Large posets

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Julian KG, fernabnor, Gregor Perčič and louismntnu.

Created on 2022-04-21.
Last modified on 2023-09-26.

module order-theory.large-posets where
open import category-theory.isomorphisms-in-large-categories
open import category-theory.isomorphisms-in-precategories
open import category-theory.large-categories
open import category-theory.large-precategories
open import category-theory.precategories

open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.large-binary-relations
open import foundation.propositions
open import foundation.sets
open import foundation.universe-levels

open import order-theory.large-preorders
open import order-theory.posets
open import order-theory.preorders


A large poset is a large preorder such that the restriction of the ordering relation to any particular universe level is antisymmetric.


Large posets

  Large-Poset (α : Level  Level) (β : Level  Level  Level) : UUω where
    large-preorder-Large-Poset : Large-Preorder α β
    antisymmetric-leq-Large-Poset :
        ( type-Large-Preorder large-preorder-Large-Poset)
        ( leq-Large-Preorder large-preorder-Large-Poset)

open Large-Poset public

module _
  {α : Level  Level} {β : Level  Level  Level} (X : Large-Poset α β)

  type-Large-Poset : (l : Level)  UU (α l)
  type-Large-Poset = type-Large-Preorder (large-preorder-Large-Poset X)

  leq-prop-Large-Poset : Large-Relation-Prop α β (type-Large-Poset)
  leq-prop-Large-Poset = leq-prop-Large-Preorder (large-preorder-Large-Poset X)

  leq-Large-Poset : Large-Relation α β (type-Large-Poset)
  leq-Large-Poset = leq-Large-Preorder (large-preorder-Large-Poset X)

  is-prop-leq-Large-Poset :
    is-prop-Large-Relation (type-Large-Poset) (leq-Large-Poset)
  is-prop-leq-Large-Poset =
    is-prop-leq-Large-Preorder (large-preorder-Large-Poset X)

  leq-eq-Large-Poset :
    {l1 : Level} {x y : type-Large-Poset l1} 
    (x  y)  leq-Large-Poset x y
  leq-eq-Large-Poset =
    leq-eq-Large-Preorder (large-preorder-Large-Poset X)

  refl-leq-Large-Poset :
    is-reflexive-Large-Relation type-Large-Poset leq-Large-Poset
  refl-leq-Large-Poset =
    refl-leq-Large-Preorder (large-preorder-Large-Poset X)

  transitive-leq-Large-Poset :
    is-transitive-Large-Relation type-Large-Poset leq-Large-Poset
  transitive-leq-Large-Poset =
    transitive-leq-Large-Preorder (large-preorder-Large-Poset X)

The predicate on a large category of being a large poset

A large category is said to be a large poset if hom X Y is a proposition for every two objects X and Y.

Lemma. Any large category of which the hom-sets are propositions is a large poset.

Proof: The condition that C is a large poset immediately gives us a large precategory. The interesting part of the claim is therefore that this large preorder is antisymmetric.

Consider two objects X and Y in C, and two morphisms f : X → Y and g : Y → X. Since there is at most one morphism between any two objects, it follows immediately that f ∘ g = id and g ∘ f = id. Therefore we have an isomorphism f : X ≅ Y. Since C was assumed to be a category, this implies that X = Y.

module _
  {α : Level  Level} {β : Level  Level  Level}
  (C : Large-Category α β)

  is-large-poset-Large-Category : UUω
  is-large-poset-Large-Category =
    is-large-preorder-Large-Precategory (large-precategory-Large-Category C)

  is-antisymmetric-is-large-poset-Large-Category :
      ( obj-Large-Category C)
      ( hom-Large-Category C)
  is-antisymmetric-is-large-poset-Large-Category H X Y f g =
    eq-iso-Large-Category C X Y
      ( f , g , eq-is-prop (H Y Y) , eq-is-prop (H X X))

  large-poset-Large-Category :
    is-large-poset-Large-Category  Large-Poset α β
  large-preorder-Large-Poset (large-poset-Large-Category H) =
    large-preorder-Large-Precategory (large-precategory-Large-Category C) H
  antisymmetric-leq-Large-Poset (large-poset-Large-Category H) =
    is-antisymmetric-is-large-poset-Large-Category H


Small posets from large posets

module _
  {α : Level  Level} {β : Level  Level  Level} (X : Large-Poset α β)

  preorder-Large-Poset : (l : Level)  Preorder (α l) (β l l)
  preorder-Large-Poset = preorder-Large-Preorder (large-preorder-Large-Poset X)

  poset-Large-Poset : (l : Level)  Poset (α l) (β l l)
  pr1 (poset-Large-Poset l) = preorder-Large-Poset l
  pr2 (poset-Large-Poset l) = antisymmetric-leq-Large-Poset X

  set-Large-Poset : (l : Level)  Set (α l)
  set-Large-Poset l = set-Poset (poset-Large-Poset l)

  is-set-type-Large-Poset : {l : Level}  is-set (type-Large-Poset X l)
  is-set-type-Large-Poset {l} = is-set-type-Poset (poset-Large-Poset l)

Large posets are large categories

module _
  {α : Level  Level} {β : Level  Level  Level} (P : Large-Poset α β)

  large-precategory-Large-Poset : Large-Precategory α β
  large-precategory-Large-Poset =
    large-precategory-Large-Preorder (large-preorder-Large-Poset P)

  precategory-Large-Poset : (l : Level)  Precategory (α l) (β l l)
  precategory-Large-Poset =
    precategory-Large-Precategory large-precategory-Large-Poset

  is-large-category-Large-Poset :
    is-large-category-Large-Precategory large-precategory-Large-Poset
  is-large-category-Large-Poset {l} x y =
      ( is-set-type-Large-Poset P x y)
      ( is-prop-iso-Precategory
        ( precategory-Large-Poset l)
        ( is-prop-leq-Large-Poset P x y))
      ( λ f 
        antisymmetric-leq-Large-Poset P x y
        ( hom-iso-Precategory (precategory-Large-Poset l) f)
        ( hom-inv-iso-Precategory (precategory-Large-Poset l) f))

  large-category-Large-Poset : Large-Category α β
  large-precategory-Large-Category large-category-Large-Poset =
  is-large-category-Large-Category large-category-Large-Poset =

  is-large-poset-large-category-Large-Poset :
    is-large-poset-Large-Category large-category-Large-Poset
  is-large-poset-large-category-Large-Poset =
    is-prop-leq-Large-Poset P

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