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|
{-# OPTIONS --without-K --safe #-}
open import Categories.Category.Monoidal.Bundle using (MonoidalCategory; SymmetricMonoidalCategory)
open import Categories.Functor.Monoidal.Symmetric using (module Lax)
open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory)
open Lax using (SymmetricMonoidalFunctor)
open FinitelyCocompleteCategory
using ()
renaming (symmetricMonoidalCategory to smc)
module Functor.Instance.DecoratedCospan.Embed
{o o′ ℓ ℓ′ e e′}
(𝒞 : FinitelyCocompleteCategory o ℓ e)
{𝒟 : SymmetricMonoidalCategory o′ ℓ′ e′}
(F : SymmetricMonoidalFunctor (smc 𝒞) 𝒟) where
import Categories.Category.Monoidal.Reasoning as ⊗-Reasoning
import Categories.Diagram.Pushout.Properties as PushoutProperties
import Categories.Morphism.Reasoning as ⇒-Reasoning
import Category.Diagram.Pushout as Pushout′
import Functor.Instance.Cospan.Embed 𝒞 as Embed
open import Categories.Category using (Category; _[_,_]; _[_≈_]; _[_∘_])
open import Categories.Category.Monoidal.Properties using (coherence-inv₃)
open import Categories.Functor.Properties using ([_]-resp-≅)
open import Category.Instance.Cospans 𝒞 using (Cospans)
open import Category.Instance.DecoratedCospans 𝒞 F using (DecoratedCospans)
import Categories.Diagram.Pushout as DiagramPushout
import Categories.Diagram.Pushout.Properties as PushoutProperties
import Categories.Morphism as Morphism
open import Categories.Category.Cocartesian using (module CocartesianMonoidal)
open import Categories.Category.Monoidal.Utilities using (module Shorthands)
open import Categories.Functor using (Functor; _∘F_)
open import Data.Product.Base using (_,_)
open import Function.Base using () renaming (id to idf)
open import Functor.Instance.DecoratedCospan.Stack using (⊗)
module 𝒞 = FinitelyCocompleteCategory 𝒞
module 𝒟 = SymmetricMonoidalCategory 𝒟
module F = SymmetricMonoidalFunctor F
module Cospans = Category Cospans
module DecoratedCospans = Category DecoratedCospans
module mc𝒞 = CocartesianMonoidal 𝒞.U 𝒞.cocartesian
open import Functor.Instance.Decorate 𝒞 F using (Decorate; Decorate-resp-⊗)
private
variable
A B C D E H : 𝒞.Obj
f : 𝒞.U [ A , B ]
g : 𝒞.U [ C , D ]
h : 𝒞.U [ E , H ]
L : Functor 𝒞.U DecoratedCospans
L = Decorate ∘F Embed.L
R : Functor 𝒞.op DecoratedCospans
R = Decorate ∘F Embed.R
B₁ : 𝒞.U [ A , C ] → 𝒞.U [ B , C ] → 𝒟.U [ 𝒟.unit , F.F₀ C ] → DecoratedCospans [ A , B ]
B₁ f g s = record
{ cospan = Embed.B₁ f g
; decoration = s
}
module _ where
module L = Functor L
module R = Functor R
module Codiagonal where
open mc𝒞 using (unitorˡ; unitorʳ; +-monoidal) public
open unitorˡ using () renaming (to to λ⇐′) public
open unitorʳ using () renaming (to to ρ⇐′) public
open 𝒞 using (U; _+_; []-cong₂; []∘+₁; ∘-distribˡ-[]; inject₁; inject₂; ¡)
renaming ([_,_] to [_,_]′; _+₁_ to infixr 10 _+₁_ )
open Category U
open Equiv
open HomReasoning
open ⇒-Reasoning 𝒞.U
μ : {X : Obj} → X + X ⇒ X
μ = [ id , id ]′
μ∘+ : {X Y Z : Obj} {f : X ⇒ Z} {g : Y ⇒ Z} → [ f , g ]′ ≈ μ ∘ f +₁ g
μ∘+ = []-cong₂ (sym identityˡ) (sym identityˡ) ○ sym []∘+₁
μ∘¡ˡ : {X : Obj} → μ ∘ ¡ +₁ id ∘ λ⇐′ ≈ id {X}
μ∘¡ˡ = begin
μ ∘ ¡ +₁ id ∘ λ⇐′ ≈⟨ pullˡ (sym μ∘+) ⟩
[ ¡ , id ]′ ∘ λ⇐′ ≈⟨ inject₂ ⟩
id ∎
μ∘¡ʳ : {X : Obj} → μ ∘ id +₁ ¡ ∘ ρ⇐′ ≈ id {X}
μ∘¡ʳ = begin
μ ∘ id +₁ ¡ ∘ ρ⇐′ ≈⟨ pullˡ (sym μ∘+) ⟩
[ id , ¡ ]′ ∘ ρ⇐′ ≈⟨ inject₁ ⟩
id ∎
μ-natural : {X Y : Obj} {f : X ⇒ Y} → f ∘ μ ≈ μ ∘ f +₁ f
μ-natural = ∘-distribˡ-[] ○ []-cong₂ (identityʳ ○ sym identityˡ) (identityʳ ○ sym identityˡ) ○ sym []∘+₁
B∘L : {A B M C : 𝒞.Obj}
→ {f : 𝒞.U [ A , B ]}
→ {g : 𝒞.U [ B , M ]}
→ {h : 𝒞.U [ C , M ]}
→ {s : 𝒟.U [ 𝒟.unit , F.₀ M ]}
→ DecoratedCospans [ DecoratedCospans [ B₁ g h s ∘ L.₁ f ] ≈ B₁ (𝒞.U [ g ∘ f ]) h s ]
B∘L {A} {B} {M} {C} {f} {g} {h} {s} = record
{ cospans-≈ = Embed.B∘L
; same-deco = same-deco
}
where
module _ where
open 𝒞 using (¡; ⊥; ¡-unique; pushout) renaming ([_,_] to [_,_]′; _+₁_ to infixr 10 _+₁_ )
open 𝒞 using (U)
open Category U
open mc𝒞 using (unitorˡ; unitorˡ-commute-to; +-monoidal) public
open unitorˡ using () renaming (to to λ⇐′) public
open ⊗-Reasoning +-monoidal
open ⇒-Reasoning 𝒞.U
open Equiv
open Pushout′ 𝒞.U using (pushout-id-g)
open PushoutProperties 𝒞.U using (up-to-iso)
open DiagramPushout 𝒞.U using (Pushout)
P P′ : Pushout 𝒞.id g
P = pushout 𝒞.id g
P′ = pushout-id-g
module P = Pushout P
module P′ = Pushout P′
open Morphism 𝒞.U using (_≅_)
open _≅_ using (from)
open P using (i₁ ; i₂; universal∘i₂≈h₂) public
open Codiagonal using (μ; μ-natural; μ∘+; μ∘¡ˡ)
≅ : P.Q ⇒ M
≅ = up-to-iso P P′ .from
≅∘[]∘¡+id : ((≅ ∘ [ i₁ , i₂ ]′) ∘ ¡ +₁ id) ∘ λ⇐′ ≈ id
≅∘[]∘¡+id = begin
((≅ ∘ [ i₁ , i₂ ]′) ∘ ¡ +₁ id) ∘ λ⇐′ ≈⟨ assoc²αε ⟩
≅ ∘ [ i₁ , i₂ ]′ ∘ ¡ +₁ id ∘ λ⇐′ ≈⟨ refl⟩∘⟨ pushˡ μ∘+ ⟩
≅ ∘ μ ∘ i₁ +₁ i₂ ∘ ¡ +₁ id ∘ λ⇐′ ≈⟨ refl⟩∘⟨ pull-center (sym split₁ʳ) ⟩
≅ ∘ μ ∘ (i₁ ∘ ¡) +₁ i₂ ∘ λ⇐′ ≈⟨ extendʳ μ-natural ⟩
μ ∘ ≅ +₁ ≅ ∘ (i₁ ∘ ¡) +₁ i₂ ∘ λ⇐′ ≈⟨ pull-center (sym ⊗-distrib-over-∘) ⟩
μ ∘ (≅ ∘ i₁ ∘ ¡) +₁ (≅ ∘ i₂) ∘ λ⇐′ ≈⟨ refl⟩∘⟨ sym (¡-unique (≅ ∘ i₁ ∘ ¡)) ⟩⊗⟨ universal∘i₂≈h₂ ⟩∘⟨refl ⟩
μ ∘ ¡ +₁ id ∘ λ⇐′ ≈⟨ μ∘¡ˡ ⟩
id ∎
open 𝒟 using (U; monoidal; _⊗₁_; unitorˡ-commute-from) renaming (module unitorˡ to λ-)
open 𝒞 using (¡; ⊥; ¡-unique; pushout) renaming ([_,_] to [_,_]′; _+₁_ to infixr 10 _+₁_ )
open Category U
open Equiv
open ⇒-Reasoning U
open ⊗-Reasoning monoidal
open F.⊗-homo using () renaming (η to φ; commute to φ-commute)
open F using (F₁; ε)
open Shorthands monoidal
same-deco : F₁ ≅ ∘ F₁ [ i₁ , i₂ ]′ ∘ φ (B , M) ∘ (F₁ ¡ ∘ ε) ⊗₁ s ∘ ρ⇐ ≈ s
same-deco = begin
F₁ ≅ ∘ F₁ [ i₁ , i₂ ]′ ∘ φ (B , M) ∘ (F₁ ¡ ∘ ε) ⊗₁ s ∘ ρ⇐ ≈⟨ pullˡ (sym F.homomorphism) ⟩
F₁ (≅ 𝒞.∘ [ i₁ , i₂ ]′) ∘ φ (B , M) ∘ (F₁ ¡ ∘ ε) ⊗₁ s ∘ ρ⇐ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ split₁ˡ ⟩
F₁ (≅ 𝒞.∘ [ i₁ , i₂ ]′) ∘ φ (B , M) ∘ F₁ ¡ ⊗₁ id ∘ ε ⊗₁ s ∘ ρ⇐ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ sym F.identity ⟩∘⟨refl ⟩
F₁ (≅ 𝒞.∘ [ i₁ , i₂ ]′) ∘ φ (B , M) ∘ F₁ ¡ ⊗₁ F₁ 𝒞.id ∘ ε ⊗₁ s ∘ ρ⇐ ≈⟨ refl⟩∘⟨ extendʳ (φ-commute (¡ , 𝒞.id)) ⟩
F₁ (≅ 𝒞.∘ [ i₁ , i₂ ]′) ∘ F₁ (¡ +₁ 𝒞.id) ∘ φ (⊥ , M) ∘ ε ⊗₁ s ∘ ρ⇐ ≈⟨ pullˡ (sym F.homomorphism) ⟩
F₁ ((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ ¡ +₁ 𝒞.id) ∘ φ (⊥ , M) ∘ ε ⊗₁ s ∘ ρ⇐ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ serialize₁₂ ⟩
F₁ ((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ ¡ +₁ 𝒞.id) ∘ φ (⊥ , M) ∘ ε ⊗₁ id ∘ id ⊗₁ s ∘ ρ⇐ ≈⟨ refl⟩∘⟨ extendʳ (switch-fromtoˡ ([ F.F ]-resp-≅ unitorˡ) F.unitaryˡ) ⟩
F₁ ((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ ¡ +₁ 𝒞.id) ∘ F₁ λ⇐′ ∘ λ⇒ ∘ id ⊗₁ s ∘ ρ⇐ ≈⟨ pullˡ (sym F.homomorphism) ⟩
F₁ (((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ ¡ +₁ 𝒞.id) 𝒞.∘ λ⇐′) ∘ λ⇒ ∘ id ⊗₁ s ∘ ρ⇐ ≈⟨ refl⟩∘⟨ extendʳ unitorˡ-commute-from ⟩
F₁ (((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ ¡ +₁ 𝒞.id) 𝒞.∘ λ⇐′) ∘ s ∘ λ⇒ ∘ ρ⇐ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ coherence-inv₃ monoidal ⟨
F₁ (((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ ¡ +₁ 𝒞.id) 𝒞.∘ λ⇐′) ∘ s ∘ λ⇒ ∘ λ⇐ ≈⟨ refl⟩∘⟨ (sym-assoc ○ cancelʳ λ-.isoʳ) ⟩
F₁ (((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ ¡ +₁ 𝒞.id) 𝒞.∘ λ⇐′) ∘ s ≈⟨ elimˡ (F.F-resp-≈ ≅∘[]∘¡+id ○ F.identity) ⟩
s ∎
R∘B : {A N B C : 𝒞.Obj}
→ {f : 𝒞.U [ A , N ]}
→ {g : 𝒞.U [ B , N ]}
→ {h : 𝒞.U [ C , B ]}
→ {s : 𝒟.U [ 𝒟.unit , F.₀ N ]}
→ DecoratedCospans [ DecoratedCospans [ R.₁ h ∘ B₁ f g s ] ≈ B₁ f (𝒞.U [ g ∘ h ]) s ]
R∘B {A} {N} {B} {C} {f} {g} {h} {s} = record
{ cospans-≈ = Embed.R∘B
; same-deco = same-deco
}
where
module _ where
open 𝒞 using (¡; ⊥; ¡-unique; pushout) renaming ([_,_] to [_,_]′; _+₁_ to infixr 10 _+₁_ )
open 𝒞 using (U)
open Category U
open mc𝒞 using (unitorʳ; unitorˡ; unitorˡ-commute-to; +-monoidal) public
open unitorˡ using () renaming (to to λ⇐′) public
open unitorʳ using () renaming (to to ρ⇐′) public
open ⊗-Reasoning +-monoidal
open ⇒-Reasoning 𝒞.U
open Equiv
open Pushout′ 𝒞.U using (pushout-f-id)
open PushoutProperties 𝒞.U using (up-to-iso)
open DiagramPushout 𝒞.U using (Pushout)
P P′ : Pushout g 𝒞.id
P = pushout g 𝒞.id
P′ = pushout-f-id
module P = Pushout P
module P′ = Pushout P′
open Morphism 𝒞.U using (_≅_)
open _≅_ using (from)
open P using (i₁ ; i₂; universal∘i₁≈h₁) public
open Codiagonal using (μ; μ-natural; μ∘+; μ∘¡ʳ)
≅ : P.Q ⇒ N
≅ = up-to-iso P P′ .from
≅∘[]∘id+¡ : ((≅ ∘ [ i₁ , i₂ ]′) ∘ id +₁ ¡) ∘ ρ⇐′ ≈ id
≅∘[]∘id+¡ = begin
((≅ ∘ [ i₁ , i₂ ]′) ∘ id +₁ ¡) ∘ ρ⇐′ ≈⟨ assoc²αε ⟩
≅ ∘ [ i₁ , i₂ ]′ ∘ id +₁ ¡ ∘ ρ⇐′ ≈⟨ refl⟩∘⟨ pushˡ μ∘+ ⟩
≅ ∘ μ ∘ i₁ +₁ i₂ ∘ id +₁ ¡ ∘ ρ⇐′ ≈⟨ refl⟩∘⟨ pull-center merge₂ʳ ⟩
≅ ∘ μ ∘ i₁ +₁ (i₂ ∘ ¡) ∘ ρ⇐′ ≈⟨ extendʳ μ-natural ⟩
μ ∘ ≅ +₁ ≅ ∘ i₁ +₁ (i₂ ∘ ¡) ∘ ρ⇐′ ≈⟨ pull-center (sym ⊗-distrib-over-∘) ⟩
μ ∘ (≅ ∘ i₁) +₁ (≅ ∘ i₂ ∘ ¡) ∘ ρ⇐′ ≈⟨ refl⟩∘⟨ universal∘i₁≈h₁ ⟩⊗⟨ sym (¡-unique (≅ ∘ i₂ ∘ ¡)) ⟩∘⟨refl ⟩
μ ∘ id +₁ ¡ ∘ ρ⇐′ ≈⟨ μ∘¡ʳ ⟩
id ∎
open 𝒟 using (U; monoidal; _⊗₁_; unitorʳ-commute-from) renaming (module unitorˡ to λ-; module unitorʳ to ρ)
open 𝒞 using (¡; ⊥; ¡-unique; pushout) renaming ([_,_] to [_,_]′; _+₁_ to infixr 10 _+₁_ )
open Category U
open Equiv
open ⇒-Reasoning U
open ⊗-Reasoning monoidal
open F.⊗-homo using () renaming (η to φ; commute to φ-commute)
open F using (F₁; ε)
open Shorthands monoidal
same-deco : F₁ ≅ ∘ F₁ [ i₁ , i₂ ]′ ∘ φ (N , B) ∘ s ⊗₁ (F₁ ¡ ∘ ε) ∘ ρ⇐ ≈ s
same-deco = begin
F₁ ≅ ∘ F₁ [ i₁ , i₂ ]′ ∘ φ (N , B) ∘ s ⊗₁ (F₁ ¡ ∘ ε) ∘ ρ⇐ ≈⟨ pullˡ (sym F.homomorphism) ⟩
F₁ (≅ 𝒞.∘ [ i₁ , i₂ ]′) ∘ φ (N , B) ∘ s ⊗₁ (F₁ ¡ ∘ ε) ∘ ρ⇐ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ split₂ˡ ⟩
F₁ (≅ 𝒞.∘ [ i₁ , i₂ ]′) ∘ φ (N , B) ∘ id ⊗₁ F₁ ¡ ∘ s ⊗₁ ε ∘ ρ⇐ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ sym F.identity ⟩⊗⟨refl ⟩∘⟨refl ⟩
F₁ (≅ 𝒞.∘ [ i₁ , i₂ ]′) ∘ φ (N , B) ∘ F₁ 𝒞.id ⊗₁ F₁ ¡ ∘ s ⊗₁ ε ∘ ρ⇐ ≈⟨ refl⟩∘⟨ extendʳ (φ-commute (𝒞.id , ¡)) ⟩
F₁ (≅ 𝒞.∘ [ i₁ , i₂ ]′) ∘ F₁ (𝒞.id +₁ ¡) ∘ φ (N , ⊥) ∘ s ⊗₁ ε ∘ ρ⇐ ≈⟨ pullˡ (sym F.homomorphism) ⟩
F₁ ((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ 𝒞.id +₁ ¡) ∘ φ (N , ⊥) ∘ s ⊗₁ ε ∘ ρ⇐ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ serialize₂₁ ⟩
F₁ ((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ 𝒞.id +₁ ¡) ∘ φ (N , ⊥) ∘ id ⊗₁ ε ∘ s ⊗₁ id ∘ ρ⇐ ≈⟨ refl⟩∘⟨ extendʳ (switch-fromtoˡ ([ F.F ]-resp-≅ unitorʳ) F.unitaryʳ) ⟩
F₁ ((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ 𝒞.id +₁ ¡) ∘ F₁ ρ⇐′ ∘ ρ⇒ ∘ s ⊗₁ id ∘ ρ⇐ ≈⟨ pullˡ (sym F.homomorphism) ⟩
F₁ (((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ 𝒞.id +₁ ¡) 𝒞.∘ ρ⇐′) ∘ ρ⇒ ∘ s ⊗₁ id ∘ ρ⇐ ≈⟨ refl⟩∘⟨ extendʳ unitorʳ-commute-from ⟩
F₁ (((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ 𝒞.id +₁ ¡) 𝒞.∘ ρ⇐′) ∘ s ∘ ρ⇒ ∘ ρ⇐ ≈⟨ refl⟩∘⟨ (sym-assoc ○ cancelʳ ρ.isoʳ) ⟩
F₁ (((≅ 𝒞.∘ [ i₁ , i₂ ]′) 𝒞.∘ 𝒞.id +₁ ¡) 𝒞.∘ ρ⇐′) ∘ s ≈⟨ elimˡ (F.F-resp-≈ ≅∘[]∘id+¡ ○ F.identity) ⟩
s ∎
open Morphism 𝒞.U using (_≅_)
open _≅_
≅-L-R : (X≅Y : A ≅ B) → DecoratedCospans [ L.₁ (to X≅Y) ≈ R.₁ (from X≅Y) ]
≅-L-R X≅Y = Decorate.F-resp-≈ (Embed.≅-L-R X≅Y)
where
module Decorate = Functor Decorate
module ⊗ = Functor (⊗ 𝒞 F)
open 𝒞 using (_+₁_)
L-resp-⊗ : DecoratedCospans [ L.₁ (f +₁ g) ≈ ⊗.₁ (L.₁ f , L.₁ g) ]
L-resp-⊗ = Decorate.F-resp-≈ Embed.L-resp-⊗ ○ Decorate-resp-⊗
where
module Decorate = Functor Decorate
open DecoratedCospans.HomReasoning
|
