Dependent products of large locales

Content created by Egbert Rijke, Fredrik Bakke, Julian KG, Maša Žaucer, fernabnor and louismntnu.

Created on 2023-05-09.
Last modified on 2023-09-21.

module order-theory.dependent-products-large-locales where
open import foundation.identity-types
open import foundation.large-binary-relations
open import foundation.sets
open import foundation.universe-levels

open import order-theory.dependent-products-large-frames
open import order-theory.greatest-lower-bounds-large-posets
open import order-theory.large-locales
open import order-theory.large-meet-semilattices
open import order-theory.large-posets
open import order-theory.large-suplattices
open import order-theory.least-upper-bounds-large-posets
open import

Given a family L : I → Large-Locale α β of large locales indexed by a type I : UU l, the product of the large locales L i is again a large locale.

module _
  {α : Level  Level} {β : Level  Level  Level} {γ : Level}
  {l1 : Level} {I : UU l1} (L : I  Large-Locale α β γ)

  Π-Large-Locale : Large-Locale  l2  α l2  l1)  l2 l3  β l2 l3  l1) γ
  Π-Large-Locale = Π-Large-Frame L

  large-poset-Π-Large-Locale :
    Large-Poset  l2  α l2  l1)  l2 l3  β l2 l3  l1)
  large-poset-Π-Large-Locale = large-poset-Π-Large-Frame L

  large-meet-semilattice-Π-Large-Locale :
    Large-Meet-Semilattice  l2  α l2  l1)  l2 l3  β l2 l3  l1)
  large-meet-semilattice-Π-Large-Locale =
    large-meet-semilattice-Π-Large-Frame L

  has-meets-Π-Large-Locale :
    has-meets-Large-Poset large-poset-Π-Large-Locale
  has-meets-Π-Large-Locale = has-meets-Π-Large-Frame L

  large-suplattice-Π-Large-Locale :
    Large-Suplattice  l2  α l2  l1)  l2 l3  β l2 l3  l1) γ
  large-suplattice-Π-Large-Locale = large-suplattice-Π-Large-Frame L

  is-large-suplattice-Π-Large-Locale :
    is-large-suplattice-Large-Poset γ large-poset-Π-Large-Locale
  is-large-suplattice-Π-Large-Locale =
    is-large-suplattice-Π-Large-Frame L

  set-Π-Large-Locale : (l : Level)  Set (α l  l1)
  set-Π-Large-Locale = set-Π-Large-Frame L

  type-Π-Large-Locale : (l : Level)  UU (α l  l1)
  type-Π-Large-Locale = type-Π-Large-Frame L

  is-set-type-Π-Large-Locale : {l : Level}  is-set (type-Π-Large-Locale l)
  is-set-type-Π-Large-Locale = is-set-type-Π-Large-Frame L

  leq-prop-Π-Large-Locale :
      ( λ l2  α l2  l1)
      ( λ l2 l3  β l2 l3  l1)
      ( type-Π-Large-Locale)
  leq-prop-Π-Large-Locale = leq-prop-Π-Large-Frame L

  leq-Π-Large-Locale :
      ( λ l2  α l2  l1)
      ( λ l2 l3  β l2 l3  l1)
      ( type-Π-Large-Locale)
  leq-Π-Large-Locale = leq-Π-Large-Frame L

  is-prop-leq-Π-Large-Locale :
    is-prop-Large-Relation type-Π-Large-Locale leq-Π-Large-Locale
  is-prop-leq-Π-Large-Locale = is-prop-leq-Π-Large-Frame L

  refl-leq-Π-Large-Locale :
    is-reflexive-Large-Relation type-Π-Large-Locale leq-Π-Large-Locale
  refl-leq-Π-Large-Locale = refl-leq-Π-Large-Frame L

  antisymmetric-leq-Π-Large-Locale :
    is-antisymmetric-Large-Relation type-Π-Large-Locale leq-Π-Large-Locale
  antisymmetric-leq-Π-Large-Locale = antisymmetric-leq-Π-Large-Frame L

  transitive-leq-Π-Large-Locale :
    is-transitive-Large-Relation type-Π-Large-Locale leq-Π-Large-Locale
  transitive-leq-Π-Large-Locale = transitive-leq-Π-Large-Frame L

  meet-Π-Large-Locale :
    {l2 l3 : Level} 
    type-Π-Large-Locale l2  type-Π-Large-Locale l3 
    type-Π-Large-Locale (l2  l3)
  meet-Π-Large-Locale = meet-Π-Large-Frame L

  is-greatest-binary-lower-bound-meet-Π-Large-Locale :
    {l2 l3 : Level}
    (x : type-Π-Large-Locale l2)
    (y : type-Π-Large-Locale l3) 
      ( large-poset-Π-Large-Locale)
      ( x)
      ( y)
      ( meet-Π-Large-Locale x y)
  is-greatest-binary-lower-bound-meet-Π-Large-Locale =
    is-greatest-binary-lower-bound-meet-Π-Large-Frame L

  top-Π-Large-Locale : type-Π-Large-Locale lzero
  top-Π-Large-Locale = top-Π-Large-Frame L

  is-top-element-top-Π-Large-Locale :
    {l1 : Level} (x : type-Π-Large-Locale l1) 
    leq-Π-Large-Locale x top-Π-Large-Locale
  is-top-element-top-Π-Large-Locale =
    is-top-element-top-Π-Large-Frame L

  has-top-element-Π-Large-Locale :
    has-top-element-Large-Poset large-poset-Π-Large-Locale
  has-top-element-Π-Large-Locale =
    has-top-element-Π-Large-Frame L

  is-large-meet-semilattice-Π-Large-Locale :
    is-large-meet-semilattice-Large-Poset large-poset-Π-Large-Locale
  is-large-meet-semilattice-Π-Large-Locale =
    is-large-meet-semilattice-Π-Large-Frame L

  sup-Π-Large-Locale :
    {l2 l3 : Level} {J : UU l2} (x : J  type-Π-Large-Locale l3) 
    type-Π-Large-Locale (γ  l2  l3)
  sup-Π-Large-Locale = sup-Π-Large-Frame L

  is-least-upper-bound-sup-Π-Large-Locale :
    {l2 l3 : Level} {J : UU l2} (x : J  type-Π-Large-Locale l3) 
      ( large-poset-Π-Large-Locale)
      ( x)
      ( sup-Π-Large-Locale x)
  is-least-upper-bound-sup-Π-Large-Locale =
    is-least-upper-bound-sup-Π-Large-Frame L

  distributive-meet-sup-Π-Large-Locale :
    {l2 l3 l4 : Level}
    (x : type-Π-Large-Locale l2)
    {J : UU l3} (y : J  type-Π-Large-Locale l4) 
    meet-Π-Large-Locale x (sup-Π-Large-Locale y) 
    sup-Π-Large-Locale  j  meet-Π-Large-Locale x (y j))
  distributive-meet-sup-Π-Large-Locale =
    distributive-meet-sup-Π-Large-Frame L

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