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{-# OPTIONS --without-K --safe #-}
open import Categories.Category using (Category)
open import Category.Dagger.Semiadditive using (IdempotentSemiadditiveDagger)
open import Level using (Level; suc; _⊔_)
module Category.Dagger.2-Poset {o ℓ e : Level} where
import Relation.Binary.Reasoning.PartialOrder as ≤-Reasoning
open import Category.Monoidal.Instance.Posets {ℓ} {e} {e} using (Posets-Monoidal)
open import Categories.Category.Dagger using (HasDagger)
open import Categories.Category.Helper using (categoryHelper)
open import Categories.Category.Instance.Posets using (Posets)
open import Categories.Enriched.Category Posets-Monoidal using () renaming (Category to 2-Poset)
open import Data.Product using (_,_)
open import Data.Unit.Polymorphic using (tt)
open import Relation.Binary using (Poset)
open import Relation.Binary.Morphism.Bundles using (PosetHomomorphism; mkPosetHomo)
open PosetHomomorphism using (⟦_⟧; cong; mono)
record Dagger-2-Poset : Set (suc (o ⊔ ℓ ⊔ e)) where
open Poset using (Carrier; _≈_; isEquivalence)
field
2-poset : 2-Poset o
open 2-Poset 2-poset hiding (id) public
open 2-Poset 2-poset using (id)
category : Category o ℓ e
category = categoryHelper record
{ Obj = Obj
; _⇒_ = λ A B → Carrier (hom A B)
; _≈_ = λ {A B} → _≈_ (hom A B)
; id = ⟦ id ⟧ tt
; _∘_ = λ f g → ⟦ ⊚ ⟧ (f , g)
; assoc = ⊚-assoc
; identityˡ = unitˡ
; identityʳ = unitʳ
; equiv = λ {A B} → isEquivalence (hom A B)
; ∘-resp-≈ = λ f≈h g≈i → cong ⊚ (f≈h , g≈i)
}
field
hasDagger : HasDagger category
private
module P {A B : Obj} = Poset (hom A B)
open P using (_≤_) public
open Category category hiding (Obj) public
open HasDagger hasDagger public
field
†-resp-≤ : {A B : Obj} {f g : A ⇒ B} → f ≤ g → f † ≤ g †
dagger-2-poset : {𝒞 : Category o ℓ e} (ISA† : IdempotentSemiadditiveDagger 𝒞) → Dagger-2-Poset
dagger-2-poset {𝒞} ISA† = record
{ 2-poset = record
{ Obj = Obj
; hom = λ A B → record
{ Carrier = A ⇒ B
; _≈_ = _≈_
; _≤_ = ISA†._≤_
; isPartialOrder = record
{ isPreorder = record
{ isEquivalence = equiv
; reflexive = λ x≈y → Equiv.trans (ISA†.+-congʳ x≈y) ISA†.≤-refl
; trans = ISA†.≤-trans
}
; antisym = ISA†.≤-antisym
}
}
; id = mkPosetHomo _ _ (λ _ → id) (λ _ → ISA†.≤-refl)
; ⊚ = mkPosetHomo _ _ (λ (f , g) → f ∘ g) (λ (≤₁ , ≤₂) → ISA†.≤-resp-∘ ≤₁ ≤₂)
; ⊚-assoc = assoc
; unitˡ = identityˡ
; unitʳ = identityʳ
}
; hasDagger = record
{ _† = ISA†._†
; †-identity = ISA†.†-identity
; †-homomorphism = ISA†.†-homomorphism
; †-resp-≈ = ISA†.⟨_⟩†
; †-involutive = ISA†.†-involutive
}
; †-resp-≤ = ISA†.†-resp-≤
}
where
module ISA† = IdempotentSemiadditiveDagger ISA†
open Category 𝒞
module _ (S : Dagger-2-Poset) where
open Dagger-2-Poset S
record IsMap {A B : Obj} (f : A ⇒ B) : Set e where
field
functional : f ∘ f † ≤ id
entire : id ≤ f † ∘ f
record Map (A B : Obj) : Set (ℓ ⊔ e) where
field
map : A ⇒ B
isMap : IsMap map
open IsMap isMap public
idMap : {A : Obj} → Map A A
idMap {A} = record
{ map = id
; isMap = record
{ functional = begin
id ∘ id † ≈⟨ identityˡ ⟩
id † ≈⟨ †-identity ⟩
id ∎
; entire = begin
id ≈⟨ †-identity ⟨
id † ≈⟨ identityʳ ⟨
id † ∘ id ∎
}
}
where
open ≤-Reasoning (hom A A)
_∘-map_ : {A B C : Obj} → Map B C → Map A B → Map A C
_∘-map_ {A} {B} {C} g f = record
{ map = g.map ∘ f.map
; isMap = record
{ functional = func
; entire = ent
}
}
where
module g = Map g
module f = Map f
func : (g.map ∘ f.map) ∘ (g.map ∘ f.map) † ≤ id
func = begin
(g.map ∘ f.map) ∘ (g.map ∘ f.map) † ≈⟨ refl⟩∘⟨ †-homomorphism ⟩
(g.map ∘ f.map) ∘ f.map † ∘ g.map † ≈⟨ assoc ⟩
g.map ∘ f.map ∘ f.map † ∘ g.map † ≈⟨ refl⟩∘⟨ assoc ⟨
g.map ∘ (f.map ∘ f.map †) ∘ g.map † ≤⟨ mono ⊚ (Poset.refl (hom B C) , mono ⊚ (f.functional , Poset.refl (hom C B))) ⟩
g.map ∘ id ∘ g.map † ≈⟨ refl⟩∘⟨ identityˡ ⟩
g.map ∘ g.map † ≤⟨ g.functional ⟩
id ∎
where
open ≤-Reasoning (hom C C)
open HomReasoning using (refl⟩∘⟨_)
open Poset (hom C C)
ent : id ≤ (g.map ∘ f.map) † ∘ g.map ∘ f.map
ent = begin
id ≤⟨ f.entire ⟩
f.map † ∘ f.map ≈⟨ refl⟩∘⟨ identityˡ ⟨
f.map † ∘ id ∘ f.map ≤⟨ mono ⊚ (Poset.refl (hom B A) , mono ⊚ (g.entire , Poset.refl (hom A B))) ⟩
f.map † ∘ (g.map † ∘ g.map) ∘ f.map ≈⟨ refl⟩∘⟨ assoc ⟩
f.map † ∘ g.map † ∘ g.map ∘ f.map ≈⟨ assoc ⟨
(f.map † ∘ g.map †) ∘ g.map ∘ f.map ≈⟨ †-homomorphism ⟩∘⟨refl ⟨
(g.map ∘ f.map) † ∘ g.map ∘ f.map ∎
where
open ≤-Reasoning (hom A A)
open HomReasoning using (refl⟩∘⟨_; _⟩∘⟨refl)
open Poset (hom A A)
infixr 9 _∘-map_
open Map
Maps : Category o (ℓ ⊔ e) e
Maps = categoryHelper record
{ Obj = Obj
; _⇒_ = Map
; _≈_ = λ a b → map a ≈ map b
; id = idMap
; _∘_ = _∘-map_
; assoc = assoc
; identityˡ = identityˡ
; identityʳ = identityʳ
; equiv = record
{ refl = Equiv.refl
; sym = Equiv.sym
; trans = Equiv.trans
}
; ∘-resp-≈ = ∘-resp-≈
}
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