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{-# OPTIONS --without-K --safe #-}
open import Level using (Level; levelOfTerm)
open import Categories.Category using (Category)
module Category.Semiadditive {o ℓ e : Level} (𝒞 : Category o ℓ e) where
import Categories.Morphism.Reasoning 𝒞 as ⇒-Reasoning
open import Algebra using (IsCommutativeMonoid; CommutativeMonoid)
open import Categories.Category.CMonoidEnriched using (CM-Category)
open import Categories.Object.Zero 𝒞 using (Zero)
open import Category.BinaryBiproducts 𝒞 using (BinaryBiproducts)
open import Data.Product using (_,_)
private module 𝒞 = Category 𝒞
-- A semiadditive category has all finite biproducts
record Semiadditive : Set (levelOfTerm 𝒞) where
field
zero : Zero
biproducts : BinaryBiproducts
open Zero zero public
open BinaryBiproducts biproducts public
open 𝒞
open HomReasoning
open ⇒-Reasoning
module _ {A B : Obj} where
_+_ _+′_ : A ⇒ B → A ⇒ B → A ⇒ B
f + g = ∇ ∘ f ×₁ g ∘ Δ
f +′ g = ∇ ∘ f +₁ g ∘ Δ
infix 8 _+_
+-cong : {x y u v : A ⇒ B} → x ≈ y → u ≈ v → x + u ≈ y + v
+-cong eq₁ eq₂ = refl⟩∘⟨ ×₁-cong₂ eq₁ eq₂ ⟩∘⟨refl
+-assoc : (x y z : A ⇒ B) → (x + y) + z ≈ x + (y + z)
+-assoc x y z = begin
∇ ∘ (∇ ∘ x ×₁ y ∘ Δ) ×₁ z ∘ Δ ≈⟨ refl⟩∘⟨ pushˡ (Equiv.sym first∘×₁) ⟩
∇ ∘ ∇ ×₁ id ∘ (x ×₁ y ∘ Δ) ×₁ z ∘ Δ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ×₁-cong₂ Equiv.refl identityʳ ⟩∘⟨refl ⟨
∇ ∘ ∇ ×₁ id ∘ (x ×₁ y ∘ Δ) ×₁ (z ∘ id) ∘ Δ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ (Equiv.sym ×₁∘×₁) ⟩
∇ ∘ ∇ ×₁ id ∘ (x ×₁ y) ×₁ z ∘ Δ ×₁ id ∘ Δ ≈⟨ refl⟩∘⟨ ×₁-+₁ ∇ id ⟩∘⟨refl ⟩
∇ ∘ ∇ +₁ id ∘ (x ×₁ y) ×₁ z ∘ Δ ×₁ id ∘ Δ ≈⟨ extendʳ ∇-assoc ⟩
∇ ∘ (id +₁ ∇ ∘ +-assocˡ) ∘ (x ×₁ y) ×₁ z ∘ Δ ×₁ id ∘ Δ ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ assocˡ≈+-assocˡ) ⟩∘⟨refl ⟨
∇ ∘ (id +₁ ∇ ∘ assocˡ) ∘ (x ×₁ y) ×₁ z ∘ Δ ×₁ id ∘ Δ ≈⟨ refl⟩∘⟨ pullʳ (extendʳ assocˡ∘×₁) ⟩
∇ ∘ id +₁ ∇ ∘ x ×₁ (y ×₁ z) ∘ assocˡ ∘ Δ ×₁ id ∘ Δ ≈⟨ refl⟩∘⟨ ×₁-+₁ id ∇ ⟩∘⟨ refl⟩∘⟨ Δ-assoc ⟨
∇ ∘ id ×₁ ∇ ∘ x ×₁ (y ×₁ z) ∘ id ×₁ Δ ∘ Δ ≈⟨ refl⟩∘⟨ pullˡ second∘×₁ ⟩
∇ ∘ x ×₁ (∇ ∘ y ×₁ z) ∘ id ×₁ Δ ∘ Δ ≈⟨ refl⟩∘⟨ pullˡ ×₁∘×₁ ⟩
∇ ∘ (x ∘ id) ×₁ ((∇ ∘ y ×₁ z) ∘ Δ) ∘ Δ ≈⟨ refl⟩∘⟨ ×₁-cong₂ identityʳ assoc ⟩∘⟨refl ⟩
∇ ∘ x ×₁ (∇ ∘ y ×₁ z ∘ Δ) ∘ Δ ∎
+-identityˡ : (x : A ⇒ B) → zero⇒ + x ≈ x
+-identityˡ x = begin
∇ ∘ zero⇒ ×₁ x ∘ Δ ≈⟨ refl⟩∘⟨ ×₁∘Δ ⟩
∇ ∘ ⟨ zero⇒ , x ⟩ ≈⟨ refl⟩∘⟨ ⟨⟩-cong₂ (zero-∘ʳ x) identityˡ ⟨
∇ ∘ ⟨ zero⇒ ∘ x , id ∘ x ⟩ ≈⟨ refl⟩∘⟨ ⟨⟩∘ ⟨
∇ ∘ ⟨ zero⇒ , id ⟩ ∘ x ≈⟨ refl⟩∘⟨ ⟨⟩-congʳ (zero-∘ʳ 𝟎⇐) ⟩∘⟨refl ⟨
∇ ∘ ⟨ zero⇒ {A} ∘ 𝟎⇐ , id ⟩ ∘ x ≈⟨ refl⟩∘⟨ ⟨⟩-cong₂ (π₁i₂-absorbˡ zero⇒) (Equiv.sym π₂∘i₂≈id) ⟩∘⟨refl ⟩
∇ ∘ ⟨ π₁ ∘ i₂ , π₂ ∘ i₂ ⟩ ∘ x ≈⟨ refl⟩∘⟨ g-η ⟩∘⟨refl ⟩
∇ ∘ i₂ ∘ x ≈⟨ cancelˡ ∇-identityˡ ⟩
x ∎
+-identityʳ : (x : A ⇒ B) → x + zero⇒ ≈ x
+-identityʳ x = begin
∇ ∘ x ×₁ zero⇒ ∘ Δ ≈⟨ refl⟩∘⟨ ×₁∘Δ ⟩
∇ ∘ ⟨ x , zero⇒ ⟩ ≈⟨ refl⟩∘⟨ ⟨⟩-cong₂ identityˡ (zero-∘ʳ x) ⟨
∇ ∘ ⟨ id ∘ x , zero⇒ ∘ x ⟩ ≈⟨ refl⟩∘⟨ ⟨⟩∘ ⟨
∇ ∘ ⟨ id , zero⇒ ⟩ ∘ x ≈⟨ refl⟩∘⟨ ⟨⟩-congˡ (zero-∘ʳ 𝟎⇒) ⟩∘⟨refl ⟨
∇ ∘ ⟨ id , zero⇒ {A} ∘ 𝟎⇒ ⟩ ∘ x ≈⟨ refl⟩∘⟨ ⟨⟩-cong₂ (Equiv.sym π₁∘i₁≈id) (π₂i₁-absorbˡ zero⇒) ⟩∘⟨refl ⟩
∇ ∘ ⟨ π₁ ∘ i₁ , π₂ ∘ i₁ ⟩ ∘ x ≈⟨ refl⟩∘⟨ g-η ⟩∘⟨refl ⟩
∇ ∘ i₁ ∘ x ≈⟨ cancelˡ ∇-identityʳ ⟩
x ∎
∇∘+-swap : {A : Obj} → ∇ ∘ +-swap {A} ≈ ∇
∇∘+-swap = begin
∇ ∘ [ i₂ , i₁ ] ≈⟨ ∘[] ⟩
[ ∇ ∘ i₂ , ∇ ∘ i₁ ] ≈⟨ []-cong₂ inject₂ inject₁ ⟩
[ id , id ] ∎
swap∘Δ : {A : Obj} → swap {A} ∘ Δ ≈ Δ
swap∘Δ = begin
⟨ π₂ , π₁ ⟩ ∘ Δ ≈⟨ ⟨⟩∘ ⟩
⟨ π₂ ∘ Δ , π₁ ∘ Δ ⟩ ≈⟨ ⟨⟩-cong₂ project₂ project₁ ⟩
⟨ id , id ⟩ ∎
+-comm : (x y : A ⇒ B) → x + y ≈ y + x
+-comm x y = begin
∇ ∘ x ×₁ y ∘ Δ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ swap∘Δ ⟨
∇ ∘ x ×₁ y ∘ swap ∘ Δ ≈⟨ refl⟩∘⟨ extendʳ swap∘×₁ ⟨
∇ ∘ swap ∘ y ×₁ x ∘ Δ ≈⟨ refl⟩∘⟨ swap≈+-swap ⟩∘⟨refl ⟩
∇ ∘ +-swap ∘ y ×₁ x ∘ Δ ≈⟨ pullˡ ∇∘+-swap ⟩
∇ ∘ y ×₁ x ∘ Δ ∎
isCM : IsCommutativeMonoid (_≈_ {A} {B}) _+_ zero⇒
isCM = record
{ isMonoid = record
{ isSemigroup = record
{ isMagma = record
{ isEquivalence = equiv
; ∙-cong = +-cong
}
; assoc = +-assoc
}
; identity = +-identityˡ , +-identityʳ
}
; comm = +-comm
}
hom : Obj → Obj → CommutativeMonoid ℓ e
hom A B = record
{ Carrier = A ⇒ B
; _≈_ = _≈_
; _∙_ = _+_
; ε = zero⇒
; isCommutativeMonoid = isCM
}
+-resp-∘
: {A B C D : Obj}
{f g : B ⇒ C}
{h : A ⇒ B}
{k : C ⇒ D}
→ k ∘ (f + g) ∘ h ≈ k ∘ f ∘ h + k ∘ g ∘ h
+-resp-∘ {f = f} {g} {h} {k} = begin
k ∘ (∇ ∘ f ×₁ g ∘ Δ) ∘ h ≈⟨ extendʳ (extendʳ ⇒∇) ⟩
∇ ∘ (k +₁ k ∘ f ×₁ g ∘ Δ) ∘ h ≈⟨ refl⟩∘⟨ (×₁-+₁ k k ⟩∘⟨refl) ⟩∘⟨refl ⟨
∇ ∘ (k ×₁ k ∘ f ×₁ g ∘ Δ) ∘ h ≈⟨ refl⟩∘⟨ pullʳ (pullʳ ⇒Δ) ⟩
∇ ∘ k ×₁ k ∘ f ×₁ g ∘ h ×₁ h ∘ Δ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ ×₁∘×₁ ⟩
∇ ∘ k ×₁ k ∘ (f ∘ h) ×₁ (g ∘ h) ∘ Δ ≈⟨ refl⟩∘⟨ pullˡ ×₁∘×₁ ⟩
∇ ∘ (k ∘ f ∘ h) ×₁ (k ∘ g ∘ h) ∘ Δ ∎
0-resp-∘
: {A C D : Obj}
{h : A ⇒ C}
{k : C ⇒ D}
→ k ∘ zero⇒ ∘ h ≈ zero⇒
0-resp-∘ {h = h} {k} = begin
k ∘ zero⇒ ∘ h ≈⟨ pullˡ (zero-∘ˡ k) ⟩
zero⇒ ∘ h ≈⟨ zero-∘ʳ h ⟩
zero⇒ ∎
cm-category : CM-Category o ℓ e
cm-category = record
{ 𝒞
; Hom = hom
; +-resp-∘ = +-resp-∘
; 0-resp-∘ = 0-resp-∘
}
|
