Global subuniverses

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-11-24.
Last modified on 2024-08-18.

module foundation.global-subuniverses where
Imports 
open import foundation.dependent-pair-types
open import foundation.iterated-dependent-product-types
open import foundation.subuniverses
open import foundation.universe-levels

open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.identity-types
open import foundation-core.propositions

Idea

Global subuniverses are subtypes of the large universe that are defined at every level, and are closed under equivalences of types. This does not follow from univalence, as equivalence induction only holds for homogeneous equivalences, i.e. equivalences in a single universe.

Definitions

The predicate on families of subuniverses of being closed under equivalences

module _
  (α : Level → Level) (P : (l : Level) → subuniverse l (α l))
  (l1 l2 : Level)
  where

  is-closed-under-equiv-subuniverses : UU (α l1 ⊔ α l2 ⊔ lsuc l1 ⊔ lsuc l2)
  is-closed-under-equiv-subuniverses =
    (X : UU l1) (Y : UU l2) → X ≃ Y → type-Prop (P l1 X) → type-Prop (P l2 Y)

  is-prop-is-closed-under-equiv-subuniverses :
    is-prop is-closed-under-equiv-subuniverses
  is-prop-is-closed-under-equiv-subuniverses =
    is-prop-iterated-Π 4 (λ X Y e x → is-prop-type-Prop (P l2 Y))

  is-closed-under-equiv-subuniverses-Prop :
    Prop (α l1 ⊔ α l2 ⊔ lsuc l1 ⊔ lsuc l2)
  pr1 is-closed-under-equiv-subuniverses-Prop =
    is-closed-under-equiv-subuniverses
  pr2 is-closed-under-equiv-subuniverses-Prop =
    is-prop-is-closed-under-equiv-subuniverses

The global type of global subuniverses

record global-subuniverse (α : Level → Level) : UUω where
  field
    subuniverse-global-subuniverse :
      (l : Level) → subuniverse l (α l)

    is-closed-under-equiv-global-subuniverse :
      (l1 l2 : Level) →
      is-closed-under-equiv-subuniverses α subuniverse-global-subuniverse l1 l2

  is-in-global-subuniverse : {l : Level} → UU l → UU (α l)
  is-in-global-subuniverse {l} X =
    is-in-subuniverse (subuniverse-global-subuniverse l) X

  is-prop-is-in-global-subuniverse :
    {l : Level} (X : UU l) → is-prop (is-in-global-subuniverse X)
  is-prop-is-in-global-subuniverse {l} X =
    is-prop-type-Prop (subuniverse-global-subuniverse l X)

  type-global-subuniverse : (l : Level) → UU (α l ⊔ lsuc l)
  type-global-subuniverse l =
    type-subuniverse (subuniverse-global-subuniverse l)

  inclusion-global-subuniverse :
    {l : Level} → type-global-subuniverse l → UU l
  inclusion-global-subuniverse {l} =
    inclusion-subuniverse (subuniverse-global-subuniverse l)

open global-subuniverse public

The predicate of essentially being in a subuniverse

We say a type is essentially in a global universe at universe level l if it is essentially in the subuniverse at level l.

module _
  {α : Level → Level} (P : global-subuniverse α)
  {l1 : Level} (l2 : Level) (X : UU l1)
  where

  is-essentially-in-global-subuniverse : UU (α l2 ⊔ l1 ⊔ lsuc l2)
  is-essentially-in-global-subuniverse =
    is-essentially-in-subuniverse (subuniverse-global-subuniverse P l2) X

  is-prop-is-essentially-in-global-subuniverse :
    is-prop is-essentially-in-global-subuniverse
  is-prop-is-essentially-in-global-subuniverse =
    is-prop-is-essentially-in-subuniverse
      ( subuniverse-global-subuniverse P l2) X

  is-essentially-in-global-subuniverse-Prop : Prop (α l2 ⊔ l1 ⊔ lsuc l2)
  is-essentially-in-global-subuniverse-Prop =
    is-essentially-in-subuniverse-Prop (subuniverse-global-subuniverse P l2) X

Properties

Global subuniverses are closed under homogenous equivalences

This is true for any family of subuniverses indexed by universe levels.

module _
  {α : Level → Level} (P : global-subuniverse α)
  {l : Level} {X Y : UU l}
  where

  is-in-global-subuniverse-homogenous-equiv :
    X ≃ Y → is-in-global-subuniverse P X → is-in-global-subuniverse P Y
  is-in-global-subuniverse-homogenous-equiv =
    is-in-subuniverse-equiv (subuniverse-global-subuniverse P l)

  is-in-global-subuniverse-homogenous-equiv' :
    X ≃ Y → is-in-global-subuniverse P Y → is-in-global-subuniverse P X
  is-in-global-subuniverse-homogenous-equiv' =
    is-in-subuniverse-equiv' (subuniverse-global-subuniverse P l)

Characterization of the identity type of global subuniverses at a universe level

module _
  {α : Level → Level} {l : Level} (P : global-subuniverse α)
  where

  equiv-global-subuniverse-Level : (X Y : type-global-subuniverse P l) → UU l
  equiv-global-subuniverse-Level =
    equiv-subuniverse (subuniverse-global-subuniverse P l)

  equiv-eq-global-subuniverse-Level :
    (X Y : type-global-subuniverse P l) →
    X ＝ Y → equiv-global-subuniverse-Level X Y
  equiv-eq-global-subuniverse-Level =
    equiv-eq-subuniverse (subuniverse-global-subuniverse P l)

  abstract
    is-equiv-equiv-eq-global-subuniverse-Level :
      (X Y : type-global-subuniverse P l) →
      is-equiv (equiv-eq-global-subuniverse-Level X Y)
    is-equiv-equiv-eq-global-subuniverse-Level =
      is-equiv-equiv-eq-subuniverse (subuniverse-global-subuniverse P l)

  extensionality-global-subuniverse-Level :
    (X Y : type-global-subuniverse P l) →
    (X ＝ Y) ≃ equiv-global-subuniverse-Level X Y
  extensionality-global-subuniverse-Level =
    extensionality-subuniverse (subuniverse-global-subuniverse P l)

  eq-equiv-global-subuniverse-Level :
    {X Y : type-global-subuniverse P l} →
    equiv-global-subuniverse-Level X Y → X ＝ Y
  eq-equiv-global-subuniverse-Level =
    eq-equiv-subuniverse (subuniverse-global-subuniverse P l)

  compute-eq-equiv-id-equiv-global-subuniverse-Level :
    {X : type-global-subuniverse P l} →
    eq-equiv-global-subuniverse-Level {X} {X} (id-equiv {A = pr1 X}) ＝ refl
  compute-eq-equiv-id-equiv-global-subuniverse-Level =
    compute-eq-equiv-id-equiv-subuniverse (subuniverse-global-subuniverse P l)

Equivalences of families of types in a global subuniverse

fam-global-subuniverse :
  {α : Level → Level} (P : global-subuniverse α)
  {l1 : Level} (l2 : Level) (X : UU l1) → UU (α l2 ⊔ l1 ⊔ lsuc l2)
fam-global-subuniverse P l2 X = X → type-global-subuniverse P l2

module _
  {α : Level → Level} (P : global-subuniverse α)
  {l1 : Level} (l2 : Level) {X : UU l1}
  (Y Z : fam-global-subuniverse P l2 X)
  where

  equiv-fam-global-subuniverse : UU (l1 ⊔ l2)
  equiv-fam-global-subuniverse =
    equiv-fam-subuniverse (subuniverse-global-subuniverse P l2) Y Z

  equiv-eq-fam-global-subuniverse :
    Y ＝ Z → equiv-fam-global-subuniverse
  equiv-eq-fam-global-subuniverse =
    equiv-eq-fam-subuniverse (subuniverse-global-subuniverse P l2) Y Z

  is-equiv-equiv-eq-fam-global-subuniverse :
    is-equiv equiv-eq-fam-global-subuniverse
  is-equiv-equiv-eq-fam-global-subuniverse =
    is-equiv-equiv-eq-fam-subuniverse (subuniverse-global-subuniverse P l2) Y Z

  extensionality-fam-global-subuniverse :
    (Y ＝ Z) ≃ equiv-fam-global-subuniverse
  extensionality-fam-global-subuniverse =
    extensionality-fam-subuniverse (subuniverse-global-subuniverse P l2) Y Z

  eq-equiv-fam-global-subuniverse : equiv-fam-global-subuniverse → Y ＝ Z
  eq-equiv-fam-global-subuniverse =
    map-inv-is-equiv is-equiv-equiv-eq-fam-global-subuniverse

See also

Recent changes