open import Cat.Functor.Subcategory
open import Cat.Prelude

open import Order.Instances.Pointwise.Diagrams
open import Order.Diagram.Lub.Reasoning
open import Order.Instances.Pointwise
open import Order.Semilattice.Join
open import Order.Semilattice.Meet
open import Order.Instances.Props
open import Order.Diagram.Meet
open import Order.Diagram.Lub
open import Order.Diagram.Top
open import Order.Lattice
open import Order.Base

import Cat.Reasoning

import Order.Diagram.Meet.Reasoning as Meets
import Order.Reasoning

module Order.Frame where

Frames🔗

A frame is a lattice with finite meets ¹ and arbitrary joins satisfying the infinite distributive law

x \cap i ⋃ f (i) = i ⋃ (x \cap f (i)) .

In the study of frames, for simplicity, we assume propositional resizing: that way, it suffices for a frame $A$ to have joins of $J -indexed$ families, for $J$ an arbitrary type in the same universe as $A,$ to have joins for arbitrary subsets of $A .$

record is-frame {o ℓ} (P : Poset o ℓ) : Type (lsuc o ⊔ ℓ) where
  no-eta-equality
  open Poset P
  field
    _∩_     : Ob → Ob → Ob
    ∩-meets : ∀ x y → is-meet P x y (x ∩ y)

    has-top : Top P

    ⋃       : ∀ {I : Type o} (k : I → Ob) → Ob
    ⋃-lubs  : ∀ {I : Type o} (k : I → Ob) → is-lub P k (⋃ k)

    ⋃-distribl : ∀ {I} x (f : I → Ob) → x ∩ ⋃ f ≡ ⋃ λ i → x ∩ f i

We have explicitly required that a frame be a meet-semilattice, but it’s worth explicitly pointing out that the infinitary join operation can also be used for more mundane purposes: By taking a join over the type of booleans (and over the empty type), we can show that all frames are also join-semilattices.

  infixr 25 _∩_

  module is-lubs {I} {k : I → Ob} = is-lub (⋃-lubs k)

  open Meets ∩-meets public
  open Top has-top using (top; !) public
  open Lubs P ⋃-lubs public

module _ {o ℓ} {P : Poset o ℓ} (f : is-frame P) where
  is-frame→is-meet-slat : is-meet-semilattice P
  is-frame→is-meet-slat = record { is-frame f }

  is-frame→is-join-slat : is-join-semilattice P
  is-frame→is-join-slat = record { is-frame f }

  is-frame→is-lattice : is-lattice P
  is-frame→is-lattice = record { is-frame f }

private variable
  o ℓ o' ℓ' : Level
  P Q R : Poset o ℓ

abstract
  is-frame-is-prop : is-prop (is-frame P)
  is-frame-is-prop {P = P} p q = path where
    open Order.Diagram.Top P using (H-Level-Top)

    module p = is-frame p
    module q = is-frame q
    open is-frame

    meetp : ∀ x y → x p.∩ y ≡ x q.∩ y
    meetp x y = meet-unique (p.∩-meets x y) (q.∩-meets x y)

    lubp : ∀ {I} (k : I → ⌞ P ⌟) → p.⋃ k ≡ q.⋃ k
    lubp k = lub-unique (p.⋃-lubs k) (q.⋃-lubs k)

    path : p ≡ q
    path i ._∩_     x y = meetp x y i
    path i .∩-meets x y = is-prop→pathp (λ i → hlevel {T = is-meet P x y (meetp x y i)} 1) (p.∩-meets x y) (q.∩-meets x y) i
    path i .has-top    = hlevel {T = Top P} 1 p.has-top q.has-top i
    path i .⋃ k        = lubp k i
    path i .⋃-lubs k = is-prop→pathp (λ i → hlevel {T = is-lub P k (lubp k i)} 1) (p.⋃-lubs k) (q.⋃-lubs k) i
    path i .⋃-distribl x f j = is-set→squarep (λ _ _ → Poset.Ob-is-set P)
      (λ i → meetp x (lubp f i) i)
      (p.⋃-distribl x f) (q.⋃-distribl x f)
      (λ i → lubp (λ e → meetp x (f e) i) i)
      i j

instance
  H-Level-is-frame : ∀ {n} → H-Level (is-frame P) (suc n)
  H-Level-is-frame = prop-instance is-frame-is-prop

Of course, a frame is not just a lattice, but a complete lattice. Since the infinite distributive law says exactly that “meet with $x$ ” preserves joins, this implies that it has a right adjoint, so frames are also complete Heyting algebras. Once again, the difference in naming reflects the morphisms we will consider these structures under: A frame homomorphism is a monotone map which preserves the finite meets and the infinitary joins, but not necessarily the infinitary meets (or the Heyting implication).

Since meets and joins are defined by a universal property, and we have assumed that homomorphisms are a priori monotone, it suffices to show the following inequalities:

For every $x, y : P,$ we have $f x \cap f y \leq f (x \cap y);$
$⊤ \leq f (⊤);$
and finally, for every family $k : I \to P,$ we have $f (⋃ k) \leq ⋃ (f \circ k)$

record
  is-frame-hom
    {P : Poset o ℓ} {Q : Poset o ℓ'}
    (f : Monotone P Q) (P-frame : is-frame P) (Q-frame : is-frame Q)
    : Type (lsuc o ⊔ ℓ') where

  private
    module P = Poset P
    module Pᶠ = is-frame P-frame
    module Q = Order.Reasoning Q
    module Qᶠ = is-frame Q-frame
    open is-lub

  field
    ∩-≤   : ∀ x y → f · x Qᶠ.∩ f · y Q.≤ f · (x Pᶠ.∩ y)
    top-≤ : Qᶠ.top Q.≤ f · Pᶠ.top
    ⋃-≤   : ∀ {I : Type o} (k : I → ⌞ P ⌟) → (f · Pᶠ.⋃ k) Q.≤ Qᶠ.⋃ (apply f ⊙ k)

If $f$ is a frame homomorphism, then it is also a homomorphism of meet semilattices.

  has-meet-slat-hom : is-meet-slat-hom f record { Pᶠ } record { Qᶠ }
  has-meet-slat-hom .is-meet-slat-hom.∩-≤ = ∩-≤
  has-meet-slat-hom .is-meet-slat-hom.top-≤ = top-≤

  open is-meet-slat-hom has-meet-slat-hom hiding (∩-≤; top-≤) public

Furthermore, we can actually show from the inequality required above that $f$ preserves all joins up to equality.

  pres-⋃ : ∀ {I : Type o} (k : I → ⌞ P ⌟) → (f · Pᶠ.⋃ k) ≡ Qᶠ.⋃ (apply f ⊙ k)
  pres-⋃ k =
    Q.≤-antisym
      (⋃-≤ k)
      (Qᶠ.⋃-universal _ (λ i → f .pres-≤ (Pᶠ.⋃-inj i)))

  pres-lubs
    : ∀ {I : Type o} {k : I → ⌞ P ⌟} {l}
    → is-lub P k l
    → is-lub Q (apply f ⊙ k) (f · l)
  pres-lubs {k = k} {l = l} l-lub .fam≤lub i = f .pres-≤ (l-lub .fam≤lub i)
  pres-lubs {k = k} {l = l} l-lub .least ub p =
    f · l              Q.≤⟨ f .pres-≤ (l-lub .least _ Pᶠ.⋃-inj) ⟩
    f · Pᶠ.⋃ k         Q.≤⟨ ⋃-≤ k ⟩
    Qᶠ.⋃ (apply f ⊙ k) Q.≤⟨ Qᶠ.⋃-universal ub p ⟩
    ub                 Q.≤∎

As a corollary, $f$ is also a homomorphism of the underlying join semilattices.

  opaque
    unfolding Lubs.∪-joins Lubs.has-bottom

    has-join-slat-hom : is-join-slat-hom f record { Pᶠ } record { Qᶠ }
    has-join-slat-hom .is-join-slat-hom.∪-≤ x y =
      Q.≤-trans (⋃-≤ _) $ Qᶠ.⋃-universal _ λ where
        (lift true) → Qᶠ.⋃-inj (lift true)
        (lift false) → Qᶠ.⋃-inj (lift false)
    has-join-slat-hom .is-join-slat-hom.bot-≤ =
      Q.≤-trans (⋃-≤ _) (Qᶠ.⋃-universal _ (λ ()))

  open is-join-slat-hom has-join-slat-hom public

open is-frame-hom

unquoteDecl H-Level-is-frame-hom = declare-record-hlevel 1 H-Level-is-frame-hom (quote is-frame-hom)

Clearly, the identity function is a frame homomorphism.

id-frame-hom
  : ∀ (Pᶠ : is-frame P)
  → is-frame-hom idₘ Pᶠ Pᶠ
id-frame-hom {P = P} Pᶠ .∩-≤ x y =
  Poset.≤-refl P
id-frame-hom {P = P} Pᶠ .top-≤ =
  Poset.≤-refl P
id-frame-hom {P = P} Pᶠ .⋃-≤ k =
  Poset.≤-refl P

Furthermore, frame homomorphisms are closed under composition.

∘-frame-hom
  : ∀ {Pₗ Qₗ Rₗ} {f : Monotone Q R} {g : Monotone P Q}
  → is-frame-hom f Qₗ Rₗ
  → is-frame-hom g Pₗ Qₗ
  → is-frame-hom (f ∘ₘ g) Pₗ Rₗ
∘-frame-hom {R = R} {f = f} {g = g} f-pres g-pres .∩-≤ x y =
  R .Poset.≤-trans (f-pres .∩-≤ (g · x) (g · y)) (f .pres-≤ (g-pres .∩-≤ x y))
∘-frame-hom {R = R} {f = f} {g = g} f-pres g-pres .top-≤ =
  R .Poset.≤-trans (f-pres .top-≤) (f .pres-≤ (g-pres .top-≤))
∘-frame-hom {R = R} {f = f} {g = g} f-pres g-pres .⋃-≤ k =
  R .Poset.≤-trans (f .pres-≤ (g-pres .⋃-≤ k)) (f-pres .⋃-≤ (λ i → g · k i))

This means that we can define the category of frames as a subcategory of the category of posets.

Frame-subcat : ∀ o ℓ → Subcat (Posets o ℓ) _ _
Frame-subcat o ℓ .Subcat.is-ob = is-frame
Frame-subcat o ℓ .Subcat.is-hom = is-frame-hom
Frame-subcat o ℓ .Subcat.is-hom-prop _ _ _ = hlevel 1
Frame-subcat o ℓ .Subcat.is-hom-id = id-frame-hom
Frame-subcat o ℓ .Subcat.is-hom-∘ = ∘-frame-hom

Frames : ∀ o ℓ → Precategory _ _
Frames o ℓ = Subcategory (Frame-subcat o ℓ)

module Frames {o} {ℓ} = Cat.Reasoning (Frames o ℓ)

Frame : ∀ o ℓ → Type _
Frame o ℓ = Frames.Ob {o} {ℓ}

Power sets as frames🔗

A canonical source of frames are power sets: The power set of any type is a frame, because it is a complete lattice satisfying the infinite distributive law; This can be seen by some computation, as is done below.

open is-frame
open is-meet-semilattice

Power-frame : ∀ {ℓ} (A : Type ℓ) → Frame ℓ ℓ
Power-frame {ℓ = ℓ} A .fst = Subsets A
Power-frame A .snd ._∩_ P Q i = P i ∧Ω Q i
Power-frame A .snd .∩-meets P Q =
  is-meet-pointwise λ _ → Props-has-meets (P _) (Q _)
Power-frame A .snd .has-top =
  has-top-pointwise λ _ → Props-has-top
Power-frame A .snd .⋃ k i = ∃Ω _ (λ j → k j i)
Power-frame A .snd .⋃-lubs k = is-lub-pointwise _ _ λ _ →
  Props-has-lubs λ i → k i _
Power-frame A .snd .⋃-distribl x f = ext λ i → Ω-ua
  (rec! λ xi j j~i → inc (j , xi , j~i))
  (rec! λ j xi j~i → xi , inc (j , j~i))

So, in addition to the $x \cap y$ operation, we have a top element↩︎