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Define the category of posets #1232

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Merged
maxsnew merged 1 commit into from
Jul 25, 2025
Merged

Define the category of posets #1232

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25 changes: 25 additions & 0 deletions Cubical/Categories/Instances/Posets.agda
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Original file line number Diff line number Diff line change
@@ -0,0 +1,25 @@
{-# OPTIONS --safe #-}
module Cubical.Categories.Instances.Posets where

open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function as Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Structure

open import Cubical.Relation.Binary.Order.Poset
open import Cubical.Relation.Binary.Order.Poset.Mappings

open import Cubical.Categories.Category

open Category hiding (_∘_)

PosetsCategory : ∀ ℓ ℓ' → Category (ℓ-max (ℓ-suc ℓ) (ℓ-suc ℓ')) (ℓ-max ℓ ℓ')
PosetsCategory ℓ ℓ' .ob = Poset ℓ ℓ'
PosetsCategory ℓ ℓ' .Hom[_,_] P Q = Σ[ f ∈ (⟨ P ⟩ → ⟨ Q ⟩) ] IsIsotone (str P) f (str Q)
PosetsCategory ℓ ℓ' .id = idfun _ , isisotone λ _ _ → idfun _
PosetsCategory ℓ ℓ' ._⋆_ (f , fMon) (g , gMon) = g ∘ f , IsIsotone-∘ _ _ _ _ _ fMon gMon
PosetsCategory ℓ ℓ' .⋆IdL _ = refl
PosetsCategory ℓ ℓ' .⋆IdR _ = refl
PosetsCategory ℓ ℓ' .⋆Assoc _ _ _ = refl
PosetsCategory ℓ ℓ' .isSetHom {x = P} {y = Q} = isSetΣSndProp (isSet→ Q.is-set) λ _ → isPropIsIsotone _ _ _
where module Q = PosetStr (str Q)
4 changes: 2 additions & 2 deletions Cubical/Relation/Binary/Order/Poset/Mappings.agda
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Expand Up @@ -38,6 +38,7 @@ record IsIsotone {A : Type ℓ₀} {B : Type ℓ₁}
(M : PosetStr ℓ₀' A) (f : A → B) (N : PosetStr ℓ₁' B)
: Type (ℓ-max (ℓ-max ℓ₀ ℓ₀') ℓ₁')
where
constructor isisotone
-- Shorter qualified names
private
module M = PosetStr M
Expand Down Expand Up @@ -67,8 +68,7 @@ record IsAntitone {A : Type ℓ₀} {B : Type ℓ₁}
(M : PosetStr ℓ₀' A) (f : A → B) (N : PosetStr ℓ₁' B)
: Type (ℓ-max (ℓ-max ℓ₀ ℓ₀') ℓ₁')
where
constructor
isantitone
constructor isantitone
-- Shorter qualified names
private
module M = PosetStr M
Expand Down