Obj} → LG.₀ n ⊗₀ LG.₀ m ⇒ LG.₀ (n + m)
++-⊗ = ++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂
open ⇒-Reasoning U
open ⊗-Reasoning monoidal
module _ {n n′ m m′ : 𝒞.Obj} (f : n 𝒞.⇒ n′) (g : m 𝒞.⇒ m′) where
LG[+₁∘i₁] : LG.₁ (f +₁ g) ∘ LG.₁ i₁ ≈ LG.₁ i₁ ∘ LG.₁ f
LG[+₁∘i₁] = [ List∘G ]-resp-square inject₁
LG[+₁∘i₂] : LG.₁ (f +₁ g) ∘ LG.₁ i₂ ≈ LG.₁ i₂ ∘ LG.₁ g
LG[+₁∘i₂] = [ List∘G ]-resp-square inject₂
++-⊗-commute : ++-⊗ ∘ LG.₁ f ⊗₁ LG.₁ g ≈ LG.₁ (f +₁ g) ∘ ++-⊗
++-⊗-commute = begin
(++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ∘ LG.₁ f ⊗₁ LG.₁ g ≈⟨ pushʳ (parallel LG[+₁∘i₁] LG[+₁∘i₂]) ⟨
++ ∘ (LG.₁ (f +₁ g) ⊗₁ LG.₁ (f +₁ g)) ∘ (LG.₁ i₁ ⊗₁ LG.₁ i₂) ≈⟨ extendʳ (++⇒ (G.₁ (f +₁ g))) ⟨
LG.₁ (f +₁ g) ∘ ++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂ ∎
open ⊗-Util.Shorthands monoidal
++-homo : NaturalTransformation (⊗ ∙ (List∘G ⁂ List∘G)) (List∘G ∙ -+-)
++-homo = ntHelper record
{ η = λ (n , m) → ++-⊗ {n} {m}
; commute = λ { {n , m} {n′ , m′} (f , g) → ++-⊗-commute f g }
}
α-++-⊗
: {n m o : 𝒞.Obj}
→ LG.₁ (+-assoc.to {n} {m} {o})
∘ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂)
∘ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ⊗₁ id
≈ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂)
∘ id ⊗₁ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂)
∘ α⇒
α-++-⊗ {n} {m} {o} = begin
LG.₁ +-α ∘ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ∘ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ⊗₁ id ≈⟨ refl⟩∘⟨ pullʳ merge₁ʳ ⟩
LG.₁ +-α ∘ ++ ∘ (LG.₁ i₁ ∘ ++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ⊗₁ LG.₁ i₂ ≈⟨ extendʳ (++⇒ (G.₁ +-α)) ⟩
++ ∘ LG.₁ +-α ⊗₁ LG.₁ +-α ∘ (LG.₁ i₁ ∘ ++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ⊗₁ LG.₁ i₂ ≈⟨ refl⟩∘⟨ ⊗-distrib-over-∘ ⟨
++ ∘ (LG.₁ +-α ∘ LG.₁ i₁ ∘ ++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ⊗₁ (LG.₁ +-α ∘ LG.₁ i₂) ≈⟨ refl⟩∘⟨ pullˡ (Equiv.sym LG.homomorphism) ⟩⊗⟨ [ List∘G ]-resp-square +-α∘i₂ ⟩
++ ∘ (LG.₁ (+-α 𝒞.∘ i₁) ∘ ++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂) ≈⟨ refl⟩∘⟨ extendʳ (++⇒ (G.₁ (+-α 𝒞.∘ i₁))) ⟩⊗⟨refl ⟩
++ ∘ (++ ∘ LG.₁ (+-α 𝒞.∘ i₁) ⊗₁ LG.₁ (+-α 𝒞.∘ i₁) ∘ _) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ ⊗-distrib-over-∘) ⟩⊗⟨refl ⟨
++ ∘ (++ ∘ _ ⊗₁ (LG.₁ (+-α 𝒞.∘ i₁) ∘ LG.₁ i₂)) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂) ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ [ List∘G ]-resp-∘ +-α∘i₁∘i₁ ⟩⊗⟨ [ List∘G ]-resp-square +-α∘i₁∘i₂) ⟩⊗⟨refl ⟩
++ ∘ (++ ∘ LG.₁ i₁ ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₁)) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂) ≈⟨ refl⟩∘⟨ split₁ˡ ⟩
++ ∘ ++ ⊗₁ id ∘ (LG.₁ i₁ ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₁)) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂) ≈⟨ extendʳ (μ-assoc (M.₀ (G.₀ (n + (m + o))))) ⟩
++ ∘ (id ⊗₁ ++ ∘ α⇒) ∘ (LG.₁ i₁ ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₁)) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂) ≈⟨ refl⟩∘⟨ assoc ⟩
++ ∘ id ⊗₁ ++ ∘ α⇒ ∘ (LG.₁ i₁ ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₁)) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ assoc-commute-from ⟩
++ ∘ id ⊗₁ ++ ∘ LG.₁ i₁ ⊗₁ ((LG.₁ i₂ ∘ LG.₁ i₁) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂)) ∘ α⇒ ≈⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩
++ ∘ LG.₁ i₁ ⊗₁ (++ ∘ (LG.₁ i₂ ∘ LG.₁ i₁) ⊗₁ (LG.₁ i₂ ∘ LG.₁ i₂)) ∘ α⇒ ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ ⊗-distrib-over-∘) ⟩∘⟨refl ⟩
++ ∘ LG.₁ i₁ ⊗₁ (++ ∘ LG.₁ i₂ ⊗₁ LG.₁ i₂ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ∘ α⇒ ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (extendʳ (++⇒ (G.₁ i₂))) ⟩∘⟨refl ⟨
++ ∘ LG.₁ i₁ ⊗₁ (LG.₁ i₂ ∘ ++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ∘ α⇒ ≈⟨ pushʳ (pushˡ split₂ʳ) ⟩
(++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ∘ id ⊗₁ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ∘ α⇒ ∎
where
+-α : (n + m) + o 𝒞.⇒ n + (m + o)
+-α = +-assoc.to {n} {m} {o}
module _ {n : 𝒞.Obj} where
++-[]ʳ : {A B : Obj} (f : A ⇒ B) → ++ ∘ List.₁ f ⊗₁ [] ≈ List.₁ f ∘ ρ⇒
++-[]ʳ {A} {B} f = begin
++ ∘ List.₁ f ⊗₁ [] ≈⟨ refl⟩∘⟨ serialize₂₁ ⟩
++ ∘ id ⊗₁ [] ∘ List.₁ f ⊗₁ id ≈⟨ pullˡ (Equiv.sym (μ-identityʳ (M.₀ B))) ⟩
ρ⇒ ∘ List.₁ f ⊗₁ id ≈⟨ unitorʳ-commute-from ⟩
List.₁ f ∘ ρ⇒ ∎
++-[]ˡ : {A B : Obj} (f : A ⇒ B) → ++ ∘ [] ⊗₁ List.₁ f ≈ List.₁ f ∘ λ⇒
++-[]ˡ {A} {B} f = begin
++ ∘ [] ⊗₁ List.₁ f ≈⟨ refl⟩∘⟨ serialize₁₂ ⟩
++ ∘ [] ⊗₁ id ∘ id ⊗₁ List.₁ f ≈⟨ pullˡ (Equiv.sym (μ-identityˡ (M.₀ B))) ⟩
λ⇒ ∘ id ⊗₁ List.₁ f ≈⟨ unitorˡ-commute-from ⟩
List.₁ f ∘ λ⇒ ∎
++-⊗-λ
: LG.₁ +-λ.from ∘ (++ ∘ LG.₁ (i₁ {⊥} {n}) ⊗₁ LG.₁ i₂) ∘ [] ⊗₁ id
≈ λ⇒
++-⊗-λ = begin
LG.₁ +-λ.from ∘ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ∘ [] ⊗₁ id ≈⟨ refl⟩∘⟨ pullʳ merge₁ʳ ⟩
LG.₁ +-λ.from ∘ ++ ∘ (LG.₁ i₁ ∘ []) ⊗₁ LG.₁ i₂ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ []⇒ (G.₁ i₁) ⟩⊗⟨refl ⟩
LG.₁ +-λ.from ∘ ++ ∘ [] ⊗₁ LG.₁ i₂ ≈⟨ refl⟩∘⟨ ++-[]ˡ (G.₁ i₂) ⟩
LG.₁ +-λ.from ∘ LG.₁ i₂ ∘ λ⇒ ≈⟨ cancelˡ ([ List∘G ]-resp-∘ +-λ∘i₂ ○ LG.identity) ⟩
λ⇒ ∎
++-⊗-ρ
: LG.₁ +-ρ.from ∘ (++ ∘ LG.₁ (i₁ {n}) ⊗₁ LG.₁ i₂) ∘ id ⊗₁ []
≈ ρ⇒
++-⊗-ρ = begin
LG.₁ +-ρ.from ∘ (++ ∘ LG.₁ i₁ ⊗₁ LG.₁ i₂) ∘ id ⊗₁ [] ≈⟨ refl⟩∘⟨ pullʳ merge₂ʳ ⟩
LG.₁ +-ρ.from ∘ ++ ∘ LG.₁ i₁ ⊗₁ (LG.₁ i₂ ∘ []) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ []⇒ (G.₁ i₂) ⟩
LG.₁ +-ρ.from ∘ ++ ∘ LG.₁ i₁ ⊗₁ [] ≈⟨ refl⟩∘⟨ ++-[]ʳ (G.₁ i₁) ⟩
LG.₁ +-ρ.from ∘ LG.₁ i₁ ∘ ρ⇒ ≈⟨ cancelˡ ([ List∘G ]-resp-∘ +-ρ∘i₁ ○ LG.identity) ⟩
ρ⇒ ∎
open IsMonoidalFunctor
ListOf,++,[] : MonoidalFunctor 𝒞-MC S
ListOf,++,[] .F = List∘G
ListOf,++,[] .isMonoidal .ε = []
ListOf,++,[] .isMonoidal .⊗-homo = ++-homo
ListOf,++,[] .isMonoidal .associativity = α-++-⊗
ListOf,++,[] .isMonoidal .unitaryˡ = ++-⊗-λ
ListOf,++,[] .isMonoidal .unitaryʳ = ++-⊗-ρ
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