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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 Categories.Category.Monoidal.Reasoning 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)
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
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 using (_⇒_) public
open HasDagger hasDagger using (_†) 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
open ⊗-Reasoning ISA†.+-monoidal
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