Unordered pairs of types

Content created by Fredrik Bakke, Egbert Rijke and Jonathan Prieto-Cubides.

Created on 2022-05-01.
Last modified on 2024-01-11.

module foundation.unordered-pairs-of-types where
open import foundation.dependent-pair-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.structure-identity-principle
open import foundation.univalence
open import foundation.universe-levels
open import foundation.unordered-pairs

open import foundation-core.equivalences
open import foundation-core.identity-types
open import foundation-core.torsorial-type-families

open import univalent-combinatorics.2-element-types


An unordered pair of types is an unordered pair of elements in a universe


Unordered pairs of types

unordered-pair-types : (l : Level)  UU (lsuc l)
unordered-pair-types l = unordered-pair (UU l)

Equivalences of unordered pairs of types

equiv-unordered-pair-types :
  {l1 l2 : Level} 
  unordered-pair-types l1  unordered-pair-types l2  UU (l1  l2)
equiv-unordered-pair-types A B =
  Σ ( type-unordered-pair A  type-unordered-pair B)
    ( λ e 
      (i : type-unordered-pair A) 
      element-unordered-pair A i  element-unordered-pair B (map-equiv e i))

module _
  {l1 l2 : Level} (A : unordered-pair-types l1) (B : unordered-pair-types l2)
  (e : equiv-unordered-pair-types A B)

  equiv-type-equiv-unordered-pair-types :
    type-unordered-pair A  type-unordered-pair B
  equiv-type-equiv-unordered-pair-types = pr1 e

  map-equiv-type-equiv-unordered-pair-types :
    type-unordered-pair A  type-unordered-pair B
  map-equiv-type-equiv-unordered-pair-types =
    map-equiv equiv-type-equiv-unordered-pair-types

  equiv-element-equiv-unordered-pair-types :
    (i : type-unordered-pair A) 
    ( element-unordered-pair A i) 
    ( element-unordered-pair B (map-equiv-type-equiv-unordered-pair-types i))
  equiv-element-equiv-unordered-pair-types = pr2 e


Extensionality of unordered pairs of types

module _
  {l : Level} (A : unordered-pair-types l)

  id-equiv-unordered-pair-types : equiv-unordered-pair-types A A
  pr1 id-equiv-unordered-pair-types = id-equiv
  pr2 id-equiv-unordered-pair-types i = id-equiv

  equiv-eq-unordered-pair-types :
    (B : unordered-pair-types l)  A  B  equiv-unordered-pair-types A B
  equiv-eq-unordered-pair-types .A refl = id-equiv-unordered-pair-types

  is-torsorial-equiv-unordered-pair-types :
    is-torsorial (equiv-unordered-pair-types A)
  is-torsorial-equiv-unordered-pair-types =
      ( is-torsorial-equiv-2-Element-Type (2-element-type-unordered-pair A))
      ( pair (2-element-type-unordered-pair A) id-equiv)
      ( is-torsorial-equiv-fam (element-unordered-pair A))

  is-equiv-equiv-eq-unordered-pair-types :
    (B : unordered-pair-types l)  is-equiv (equiv-eq-unordered-pair-types B)
  is-equiv-equiv-eq-unordered-pair-types =

  extensionality-unordered-pair-types :
    (B : unordered-pair-types l)  (A  B)  equiv-unordered-pair-types A B
  pr1 (extensionality-unordered-pair-types B) =
    equiv-eq-unordered-pair-types B
  pr2 (extensionality-unordered-pair-types B) =
    is-equiv-equiv-eq-unordered-pair-types B

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