Preunivalent type families

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2024-01-31.
Last modified on 2024-04-11.

module foundation.preunivalent-type-families where
open import foundation.0-maps
open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.equivalence-injective-type-families
open import foundation.faithful-maps
open import foundation.function-types
open import foundation.injective-maps
open import foundation.preunivalence
open import foundation.retractions
open import foundation.sets
open import foundation.subuniverses
open import foundation.transport-along-identifications
open import foundation.universe-levels

open import foundation-core.equivalences
open import foundation-core.identity-types
open import foundation-core.univalence


A type family B over A is said to be preunivalent if the map

  equiv-tr B : x = y → B x ≃ B y

is an embedding for every x y : A.


is-preunivalent :
  {l1 l2 : Level} {A : UU l1}  (A  UU l2)  UU (l1  l2)
is-preunivalent {A = A} B = (x y : A)  is-emb  (p : x  y)  equiv-tr B p)


The preunivalence axiom states that the identity family id : 𝒰 → 𝒰 is preunivalent

is-preunivalent-UU :
  (l : Level)  is-preunivalent (id {A = UU l})
is-preunivalent-UU l = preunivalence

Assuming the preunivalence axiom, type families are preunivalent if and only if they are faithful as maps

Proof: We have the commuting triangle of maps

                ap B
       (x = y) -----> (B x = B y)
           \               /
            \             /
  equiv-tr B \           / equiv-eq
              v         v
              (B x ≃ B y)

where the right edge is an embedding by the preunivalence axiom. Hence, the top map is an embedding if and only if the left map is.

module _
  {l1 l2 : Level} {A : UU l1} {B : A  UU l2}

    is-faithful-is-preunivalent :
      is-preunivalent B  is-faithful B
    is-faithful-is-preunivalent U x y =
        ( equiv-tr B)
        ( equiv-eq)
        ( ap B)
        ( λ where refl  refl)
        ( preunivalence (B x) (B y))
        ( U x y)

    is-preunivalent-is-faithful :
      is-faithful B  is-preunivalent B
    is-preunivalent-is-faithful is-faithful-B x y =
        ( equiv-tr B)
        ( equiv-eq)
        ( ap B)
        ( λ where refl  refl)
        ( preunivalence (B x) (B y))
        ( is-faithful-B x y)

    is-0-map-is-preunivalent :
      is-preunivalent B  is-0-map B
    is-0-map-is-preunivalent U =
      is-0-map-is-faithful (is-faithful-is-preunivalent U)

    is-preunivalent-is-0-map :
      is-0-map B  is-preunivalent B
    is-preunivalent-is-0-map H =
      is-preunivalent-is-faithful (is-faithful-is-0-map H)

Families of sets are preunivalent if equiv-tr is fiberwise injective

module _
  {l1 l2 : Level} {A : UU l1} (B : A  UU l2)
  (is-set-B : (x : A)  is-set (B x))

  is-preunivalent-is-injective-equiv-tr-is-set :
    ((x y : A)  is-injective (equiv-tr B {x} {y})) 
    is-preunivalent B
  is-preunivalent-is-injective-equiv-tr-is-set is-inj-B x y =
      ( is-set-equiv-is-set (is-set-B x) (is-set-B y))
      ( is-inj-B x y)

  is-preunivalent-retraction-equiv-tr-is-set :
    ((x y : A)  retraction (equiv-tr B {x} {y})) 
    is-preunivalent B
  is-preunivalent-retraction-equiv-tr-is-set R =
      ( λ x y  is-injective-retraction (equiv-tr B) (R x y))

module _
  {l1 l2 : Level} {A : UU l1} (B : A  Set l2)

  is-preunivalent-is-injective-equiv-tr-Set :
    ((x y : A)  is-injective (equiv-tr (type-Set  B) {x} {y})) 
    is-preunivalent (type-Set  B)
  is-preunivalent-is-injective-equiv-tr-Set =
      ( type-Set  B)
      ( is-set-type-Set  B)

  is-preunivalent-retraction-equiv-tr-Set :
    ((x y : A)  retraction (equiv-tr (type-Set  B) {x} {y})) 
    is-preunivalent (type-Set  B)
  is-preunivalent-retraction-equiv-tr-Set =
      ( type-Set  B)
      ( is-set-type-Set  B)

Inclusions of subuniverses into the universe are preunivalent

Note. These proofs rely on essential use of the preunivalence axiom.

is-preunivalent-projection-Type-With-Set-Element :
  {l1 l2 : Level} (S : UU l1  Set l2) 
  is-preunivalent (pr1 {A = UU l1} {B = type-Set  S})
is-preunivalent-projection-Type-With-Set-Element S =
  is-preunivalent-is-0-map (is-0-map-pr1 (is-set-type-Set  S))

is-preunivalent-inclusion-subuniverse :
  {l1 l2 : Level} (S : subuniverse l1 l2) 
  is-preunivalent (inclusion-subuniverse S)
is-preunivalent-inclusion-subuniverse S =
  is-preunivalent-projection-Type-With-Set-Element (set-Prop  S)

See also

Recent changes