Homomorphisms of large frames

Content created by Egbert Rijke, Fredrik Bakke and Maša Žaucer.

Created on 2023-05-12.
Last modified on 2023-11-24.

module order-theory.homomorphisms-large-frames where
Imports 
open import foundation.identity-types
open import foundation.universe-levels

open import order-theory.homomorphisms-large-meet-semilattices
open import order-theory.homomorphisms-large-suplattices
open import order-theory.large-frames

Idea

A homomorphism of large frames from K to L is an order preserving map from K to L which preserves meets, the top element, and suprema.

Definitions

Homomorphisms of frames

module _
  {αK αL : Level → Level} {βK βL : Level → Level → Level} {γ : Level}
  (K : Large-Frame αK βK γ) (L : Large-Frame αL βL γ)
  where

  record
    hom-Large-Frame : UUω
    where
    field
      hom-large-meet-semilattice-hom-Large-Frame :
        hom-Large-Meet-Semilattice
          ( large-meet-semilattice-Large-Frame K)
          ( large-meet-semilattice-Large-Frame L)
      preserves-sup-hom-Large-Frame :
        preserves-sup-hom-Large-Poset
          ( large-suplattice-Large-Frame K)
          ( large-suplattice-Large-Frame L)
          ( hom-large-poset-hom-Large-Meet-Semilattice
            ( hom-large-meet-semilattice-hom-Large-Frame))

  open hom-Large-Frame public

  module _
    (f : hom-Large-Frame)
    where

    map-hom-Large-Frame :
      {l1 : Level} → type-Large-Frame K l1 → type-Large-Frame L l1
    map-hom-Large-Frame =
      map-hom-Large-Meet-Semilattice
        ( large-meet-semilattice-Large-Frame K)
        ( large-meet-semilattice-Large-Frame L)
        ( hom-large-meet-semilattice-hom-Large-Frame f)

    preserves-order-hom-Large-Frame :
      {l1 l2 : Level} (x : type-Large-Frame K l1) (y : type-Large-Frame K l2) →
      leq-Large-Frame K x y →
      leq-Large-Frame L (map-hom-Large-Frame x) (map-hom-Large-Frame y)
    preserves-order-hom-Large-Frame =
      preserves-order-hom-Large-Meet-Semilattice
        ( large-meet-semilattice-Large-Frame K)
        ( large-meet-semilattice-Large-Frame L)
        ( hom-large-meet-semilattice-hom-Large-Frame f)

    preserves-meets-hom-Large-Frame :
      {l1 l2 : Level} (x : type-Large-Frame K l1) (y : type-Large-Frame K l2) →
      map-hom-Large-Frame (meet-Large-Frame K x y) ＝
      meet-Large-Frame L (map-hom-Large-Frame x) (map-hom-Large-Frame y)
    preserves-meets-hom-Large-Frame =
      preserves-meets-hom-Large-Meet-Semilattice
        ( hom-large-meet-semilattice-hom-Large-Frame f)

    preserves-top-hom-Large-Frame :
      map-hom-Large-Frame (top-Large-Frame K) ＝ top-Large-Frame L
    preserves-top-hom-Large-Frame =
      preserves-top-hom-Large-Meet-Semilattice
        ( hom-large-meet-semilattice-hom-Large-Frame f)

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