Galois connections between large posets

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

Created on 2023-05-10.
Last modified on 2025-05-02.

module order-theory.galois-connections-large-posets where

Imports

open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.function-types
open import foundation.homotopies
open import foundation.logical-equivalences
open import foundation.universe-levels

open import order-theory.large-posets
open import order-theory.least-upper-bounds-large-posets
open import order-theory.order-preserving-maps-large-posets
open import order-theory.principal-lower-sets-large-posets
open import order-theory.principal-upper-sets-large-posets
open import order-theory.similarity-of-elements-large-posets
open import order-theory.similarity-of-order-preserving-maps-large-posets

Idea

A galois connection between large posets P and Q consists of order preserving maps f : hom-Large-Poset (λ l → l) P Q and hom-Large-Poset id Q P such that the adjoint relation

  (f x ≤ y) ↔ (x ≤ g y)

holds for every two elements x : P and y : Q.

The following table lists the Galois connections that have been formalized in the agda-unimath library

Galois connection	File
Ideals generated by subsets of rings	`ring-theory.ideals-generated-by-subsets-rings`
Images of subgroups under group homomorphisms	`group-theory.images-of-grouphomomorphisms`
Images of subsemigroups under semigroup homomorphisms	`group-theory.images-of-semigroup-homomorphisms`
Images of subtypes	`foundation.images-subtypes`
Left ideals generated by subsets of rings	`ring-theory.left-ideals-generated-by-subsets-rings`
Normal closures of subgroups	`group-theory.normal-closures-subgroups`
Normal cores of subgroups	`group-theory.normal-cores-subgroups`
Radicals of ideals of commutative rings	`commutative-algebra.radicals-of-ideals-commutative-rings`
Right ideals generated by subsets of rings	`ring-theory.right-ideals-generated-by-subsets-rings`
Subgroups generated by subsets of groups	`group-theory.subgroups-generated-by-subsets-groups`

Definitions

The adjoint relation between order preserving maps between large posets

module _
  {αP αQ γ δ : Level → Level} {βP βQ : Level → Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (F : hom-Large-Poset γ P Q) (G : hom-Large-Poset δ Q P)
  where

  forward-implication-adjoint-relation-hom-Large-Poset : UUω
  forward-implication-adjoint-relation-hom-Large-Poset =
    {l1 l2 : Level}
    {x : type-Large-Poset P l1} {y : type-Large-Poset Q l2} →
    leq-Large-Poset Q (map-hom-Large-Poset P Q F x) y →
    leq-Large-Poset P x (map-hom-Large-Poset Q P G y)

  backward-implication-adjoint-relation-hom-Large-Poset : UUω
  backward-implication-adjoint-relation-hom-Large-Poset =
    {l1 l2 : Level}
    {x : type-Large-Poset P l1} {y : type-Large-Poset Q l2} →
    leq-Large-Poset P x (map-hom-Large-Poset Q P G y) →
    leq-Large-Poset Q (map-hom-Large-Poset P Q F x) y

  adjoint-relation-hom-Large-Poset : UUω
  adjoint-relation-hom-Large-Poset =
    {l1 l2 : Level}
    (x : type-Large-Poset P l1) (y : type-Large-Poset Q l2) →
    leq-Large-Poset Q (map-hom-Large-Poset P Q F x) y ↔
    leq-Large-Poset P x (map-hom-Large-Poset Q P G y)

Galois connections between large posets

module _
  {αP αQ : Level → Level} {βP βQ : Level → Level → Level}
  (γ : Level → Level) (δ : Level → Level)
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  where

  record
    galois-connection-Large-Poset : UUω
    where
    constructor
      make-galois-connection-Large-Poset
    field
      lower-adjoint-galois-connection-Large-Poset :
        hom-Large-Poset γ P Q
      upper-adjoint-galois-connection-Large-Poset :
        hom-Large-Poset δ Q P
      adjoint-relation-galois-connection-Large-Poset :
        adjoint-relation-hom-Large-Poset P Q
          lower-adjoint-galois-connection-Large-Poset
          upper-adjoint-galois-connection-Large-Poset

  open galois-connection-Large-Poset public

module _
  {αP αQ : Level → Level} {βP βQ : Level → Level → Level}
  {γ : Level → Level} {δ : Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G : galois-connection-Large-Poset γ δ P Q)
  where

  map-lower-adjoint-galois-connection-Large-Poset :
    {l1 : Level} → type-Large-Poset P l1 → type-Large-Poset Q (γ l1)
  map-lower-adjoint-galois-connection-Large-Poset =
    map-hom-Large-Poset P Q
      ( lower-adjoint-galois-connection-Large-Poset G)

  preserves-order-lower-adjoint-galois-connection-Large-Poset :
    {l1 l2 : Level} (x : type-Large-Poset P l1) (y : type-Large-Poset P l2) →
    leq-Large-Poset P x y →
    leq-Large-Poset Q
      ( map-lower-adjoint-galois-connection-Large-Poset x)
      ( map-lower-adjoint-galois-connection-Large-Poset y)
  preserves-order-lower-adjoint-galois-connection-Large-Poset =
    preserves-order-hom-Large-Poset P Q
      ( lower-adjoint-galois-connection-Large-Poset G)

  map-upper-adjoint-galois-connection-Large-Poset :
    {l1 : Level} → type-Large-Poset Q l1 → type-Large-Poset P (δ l1)
  map-upper-adjoint-galois-connection-Large-Poset =
    map-hom-Large-Poset Q P
      ( upper-adjoint-galois-connection-Large-Poset G)

  preserves-order-upper-adjoint-galois-connection-Large-Poset :
    {l1 l2 : Level} (x : type-Large-Poset Q l1) (y : type-Large-Poset Q l2) →
    leq-Large-Poset Q x y →
    leq-Large-Poset P
      ( map-upper-adjoint-galois-connection-Large-Poset x)
      ( map-upper-adjoint-galois-connection-Large-Poset y)
  preserves-order-upper-adjoint-galois-connection-Large-Poset =
    preserves-order-hom-Large-Poset Q P
      ( upper-adjoint-galois-connection-Large-Poset G)

  forward-implication-adjoint-relation-galois-connection-Large-Poset :
    forward-implication-adjoint-relation-hom-Large-Poset P Q
      ( lower-adjoint-galois-connection-Large-Poset G)
      ( upper-adjoint-galois-connection-Large-Poset G)
  forward-implication-adjoint-relation-galois-connection-Large-Poset =
    forward-implication (adjoint-relation-galois-connection-Large-Poset G _ _)

  backward-implication-adjoint-relation-galois-connection-Large-Poset :
    backward-implication-adjoint-relation-hom-Large-Poset P Q
      ( lower-adjoint-galois-connection-Large-Poset G)
      ( upper-adjoint-galois-connection-Large-Poset G)
  backward-implication-adjoint-relation-galois-connection-Large-Poset =
    backward-implication (adjoint-relation-galois-connection-Large-Poset G _ _)

Composition of Galois connections

module _
  {αP αQ αR : Level → Level} {βP βQ βR : Level → Level → Level}
  {γG γH : Level → Level} {δG δH : Level → Level}
  (P : Large-Poset αP βP)
  (Q : Large-Poset αQ βQ)
  (R : Large-Poset αR βR)
  (H : galois-connection-Large-Poset γH δH Q R)
  (G : galois-connection-Large-Poset γG δG P Q)
  where

  lower-adjoint-comp-galois-connection-Large-Poset :
    hom-Large-Poset (λ l → γH (γG l)) P R
  lower-adjoint-comp-galois-connection-Large-Poset =
    comp-hom-Large-Poset P Q R
      ( lower-adjoint-galois-connection-Large-Poset H)
      ( lower-adjoint-galois-connection-Large-Poset G)

  map-lower-adjoint-comp-galois-connection-Large-Poset :
    {l : Level} → type-Large-Poset P l → type-Large-Poset R (γH (γG l))
  map-lower-adjoint-comp-galois-connection-Large-Poset =
    map-comp-hom-Large-Poset P Q R
      ( lower-adjoint-galois-connection-Large-Poset H)
      ( lower-adjoint-galois-connection-Large-Poset G)

  preserves-order-lower-adjoint-comp-galois-connection-Large-Poset :
    preserves-order-map-Large-Poset P R
      map-lower-adjoint-comp-galois-connection-Large-Poset
  preserves-order-lower-adjoint-comp-galois-connection-Large-Poset =
    preserves-order-comp-hom-Large-Poset P Q R
      ( lower-adjoint-galois-connection-Large-Poset H)
      ( lower-adjoint-galois-connection-Large-Poset G)

  upper-adjoint-comp-galois-connection-Large-Poset :
    hom-Large-Poset (λ l → δG (δH l)) R P
  upper-adjoint-comp-galois-connection-Large-Poset =
    comp-hom-Large-Poset R Q P
      ( upper-adjoint-galois-connection-Large-Poset G)
      ( upper-adjoint-galois-connection-Large-Poset H)

  map-upper-adjoint-comp-galois-connection-Large-Poset :
    {l : Level} → type-Large-Poset R l → type-Large-Poset P (δG (δH l))
  map-upper-adjoint-comp-galois-connection-Large-Poset =
    map-comp-hom-Large-Poset R Q P
      ( upper-adjoint-galois-connection-Large-Poset G)
      ( upper-adjoint-galois-connection-Large-Poset H)

  preserves-order-upper-adjoint-comp-galois-connection-Large-Poset :
    preserves-order-map-Large-Poset R P
      map-upper-adjoint-comp-galois-connection-Large-Poset
  preserves-order-upper-adjoint-comp-galois-connection-Large-Poset =
    preserves-order-comp-hom-Large-Poset R Q P
      ( upper-adjoint-galois-connection-Large-Poset G)
      ( upper-adjoint-galois-connection-Large-Poset H)

  forward-implication-adjoint-relation-comp-galois-connection-Large-Poset :
    forward-implication-adjoint-relation-hom-Large-Poset P R
      lower-adjoint-comp-galois-connection-Large-Poset
      upper-adjoint-comp-galois-connection-Large-Poset
  forward-implication-adjoint-relation-comp-galois-connection-Large-Poset =
    forward-implication-adjoint-relation-galois-connection-Large-Poset P Q G ∘
    forward-implication-adjoint-relation-galois-connection-Large-Poset Q R H

  backward-implication-adjoint-relation-comp-galois-connection-Large-Poset :
    backward-implication-adjoint-relation-hom-Large-Poset P R
      lower-adjoint-comp-galois-connection-Large-Poset
      upper-adjoint-comp-galois-connection-Large-Poset
  backward-implication-adjoint-relation-comp-galois-connection-Large-Poset =
    backward-implication-adjoint-relation-galois-connection-Large-Poset Q R H ∘
    backward-implication-adjoint-relation-galois-connection-Large-Poset P Q G

  adjoint-relation-comp-galois-connection-Large-Poset :
    adjoint-relation-hom-Large-Poset P R
      lower-adjoint-comp-galois-connection-Large-Poset
      upper-adjoint-comp-galois-connection-Large-Poset
  pr1 (adjoint-relation-comp-galois-connection-Large-Poset x y) =
    forward-implication-adjoint-relation-comp-galois-connection-Large-Poset
  pr2 (adjoint-relation-comp-galois-connection-Large-Poset x y) =
    backward-implication-adjoint-relation-comp-galois-connection-Large-Poset

  comp-galois-connection-Large-Poset :
    galois-connection-Large-Poset (λ l → γH (γG l)) (λ l → δG (δH l)) P R
  lower-adjoint-galois-connection-Large-Poset
    comp-galois-connection-Large-Poset =
    lower-adjoint-comp-galois-connection-Large-Poset
  upper-adjoint-galois-connection-Large-Poset
    comp-galois-connection-Large-Poset =
    upper-adjoint-comp-galois-connection-Large-Poset
  adjoint-relation-galois-connection-Large-Poset
    comp-galois-connection-Large-Poset =
    adjoint-relation-comp-galois-connection-Large-Poset

Homotopies of Galois connections

Homotopies of Galois connections are pointwise identifications between either their lower adjoints or their upper adjoints. We will show below that homotopies between lower adjoints induce homotopies between upper adjoints and vice versa.

Note: We can only have homotopies between Galois connections with the same universe level reindexing functions.

module _
  {αP αQ γ δ : Level → Level} {βP βQ : Level → Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G H : galois-connection-Large-Poset γ δ P Q)
  where

  lower-htpy-galois-connection-Large-Poset : UUω
  lower-htpy-galois-connection-Large-Poset =
    htpy-hom-Large-Poset P Q
      ( lower-adjoint-galois-connection-Large-Poset G)
      ( lower-adjoint-galois-connection-Large-Poset H)

  upper-htpy-galois-connection-Large-Poset : UUω
  upper-htpy-galois-connection-Large-Poset =
    htpy-hom-Large-Poset Q P
      ( upper-adjoint-galois-connection-Large-Poset G)
      ( upper-adjoint-galois-connection-Large-Poset H)

Similarity of Galois connections

Similarities of Galois connections are pointwise similarities between either their lower or their upper adjoints. We will show below that similarities between lower adjoints induce similarities between upper adjoints and vice versa.

Note: Since the notion of similarity applies to galois connections with not necessarily the same universe level reindexing function, it is slightly more flexible. For this reason, it may be easier to work with similarity of galois connections.

module _
  {αP αQ γG γH δG δH : Level → Level} {βP βQ : Level → Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G : galois-connection-Large-Poset γG δG P Q)
  (H : galois-connection-Large-Poset γH δH P Q)
  where

  sim-lower-galois-connection-Large-Poset : UUω
  sim-lower-galois-connection-Large-Poset =
    sim-hom-Large-Poset P Q
      ( lower-adjoint-galois-connection-Large-Poset G)
      ( lower-adjoint-galois-connection-Large-Poset H)

  sim-upper-galois-connection-Large-Poset : UUω
  sim-upper-galois-connection-Large-Poset =
    sim-hom-Large-Poset Q P
      ( upper-adjoint-galois-connection-Large-Poset G)
      ( upper-adjoint-galois-connection-Large-Poset H)

Lower universal objects of galois connections

Consider a Galois connection G : P → Q and an element x : P. An element x' : Q is then said to satisfy the lower universal property with respect to x if the logical equivalence

  (x' ≤-Q y) ↔ (x ≤-P UG y)

holds for every element y : Q.

module _
  {αP αQ γ δ : Level → Level} {βP βQ : Level → Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G : galois-connection-Large-Poset γ δ P Q)
  {l1 l2 : Level} (x : type-Large-Poset P l1)
  (x' : type-Large-Poset Q l2)
  where

  is-lower-element-galois-connection-Large-Poset : UUω
  is-lower-element-galois-connection-Large-Poset =
    {l : Level} (y : type-Large-Poset Q l) →
    leq-Large-Poset Q x' y ↔
    leq-Large-Poset P x
      ( map-upper-adjoint-galois-connection-Large-Poset P Q G y)

Upper universal objects of galois connections

Consider a Galois connection G : P → Q and an element y : Q. An element y' : P is then said to satisfy the upper universal property with respect to y if the logical equivalence

  (LG x ≤-Q y) ↔ (x ≤-P y')

holds for every element x : P.

module _
  {αP αQ γ δ : Level → Level} {βP βQ : Level → Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G : galois-connection-Large-Poset γ δ P Q)
  {l1 l2 : Level} (y : type-Large-Poset Q l1)
  (y' : type-Large-Poset P l2)
  where

  is-upper-element-galois-connection-Large-Poset : UUω
  is-upper-element-galois-connection-Large-Poset =
    {l : Level} (x : type-Large-Poset P l) →
    leq-Large-Poset Q
      ( map-lower-adjoint-galois-connection-Large-Poset P Q G x)
      ( y) ↔
    leq-Large-Poset P x y'

Properties

A similarity between lower adjoints of a Galois connection induces a similarity between upper adjoints, and vice versa

Proof: Consider two Galois connections LG ⊣ UG and LH ⊣ UH between P and Q, and suppose that LG(x) ~ LH(x) for all elements x : P. Then it follows that

  (x ≤ UG(y)) ↔ (LG(x) ≤ y) ↔ (LH(x) ≤ y) ↔ (x ≤ UH(y)).

Therefore it follows that UG(y) and UH(y) have the same lower sets, and hence they must be equal.

module _
  {αP αQ γG γH δG δH : Level → Level} {βP βQ : Level → Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G : galois-connection-Large-Poset γG δG P Q)
  (H : galois-connection-Large-Poset γH δH P Q)
  where

  sim-upper-sim-lower-galois-connection-Large-Poset :
    sim-lower-galois-connection-Large-Poset P Q G H →
    sim-upper-galois-connection-Large-Poset P Q H G
  sim-upper-sim-lower-galois-connection-Large-Poset
    p x =
    sim-has-same-elements-principal-lower-set-element-Large-Poset P
      ( λ y →
        logical-equivalence-reasoning
          leq-Large-Poset P y
            ( map-upper-adjoint-galois-connection-Large-Poset P Q H x)
          ↔ leq-Large-Poset Q
              ( map-lower-adjoint-galois-connection-Large-Poset P Q H y)
              ( x)
            by inv-iff (adjoint-relation-galois-connection-Large-Poset H y x)
          ↔ leq-Large-Poset Q
              ( map-lower-adjoint-galois-connection-Large-Poset P Q G y)
              ( x)
            by
            inv-iff
              ( has-same-elements-principal-upper-set-element-sim-Large-Poset Q
                ( p y)
                ( x))
          ↔ leq-Large-Poset P y
              ( map-upper-adjoint-galois-connection-Large-Poset P Q G x)
            by adjoint-relation-galois-connection-Large-Poset G y x)

  sim-lower-sim-upper-galois-connection-Large-Poset :
    sim-upper-galois-connection-Large-Poset P Q H G →
    sim-lower-galois-connection-Large-Poset P Q G H
  sim-lower-sim-upper-galois-connection-Large-Poset
    p y =
    sim-has-same-elements-principal-upper-set-element-Large-Poset Q
      ( λ x →
        logical-equivalence-reasoning
          leq-Large-Poset Q
            ( map-lower-adjoint-galois-connection-Large-Poset P Q G y)
            ( x)
          ↔ leq-Large-Poset P y
              ( map-upper-adjoint-galois-connection-Large-Poset P Q G x)
            by adjoint-relation-galois-connection-Large-Poset G y x
          ↔ leq-Large-Poset P y
              ( map-upper-adjoint-galois-connection-Large-Poset P Q H x)
            by
            inv-iff
              ( has-same-elements-principal-lower-set-element-sim-Large-Poset P
                ( p x)
                ( y))
          ↔ leq-Large-Poset Q
              ( map-lower-adjoint-galois-connection-Large-Poset P Q H y)
              ( x)
            by inv-iff (adjoint-relation-galois-connection-Large-Poset H y x))

A homotopy between lower adjoints of a Galois connection induces a homotopy between upper adjoints, and vice versa

Proof: Consider two Galois connections LG ⊣ UG and LH ⊣ UH between P and Q, and suppose that LG ~ LH. Then there is a similarity LG ≈ LH, and this induces a similarity UG ≈ UH. In other words, we obtain that

  UG y ~ UH y

for any element y : Q. Since UG y and UH y are of the same universe level, it follows that they are equal.

module _
  {αP αQ γ δ : Level → Level} {βP βQ : Level → Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G H : galois-connection-Large-Poset γ δ P Q)
  where

  upper-htpy-lower-htpy-galois-connection-Large-Poset :
    lower-htpy-galois-connection-Large-Poset P Q G H →
    upper-htpy-galois-connection-Large-Poset P Q G H
  upper-htpy-lower-htpy-galois-connection-Large-Poset p =
    htpy-sim-hom-Large-Poset Q P
      ( upper-adjoint-galois-connection-Large-Poset G)
      ( upper-adjoint-galois-connection-Large-Poset H)
      ( sim-upper-sim-lower-galois-connection-Large-Poset P Q H G
        ( sim-htpy-hom-Large-Poset P Q
          ( lower-adjoint-galois-connection-Large-Poset H)
          ( lower-adjoint-galois-connection-Large-Poset G)
          ( inv-htpy p)))

  lower-htpy-upper-htpy-galois-connection-Large-Poset :
    upper-htpy-galois-connection-Large-Poset P Q H G →
    lower-htpy-galois-connection-Large-Poset P Q G H
  lower-htpy-upper-htpy-galois-connection-Large-Poset p =
    htpy-sim-hom-Large-Poset P Q
      ( lower-adjoint-galois-connection-Large-Poset G)
      ( lower-adjoint-galois-connection-Large-Poset H)
      ( sim-lower-sim-upper-galois-connection-Large-Poset P Q G H
        ( sim-htpy-hom-Large-Poset Q P
          ( upper-adjoint-galois-connection-Large-Poset H)
          ( upper-adjoint-galois-connection-Large-Poset G)
          ( p)))

An element `x' : Q` satisfies the lower universal property with respect to `x : P` if and only if it is similar to the element `LG x`

module _
  {αP αQ : Level → Level} {βP βQ : Level → Level → Level}
  {γ δ : Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G : galois-connection-Large-Poset γ δ P Q)
  {l1 l2 : Level} (x : type-Large-Poset P l1) (x' : type-Large-Poset Q l2)
  where

  is-lower-element-sim-galois-connection-Large-Poset :
    sim-Large-Poset Q
      ( map-lower-adjoint-galois-connection-Large-Poset P Q G x)
      ( x') →
    is-lower-element-galois-connection-Large-Poset P Q G x x'
  pr1 (is-lower-element-sim-galois-connection-Large-Poset s y) p =
    forward-implication-adjoint-relation-galois-connection-Large-Poset P Q G
      ( transitive-leq-Large-Poset Q _ x' y p (pr1 s))
  pr2 (is-lower-element-sim-galois-connection-Large-Poset s y) p =
    transitive-leq-Large-Poset Q x' _ y
      ( backward-implication-adjoint-relation-galois-connection-Large-Poset
        P Q G p)
      ( pr2 s)

  sim-is-lower-element-galois-connection-Large-Poset :
    is-lower-element-galois-connection-Large-Poset P Q G x x' →
    sim-Large-Poset Q
      ( map-lower-adjoint-galois-connection-Large-Poset P Q G x)
      ( x')
  sim-is-lower-element-galois-connection-Large-Poset l =
    sim-has-same-elements-principal-upper-set-element-Large-Poset Q
      ( λ y →
        logical-equivalence-reasoning
          leq-Large-Poset Q _ y
          ↔ leq-Large-Poset P x _
            by adjoint-relation-galois-connection-Large-Poset G x y
          ↔ leq-Large-Poset Q x' y
            by inv-iff (l y))

An element `y' : P` satisfies the upper universal property with respect to `y : Q` if and only if it is similar to the element `UG y`

module _
  {αP αQ : Level → Level} {βP βQ : Level → Level → Level}
  {γ δ : Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G : galois-connection-Large-Poset γ δ P Q)
  {l1 l2 : Level} (y : type-Large-Poset Q l1) (y' : type-Large-Poset P l2)
  where

  is-upper-element-sim-galois-connection-Large-Poset :
    sim-Large-Poset P
      ( map-upper-adjoint-galois-connection-Large-Poset P Q G y)
      ( y') →
    is-upper-element-galois-connection-Large-Poset P Q G y y'
  pr1 (is-upper-element-sim-galois-connection-Large-Poset s x) p =
    transitive-leq-Large-Poset P x _ y'
      ( pr1 s)
      ( forward-implication-adjoint-relation-galois-connection-Large-Poset
        P Q G p)
  pr2 (is-upper-element-sim-galois-connection-Large-Poset s x) p =
    backward-implication-adjoint-relation-galois-connection-Large-Poset P Q G
      ( transitive-leq-Large-Poset P x y' _ (pr2 s) p)

  sim-is-upper-element-galois-connection-Large-Poset :
    is-upper-element-galois-connection-Large-Poset P Q G y y' →
    sim-Large-Poset P
      ( map-upper-adjoint-galois-connection-Large-Poset P Q G y)
      ( y')
  sim-is-upper-element-galois-connection-Large-Poset u =
    sim-has-same-elements-principal-lower-set-element-Large-Poset P
      ( λ x →
        logical-equivalence-reasoning
          leq-Large-Poset P x _
          ↔ leq-Large-Poset Q _ y
            by inv-iff (adjoint-relation-galois-connection-Large-Poset G x y)
          ↔ leq-Large-Poset P x y'
            by u x)

The lower adjoint of a Galois connection preserves all existing joins

module _
  {αP αQ : Level → Level} {βP βQ : Level → Level → Level}
  {γ δ : Level → Level}
  (P : Large-Poset αP βP) (Q : Large-Poset αQ βQ)
  (G : galois-connection-Large-Poset γ δ P Q)
  where

  private
    _≤-P_ :
      {l1 l2 : Level} → type-Large-Poset P l1 → type-Large-Poset P l2 →
      UU (βP l1 l2)
    _≤-P_ = leq-Large-Poset P

    _≤-Q_ :
      {l1 l2 : Level} → type-Large-Poset Q l1 → type-Large-Poset Q l2 →
      UU (βQ l1 l2)
    _≤-Q_ = leq-Large-Poset Q

    hom-f : hom-Large-Poset _ P Q
    hom-f = lower-adjoint-galois-connection-Large-Poset G

    f : {l : Level} → type-Large-Poset P l → type-Large-Poset Q (γ l)
    f = map-hom-Large-Poset P Q hom-f

    hom-g : hom-Large-Poset _ Q P
    hom-g = upper-adjoint-galois-connection-Large-Poset G

    g : {l : Level} → type-Large-Poset Q l → type-Large-Poset P (δ l)
    g = map-hom-Large-Poset Q P hom-g

    adjoint-relation-G :
      {l1 l2 : Level}
      (x : type-Large-Poset P l1) (y : type-Large-Poset Q l2) →
      leq-Large-Poset Q (f x) y ↔
      leq-Large-Poset P x (g y)
    adjoint-relation-G = adjoint-relation-galois-connection-Large-Poset G

  preserves-join-lower-adjoint-galois-connection-Large-Poset :
    {l1 l2 l3 : Level} {U : UU l1} (x : U → type-Large-Poset P l2)
    (y : type-Large-Poset P l3) →
    is-least-upper-bound-family-of-elements-Large-Poset P x y →
    is-least-upper-bound-family-of-elements-Large-Poset Q
      ( λ α → f (x α))
      ( f y)
  preserves-join-lower-adjoint-galois-connection-Large-Poset x y H z =
    logical-equivalence-reasoning
      ((α : _) → f (x α) ≤-Q z)
      ↔ ((α : _) → (x α) ≤-P g z)
        by iff-Π-iff-family (λ α → adjoint-relation-G (x α) z)
      ↔ y ≤-P g z by H (g z)
      ↔ f y ≤-Q z by inv-iff (adjoint-relation-G y z)

Recent changes

2025-05-02. Fredrik Bakke. Strict orders (#1419).
2023-11-24. Egbert Rijke. Abelianization (#877).
2023-10-19. Fredrik Bakke. Level-universe compatibility patch (#862).
2023-09-29. Vojtěch Štěpančík. Extract tables (#806).
2023-09-26. Fredrik Bakke and Egbert Rijke. Maps of categories, functor categories, and small subprecategories (#794).