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{-# OPTIONS --without-K --safe #-}
open import Level using (Level; suc; _⊔_)
open import Categories.Category using (Category)
module Category.Dagger.Semiadditive {o ℓ e : Level} (𝒞 : Category o ℓ e) where
import Categories.Morphism.Reasoning 𝒞 as ⇒-Reasoning
open import Categories.Category.Dagger using (HasDagger)
open import Category.Semiadditive using (Semiadditive)
open import Relation.Binary using (Rel)
record SemiadditiveDagger : Set (suc (o ⊔ ℓ ⊔ e)) where
field
semiadditive : Semiadditive 𝒞
dagger : HasDagger 𝒞
open Category 𝒞
open HasDagger dagger public
open Semiadditive semiadditive public
field
π₁† : {A B : Obj} → π₁ {A} {B} † ≈ i₁
π₂† : {A B : Obj} → π₂ {A} {B} † ≈ i₂
⟨⟩-† : {A B C : Obj} {f : A ⇒ B} {g : A ⇒ C} → ⟨ f , g ⟩ † ≈ [ f † , g † ]
open HomReasoning
open ⇒-Reasoning
Δ† : {A : Obj} → Δ {A} † ≈ ∇
Δ† = begin
⟨ id , id ⟩ † ≈⟨ ⟨⟩-† ⟩
[ id † , id † ] ≈⟨ []-cong₂ †-identity †-identity ⟩
[ id , id ] ∎
∇† : {A : Obj} → ∇ {A} † ≈ Δ
∇† = begin
∇ † ≈⟨ ⟨ Δ† ⟩† ⟨
Δ † † ≈⟨ †-involutive Δ ⟩
Δ ∎
†-resp-×₁ : {A B C D : Obj} {f : A ⇒ B} {g : C ⇒ D} → (f ×₁ g) † ≈ (f †) ×₁ (g †)
†-resp-×₁ {f = f} {g} = begin
⟨ f ∘ π₁ , g ∘ π₂ ⟩ † ≈⟨ ⟨⟩-† ⟩
[ (f ∘ π₁) † , (g ∘ π₂) † ] ≈⟨ []-cong₂ †-homomorphism †-homomorphism ⟩
[ π₁ † ∘ f † , π₂ † ∘ g † ] ≈⟨ []-cong₂ (π₁† ⟩∘⟨refl) (π₂† ⟩∘⟨refl) ⟩
[ i₁ ∘ f † , i₂ ∘ g † ] ≈⟨ ×₁-+₁ (f †) (g †) ⟨
⟨ f † ∘ π₁ , g † ∘ π₂ ⟩ ∎
+-congˡ : {A B : Obj} {f g h : A ⇒ B} → g ≈ h → f + g ≈ f + h
+-congˡ g≈h = +-cong Equiv.refl g≈h
+-congʳ : {A B : Obj} {f g h : A ⇒ B} → f ≈ g → f + h ≈ g + h
+-congʳ f≈g = +-cong f≈g Equiv.refl
+-† : {A B : Obj} {f g : A ⇒ B} → (f + g) † ≈ (f †) + (g †)
+-† {f = f} {g} = begin
(∇ ∘ f ×₁ g ∘ Δ) † ≈⟨ †-homomorphism ⟩
(f ×₁ g ∘ Δ) † ∘ ∇ † ≈⟨ pushˡ †-homomorphism ⟩
Δ † ∘ (f ×₁ g) † ∘ ∇ † ≈⟨ Δ† ⟩∘⟨ †-resp-×₁ ⟩∘⟨ ∇† ⟩
∇ ∘ (f †) ×₁ (g †) ∘ Δ ∎
-- bilinearity of composition
∘-distribˡ : {A B C : Obj} {f : B ⇒ C} {g h : A ⇒ B} → f ∘ (g + h) ≈ f ∘ g + f ∘ h
∘-distribˡ {f = f} {g} {h} = begin
f ∘ (g + h) ≈⟨ refl⟩∘⟨ identityʳ ⟨
f ∘ (g + h) ∘ id ≈⟨ +-resp-∘ ⟩
f ∘ g ∘ id + f ∘ h ∘ id ≈⟨ +-cong (refl⟩∘⟨ identityʳ) (refl⟩∘⟨ identityʳ) ⟩
f ∘ g + f ∘ h ∎
∘-distribʳ : {A B C : Obj} {f g : B ⇒ C} {h : A ⇒ B} → (f + g) ∘ h ≈ f ∘ h + g ∘ h
∘-distribʳ {f = f} {g} {h} = begin
(f + g) ∘ h ≈⟨ pushˡ (Equiv.sym identityˡ) ⟩
id ∘ (f + g) ∘ h ≈⟨ +-resp-∘ ⟩
id ∘ f ∘ h + id ∘ g ∘ h ≈⟨ +-cong (pullˡ identityˡ) (pullˡ identityˡ) ⟩
f ∘ h + g ∘ h ∎
record IdempotentSemiadditiveDagger : Set (suc (o ⊔ ℓ ⊔ e)) where
field
semiadditiveDagger : SemiadditiveDagger
open SemiadditiveDagger semiadditiveDagger public
open Category 𝒞
open HomReasoning
open ⇒-Reasoning
field
idempotent : {A B : Obj} {f : A ⇒ B} → f + f ≈ f
_≤_ : {A B : Obj} → Rel (A ⇒ B) e
_≤_ {A} {B} f g = f + g ≈ g
≤-refl : {A B : Obj} {f : A ⇒ B} → f ≤ f
≤-refl = idempotent
≤-antisym : {A B : Obj} {f g : A ⇒ B} → f ≤ g → g ≤ f → f ≈ g
≤-antisym {A} {B} {f} {g} f≤g g≤f = begin
f ≈⟨ g≤f ⟨
g + f ≈⟨ +-comm g f ⟩
f + g ≈⟨ f≤g ⟩
g ∎
≤-trans : {A B : Obj} {f g h : A ⇒ B} → f ≤ g → g ≤ h → f ≤ h
≤-trans {A} {B} {f} {g} {h} f≤g g≤h = begin
f + h ≈⟨ refl⟩∘⟨ ×₁-congˡ g≤h ⟩∘⟨refl ⟨
f + (g + h) ≈⟨ +-assoc f g h ⟨
(f + g) + h ≈⟨ refl⟩∘⟨ ×₁-congʳ f≤g ⟩∘⟨refl ⟩
g + h ≈⟨ g≤h ⟩
h ∎
≤-resp-+
: {A B : Obj}
{f g h i : A ⇒ B}
→ f ≤ h
→ g ≤ i
→ (f + g) ≤ (h + i)
≤-resp-+ {f = f} {g} {h} {i} f≤h g≤i = begin
(f + g) + (h + i) ≈⟨ +-assoc f g (h + i) ⟩
f + (g + (h + i)) ≈⟨ +-congˡ (+-assoc g h i) ⟨
f + ((g + h) + i) ≈⟨ +-congˡ (+-congʳ (+-comm g h)) ⟩
f + ((h + g) + i) ≈⟨ +-congˡ (+-assoc h g i) ⟩
f + (h + (g + i)) ≈⟨ +-assoc f h (g + i) ⟨
(f + h) + (g + i) ≈⟨ +-cong f≤h g≤i ⟩
h + i ∎
≤-resp-∘
: {A B C : Obj}
{f h : B ⇒ C}
{g i : A ⇒ B}
→ f ≤ h
→ g ≤ i
→ (f ∘ g) ≤ (h ∘ i)
≤-resp-∘ {f = f} {h} {g} {i} f≤h g≤i = begin
f ∘ g + (h ∘ i) ≈⟨ +-congˡ (f≤h ⟩∘⟨refl) ⟨
f ∘ g + ((f + h) ∘ i) ≈⟨ +-congˡ ∘-distribʳ ⟩
f ∘ g + (f ∘ i + h ∘ i) ≈⟨ +-assoc (f ∘ g) (f ∘ i) (h ∘ i) ⟨
(f ∘ g + f ∘ i) + h ∘ i ≈⟨ +-congʳ ∘-distribˡ ⟨
f ∘ (g + i) + h ∘ i ≈⟨ +-congʳ (refl⟩∘⟨ g≤i) ⟩
f ∘ i + h ∘ i ≈⟨ ∘-distribʳ ⟨
(f + h) ∘ i ≈⟨ f≤h ⟩∘⟨refl ⟩
h ∘ i ∎
†-resp-≤ : {A B : Obj} {f g : A ⇒ B} → f ≤ g → (f †) ≤ (g †)
†-resp-≤ {A} {B} {f} {g} f≤g = begin
(f †) + (g †) ≈⟨ +-† ⟨
(f + g) † ≈⟨ ⟨ f≤g ⟩† ⟩
g † ∎
-- special law
∇∘Δ : {A : Obj} → ∇ ∘ Δ ≈ id {A}
∇∘Δ = begin
∇ ∘ Δ ≈⟨ refl⟩∘⟨ introˡ id×₁id ⟩
∇ ∘ id ×₁ id ∘ Δ ≈⟨ idempotent ⟩
id ∎
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