diff options
Diffstat (limited to 'Functor/Instance/Cospan')
| -rw-r--r-- | Functor/Instance/Cospan/Embed.agda | 103 | ||||
| -rw-r--r-- | Functor/Instance/Cospan/Stack.agda | 98 |
2 files changed, 97 insertions, 104 deletions
diff --git a/Functor/Instance/Cospan/Embed.agda b/Functor/Instance/Cospan/Embed.agda index 77f0361..6dbc04a 100644 --- a/Functor/Instance/Cospan/Embed.agda +++ b/Functor/Instance/Cospan/Embed.agda @@ -1,4 +1,5 @@ {-# OPTIONS --without-K --safe #-} +{-# OPTIONS --hidden-argument-puns #-} open import Category.Cocomplete.Finitely.Bundle using (FinitelyCocompleteCategory) @@ -14,9 +15,10 @@ open import Categories.Category using (_[_,_]; _[_∘_]; _[_≈_]) open import Categories.Category.Core using (Category) open import Categories.Functor.Core using (Functor) open import Category.Instance.Cospans 𝒞 using (Cospans) +open import Category.Diagram.Cospan 𝒞 using (cospan) open import Data.Product.Base using (_,_) open import Function.Base using (id) -open import Functor.Instance.Cospan.Stack using (⊗) +open import Functor.Instance.Cospan.Stack 𝒞 using (⊗) module 𝒞 = FinitelyCocompleteCategory 𝒞 module Cospans = Category Cospans @@ -28,24 +30,26 @@ open Morphism U using (module ≅; _≅_) open PushoutProperties U using (up-to-iso) open Pushout′ U using (pushout-id-g; pushout-f-id) -L₁ : {A B : 𝒞.Obj} → U [ A , B ] → Cospans [ A , B ] -L₁ f = record - { f₁ = f - ; f₂ = 𝒞.id - } +private + variable + A B C : 𝒞.Obj + W X Y Z : 𝒞.Obj + +L₁ : U [ A , B ] → Cospans [ A , B ] +L₁ f = cospan f 𝒞.id -L-identity : {A : 𝒞.Obj} → L₁ 𝒞.id ≈ Cospans.id {A} +L-identity : L₁ 𝒞.id ≈ Cospans.id {A} L-identity = record { ≅N = ≅.refl - ; from∘f₁≈f₁′ = 𝒞.identity² - ; from∘f₂≈f₂′ = 𝒞.identity² + ; from∘f₁≈f₁ = 𝒞.identity² + ; from∘f₂≈f₂ = 𝒞.identity² } -L-homomorphism : {X Y Z : 𝒞.Obj} {f : U [ X , Y ]} {g : U [ Y , Z ]} → L₁ (U [ g ∘ f ]) ≈ Cospans [ L₁ g ∘ L₁ f ] +L-homomorphism : {f : U [ X , Y ]} {g : U [ Y , Z ]} → L₁ (U [ g ∘ f ]) ≈ Cospans [ L₁ g ∘ L₁ f ] L-homomorphism {X} {Y} {Z} {f} {g} = record { ≅N = up-to-iso P′ P - ; from∘f₁≈f₁′ = pullˡ (P′.universal∘i₁≈h₁ {eq = P.commute}) - ; from∘f₂≈f₂′ = P′.universal∘i₂≈h₂ {eq = P.commute} ○ sym 𝒞.identityʳ + ; from∘f₁≈f₁ = pullˡ (P′.universal∘i₁≈h₁ {eq = P.commute}) + ; from∘f₂≈f₂ = P′.universal∘i₂≈h₂ {eq = P.commute} ○ sym 𝒞.identityʳ } where open ⇒-Reasoning U @@ -57,11 +61,11 @@ L-homomorphism {X} {Y} {Z} {f} {g} = record module P = Pushout P module P′ = Pushout P′ -L-resp-≈ : {A B : 𝒞.Obj} {f g : U [ A , B ]} → U [ f ≈ g ] → Cospans [ L₁ f ≈ L₁ g ] +L-resp-≈ : {f g : U [ A , B ]} → U [ f ≈ g ] → Cospans [ L₁ f ≈ L₁ g ] L-resp-≈ {A} {B} {f} {g} f≈g = record { ≅N = ≅.refl - ; from∘f₁≈f₁′ = 𝒞.identityˡ ○ f≈g - ; from∘f₂≈f₂′ = 𝒞.identity² + ; from∘f₁≈f₁ = 𝒞.identityˡ ○ f≈g + ; from∘f₂≈f₂ = 𝒞.identity² } where open 𝒞.HomReasoning @@ -75,24 +79,21 @@ L = record ; F-resp-≈ = L-resp-≈ } -R₁ : {A B : 𝒞.Obj} → U [ B , A ] → Cospans [ A , B ] -R₁ g = record - { f₁ = 𝒞.id - ; f₂ = g - } +R₁ : U [ B , A ] → Cospans [ A , B ] +R₁ g = cospan 𝒞.id g -R-identity : {A : 𝒞.Obj} → R₁ 𝒞.id ≈ Cospans.id {A} +R-identity : R₁ 𝒞.id ≈ Cospans.id {A} R-identity = record { ≅N = ≅.refl - ; from∘f₁≈f₁′ = 𝒞.identity² - ; from∘f₂≈f₂′ = 𝒞.identity² + ; from∘f₁≈f₁ = 𝒞.identity² + ; from∘f₂≈f₂ = 𝒞.identity² } -R-homomorphism : {X Y Z : 𝒞.Obj} {f : U [ Y , X ]} {g : U [ Z , Y ]} → R₁ (U [ f ∘ g ]) ≈ Cospans [ R₁ g ∘ R₁ f ] -R-homomorphism {X} {Y} {Z} {f} {g} = record +R-homomorphism : {f : U [ Y , X ]} {g : U [ Z , Y ]} → R₁ (U [ f ∘ g ]) ≈ Cospans [ R₁ g ∘ R₁ f ] +R-homomorphism {f} {g} = record { ≅N = up-to-iso P′ P - ; from∘f₁≈f₁′ = P′.universal∘i₁≈h₁ {eq = P.commute} ○ sym 𝒞.identityʳ - ; from∘f₂≈f₂′ = pullˡ (P′.universal∘i₂≈h₂ {eq = P.commute}) + ; from∘f₁≈f₁ = P′.universal∘i₁≈h₁ {eq = P.commute} ○ sym 𝒞.identityʳ + ; from∘f₂≈f₂ = pullˡ (P′.universal∘i₂≈h₂ {eq = P.commute}) } where open ⇒-Reasoning U @@ -104,11 +105,11 @@ R-homomorphism {X} {Y} {Z} {f} {g} = record module P = Pushout P module P′ = Pushout P′ -R-resp-≈ : {A B : 𝒞.Obj} {f g : U [ A , B ]} → U [ f ≈ g ] → Cospans [ R₁ f ≈ R₁ g ] -R-resp-≈ {A} {B} {f} {g} f≈g = record +R-resp-≈ : {f g : U [ A , B ]} → U [ f ≈ g ] → Cospans [ R₁ f ≈ R₁ g ] +R-resp-≈ {f} {g} f≈g = record { ≅N = ≅.refl - ; from∘f₁≈f₁′ = 𝒞.identity² - ; from∘f₂≈f₂′ = 𝒞.identityˡ ○ f≈g + ; from∘f₁≈f₁ = 𝒞.identity² + ; from∘f₂≈f₂ = 𝒞.identityˡ ○ f≈g } where open 𝒞.HomReasoning @@ -122,17 +123,11 @@ R = record ; F-resp-≈ = R-resp-≈ } -B₁ : {A B C : 𝒞.Obj} → U [ A , C ] → U [ B , C ] → Cospans [ A , B ] -B₁ f g = record - { f₁ = f - ; f₂ = g - } - -B∘L : {W X Y Z : 𝒞.Obj} {f : U [ W , X ]} {g : U [ X , Y ]} {h : U [ Z , Y ]} → Cospans [ B₁ g h ∘ L₁ f ] ≈ B₁ (U [ g ∘ f ]) h -B∘L {W} {X} {Y} {Z} {f} {g} {h} = record +B∘L : {f : U [ W , X ]} {g : U [ X , Y ]} {h : U [ Z , Y ]} → Cospans [ cospan g h ∘ L₁ f ] ≈ cospan (U [ g ∘ f ]) h +B∘L {f} {g} {h} = record { ≅N = up-to-iso P P′ - ; from∘f₁≈f₁′ = pullˡ (P.universal∘i₁≈h₁ {eq = P′.commute}) - ; from∘f₂≈f₂′ = pullˡ (P.universal∘i₂≈h₂ {eq = P′.commute}) ○ 𝒞.identityˡ + ; from∘f₁≈f₁ = pullˡ (P.universal∘i₁≈h₁ {eq = P′.commute}) + ; from∘f₂≈f₂ = pullˡ (P.universal∘i₂≈h₂ {eq = P′.commute}) ○ 𝒞.identityˡ } where open ⇒-Reasoning U @@ -144,11 +139,11 @@ B∘L {W} {X} {Y} {Z} {f} {g} {h} = record module P = Pushout P module P′ = Pushout P′ -R∘B : {W X Y Z : 𝒞.Obj} {f : U [ W , X ]} {g : U [ Y , X ]} {h : U [ Z , Y ]} → Cospans [ R₁ h ∘ B₁ f g ] ≈ B₁ f (U [ g ∘ h ]) -R∘B {W} {X} {Y} {Z} {f} {g} {h} = record +R∘B : {f : U [ W , X ]} {g : U [ Y , X ]} {h : U [ Z , Y ]} → Cospans [ R₁ h ∘ cospan f g ] ≈ cospan f (U [ g ∘ h ]) +R∘B {f} {g} {h} = record { ≅N = up-to-iso P P′ - ; from∘f₁≈f₁′ = pullˡ (P.universal∘i₁≈h₁ {eq = P′.commute}) ○ 𝒞.identityˡ - ; from∘f₂≈f₂′ = pullˡ (P.universal∘i₂≈h₂ {eq = P′.commute}) + ; from∘f₁≈f₁ = pullˡ (P.universal∘i₁≈h₁ {eq = P′.commute}) ○ 𝒞.identityˡ + ; from∘f₂≈f₂ = pullˡ (P.universal∘i₂≈h₂ {eq = P′.commute}) } where open ⇒-Reasoning U @@ -164,20 +159,18 @@ module _ where open _≅_ - ≅-L-R : ∀ {X Y : 𝒞.Obj} (X≅Y : X ≅ Y) → L₁ (to X≅Y) ≈ R₁ (from X≅Y) - ≅-L-R {X} {Y} X≅Y = record + ≅-L-R : (X≅Y : X ≅ Y) → L₁ (to X≅Y) ≈ R₁ (from X≅Y) + ≅-L-R X≅Y = record { ≅N = X≅Y - ; from∘f₁≈f₁′ = isoʳ X≅Y - ; from∘f₂≈f₂′ = 𝒞.identityʳ + ; from∘f₁≈f₁ = isoʳ X≅Y + ; from∘f₂≈f₂ = 𝒞.identityʳ } -module ⊗ = Functor (⊗ 𝒞) - -L-resp-⊗ : {X Y X′ Y′ : 𝒞.Obj} {a : U [ X , X′ ]} {b : U [ Y , Y′ ]} → L₁ (a +₁ b) ≈ ⊗.₁ (L₁ a , L₁ b) -L-resp-⊗ {X} {Y} {X′} {Y′} {a} {b} = record +L-resp-⊗ : {a : U [ W , X ]} {b : U [ Y , Z ]} → L₁ (a +₁ b) ≈ ⊗.₁ (L₁ a , L₁ b) +L-resp-⊗ {a} {b} = record { ≅N = ≅.refl - ; from∘f₁≈f₁′ = 𝒞.identityˡ - ; from∘f₂≈f₂′ = 𝒞.identityˡ ○ sym +-η ○ sym ([]-cong₂ identityʳ identityʳ) + ; from∘f₁≈f₁ = 𝒞.identityˡ + ; from∘f₂≈f₂ = 𝒞.identityˡ ○ sym +-η ○ sym ([]-cong₂ identityʳ identityʳ) } where open 𝒞.HomReasoning diff --git a/Functor/Instance/Cospan/Stack.agda b/Functor/Instance/Cospan/Stack.agda index b7664dc..b72219b 100644 --- a/Functor/Instance/Cospan/Stack.agda +++ b/Functor/Instance/Cospan/Stack.agda @@ -9,11 +9,13 @@ import Categories.Diagram.Pushout.Properties as PushoutProperties import Categories.Morphism as Morphism import Categories.Morphism.Reasoning as ⇒-Reasoning -open import Categories.Category.Core using (Category) +open import Categories.Category using (Category) +open import Categories.Functor using (Functor) open import Categories.Functor.Bifunctor using (Bifunctor) -open import Category.Instance.Cospans 𝒞 using (Cospan; Cospans; Same; id-Cospan; compose) +open import Category.Instance.Cospans 𝒞 using (Cospans) +open import Category.Diagram.Cospan 𝒞 as Cospan using (Cospan; identity; compose; _⊗_) open import Category.Instance.FinitelyCocompletes {o} {ℓ} {e} using () renaming (_×_ to _×′_) -open import Category.Instance.Properties.FinitelyCocompletes {o} {ℓ} {e} using (-+-; FinitelyCocompletes-CC) +open import Category.Cartesian.Instance.FinitelyCocompletes {o} {ℓ} {e} using (-+-; FinitelyCocompletes-CC) open import Data.Product.Base using (Σ; _,_; _×_; proj₁; proj₂) open import Functor.Exact using (RightExactFunctor; IsPushout⇒Pushout) open import Level using (Level; _⊔_; suc) @@ -32,27 +34,19 @@ open DiagramPushout U×U using () renaming (Pushout to Pushout′) open import Categories.Category.Monoidal.Utilities monoidal using (_⊗ᵢ_) -together : {A A′ B B′ : Obj} → Cospan A B → Cospan A′ B′ → Cospan (A + A′) (B + B′) -together A⇒B A⇒B′ = record - { f₁ = f₁ A⇒B +₁ f₁ A⇒B′ - ; f₂ = f₂ A⇒B +₁ f₂ A⇒B′ - } - where - open Cospan - -id⊗id≈id : {A B : Obj} → Same (together (id-Cospan {A}) (id-Cospan {B})) (id-Cospan {A + B}) +id⊗id≈id : {A B : Obj} → identity {A} ⊗ identity {B} Cospan.≈ identity {A + B} id⊗id≈id {A} {B} = record { ≅N = ≅.refl - ; from∘f₁≈f₁′ = from∘f≈f′ - ; from∘f₂≈f₂′ = from∘f≈f′ + ; from∘f₁≈f₁ = from∘f≈f + ; from∘f₂≈f₂ = from∘f≈f } where open Morphism U using (module ≅) open HomReasoning open 𝒞 using (+-η; []-cong₂) open coproduct {A} {B} using (i₁; i₂) - from∘f≈f′ : id ∘ [ i₁ ∘ id , i₂ ∘ id ] 𝒞.≈ id - from∘f≈f′ = begin + from∘f≈f : id ∘ [ i₁ ∘ id , i₂ ∘ id ] 𝒞.≈ id + from∘f≈f = begin id ∘ [ i₁ ∘ id , i₂ ∘ id ] ≈⟨ identityˡ ⟩ [ i₁ ∘ id , i₂ ∘ id ] ≈⟨ []-cong₂ identityʳ identityʳ ⟩ [ i₁ , i₂ ] ≈⟨ +-η ⟩ @@ -64,14 +58,14 @@ homomorphism → (B⇒C : Cospan B C) → (A⇒B′ : Cospan A′ B′) → (B⇒C′ : Cospan B′ C′) - → Same (together (compose A⇒B B⇒C) (compose A⇒B′ B⇒C′)) (compose (together A⇒B A⇒B′) (together B⇒C B⇒C′) ) + → compose A⇒B B⇒C ⊗ compose A⇒B′ B⇒C′ Cospan.≈ compose (A⇒B ⊗ A⇒B′) (B⇒C ⊗ B⇒C′) homomorphism A⇒B B⇒C A⇒B′ B⇒C′ = record { ≅N = ≅N - ; from∘f₁≈f₁′ = from∘f₁≈f₁′ - ; from∘f₂≈f₂′ = from∘f₂≈f₂′ + ; from∘f₁≈f₁ = from∘f₁≈f₁ + ; from∘f₂≈f₂ = from∘f₂≈f₂ } where - open Cospan + open Cospan.Cospan open Pushout open HomReasoning open ⇒-Reasoning U @@ -89,56 +83,62 @@ homomorphism A⇒B B⇒C A⇒B′ B⇒C′ = record P₃′ = IsPushout⇒Pushout (-+-.F-resp-pushout P₁×P₂.isPushout) ≅N : Q P₃′ ≅ Q P₃ ≅N = up-to-iso P₃′ P₃ - from∘f₁≈f₁′ : from ≅N ∘ (f₁ (compose A⇒B B⇒C) +₁ f₁ (compose A⇒B′ B⇒C′)) ≈ f₁ (compose (together A⇒B A⇒B′) (together B⇒C B⇒C′)) - from∘f₁≈f₁′ = begin + from∘f₁≈f₁ : from ≅N ∘ (f₁ (compose A⇒B B⇒C) +₁ f₁ (compose A⇒B′ B⇒C′)) ≈ f₁ (compose (A⇒B ⊗ A⇒B′) (B⇒C ⊗ B⇒C′)) + from∘f₁≈f₁ = begin from ≅N ∘ (f₁ (compose A⇒B B⇒C) +₁ f₁ (compose A⇒B′ B⇒C′)) ≈⟨ Equiv.refl ⟩ from ≅N ∘ ((i₁ P₁ ∘ f₁ A⇒B) +₁ (i₁ P₂ ∘ f₁ A⇒B′)) ≈⟨ refl⟩∘⟨ +₁∘+₁ ⟨ from ≅N ∘ (i₁ P₁ +₁ i₁ P₂) ∘ (f₁ A⇒B +₁ f₁ A⇒B′) ≈⟨ Equiv.refl ⟩ - from ≅N ∘ i₁ P₃′ ∘ f₁ (together A⇒B A⇒B′) ≈⟨ pullˡ (universal∘i₁≈h₁ P₃′) ⟩ - i₁ P₃ ∘ f₁ (together A⇒B A⇒B′) ∎ - from∘f₂≈f₂′ : from ≅N ∘ (f₂ (compose A⇒B B⇒C) +₁ f₂ (compose A⇒B′ B⇒C′)) ≈ f₂ (compose (together A⇒B A⇒B′) (together B⇒C B⇒C′)) - from∘f₂≈f₂′ = begin + from ≅N ∘ i₁ P₃′ ∘ f₁ (A⇒B ⊗ A⇒B′) ≈⟨ pullˡ (universal∘i₁≈h₁ P₃′) ⟩ + i₁ P₃ ∘ f₁ (A⇒B ⊗ A⇒B′) ∎ + from∘f₂≈f₂ : from ≅N ∘ (f₂ (compose A⇒B B⇒C) +₁ f₂ (compose A⇒B′ B⇒C′)) ≈ f₂ (compose (A⇒B ⊗ A⇒B′) (B⇒C ⊗ B⇒C′)) + from∘f₂≈f₂ = begin from ≅N ∘ (f₂ (compose A⇒B B⇒C) +₁ f₂ (compose A⇒B′ B⇒C′)) ≈⟨ Equiv.refl ⟩ from ≅N ∘ ((i₂ P₁ ∘ f₂ B⇒C) +₁ (i₂ P₂ ∘ f₂ B⇒C′)) ≈⟨ refl⟩∘⟨ +₁∘+₁ ⟨ from ≅N ∘ (i₂ P₁ +₁ i₂ P₂) ∘ (f₂ B⇒C +₁ f₂ B⇒C′) ≈⟨ Equiv.refl ⟩ - from ≅N ∘ i₂ P₃′ ∘ f₂ (together B⇒C B⇒C′) ≈⟨ pullˡ (universal∘i₂≈h₂ P₃′) ⟩ - i₂ P₃ ∘ f₂ (together B⇒C B⇒C′) ∎ + from ≅N ∘ i₂ P₃′ ∘ f₂ (B⇒C ⊗ B⇒C′) ≈⟨ pullˡ (universal∘i₂≈h₂ P₃′) ⟩ + i₂ P₃ ∘ f₂ (B⇒C ⊗ B⇒C′) ∎ ⊗-resp-≈ : {A A′ B B′ : Obj} {f f′ : Cospan A B} {g g′ : Cospan A′ B′} - → Same f f′ - → Same g g′ - → Same (together f g) (together f′ g′) + → f Cospan.≈ f′ + → g Cospan.≈ g′ + → f ⊗ g Cospan.≈ f′ ⊗ g′ ⊗-resp-≈ {_} {_} {_} {_} {f} {f′} {g} {g′} ≈f ≈g = record { ≅N = ≈f.≅N ⊗ᵢ ≈g.≅N - ; from∘f₁≈f₁′ = from∘f₁≈f₁′ - ; from∘f₂≈f₂′ = from∘f₂≈f₂′ + ; from∘f₁≈f₁ = from∘f₁≈f₁ + ; from∘f₂≈f₂ = from∘f₂≈f₂ } where open 𝒞 using (-+-) - module ≈f = Same ≈f - module ≈g = Same ≈g + module ≈f = Cospan._≈_ ≈f + module ≈g = Cospan._≈_ ≈g open HomReasoning - open Cospan + open Cospan.Cospan open 𝒞 using (+₁-cong₂; +₁∘+₁) - from∘f₁≈f₁′ : (≈f.from +₁ ≈g.from) ∘ (f₁ f +₁ f₁ g) ≈ f₁ f′ +₁ f₁ g′ - from∘f₁≈f₁′ = begin + from∘f₁≈f₁ : (≈f.from +₁ ≈g.from) ∘ (f₁ f +₁ f₁ g) ≈ f₁ f′ +₁ f₁ g′ + from∘f₁≈f₁ = begin (≈f.from +₁ ≈g.from) ∘ (f₁ f +₁ f₁ g) ≈⟨ +₁∘+₁ ⟩ - (≈f.from ∘ f₁ f) +₁ (≈g.from ∘ f₁ g) ≈⟨ +₁-cong₂ (≈f.from∘f₁≈f₁′) (≈g.from∘f₁≈f₁′) ⟩ + (≈f.from ∘ f₁ f) +₁ (≈g.from ∘ f₁ g) ≈⟨ +₁-cong₂ ≈f.from∘f₁≈f₁ ≈g.from∘f₁≈f₁ ⟩ f₁ f′ +₁ f₁ g′ ∎ - from∘f₂≈f₂′ : (≈f.from +₁ ≈g.from) ∘ (f₂ f +₁ f₂ g) ≈ f₂ f′ +₁ f₂ g′ - from∘f₂≈f₂′ = begin + from∘f₂≈f₂ : (≈f.from +₁ ≈g.from) ∘ (f₂ f +₁ f₂ g) ≈ f₂ f′ +₁ f₂ g′ + from∘f₂≈f₂ = begin (≈f.from +₁ ≈g.from) ∘ (f₂ f +₁ f₂ g) ≈⟨ +₁∘+₁ ⟩ - (≈f.from ∘ f₂ f) +₁ (≈g.from ∘ f₂ g) ≈⟨ +₁-cong₂ (≈f.from∘f₂≈f₂′) (≈g.from∘f₂≈f₂′) ⟩ + (≈f.from ∘ f₂ f) +₁ (≈g.from ∘ f₂ g) ≈⟨ +₁-cong₂ ≈f.from∘f₂≈f₂ ≈g.from∘f₂≈f₂ ⟩ f₂ f′ +₁ f₂ g′ ∎ +private + ⊗′ : Bifunctor Cospans Cospans Cospans + ⊗′ = record + { F₀ = λ (A , A′) → A + A′ + ; F₁ = λ (f , g) → f ⊗ g + ; identity = λ { {x , y} → id⊗id≈id {x} {y} } + ; homomorphism = λ { {_} {_} {_} {A⇒B , A⇒B′} {B⇒C , B⇒C′} → homomorphism A⇒B B⇒C A⇒B′ B⇒C′ } + ; F-resp-≈ = λ (≈f , ≈g) → ⊗-resp-≈ ≈f ≈g + } + ⊗ : Bifunctor Cospans Cospans Cospans -⊗ = record - { F₀ = λ { (A , A′) → A + A′ } - ; F₁ = λ { (f , g) → together f g } - ; identity = λ { {x , y} → id⊗id≈id {x} {y} } - ; homomorphism = λ { {_} {_} {_} {A⇒B , A⇒B′} {B⇒C , B⇒C′} → homomorphism A⇒B B⇒C A⇒B′ B⇒C′ } - ; F-resp-≈ = λ { (≈f , ≈g) → ⊗-resp-≈ ≈f ≈g } - } +⊗ = ⊗′ + +module ⊗ = Functor ⊗ |